during the fourth year of my Ph.D. Furthermore, I would like to thank the French-German University (UFA-DFH) for offering me a

List of Tables

Introduction

Common sense suggests that a certain strategic ambiguity can be useful in conflicts. 1 "Many different strategies are used to orient toward conflicting interactional goals; some examples include avoiding interaction altogether, remaining silent, or changing the topic", says Eric Eisenberg in his famous article "Ambiguity as strategy in organizational communication" [START_REF] Eisenberg | Ambiguity as strategy in organizational communication[END_REF]), and he points out that applying one's resources of ambiguity is key in successful communication when conflicts of interest are present.

My doctoral thesis introduces such strategic use of ambiguity into games. Although game theory was invented to model conflicts of interest, so far the theory does not allow players to intentionally choose ambiguity as a strategy. To this end, I am going back to the beginnings of game theory. Von [START_REF] Von Neumann | Theory of Games and Economic Behavior[END_REF] introduced mixed strategies as random devices that are used to conceal one's behavior. I take up this interpretation and propose a generalization. In short, I allow players to use Ellsberg urns in addition to probabilistic devices like a roulette wheel or a die. For example, a player can base his action on the draw from an urn that contains hundred red and blue balls and it is only known that the number of red balls is between thirty and fifty. Such urns are objectively ambiguous, by design; players can thus create ambiguity. The recent advances in decision theory, which were motivated by Ellsberg's famous experiments and Knight's distinction between risk and uncertainty, allow to model such strategic behavior formally. probability distribution. 2 While Knight's point was mainly to draw attention to the role of experience in making good predictions in complex circumstances 3 , decision theorists incorporated his distinction between risk and ambiguity into decision models, and this with great success. One major achievement of ambiguity theory is to provide decision models which resolve the famous Ellsberg paradox [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF]). Most prominent models are by now the multiple-prior expected utility by [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF], Choquet expected utility by [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF] and incomplete preferences by [START_REF] Bewley | Knightian decision theory. Part I[END_REF]. Many other authors have added to these models by refining and extending them, others by applying them to economic and behavioral problems. Interesting applications include, e.g., explanation of incomplete contracts [START_REF] Mukerji | Ambiguity aversion and incompleteness of contractual form[END_REF]), ambiguity in financial markets [START_REF] Rigotti | Uncertainty and risk in financial markets[END_REF]), asset pricing under ambiguity [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF]), optimal stopping with ambiguity [START_REF] Riedel | Optimal stopping with multiple priors[END_REF]), strategies in insurance fraud detection [START_REF] Lang | The fog of fraud-mitigating fraud by strategic ambiguity[END_REF]), and strategic voting [START_REF] Ellis | Condorcet meets Ellsberg[END_REF]). See [START_REF] Luo | Recent advancements in the theory of choice under Knightian uncertainty and their applications in economics[END_REF] and [START_REF] Mukerji | An overview of economic applications of David Schmeidler's models of decision making under uncertainty[END_REF] for an overview of applications and [START_REF] Etner | Decision theory under ambiguity[END_REF] for a comprehensive recent survey on decision theory under ambiguity. A thorough review of philosophical and mathematical aspects of probability and its influence on decision theory is presented in [START_REF] Binmore | Rational decisions[END_REF].

Ambiguity in Games

Ambiguity is naturally present in strategic contexts. A human is a complex opponent and hence strategic interaction can be fruitfully modeled using ambiguity in the Knightian sense. The existing literature on ambiguity in games can be devided into two branches, based on the co-existing interpretations of mixed strategies in the literature. The Bayesian view of mixed strategies has been generalized to ambiguity starting with [START_REF] Dow | Nash equilibrium under Knightian uncertainty: breaking down backward induction[END_REF], [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF], and [START_REF] Marinacci | Ambiguous games[END_REF], and followed by many other authors. There, the interpretation of mixed strategies as beliefs about others' actions is generalized to uncertain beliefs in the sense of the decision-theoretic literature on Knightian uncertainty [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF], [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]). The other interpretation, where a mixed strategy is seen as an objective randomizing device, has to my knowledge not been generalized to ambiguity until now, with one exception. [START_REF] Bade | Ambiguous act equilibria[END_REF] allows for ambiguous Anscombe-Aumann acts, and is closer to my approach in that sense. However, she assumes that ambiguity is a subjective part of players' preferences, and thus generalizes Aumann's subjective equilibria [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF]). In opposition to her concept using subjective preferences, my approach takes purely the Von Neumann and Morgenstern point of view. Players can use ambiguity-creating devices modeled as sets of probabilities over the pure strategies, and players have ambiguity-averse preferences over these sets. Thus, I generalize the classical model to incorporate Knightian uncertainty in an objective sense, as it was recently axiomatized from a decision-theoretic point of view by [START_REF] Gajdos | Attitude toward imprecise information[END_REF]. Ellsberg urns in addition to mixed strategies; these Ellsberg urns are modeled as convex and compact sets of mixed strategies. A remarkable consequence is that players, once offered the possibility to use these Ellsberg urns, actually use them, even though they are ambiguity-averse. 4 New equilibria emerge, which are not Nash equilibria in the original game, with outcomes that are not in the support of the original Nash equilibrium. I explain this in Section 1.4 with an example of a peace negotiation taken from [START_REF] Greenberg | The right to remain silent[END_REF].

This game has a unique Nash equilibrium in which war is the outcome; there is another, as I call it, Ellsberg equilibrium, in which peace is the outcome. Other examples show that, as in the classical case, my approach will have the most fruitful applications in games of conflict where it is in players' own interest to conceal their behavior. The approach might seem less plausible in common interest games. However, the theory in its abstract form applies there as well, of course, and I think it is useful to study the consequences in such games, too. Interestingly, the concealment of Nash equilibrium and maximin behavior that I observe in two-player Ellsberg games in Section 2.6, has its strongest realization in symmetric coordination games (Section 3.2).

Objective Ambiguity as a Strategy

An important modeling aspect of Ellsberg games is that ambiguity is present in an objective sense, and not in the subjective sense mostly encountered in applications of ambiguity.

I do not use ambiguity merely in the beliefs of the players, but in the actual strategies.

This has an important implication: using objective ambiguity as a strategy means that players cannot have a look into their own urn. They intentionally choose a device that leaves themselves uncertain about the pure strategy eventually played, once uncertainty is resolved. Players choose to know less about their own play than they could know. This may sound surprising or implausible at first. It is, however, the natural generalization of the usual mixed strategy when it is interpreted as an objective device to conceal one's behavior, as in the classical justification by Von Neumann and Morgenstern. The founders of game theory justify the use of random devices in zero-sum games by a thought experiment.

If your opponent might find out your strategy, then it is optimal to conceal your behavior by using a random device. I just go one step further and allow players to use Ellsberg urns to conceal their strategy in the objective sense of Von Neumann and Morgenstern.

Notably, this modeling approach also specifies the type of strategic ambiguity we deal with in Ellsberg games. One could imagine to model strategic ambiguity differently, Many times during my presentations of the working paper [START_REF] Riedel | The strategic use of ambiguity[END_REF], the question came up what Ellsberg strategies could actually be in real life. This question always led to vivid discussions, I want to present two of the suggestions made5 as examples of Ellsberg strategies.

For the first example imagine a tax authority that is interested in minimizing tax fraud.

To this end, it uses a mechanism of selective inspection of tax payers, which is too complicated to completely understand and to form a precise prior about the probability of being inspected. Even the tax authority has designed it in a way that does not allow it to form a precise prior, in order to protect its employees from bribery. This too complex mechanism is an example for an Ellsberg urn.

As a second example, think of a state minister of, e.g., environment who promotes a new law of environmental protection. He is head of a large machinery of civil servants and if he suggests a new law, the passing of this law depends (in Germany) on the decisions of Bundesrat and Bundespräsident. When the minister makes a public announcement about the characteristics of the new law, the citizens as well as himself do not know what the law will actually be like when it has gone through the institutional process of being implemented. In that sense the minister is playing with an Ellsberg strategy. This interpretation of Ellsberg strategies is close to the notion of imprecise information, see for example [START_REF] Giraud | Are beliefs a matter of taste? A case for objective imprecise information[END_REF], which is a reason why I adapted the axiomatization of attitude towards imprecise information from [START_REF] Gajdos | Attitude toward imprecise information[END_REF].

Having the above interpretations and examples in mind, I suggest the following, for the intent and purpose of this work practical, way how to think of Ellsberg urns while reading this thesis: the players have the possibility to delegate the creation of ambiguity to an independent laboratory. Players can order any kind of Ellsberg urn that they might want to use in the Ellsberg game they are facing. When it comes to playing, the players draw a colored ball from their ambiguous urn and choose their strategy according to the color drawn. Of course, in some games the same kind of commitment problems as with classical mixed strategies can arise.

Why Ellsberg Games?

The important contribution of Ellsberg games to the existing game theory literature lies, to my mind, mainly in providing a model in which people can deliberately choose how ambiguous they want their strategy to be. In fact, in other models ambiguity has been found, already some time ago, to be a most natural strategy in human interaction. For example, in strategic information transmission models such as suggested by [START_REF] Crawford | Strategic information transmission[END_REF], ambiguity (or vagueness, or concealment) was found to be the prevalent strategy when conflict of interest is present. The partition equilibria in their model, i.e., sending the same message for an interval of types, are in interpretation closely connected to Ellsberg equilibrium. I want to draw attention to the fact that also in games ambiguity is a most natural strategy.

With my thesis I seek to set up such a model which allows the strategic use of ambiguity in games, and explore its basic properties and implications. Of course, there are many aspects which I do not cover in the present text. Furthermore, certain facets of strategic ambiguity in games cannot be explained with the way I define Ellsberg games. However, the modeling choices I make seem to me a good starting point in oder to explore further how ambiguity can be used as a strategy in interactive contexts.

Topics and Main Results of this Work

My doctoral thesis consists of six chapters, Chapter 1 -5 on strategic use of ambiguity in normal form games and Chapter 6 on dynamic Ellsberg games. I summarize the main results of each chapter in the following.

In Chapter 1, I first discuss the conceptual foundations of my approach. To this end, I compare the two interpretations of mixed strategies as objective random devices (Von Neumann and Morgenstern) or as beliefs about other players' pure actions (the Bayesian view). I discuss in Section 1.2 how the ambiguity-averse players in Ellsberg games evaluate the objective ambiguity inherent in the Ellsberg strategies. Section 1.3 deals with the definition and first characterization of Ellsberg games in normal form. I define the theoretical framework for Ellsberg games in Section 1.3.2., followed by the introduction of Ellsberg equilibrium. Subsequently in Section 1.3.3, I show the equivalence of Ellsberg equilibrium to a reduced form Ellsberg equilibrium, where the state space is represented by the set of pure strategies. This leads to a simplification of Ellsberg equilibrium which facilitates the calculation of equilibria. The proposition on equivalence reads Proposition 1.3. Ellsberg equilibrium and reduced form Ellsberg equilibrium are equivalent in the sense that every Ellsberg equilibrium ((Ω * , F * , P * ), f * ) induces a payoffequivalent reduced form Ellsberg equilibrium on Ω * = S; and every reduced form Ellsberg equilibrium Q * is an Ellsberg equilibrium ((S, F, Q * ), f * ) with f * the constant act.

Most importantly, Ellsberg equilibria generalize Nash equilibria, hence Ellsberg equilibria exist. This is formulated in the following theorem in Section 1.3.4.

Theorem 1.4. Let G = �N, (S i ), (u i )� be a normal form game. Then a mixed strategy profile (π 1 , . . . , π n ) of G is a Nash equilibrium of G if and only if the corresponding profile of singletons (P 1 , . . . , P n ) with P i = {δ π i } is an Ellsberg equilibrium. In particular, Ellsberg equilibria exist when the strategy sets S i are finite.

I prove a Principle of Indifference in Distributions in Section 1.3.5. It shows that a player is indifferent between all the mixed strategies which are part of his Ellsberg equilibrium strategy.

Theorem 1.14 (The Principle of Indifference in Distributions). Let (P * 1 , . . . , P * n ) be an Ellsberg equilibrium of a normal form game G = �N, (S i ), (u i )�. Then for all P i ∈ P * i , min

P -i ∈P * -i u i (P i , P -i ) = c * for some c * ∈ R .
In Section 1.3.6, I show that strictly dominated strategies are never used in Ellsberg equilibrium. Precisely, Proposition 1.16. Any Ellsberg equilibrium strategy profile (P * i ) i∈N must put weight only on strategies that are not strictly dominated.

Section 1.4 provides the leading example why Ellsberg games are an interesting object to study, namely the peace negotiation example from [START_REF] Greenberg | The right to remain silent[END_REF] which I mentioned earlier. The section closes with a discussion of the existing literature and the relation of the different approaches to my model in Section 1.5.

Chapter 2 provides insight into immunization against strategic ambiguity. In normal form Ellsberg games players often have a strategy available that immunizes them against any ambiguity played by their opponents. The hedging effect of ambiguity aversion leads to this phenomenon. In this chapter I look at these immunization strategies more closely. I start by giving an example in Section 2.1 and then, after defining immunization strategies, calculate the immunization strategies of a large class of 2 × 2 games in Section 2.3. I then provide some examples which show that in certain games players immunize themselves against ambiguity by playing their maximin strategy. To generalize this observation, I prove in Section 2.5 under which circumstances in general two-player games the immunization strategy is the maximin strategy of the players. The result is formulated in the following theorem Theorem 2.8. Let G be a square two-person normal form game with a completely mixed Nash equilibrium (P * , Q * ). If player 1 (2) has an immunization strategy P ( Q) in G, then P ( Q) is a maximin strategy of player 1 (2).

The last section, Section 2.6, uses the results of the chapter to characterize the role of immunization strategies in Ellsberg equilibria. I explain that in many 2 × 2 games, Ellsberg equilibria are bounded by the immunization and the Nash equilibrium strategy.

In Chapter 3, I present a number of examples and properties of Ellsberg equilibria of normal form games with two players. I start with an explanation on how to compute Ellsberg equilibria of two-person 2 × 2 games. Subsequently, in Sections 3.2-3.5, I look at different classes of games: coordination games, conflict games and zero-sum games. I analyze examples and derive general results on the Ellsberg equilibria of these classes of normal form games, supplemented by tables (summarized in Table 3.1) for the Ellsberg equilibria of general coordination and conflict games. Following this, I prove in Section 3.5 that two-person zero-sum games are value preserving.

Theorem 3.18. Let G be a two-person zero-sum game. Then for all i ∈ {1, 2} and every Ellsberg equilibrium (P * , Q * ) we have max

P i ⊆∆S i min P -i ⊆∆S -i U i (P, Q) = min P -i ⊆∆S -i max P i ⊆∆S i U i (P, Q) = U i (P * , Q * ) .
In Section 3.6, I calculate the Ellsberg equilibria of some classic 2 × 2 games such as, e.g., Hawk and Dove and the Prisoners' Dilemma. I explain with an example by Myerson, that also games with linear (as opposed to piecewise linear) payoff functions can have proper Ellsberg equilibria. I close the chapter in Section 3.7 with an analysis of a special class of 3 × 3 games. I consider a slightly modified version of Rock Scissors Paper, and with the help of the experience with 2 × 2 games I succeed in calculating the Ellsberg equilibria of these games. For a better understanding of how the equilibria of modified circulant games can be determined, I provide a geometric analysis of the equilibrium strategies in the two-dimensional probability simplex.

In Chapter 4, I present some thoughts on a general interpretation of Ellsberg equilibria, and I provide a classification of when ambiguity is an option in two-player Ellsberg equilibria. I am on the one hand interested in human behavior in two-person games and its connection to Ellsberg equilibria. In an experiment with a modified version of Matching Pennies similar to mine, [START_REF] Goeree | Ten little treasures of game theory and ten intuitive contradictions[END_REF] observe that Nash equilibrium prediction was frequently violated; interestingly, Ellsberg equilibria can explain this be-havior. In Section 4.2 on the other hand, I classify Ellsberg equilibria that are supported by strategies which are not in the support of any Nash equilibrium of the game. These Ellsberg equilibria arise frequently in games with weakly dominated strategies and have an interesting behavioral interpretation. Finally, in Section 4.3, I characterize in which circumstances and how ambiguity is used in Ellsberg equilibria in two-person games. This

analysis provides an answer to the question in which two-person games proper Ellsberg equilibria exist. The result is summarized by the following theorem.

Theorem 4.2. Let G = �{1, 2}, (S i ), (u i )� be a normal form game with a completely mixed Nash equilibrium (P * , Q * ). Let M 1 and M 2 denote the immunization strategies of player 1 and 2, respectively. Then there exist Ellsberg equilibria with unilateral full ambiguity if and only if P * = M 1 or Q * = M 2 . The Ellsberg equilibria are of the following form:

(P * , [Q 0 , Q 1 ]), where Q 0 < Q * < Q 1 , if P * = M 1 ,
and ([P 0 , P 1 ] , Q * ), where P 0 < P * < P 1 , if Q * = M 2 .

Chapter 5 analyzes the relation of Ellsberg equilibria with subjective equilibria defined by [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF]. In the preceding chapters I presented the basic properties of Ellsberg equilibria and analyzed how players behave when one allows them to use objective ambiguity in two-player games. Naturally, I want to characterize what can happen in games with more than two players. To tackle this problem, I first investigate the relation of Ellsberg equilibrium to the concept of subjective equilibrium. Since there exists no other solution concept that allows players to use sets of probabilities as their strategy, the question of comparison has to be answered with regard to the attainability of certain outcomes. The objective is to find out under which conditions subjective equilibria can (or cannot) have the same support as Ellsberg equilibria. I start in Section 5.1 with an example of a three-player game, a mediated Prisoners' Dilemma, where an Ellsberg equilibrium exists that attains an outcome that is not in the support of any Nash equilibria of the game, namely cooperation of the two prisoners. This suggests that in very simple games with more than two players the prediction of Ellsberg equilibrium can be quite different from the Nash equilibrium prediction. Subsequently, in Section 5.2.1, I define subjective equilibrium in the spirit of [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] and [START_REF] Hallin | Subjectively mixed strategies[END_REF] and subjective beliefs equilibrium like [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF]. I show in Section 5.2.2 that every Ellsberg equilibrium contains a subjective equilibrium and a subjective beliefs equilibrium.

In two-player games, subjective beliefs equilibria have the same support as Nash equilibria, but this changes for three players. This explains that in Ellsberg games with more than two players outcomes outside the Nash equilibrium support can be attained; this is treated in Section 5.2.3. To this end I prove the following theorem.

Theorem 5.8. If (P * 1 , . . . , P * n ) is an Ellsberg equilibrium, then there exist µ i -i ∈ P * -i such that (µ 1 -1 , . . . , µ n -n ) is a subjective beliefs equilibrium and arg max

P i ∈∆S i U i (P i , P * -i ) ⊆ arg max P i ∈∆S i u i (P i , µ i -i ) .
Finally, in Chapter 6, the notion of Ellsberg game and Ellsberg equilibrium is extended to dynamic games. It is straightforward that Ellsberg urns can also be used as strategies in dynamic games. In this chapter I develop the theoretical framework to analyze such extensive form Ellsberg games. As in classic extensive form theory, players can use an urn to create ambiguity over their set of pure strategies, or they can place an Ellsberg urn at every decision point in the process of the game. In the first section of the chapter, Section 6.1, I thus extend the notions of mixed strategy and behavioral strategy to extensive form Ellsberg games.

Considering dynamic games leads directly to the question of dynamic consistency of the players' preferences. I discuss this in Section 6.2 and develop a formalism to translate the notion of rectangularity by [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] to extensive form Ellsberg games. This property of Ellsberg strategies is then used to prove a version of Kuhn's Theorem [START_REF] Kuhn | Extensive games and the problem of information[END_REF]) for extensive form Ellsberg games. My theorem reads Theorem 6.7. In an extensive form Ellsberg game (F, U ) with F = (T , C) satisfying perfect recall, every rectangular Ellsberg strategy profile P induces an Ellsberg behavior strategy profile Θ P via prior-by-prior updating; every Ellsberg behavior strategy profile Θ induces a rectangular Ellsberg strategy profile P Θ such that prior-by-prior updating of P Θ yields Θ. The induced strategy profiles are payoff-equivalent, i.e., U i (P) = U i (Θ P ) and U i (Θ) = U i (P Θ ) .

I present a simple two-player example for intuition, and I provide a picture of the rectangular Ellsberg strategy derived in the example in the two-dimensional probability simplex.

Thereafter in Section 6. 1 Strategic Use of Ambiguity in Normal Form Games

In the first chapter we present the fundamentals of Ellsberg games. 1 We start by discussing the major aspect in which Ellsberg games differ from other models of ambiguity in games:

the difference between objective and subjective ambiguity and the historic development of these notions. In Ellsberg games, players can use additional devices which create ambiguity and these are treated differently than classic randomizing devices. Hence, in Section 1.2 we present the decision-theoretic framework of how to evaluate the utility of the strategies used in Ellsberg games. In Section 1.3 we define the model and show some basic properties of Ellsberg games. These include the possibility to reduce Ellsberg equilibria to the state space of pure strategies, we show the existence of Ellsberg equilibria, and explain and prove the Principle of Indifference in Distributions. Section 1.4 analyzes an important example of an Ellsberg game, where the Ellsberg equilibrium predicts an outcome that is not in the support of any Nash equilibrium of the game. Finally we discuss related literature.

Ellsberg Urns and Mixed Strategies: Concealment

Device versus Beliefs about Opponents' Behavior

Let us go back to the very foundations of game theory. A game consists of a finite set N of players, a finite set of (pure) strategies S i , i ∈ N for each player, as well as a collection of payoff functions u i : S → R defined over strategy profiles S = × i∈N S i .

Von [START_REF] Von Neumann | Theory of Games and Economic Behavior[END_REF] introduce mixed strategies as probability vectors P i over pure strategies S i . The question then emerges how players evaluate profiles of such mixed strategies P = (P 1 , . . . , P n ); as the reader knows, Von Neumann and Morgenstern adopt expected utility (and axiomatize their choice). Here we are going back to these foundations and propose a generalization. We allow players to use Ellsberg urns in addition to probabilistic devices like a roulette wheel or a die. So we imagine that a player can 1 Parts of this chapter, namely Sections 1.1, 1.3.1-1.3.3, 1.3.5, 1.4 and 1.5, were published in the IMW working paper [START_REF] Riedel | The strategic use of ambiguity[END_REF].

credibly commit his behavior on the outcome of an Ellsberg urn whose parameters he has chosen. To give an example, in a Matching Pennies game he would play HEAD if the draw from an urn with 100 red and blue balls yields a red ball, while he himself and the other players only know that the proportion of red balls lies between 30 and 50 percent.

The Ellsberg urn thus displays objective, common knowledge of ambiguity. All players know the possible probability distributions of outcomes, but no player has an informational advantage over others. We want to find out what the consequences for game theory are if we change the foundations in such a way.

Before we justify our new approach conceptually, let us go back again to classical game theory and ask how mixed strategies are justified and interpreted there. Our discussion follows closely the excellent account delivered in [START_REF] Reny | Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views[END_REF].

From a mathematical point of view, mixed strategies lead to convex strategy sets, and if one wants to assign a unique value to zero-sum games, e.g., such convexity is needed.

Convexity and linearity of the payoff functions are useful in many other respects as well, of course. Just think about the indifference principle by which we usually find Nash equilibria.

The purely mathematical aspect would not be very compelling, of course, had it not a plausible interpretation. In the words of Von Neumann and Morgenstern (1953), p.

144: "In playing Matching Pennies against an at least moderately intelligent opponent, the player will not attempt to find out the opponent's intentions, but will concentrate on avoiding having his own intentions found out, by playing irregularly 'heads' and 'tails' in successive games. Since we wish to describe the strategy in one play -indeed we must discuss the course in one play and not that of a sequence of plays -it is preferable to express this as follows: The player's strategy consists neither of playing 'tails' and 'heads', but of playing 'tails' with the probability of 1/2 and 'heads' with the probability of 1/2."

They then point out that this strategy protects the player against losses as his expected gain is always zero regardless how the opponent plays.

Von Neumann and Morgenstern always interpret these strategies as objective random devices like a fair coin or die. In particular, all players assign the same probabilities to the device's outcomes. In zero-sum games, the use of such mixing can be justified as an attempt to conceal your behavior from your opponents. These arguments run into problems in common interest games as one would prefer one's own strategy to be found out by the opponent in simple coordination games (as has been pointed out by [START_REF] Schelling | The Strategy of Conflict[END_REF] and [START_REF] Lewis | Convention: A Philosophical Study[END_REF] already). [START_REF] Harsanyi | Games with incomplete information played by "Bayesian" players, I-III[END_REF]'s construction allows to resolve this plausibility problem. He shows that mixed strategy equilibria can be interpreted as pure strategy equilibria in nearby incomplete information games where the payoffs are suitably perturbed, and players have private information. A common interpretation of mixed strategies nowadays goes even a step further. Several authors, including [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF], [START_REF] Armbruster | Bayesian Game Theory[END_REF], [START_REF] Tan | The Bayesian foundations of solution concepts of games[END_REF], [START_REF] Aumann | Epistemic conditions for Nash equilibrium[END_REF], propose to forgo Harsanyis's construction and to interpret mixed strategies directly as the belief about which pure strategies the other players are going to use. Players are assumed to choose a definite action, and as other players do not know exactly which one, the mixed strategy represents their uncertainty. For more than two players, this requires some consistency among beliefs in equilibrium, of course. We will come back to this below when we discuss beliefs equilibria.

Summing up, we have here two opposing interpretations of mixed strategies: on the one hand, the "objective" interpretation by Von Neumann and Morgenstern where players deliberately use random devices with known probabilities to conceal their behavior, on the other hand, the "subjective" beliefs interpretation where the probability distributions represent players' uncertainty about other players' pure strategy choice. 2 The literature on ambiguity in games has mainly focused on the beliefs interpretation of mixed strategies.

So far, it has usually been assumed that the players choose pure strategies and that the opponents are uncertain in the Knightian sense about their choice. Our approach differs from that in the literature as we take up the interpretation of mixed strategies by Von

Neumann and Morgenstern and allow players to use objective devices that create Knightian uncertainty.

2 Both approaches are merged in the framework of [START_REF] Reny | Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views[END_REF]. Reny and Robson unify both views of mixed equilibria with the help of another construction. In their perturbed game, every player i is characterized by a privately known subjective probability ti ∈ [a, b], 0 < a < b < 1 according to which he believes that a spy finds out his strategy. The payoff of the perturbed game is then

(1 -ti)ui(mi, m-i) + tiui(mi, bestreplyto(mi)) .

So with probability 1 -ti the normal static payoff is obtained, whereas with probability ti, the other player finds out one's strategy and is allowed to play a best reply to it. In such a situation, for a generic class of games, one can approximate all Nash equilibria by suitable pure strategy equilibria of the perturbed game, with a and b close to zero. In contrast to [START_REF] Harsanyi | Games with incomplete information played by "Bayesian" players, I-III[END_REF], players sometimes do use mixed strategies in the perturbed game, e.g., in zero-sum games. We refer to their insightful paper for a more detailed discussion which is beyond the scope of our aims here.

Evaluating Objective Ambiguity

Before we come to formalize the possibility of using Ellsberg urns to create ambiguity in games, we show how to represent players' preferences over these objectively ambiguous devices. [START_REF] Knight | Risk, Uncertainty and Profit[END_REF] is one of the earliest references for the distinction between risk and (Knightian) uncertainty, see [START_REF] Arrow | Alternative approaches to the theory of choice in risk-taking situations[END_REF] p. 417. In fact, Knight made the threefold distinction between different types of uncertain outcomes into those predicted by

(1) a priori probability (deductive, "mathematical law")

(2) statistical probability (relative frequency)

(3) estimates ("The instance in question is so entirely unique that there are no others or not a sufficient number to make it possible to tabulate enough like it to form a basis for any inference of value about any real probability in the case we are interested in", Knight (1921) p. 226).

(1) and ( 2) are, in the subsequent literature, summarized as "risk" and ( 3) is what we refer to as uncertainty or ambiguity. There has been an ongoing debate about what implications Knight's observation has for preferences and decision making under uncertainty, which is at the heart of microeconomic theory. The discussion has been fueled by the famous Ellsberg experiment published in [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] (to which the Ellsberg games treated in this thesis owe their name) and has then gained new momentum after the publication of [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF] and [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF], two articles which axiomatize and represent ambiguity-averse preferences in a way that "solves" the Ellsberg paradox.

Maxmin expected utility by [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF] says, that an act f is preferred to an act g if and only if there exists a set of priors C and a utility function u such that

min p∈C � f (u) dp ≥ min p∈C � g(u) dp .
Since this representation is derived from subjective ambiguity-averse preferences, the set of priors C can be interpreted as the subjective beliefs of the decision maker. In effect, this approach to ambiguity-averse preferences is entirely subjective. It generalizes the expected utility approach with subjective probabilities by [START_REF] Savage | Foundations of Statistics[END_REF] that was turned into the language of horse-races and lotteries and thus made more tractable by [START_REF] Anscombe | A definition of subjective probability[END_REF]. A number of articles provide other subjective representations of ambiguityaverse preferences, e.g., [START_REF] Hansen | Robust control and model uncertainty[END_REF], [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF], Maccheroni, Marinacci, andRustichini (2006), Cerreia-Vioglio, Maccheroni, Marinacci, and[START_REF] Cerreia-Vioglio | Uncertainty averse preferences[END_REF].

After our explanations in Section 1.1 it is clear, that the maxmin expected utility representation derived from subjective preferences is not suitable to represent the ambiguityaverse preferences over the objectively ambiguous devices used in Ellsberg urns. What we need to specify for Ellsberg games is rather an attitude towards imprecise risk, a term coined by [START_REF] Jaffray | Généralisation du critère de l'utilité à l'incertain régulier[END_REF]. With the same motivation as the subjective representations mentioned above, he seeks to distinguish attitude towards risk and uncertainty, where he understands uncertainty objectively: due to unavailable or demonstrably bad information the decision maker is not facing a precise probability distribution. In the same spirit [START_REF] Stinchcombe | Choice with ambiguity as sets of probabilities[END_REF], [START_REF] Olszewski | Preferences over sets of lotteries[END_REF] and [START_REF] Ahn | Ambiguity without a state space[END_REF] model imprecise risk directly by sets of lotteries. We use a generalization of this approach by [START_REF] Gajdos | Attitude toward imprecise information[END_REF], pioneered by [START_REF] Wang | Two classes of multi-prior preferences[END_REF], that compares pairs (P, f ) where f is an act and P is a set of probability distributions over the state space. In their representation,

(P, f ) is preferred to (Q, g) if and only if min p∈φ(P) � f (u) dp ≥ min p∈φ(Q) � g(u) dp . (1.1)
For φ the identity (φ : P → P selects the probability-possibility set and captures the subjective part of the preferences) and a fixed set of probability distributions P, that is, when we evaluate a pair (P, f ) through the induced set of distributions on outcomes, this yields maxmin expected utility by [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF], see Section 3.3 in [START_REF] Gajdos | Attitude toward imprecise information[END_REF] for details. For a more detailed review on literature on objective ambiguity aversion see [START_REF] Giraud | Are beliefs a matter of taste? A case for objective imprecise information[END_REF] and [START_REF] Gajdos | Attitude toward imprecise information[END_REF].

We apply the utility representation (1.1) to Ellsberg strategies ((Ω, F , P), f ) which we define in the following section. An Ellsberg strategy consists of a convex and compact set of probability distributions P on a state space Ω, and an act f . The utility of an Ellsberg strategy ((Ω, F, P), f ) is defined using (1.1) with φ = id. The reduced form of an Ellsberg strategy, which will just be a convex and compact set of probability distributions Q, is evaluated with maxmin expected utility derived from (1.1). This representation is mathematically identical to the representation by [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]: An Ellsberg strategy P is preferred to an Ellsberg strategy Q if and only if min

P ∈P � u dP ≥ min Q∈Q � u dQ .
We thus use the terms maxmin expected utility and maxmin rule, having in mind that we derived this representation from attitude towards objective ambiguity. Let us formalize the intuitive idea that players can create ambiguity with the help of Ellsberg urns. An Ellsberg urn is, for us, a triple (Ω, F, P) of a non-empty set Ω of states of the world, a σ-field F on Ω (where one can take the power set in case of a finite Ω), and a set of probability measures P on the measurable space (Ω, F). This set of probability measures represents the Knightian uncertainty of the strategy. A typical example is the classical Ellsberg urn that contains 30 red balls, and 60 balls that are either black or yellow.

One ball is drawn from that urn. The state space consists of three elements {R, B, Y }, F is the power set, and P the set of probability vectors (P 1 , P 2 , P 3 ) such3 that P 1 = 1/3, P 2 = k/90, P 3 = (60k)/90 for any k = 0, . . . , 60.

We assume that the players of our game have access to and can design the parameters of such Ellsberg urns: imagine that there is an independent, trustworthy laboratory that sets up such urns and reports the outcome truthfully. Note that we allow the player to choose the degree of ambiguity of his urn. He tells the experimentalists of his laboratory to set up such and such an Ellsberg experiment that generates exactly the set of distributions P i . In this sense, the ambiguity in our formulation of the game is "objective": it is not a matter of agents' beliefs about the actions of other players, but rather a property of the device used to determine his action.

Definition of Ellsberg Games

We come now to the game where players can use such urns in addition to the usual mixed strategies (that correspond to roulette wheels or dice). Let N = {1, . . . , n} be the set of players. Each player i has a finite strategy set S i . Let S = × n i=1 S i be the set of pure strategy profiles. Players' payoffs are given by functions

u i : S → R (i ∈ N ) .
The normal form game is denoted G = �N, (S i ), (u i )�.

Players can now use different devices. On the one hand, we assume that they have "roulette wheels" or "dice" at their disposal, i.e., randomizing devices with objectively known probabilities.

f i : (Ω i , F i ) → ∆S i
which specifies the classical mixed strategy played once the outcome of the Ellsberg urn is revealed. An Ellsberg strategy for player i is then a pair

((Ω i , F i , P i ), f i )
of an Ellsberg urn and an act. A profile of Ellsberg strategies is then as usual

((Ω, F, P), f ) = (((Ω 1 , F 1 , P 1 ), f 1 ), . . . , ((Ω n , F n , P n ), f n )) .
To finish the description of our Ellsberg game, we have to determine players' payoffs. We suppose that all players are ambiguity-averse: in the face of ambiguous events (as opposed to simply random events) they evaluate their utility in a cautious and pessimistic way. This behavior in response to ambiguity has been observed in the famous experiments of [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] and confirmed in further experiments, for example of [START_REF] Pulford | Is luck on my side? Optimism, pessimism, and ambiguity aversion[END_REF] and [START_REF] Camerer | Recent developments in modeling preferences: uncertainty and ambiguity[END_REF], see also [START_REF] Etner | Decision theory under ambiguity[END_REF] for references. For our purpose we follow the axiomatization of attitude towards objective but imprecise information in [START_REF] Gajdos | Attitude toward imprecise information[END_REF] which we introduced and discussed in Section 1.2. Utility is evaluated as maxmin expected utility like in the axiomatization of [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF], but with the difference of the decision maker facing objective instead of subjective ambiguity.

The payoff of player i ∈ N at an Ellsberg strategy profile ((Ω, F, P), f ) is thus the minimal expected utility with respect to all different probability distributions in the convex and compact set P,

U i (((Ω, F , P), f )) := min P 1 ∈P 1 ,...,Pn∈Pn � Ω 1 • • • � Ωn u i (f (ω)) dP n . . . dP 1 .
We call the described larger game an Ellsberg game. An Ellsberg equilibrium is, in the same spirit as Nash equilibrium, a profile of Ellsberg strategies

(((Ω * 1 , F * 1 , P * 1 ), f * 1 ), . . . , ((Ω * n , F * n , P * n ), f * n )) ,
where no player has an incentive to deviate, i.e., for all players i ∈ N , all Ellsberg urns (Ω i , F i , P i ), and all acts f i for player i we have

U i (((Ω * , F * , P * ), f * )) ≥ U i � ((Ω i , F i , P i ), f i ), ((Ω * -i , F * -i , P * -i ), f * -i ) � . 5 Definition 1.1. Let G = �N, (S i ), (u i )� be a normal form game. A profile (((Ω * 1 , F * 1 , P * 1 ), f * 1 ), . . . , ((Ω * n , F * n , P * n ), f * n ))
of Ellsberg strategies is an Ellsberg equilibrium of G if no player has an incentive to deviate from ((Ω * , F * , P * ), f * ), i.e., for all players i ∈ N , all Ellsberg urns (Ω i , F i , P i ) and all acts f i for player i we have

U i (((Ω * , F * , P * ), f * )) ≥ U i (((Ω i , F i , P i ), f i ), ((Ω * -i , F * -i , P * -i ), f * -i )) , that is min P i ∈P * i ,P -i ∈P * -i � Ω * i � Ω * -i u i (f * i (ω i ), f * -i (ω -i )) dP -i dP i ≥ min P i ∈P i ,P -i ∈P * -i � Ω i � Ω * -i u i (f i (ω i ), f * -i (ω -i )) dP -i dP i .

Reduced Form Ellsberg Games

This definition of an Ellsberg game depends on the particular Ellsberg urn used by each player i. As there are arbitrarily many possible state spaces 6 , the definition of Ellsberg equilibrium might not seem very tractable. Fortunately, there is a more concise way to define Ellsberg equilibrium. The procedure is similar to the reduced form of a correlated equilibrium, see [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] or [START_REF] Fudenberg | Game Theory[END_REF]. Instead of working with arbitrary Ellsberg urns, we note that the players' payoff depends, in the end, on the set of distributions that the Ellsberg urns and the associated acts induce on the set of strategies.

One can then work with that set of distributions directly.

5 Throughout the text, we follow the notational convention that (fi, f *

-i ) := (f * 1 , . . . , f * i-1 , fi, f * i+1 , . . . , f * n ).
The same convention is used for profiles of pure strategies (si, s-i) and probability distributions (Pi, P-i). 6 In fact, the class of all state spaces is too large to be a well-defined set according to set theory. Definition 1.2. Let G = �N, (S i ), (u i )� be a normal form game. A reduced form Ellsberg equilibrium of the game G is a profile of sets of probability measures Q * i ⊆ ∆S i , such that for all players i ∈ N and all sets of probability measures Q i ⊆ ∆S i we have

min P i ∈Q * i ,P -i ∈Q * -i � S i � S -i u i (s i , s -i ) dP -i dP i ≥ min P i ∈Q i ,P -i ∈Q * -i � S i � S -i u i (s i , s -i ) dP -i dP i .
The two definitions of Ellsberg equilibrium are equivalent in the following sense. Proof. " ⇐ " Let Q * be an Ellsberg equilibrium according to Definition 1.2. We choose the states of the world Ω = S to be the set of pure strategy profiles, thereby we see that player i uses the Ellsberg urn (S i , F i , Q * i ). We define the act f * i : (S i , F i ) → ∆S i to be the constant act that maps f * i (s i ) = {δ s i }. {δ s i } ∈ ∆S i is the degenerate mixed strategy which puts all weight on the pure strategy s i . Each measure

Q i ∈ Q * i has an image measure under f * i , Q i • f * -1 i : {δ s i } � → Q i (f * -1 i ({δ s i })) . Q i • f * -1 i can be identified with Q i ∈ Q * i .
Thus, the reduced form Ellsberg equilibrium strategy Q * i can be written as the Ellsberg strategy ((S, F, Q * ), f * ). This strategy is an Ellsberg equilibrium according to Definition 1.1. " ⇒ " Let now ((Ω * , F * , P * ), f * ) be an Ellsberg equilibrium according to Definition 1.1.

Every P i ∈ P * i induces an image measure P i • f * -1 i on ∆S i that assigns a probability to a distribution f * i (ω i ) ∈ ∆S i to occur.
To describe the probability that a pure strategy s i is played, given a distribution P i and an Ellsberg strategy ((

Ω * i , F * i , P * i ), f * i ), we integrate (f * i (ω i ))(s i ) over all states ω i ∈ Ω i . Thus, we can define Q i to be Q i (s i ) := � Ω * i (f * i (ω i ))(s i ) dP i . (1.2)
Recall that P i is a closed and convex set of probability distributions. We get a measure Q i on S i for each P i ∈ P i ⊆ ∆Ω i . We call the resulting set of probability measures Q * i .

Q * i (s i ) := � Q i (s i ) = � Ω * i (f * i (ω i ))(s i ) dP i � � � P i ∈ P * i � .
Q * i is closed and convex, since P * i is.

Now suppose Q * was not a reduced form Ellsberg equilibrium. Then for some player i ∈ N there existed a set Q i of probability measures on S i that yields a higher minimal expected utility. This means we would have min

Q i ∈Q i ,Q -i ∈Q * -i � S i � S -i u i (s i , s -i ) dQ -i dQ i > min Q i ∈Q * i ,Q -i ∈Q * -i � S i � S -i u i (s i , s -i ) dQ -i dQ i (1.3) for some Q i � = Q * i .
Let Q � i be the minimizer of the first expression, then it must be that

Q � i / ∈ Q * i .
We know that Q � i is derived from some P � i under the equilibrium act,

Q � i (s i ) = � Ω * i (f * i (ω i ))(s i ) dP � i .
(1.4) It follows that P � i is not element of the equilibrium Ellsberg urn (Ω * i , F * i , P * i ), that is P � i / ∈ P * i . Now it remains to show that in the original game P � i yields a higher minimal expected utility than using P * i . In that case ((Ω * , F * , P * ), f * ) is not an Ellsberg equilibrium and the proof is complete.

Let player i use P � i in his maxmin expected utility evaluation in the original game. This yields (recall that we let u i denote the utility function on S i as well as on ∆S i ) min

P -i ∈P * -i � Ω * i � Ω * -i u i (f * i (ω i ), f * -i (ω -i )) dP -i dP � i (1.5) = min P -i ∈P * -i � Ω * i � Ω * -i � S i � S -i u i (s i , s -i ) df * -i (ω -i )df * i (ω i ) dP -i dP � i .
We use equations (1.2) and (1.4) to rewrite the expression and get min

Q -i ∈Q * -i � S i � S -i u i (s i , s -i ) dQ -i dQ � i . (1.6)
We know by equation (1.3) that this is larger than the minimal expected utility over Q * i and this gives (1.6) > min

Q i ∈Q * i ,Q -i ∈Q * -i � S i � S -i u i (s i , s -i ) dQ -i dQ i = min P i ∈P * i ,P -i ∈P * -i � Ω * i � Ω * -i u i (f * i (ω i ), f * -i (ω -i )) dP -i dP i .
Using (1.5) we see that this contradicts the assumption that ((Ω * , F * , P * ), f * ) was an Ellsberg equilibrium. Thus, Q * is a reduced form Ellsberg equilibrium.

We henceforth call a set Q i ⊆ ∆S i an Ellsberg strategy whenever it is clear that we are in the reduced form context. The utility of a (reduced) Ellsberg strategy profile is evaluated as U i (P i , P -i ) = min

P i ∈P i ,P -i ∈P -i � S i � S -i u i (s i , s -i ) dP -i dP i .
In the definition of an Ellsberg game we assume that the Ellsberg urns (Ω i , F i , P i ) of all players i ∈ N are stochastically independent. This is done by using product spaces as first suggested by [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]. 7 We define the product (Ω, F, P) of n

Ellsberg urns (Ω i , F i , P i ), i = 1, . . . , n, as follows.

Ω := Ω 1 × . . . × Ω n , F := F 1 ⊗ . . . ⊗ F n ,
and P is the closed convex hull of the set of product measures,

P := co {P 1 ⊗ . . . ⊗ P n | P 1 ∈ P 1 , . . . , P n ∈ P n } . (1.7)
This way the Ellsberg urns are stochastically independent. We usually denote by ∆S i the set of lotteries on the set of pure strategies S i of player i. If not otherwise specified, by

∆S := × N i=1 ∆S i , or ∆S -i := × N j=1
,j� =i ∆S j we mean the profiles of mixed strategies where each player mixes independently.

Different notions of stochastic independence in the context of ambiguity aversion have been discussed in the literature, see for example [START_REF] Klibanoff | Stochastically independent randomization and uncertainty aversion[END_REF], [START_REF] Bade | Ambiguous act equilibria[END_REF] and Bade (2011a). In the present context of objective ambiguity in the form of Ellsberg urns the above notion seems the most natural.

Ellsberg Equilibria Generalize Nash Equilibria

Note that the classical game is contained in our formulation: players just choose a singleton P i = {δ π i } that puts all weight on a particular (classical) mixed strategy π i . Now let (π 1 , . . . , π n ) be a Nash equilibrium of the game G. Can any player unilaterally gain by creating ambiguity in such a situation? The answer is no. Take the game in Figure 1.1 and look at the pure strategy Nash equilibrium (D, R) with equilibrium payoff 1 for both players.

Player 1 Player 2 L R U 3, 3 0, 0 D 0, 0 1, 1
Figure 1.1: Strategic ambiguity does not unilaterally make a player better off.

If player 1 introduces ambiguity, he will play U in some states of the world (without knowing the exact probability of those states). But this does not help here because player 2 sticks to his strategy R, so playing U just leads to a payoff of zero. Unilateral introduction of ambiguity does not increase one's own payoff. We think that this is an important property of our formulation.

Theorem 1.4. Let G = �N, (S i ), (u i )� be a normal form game. Then a mixed strategy profile (π 1 , . . . , π n ) of G is a Nash equilibrium of G if and only if the corresponding profile of singletons (P 1 , . . . , P n ) with P i = {δ π i } is an Ellsberg equilibrium. In particular, Ellsberg equilibria exist when the strategy sets S i are finite.

We show in Theorem 1.4 that a mixed strategy profile is a Nash equilibrium of a normal form game if and only if it is an Ellsberg equilibrium of that game. The main part of the proof is that every player can find for every profile of Ellsberg strategies a mixed strategy that gives him at least the same utility as his Ellsberg strategy. We first prove the latter result in the following lemma. We let π * i abbreviate the constant act that maps every state of the world to the mixed strategy π * i ∈ ∆S i .

Lemma 1.5. Let G = �N, (S i ), (u i )� be a normal form game. Then for any profile of Ellsberg strategies (P 1 , . . . , P n ) and every i ∈ N there exists a mixed strategy

π * i ∈ ∆S i such that U i ({δ π * i }, P -i ) ≥ U i (P i , P -i ).
Proof. If P i is a singleton {δ π i }, we take π * i := π i and we are done. Let P i not be a singleton. Recall that U i (P i , P -i ) is defined to be the minimal expected payoff according to all distributions P ∈ P. Thus, there must exist a π * i ∈ ∆S i for which {δ π * i } ∈ P i with U i ({δ π * i }, P -i ) ≥ U i (P i , P -i ) .

Proof of Theorem 1.4. Let P with

P i = {δ π i } be an Ellsberg equilibrium of G. So U i ({δ π i }, {δ π -i }) ≥ U i (P � i , {δ π -i })
holds for all i ∈ N for all Ellsberg strategies P � i ⊆ ∆S i . In particular this holds for all singletons {δ π � i }, so

U i ({δ π i }, {δ π -i }) ≥ U i ({δ π � i }, {δ π -i })
and thus

u i (π i , π -i ) ≥ u i (π � i , π -i ) for all π � i ∈ ∆S i .
Therefore the profile (π 1 , . . . , π n ) is a Nash equilibrium of G.

Next, assume that (π 1 , . . . , π n ) is a Nash equilibrium of G. Suppose it was not an Ellsberg equilibrium of G, that is there exists an Ellsberg strategy P i for some player i such that

U i (P i , {δ π -i }) > U i ({δ π i }, {δ π -i })
. By Lemma 1.5 we have that there exists a π * i such that

U i ({δ π * i }, {δ π -i }) ≥ U i (P i , {δ π -i }) > U i ({δ π i }, {δ π -i }) .
This is a contradiction with the assumption that (π 1 , . . . , π n ) was a Nash equilibrium of the game G.

Theorem 1.4 proves that Ellsberg equilibria generalize Nash equilibria. Hence our formulation avoids the existence pitfalls that one encounters when players are assumed to play pure strategies and beliefs are uncertain about those pure actions.

We show that the support of an Ellsberg equilibrium, i.e., the strategies played with probability greater than zero, is identical to the support of the corresponding reduced form Ellsberg equilibrium. First we define what is the support of a (reduced) Ellsberg strategy.

We understand null-events in the sense of [START_REF] Savage | Foundations of Statistics[END_REF].8

Definition 1.6. Let ((Ω i , F i , P i ), f i ) be an Ellsberg strategy. The support of ((Ω i , F i , P i ), f i ) is defined as the support of f i , that is

supp f i := {s i ∈ S i | (f i (ω i ))(s i ) > 0 for some non-null ω i ∈ Ω i } .
Let Q i ⊆ ∆S i be a reduced Ellsberg strategy. The support of Q i is then defined as

supp Q i := � Q i ∈Q i {s i ∈ S i | Q i (s i ) > 0} .
Proposition 1.7. Let ((Ω * , F * , P * ), f * ) be an Ellsberg equilibrium of a normal form game G, Q * ⊆ ∆S the equivalent reduced form Ellsberg equilibrium. Then

supp f * = supp Q * .
Proof. It suffices to show the equality for supp f * i and supp Q * i for some player i ∈ N . Recall that Q * i is the set of measures on S i induced by P i ∈ P * i under f * i , that is

Q * i (s i ) := � Q i (s i ) = � Ω * i f * i (ω i )(s i ) dP i � � � P i ∈ P * i � .
We first show that supp

f * i ⊆ supp Q * i . Let s i ∈ supp f * i , then f * i (ω i )(s i ) > 0 for some ω i ∈ Ω i . Suppose, s i / ∈ supp Q * i . Then Q i (s i ) = 0 for all Q i ∈ Q * i ⇒ Q i (s i ) = � Ω i f * i (ω i )(s i ) dP i = 0 for all P i ∈ P * i ⇒ f * i (ω i )(s i ) = 0 for all ω i ∈ Ω i
This contradicts the fact that s i was in the support of f * i .

Let now

s i ∈ supp Q * i , then Q i (s i ) > 0 for some Q i ∈ Q * i . Suppose s i / ∈ supp f * i . Then f * i (ω i )(s i ) = 0 for all ω i ∈ Ω i ⇒ Q i (s i ) = � Ω i f * i (ω i )(s i ) dP i = 0 for all ω i ∈ Ω i
This contradicts the fact that s i was in the support of Q * i .

We have seen that Nash equilibria are special cases of Ellsberg equilibria. To simplify reference to different types of Ellsberg equilibria we introduce proper Ellsberg equilibria.

The notion serves to address exclusively those Ellsberg equilibria which are not Nash equilibria, i.e., those in which ambiguity prevails. This is the result of Corollary 1.9 which follows directly from Theorem 1.4.

Definition 1.8. Let G = �N, (S i ), (u i )� be a normal form game with i ∈ N . G has proper Ellsberg equilibria when the set of Ellsberg equilibria that are not Nash equilibria is non-empty.

Corollary 1.9 (of Theorem 1.4). Let G = �N, (S i ), (u i )� be a normal form game with i ∈ N . The set of proper Ellsberg equilibria of G coincides with the set of Ellsberg equilibria of G in which at least one player i ∈ N uses an Ellsberg strategy P i ⊆ ∆S i which is not a singleton.

In the same spirit we sometimes speak of proper Ellsberg strategies when they are not identical to a pure or mixed strategy.

Principle of Indifference in Distributions

An important characteristic of Nash equilibrium in mixed strategies is what we call here the Principle of Indifference: in a Nash equilibrium in mixed strategies players are indifferent between playing the randomized strategy or playing any pure strategy that is in its support.

We restate here the corresponding lemma from [START_REF] Osborne | A Course in Game Theory[END_REF], Lemma

33.2 therein.

Lemma. Let G be a finite normal form game. Then P * ∈ × N i=1 ∆S i is a mixed strategy Nash equilibrium of G if and only if for every player i ∈ N every pure strategy in the support of P * i is a best response to P * -i .

Osborne and Rubinstein (1994) conclude: "It follows that every action in the support of any player's equilibrium mixed strategy yields that player the same payoff."9 In the same section the authors point out, that this may no longer be true when players are not Von

Neumann and Morgenstern expected utility maximizers. We confirm this observation with an example and show that an alternative version of the Principle of Indifference holds for Ellsberg games.

In Ellsberg games players are no longer expected utility maximizers but ambiguityaverse. Ambiguity-averse preferences have the property that the utility functions often have kinks: in many cases the decision maker switches at some point from one boundary of the set of probability distributions to the other and thus changes the slope of the utility function. In the analysis of Ellsberg games this causes that the Principle of Indifference no longer prevails. In an Ellsberg equilibrium players are in general not indifferent between playing the Ellsberg strategy or playing the pure strategies in its support. We illustrate this with an example.

Example 1.10. We consider a similar coordination game as before, the payoff matrix is as in Figure 1 To make this argument more precise we present the utility analysis. We first fix a distribution P ∈ [P 0 , P 1 ] for player 1 and consider his payoff function

U 1 (P, [Q 0 , Q 1 ]).
Player 1 evaluates his minimal expected payoff given player

2 plays Q ∈ [Q 0 , Q 1 ]: U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 P Q + 3(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(4P -3) -3P + 3 =          Q 0 (4P -3) -3P + 3 if P > 3/4 , 3/4 if P = 3/4 , Q 1 (4P -3) -3P + 3 if P < 3/4 .
In the given Ellsberg equilibrium player 1 thus evaluates

U 1 ([3/4, 1] , [3/4, 1]) =    3/4(4P -3) -3P + 3 if P > 3/4 , 3/4 if P = 3/4 = 3/4 for all P ∈ [3/4, 1] .
Now consider the two cases where player 1 plays U, that is P 0 = P 1 = 1, or D, that is

P 0 = P 1 = 0. Then U 1 (1, [3/4, 1]) = 3/4(4 -3) = 3/4 ,
and U 1 (0, [3/4, 1]) = 1(-3) + 3 = 0 .

The last two equations show that the indifference in pure strategies does in general not hold.

Example 1.10 shows violation of the classical Principle of Indifference. However, we can show that in Ellsberg equilibrium players are indifferent between all distributions that are part of their Ellsberg equilibrium strategy. We come to this next.

Recall that a profile (P

P i ⊆∆S i U i (P i , P * -i ) .
(1.9)

Conditions (1.8) and (1.9) are obviously equivalent. This is formulated in the following proposition.

Proposition 1.11. Let G = �N, (S i ), (u i )� be a normal form game. A profile of Ellsberg strategies (P * 1 , . . . , P * n ) satisfies (1.8) if an only if it satisfies (1.9).

For the following theorems we need a lemma concerning the maximization and minimization of a linear function u on some convex set ∆S (for the moment, S does not denote a set of strategy profiles, but simply a set with m elements).

Lemma 1.12. P is a probability distribution on the set S = {s 1 , . and the equality holds.

Since the payoff functions in Ellsberg games are multilinear, the preceding lemma gives us the following corollary of which we will make use also in later chapters.

Corollary 1.13. Let G = �N, (S i ), (u i )� be a normal form game. Then for all i ∈ N , max

P i ⊆∆S i U i (P i , P -i ) = max P i ∈∆S i min P -i ∈P -i u i (P i , P -i ) ,
and min

P i ⊆∆S i U i (P i , P -i ) = min P i ∈∆S i min P -i ∈P -i u i (P i , P -i ) .
With Corollary 1.13 we thus derive from Proposition 1.11 and Lemma 1.12 that in an Ellsberg equilibrium (P * 1 , . . . , P * n ), for all i ∈ N , P i ∈ arg max

P � i ∈∆S i U i (P � i , P * -i ) for all P i ∈ P * i ,
or, put differently, U i (P * i , P * -i ) = max

P i ∈∆S i min P -i ∈P * -i u i (P i , P -i ) .
(1.12)

Suppose the left hand side of (1.12) was strictly smaller than the right hand side. Then there exists a P � i ∈ ∆S i which maximizes the right hand side, but is not the minimizer of the left hand side. That means player i could deviate from P * i to a set of probability distributions P � i with P � i ∈ P � i for which the minimizer of the left hand side of (1.12) was P � i and obtain a minimal expected payoff that was higher than U i (P * i , P * -i ). This contradicts the assumption that (P * i , P * -i ) was an Ellsberg equilibrium. Thence, although U i is defined as the minimal expected utility over all P 1 , . . . , P n , de facto player i uses only maximizers in his own Ellsberg equilibrium strategy P * i , any other distribution would reduce his utility. In other words, all distributions P i which player i uses in his Ellsberg equilibrium strategy yield the same utility, that is, player i is indifferent between any P i ∈ P * i . This is formulated in the following proposition.

Theorem 1.14 (The Principle of Indifference in Distributions). Let (P * 1 , . . . , P * n ) be an Ellsberg equilibrium of a normal form game G = �N, (S i ), (u i )�. Then for all P i ∈ P * i , min

P -i ∈P * -i u i (P i , P -i ) = c * for some c * ∈ R .
Furthermore, a necessary condition for his equilibrium strategy not to be a singleton is therefore that U i is constant for some strategy P -i . However, this is not a sufficient condition for a strategy to be an Ellsberg equilibrium strategy. When P i is a best response to some P -i then all P i ∈ P i yield the same minimal expected utility, but P i is only an equilibrium strategy, if P -i is one, too.

An important property of the payoff functions is that they are linear in probabilities. In particular, with the equality established in Proposition 1.11, U i (P * i , P * -i ) is linear on ∆S i for each P i and linear on P * j for each P j ∈ P * j , j � = i. This observation suffices to fulfill the assumptions (multilinear functions are convex in each variable) to Fan's Minimax Theorem in Fan (1952). Thus, we have the following minimax theorem for Ellsberg games.

Theorem 1.15 (Minimax Theorem 1). In an Ellsberg equilibrium (P * i , P * -i ), for all i ∈ N , max

P i ∈∆S i min P -i ∈P * -i u i (P i , P -i ) = min P -i ∈P * -i max P i ∈∆S i u i (P i , P -i ) .
We present more minimax results in Section 3.5 on zero-sum games.

Strict Domination in Ellsberg Games

In this section we show that strictly dominated strategies are never used in Ellsberg equilibria. The question under which conditions weakly dominated strategies can be in the support of an Ellsberg equilibrium is postponed to Section 4.2.2. In a normal form game G = �N, (S i ), (u i )� a pure strategy s i ∈ S i is strictly dominated if there exists a mixed strategy P i ∈ ∆S i that yields a higher utility for player i for any strategy profile s -i ∈ S -i .

Proposition 1.16. Any Ellsberg equilibrium strategy profile (P * i ) i∈N must put weight only on strategies that are not strictly dominated.

Proof. Let si be strictly dominated. Then there exists a mixed strategy P i ∈ ∆S i with U i (P i , P -i ) > U i ( si , P -i ) for all P -i ⊆ ∆S -i . Let (P * i , P * -i ) be an Ellsberg equilibrium profile. Suppose for some player i ∈ N , P * i (s i ) > 0 for some P * i ∈ P * i . Then we have:

U i (P * i , P * -i ) = min P * -i ∈P * -i U i (P * i , P * -i ) = min P * -i ∈P * -i u i (P * i , P * -i ) = min P * -i ∈P * -i � s i ∈S i u i (s i , P * -i ) P * i (s i ) = min P * -i ∈P * -i � s i � =s i u i (s i , P * -i )P * i (s i ) + u i (s i , P * -i )P * i (s i ) < min P * -i ∈P * -i � s i � =s i u i (s i , P * -i )P * i (s i ) + u i (P i , P * -i ) = min P * -i ∈P * -i � s i � =s i u i (s i , P * -i )P * i (s i ) + � s i ∈S i u i (s i , P * -i )P i (s i ) .
(1.13)

We construct a distribution P � i ∈ ∆S i :

P � i (s i ) :=    P * i (s i )P i (s i ) for s i � = si , P i (s i ) for s i = si .
With this distribution we can write (1.13) = min

P * -i ∈P * -i � s i ∈S i u i (s i , P * -i )P � i (s i ) = U i (P � i , P -i * ) .
This is a contradiction to the assumption that (P * i , P * -i ) was an Ellsberg equilibrium.

From Proposition 1.16 we get an immediate corollary.

Corollary 1.17. In two-person games, when one round of elimination of dominated strategies yields a unique strategy profile, this is the unique Ellsberg equilibrium.

The iterated elimination of dominated strategies raises the question of rationalizability in Ellsberg games. It is subject to further research to determine the exact characterization and interdependence of rationalizability and dominance in preference models that account for aversion to objective ambiguity, but we give a brief overview over the main existing contributions in the area of ambiguity-averse preferences.

In classic game theory, the concept of Nash equilibrium has often been criticized in that it assumes correct beliefs in equilibrium. A weaker solution concept that circumvents this criticism is rationalizability. If one allows for correlation in the definition of rationalizability, then in finite normal form games the set of correlated rationalizable strategies coincides with the set of strategies that survive iterated elimination of strictly dominated strategies. [START_REF] Brandenburger | Rationalizability and correlated equilibria[END_REF] show that these solution concepts are equivalent, in some suitable sense, to a posteriori equilibrium introduced by Aumann (1974) as a strengthening of subjective correlated equilibrium.

Naturally, it is interesting how these solution concepts relate to each other when we depart from the Bayesian framework and allow for non-expected utility preferences. Notably, four papers provide results on this relation. [START_REF] Epstein | Preference, rationalizability and equilibrium[END_REF] analyzes finite normal form games with two players, where he excludes explicit randomization over pure strategies. He finds that in this case, iterated deletion of strictly dominated strategies and multiple-prior-rationalizability do in general not lead to the same set of strategies. The equivalence is obtained when one allows for explicit randomization, as shown by [START_REF] Klibanoff | Stochastically independent randomization and uncertainty aversion[END_REF]. This equivalence is generalized by Lo (2000), who shows that rationalizable behavior is indistinguishable for any model of preference satisfying Savage's axiom P3 (eventwise 1 Strategic Use of Ambiguity in Normal Form Games monotonicity). Finally, [START_REF] Chen | An indistinguishability result on rationalizability under general preferences[END_REF] show indistinguishability for another class of payoff functions and ambiguity-averse preferences.

1.4 Non-Nash Outcomes: Strategic Use of Ambiguity in

Negotiation Games

Strategic ambiguity can lead to new phenomena that lie outside the scope of classical game theory. To give the reader a first impression of interesting Ellsberg equilibria, we consider the following peace negotiation game taken from [START_REF] Greenberg | The right to remain silent[END_REF]. There are two small countries who can either opt for peace, or war. If both countries opt for peace, all three This game possesses a unique Nash equilibrium where country A mixes with equal probabilities, and country B opts for war. The superpower has no clue who started the war given these strategies. It is thus indifferent about whom to punish and mixes with equal probabilities, as well. War occurs with probability 1. The resulting equilibrium payoff vector is (4.5, 4.5, 0.5).

war peace war 0,9,1 0,9,1 peace 3,9,0 4,4,4 punish A war peace war 9,0,0 9,0,0 peace 6,0,1 4,4, If the superpower can create ambiguity (and if the countries A and B are ambiguityaverse), the picture changes. Suppose for simplicity, that the superpower creates maximal ambiguity by using a device that allows for any probability between 0 and 1 for its strategy punish A. The pessimistic players A and B are ambiguity-averse and thus maximize against the worst case. For both of them, the worst case is to be punished by the superpower, with a payoff of 0. Hence, both prefer to opt for peace given that the superpower creates ambiguity. As this leads to a very desirable outcome for the superpower, it has no incentive to deviate from this strategy. We have thus found an equilibrium where the strategic use of ambiguity leads to an equilibrium outcome outside the support of the Nash equilibrium outcome.

Let us formalize the above considerations. We claim that there is the following type of Ellsberg equilibria. The superpower creates ambiguity about its decision. If this ambiguity is sufficiently large, both players fear to be punished by the superpower in case of war. As a consequence, they opt for peace. In our game with just two actions for the superpower, we can identify an Ellsberg strategy with an interval [P 0 , P 1 ] where P ∈ [P 0 , P 1 ] is the probability that the superpower punishes country A. Suppose the superpower plays so with P 0 < 4/9 and P 1 > 5/9. Assume also that country B opts for peace. If A goes for war, it uses that prior in [P 0 , P 1 ] which minimizes its expected payoff, which is P 1 . This yields U A (war, war,

[P 0 , P 1 ]) = P 1 • 0 + (1 -P 1 ) • 9 < 4.
Hence, opting for peace is country A's best reply. The reasoning for country B is similar, but with the opposite probability P 0 . If both countries A and B go for peace, the superpower gets 4 regardless of what it does; in particular, the ambiguous strategy described above is optimal. We conclude that (peace, peace, [P 0 , P 1 ]) is a (reduced form) Ellsberg equilibrium.

Proposition 1.18. In Greenberg's game, the strategies (peace, peace, [P 0 , P 1 ]) with P 0 < 4/9 and P 1 > 5/9 form an Ellsberg equilibrium.

Note that this Ellsberg equilibrium is very different from the game's unique Nash equilibrium. In Nash equilibrium, war occurs in every play of the game. However, in our Ellsberg equilibrium, peace is the unique outcome. By using the strategy [P 0 , P 1 ] which is a set of probability distributions, the superpower creates ambiguity. This supports an

Ellsberg equilibrium where players' strategies do not lie in the support of the unique Nash equilibrium. We also point out that the countries A and B use different worst-case priors in equilibrium, this is a typical phenomenon in Ellsberg equilibria that are supported by strategies which are not in the support of any Nash equilibrium of the game.

Other equilibrium concepts for extensive form games (without Knightian uncertainty) such as conjectural equilibrium [START_REF] Battigalli | Conjectural equilibria and rationalizability in a macroeconomic game with incomplete information[END_REF], self-confirming equilibrium [START_REF] Fudenberg | Self-confirming equilibrium[END_REF], subjective equilibrium [START_REF] Kalai | Subjective games and equilibria[END_REF] and mutually acceptable courses of action [START_REF] Greenberg | Mutually acceptable courses of action[END_REF] can also assure the peace equilibrium outcome in the example by Greenberg. Other equillibrium concepts for extensive form games with Knightian uncertainty are, e.g., [START_REF] Battigalli | Selfconfirming equilibrium and model uncertainty[END_REF] and [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF]. Postponing the analysis of the relation of these equilibrium concepts to Ellsberg equilibrium to Chapter 6 on dynamic Ellsberg games, we only want to stress here that in difference to the existing concepts the driving factor in Ellsberg equilibrium is that ambiguity is employed strategically and objectively.

Greenberg refers to historic peace negotiations between Israel and Egypt (countries A and B in the negotiation example) mediated by the USA (superpower C) after the 1973 war.

As explained by [START_REF] Kissinger | Years of Upheaval[END_REF] 11 , the fact that both Egypt and Israel were too afraid to be punished if negotiations broke down partly contributed to the success of the peace negotiations. This story is supported by our Ellsberg equilibrium, a first evidence that Ellsberg equilibria might capture some real world phenomena better than Nash equilibria.

Related Literature

Several authors introduce Knightian uncertainty into complete-information normal form games. We discuss their concepts and compare them to our approach. [START_REF] Dow | Nash equilibrium under Knightian uncertainty: breaking down backward induction[END_REF], [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF], [START_REF] Marinacci | Ambiguous games[END_REF], [START_REF] Eichberger | Non-additive beliefs and strategic equilibria[END_REF] and [START_REF] Eichberger | Ambiguity and social interaction[END_REF] all extend the interpretation of Nash equilibrium as an equilibrium in beliefs. For example, [START_REF] Dow | Nash equilibrium under Knightian uncertainty: breaking down backward induction[END_REF] interpret their

non-additive (Choquet) probabilities as uncertain beliefs about the other player's action.

A pair (P 1 , P 2 ) of non-additive probabilities is then a Nash equilibrium under Knightian uncertainty if each action in a support of player 1's belief P 1 is optimal given that he uses P 2 to evaluate his expected payoff, and similarly for player 2. We thus have here a first version of an equilibrium in beliefs. This approach is refined by [START_REF] Marinacci | Ambiguous games[END_REF] and extended to n-person games by [START_REF] Eichberger | Non-additive beliefs and strategic equilibria[END_REF]. [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF] Note that all the equilibrium concepts discussed above do not specify which action will actually be played in equilibrium. In [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF] players can play any pure or mixed strategy that is a best response to their belief set, in the other equilibrium notions mentioned, players only have access to pure strategies in the support of the capacities. This stands in contrast to Ellsberg equilibrium, where the equilibrium strategy is fixed by the Ellsberg urn chosen. The strategy is a best response to the belief, and the belief coincides with the strategy played. [START_REF] Klibanoff | Stochastically independent randomization and uncertainty aversion[END_REF], [START_REF] Lehrer | Partially-specified probabilities: decisions and games[END_REF] and Lo (2009) propose an approach similar to beliefs equilibrium. Uncertainty is present in players' beliefs that are represented by sets of distributions. Equilibrium is defined as a profile of beliefs and an objectively mixed (or pure) strategy for each player, which is the strategy that he plays in equilibrium. These strategies need to be contained in the belief sets. Accordingly, players have to anticipate their opponents' strategy correctly in the sense that the truth is part of their belief. This consistency requirement is weaker than in Nash equilibrium (and weaker than in Ellsberg equilibrium!) and typically the strategies in equilibrium are not best responses to the actual strategies played. [START_REF] Klibanoff | Stochastically independent randomization and uncertainty aversion[END_REF] proposes a refinement where only correlated rationalizable beliefs are allowed.12 Lehrer (2008) develops a model of decision making under uncertainty with partially-specified probabilities, these are used to represent the players' uncertain beliefs about their opponents. Lo (2009) establishes formal epistemic foundations for an equilibrium concept with ambiguity-averse preferences. He finds that epistemically stochastic independence is not necessary for a generalized Nash equilibrium concept. A correlated Nash equilibrium is a pair �σ, Φ� consisting of a profile of beliefs Φ i and a profile of mixed strategies σ i where, for consistency, each strategy a i in the support of σ i is a best response to the belief Φ i .

Bade (2011b) goes a first step in another direction, away from the beliefs interpretation of Nash equilibrium. She allows players to use acts in the sense of Anscombe-Aumann and players are uncertainty-averse over such acts. In an ambiguous act equilibrium, players play best responses as in Nash equilibrium, but under the generalized framework. A large class of ambiguity-averse preferences are covered. The possible priors for an ambiguous act are part of the players' preferences in her setup. Bade then adds some appropriate consistency properties (agreement on null events) to exclude unreasonably divergent beliefs, and she imposes the rather strong assumption that preferences are strictly monotone, following [START_REF] Klibanoff | Stochastically independent randomization and uncertainty aversion[END_REF] here. This excludes beliefs on the boundary of strategy sets; such degenerated beliefs are sometimes important, though. For example, it excludes Ellsberg urns with full ambiguity where it is only known that the probability for a red ball is between 0 and 1. Bade's main theorem establishes that under her assumptions, in two-person games the support of ambiguous act equilibria and the support of Nash equilibria coincide.

Note that [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF], [START_REF] Epstein | Preference, rationalizability and equilibrium[END_REF] and [START_REF] Azrieli | Uncertainty aversion and equilibrium existence in games with incomplete information[END_REF] (amongst others) have also defined games that have Anscombe-Aumann acts as strategies, but to different ends. [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] defines such a general game, then imposes Savage expected utility and analyses properties of correlated and subjective equilibrium. [START_REF] Epstein | Preference, rationalizability and equilibrium[END_REF] analyses games very similar to Bade's, but is mainly interested in rationalizability and iterated deletion of strictly dominated strategies in the generalized framework. [START_REF] Azrieli | Uncertainty aversion and equilibrium existence in games with incomplete information[END_REF] define an extension of an incomplete-information game.

In difference to Bade (2011b)'s setup, we let ambiguity be an objective instrument that is not derived from subjective preferences. Players can credibly commit to play an Ellsberg urn with a given and known degree of ambiguity. In Ellsberg games players use devices that create ambiguity, thus we extend the objective random devices interpretation of Nash equilibrium. The articles cited above impose non-expected utility representations derived from subjective preferences, like maxmin expected utility by [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF],

Choquet expected utility by [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF], or they fix only certain axioms to allow for a large class of ambiguity-averse preferences. To model the preferences in Ellsberg games we use the representation results by [START_REF] Gajdos | Attitude toward imprecise information[END_REF] on attitude towards imprecise information which capture the objective ambiguity we have in mind.

We want to mention some other approaches that introduce Knightian uncertainty into games which are not as closely related to the concept of Ellsberg games. [START_REF] Crawford | Equilibrium without independence[END_REF] was one of the earliest to define an equilibrium concept which incorporated the relaxations of Von Neumann and Morgenstern axioms which were introduced to explain phenomena like the Ellsberg paradox. To this end, [START_REF] Crawford | Equilibrium without independence[END_REF] analyzes an equilibrium concept without the assumption of independence on the underlying preference relation. Stauber

(2011) considers incomplete information games with ambiguous beliefs and analyzes the robustness of equilibria of these games. Finally, [START_REF] Perchet | Nash equilibria with uncertainties; generalization of Lemke-Howson algorithm[END_REF] works in the context of repeated games with the possibility of monitoring. The information about the opponents' play is "ambiguously" disturbed and hence the notion of Nash equilibrium generalized. He also generalizes the Lemke-Howson algorithm to compute equilibria of two-player games.

Of course, the literature reviewed here is not exhaustive, but we hope to have mentioned all closely related concepts.

Immunization Against Strategic Ambiguity

The hedging effect of ambiguity aversion leads to what we call immunization against strategic ambiguity. 1 In normal form Ellsberg games players often have a strategy available that immunizes them against any ambiguity played by their opponents. In this chapter we look at these immunization strategies more closely. We start by giving an example and then, after defining immunization strategies, calculate the immunization strategies of a large class of 2 × 2 games. We then provide some examples which show that in certain games players immunize themselves against ambiguity by playing their maximin strategy. To generalize this observation, we prove under which circumstances in general two-player games the immunization strategy is the maximin strategy of the players. The last section uses the results in the chapter to characterize the role of immunization strategies in Ellsberg equilibria. We see that in many 2 × 2 games, Ellsberg equilibria are bounded by the immunization and the Nash equilibrium strategy.

Example of an Immunization Strategy

To get an intuition for immunization we take a slightly modified version of Matching Pennies as our example. The results generalize to a large class of two-person 2 × 2 games as we show below in Proposition 2.3. The payoff matrix for the modified Matching Pennies

game is in Figure 2.1. Player 1 Player 2 HEAD T AIL HEAD 3, -1 -1, 1 T AIL -1, 1 1, -1 Figure 2.1: Modified Matching Pennies I.
In our modified version of Matching Pennies, the unique Nash equilibrium is that player 1 mixes uniformly over his strategies, and player 2 mixes with (1/3, 2/3). This yields the equilibrium payoffs 1/3 and 0. One might guess that one can get an Ellsberg equilibrium where both players use a set of probability measures around the Nash equilibrium distribution as their strategy. This is, somewhat surprisingly, at least to us, not true.

The crucial point to understand here is the following. Players can immunize themselves against ambiguity; in the modified Matching Pennies example, player 1 can use the mixed strategy (1/3, 2/3) to make himself independent of any ambiguity used by the opponent.

Indeed, with this strategy, his expected payoff is 1/3 against any mixed strategy of the opponent, and a fortiori against Ellsberg strategies as well. This strategy is also the unique best reply of player 1 to Ellsberg strategies with ambiguity around the Nash equilibrium, in particular, such strategic ambiguity is not part of an Ellsberg equilibrium.

Let us explain this more formally. An Ellsberg strategy for player 2 can be identified

with an interval [Q 0 , Q 1 ] ⊆ [0, 1] where Q ∈ [Q 0 , Q 1 ]
is the probability to play HEAD.

Suppose player 2 uses many probabilities around 1/3, so Q 0 < 1/3 < Q 1 . The minimal expected payoff for player 1 when he uses the mixed strategy with probability P for HEAD is then

U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 3P Q -P (1 -Q) -(1 -P )Q + (1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 6P Q -2P -2Q + 1 = min {Q 0 (6P -2), Q 1 (6P -2)} + 1 -2P =          Q 1 (6P -2) + 1 -2P if P < 1/3 , 1/3 if P = 1/3 , Q 0 (6P -2) + 1 -2P if P > 1/3 . (2.1)
We plot the payoff function in Figure 2.2 for Q 0 = 1/4 and Q 1 = 1/2. By choosing the mixed strategy P = 1/3, player 1 becomes immune against any ambiguity and ensures the (Nash) equilibrium payoff of 1/3. If there was an Ellsberg equilibrium with P 0 < 1/2 < P 1 and Q 0 < 1/3 < Q 1 , then the minimal expected payoff would be below 1/3. Hence, such Ellsberg equilibria do not exist.

Immunization Strategies

We call the strategy that renders the player immune against any ambiguity used by the opponents immunization strategy. Definition 2.1.

Let G = �N, (S i ), (u i )� be a normal form game. A mixed strategy Pi ∈ ∆S i for player i is called an immunization strategy, if there exists a v i ∈ R such that U i ( Pi , P -i ) = v i for all sets P -i ⊆ ∆S -i .
Such immunization plays frequently a role in two-person games, and it need not always be the Nash equilibrium strategy that is used to render oneself immune. In fact, Nash equilibrium and immunization are in some sense opposite concepts: with a Nash equilibrium strategy the player wants to make his opponent indifferent between all his strategies, and with an immunization strategy the player wants to make himself indifferent. Consider, e.g., the slightly changed payoff matrix in Figure 2.3.

Player 1

Player 2

HEAD T AIL HEAD 1, -1 -1, 1 T AIL -2, 1 1, -1 Figure 2.3: Modified Matching Pennies II.
In the unique Nash equilibrium, player 1 still plays both strategies with probability 1/2 (to render player 2 indifferent); however, in order to be immune against Ellsberg strategies, he has to play HEAD with probability 3/5. Then his payoff is -1/5 regardless of what player 2 does. This strategy does not play any role in Nash equilibrium, but note that the payoff to the strategy is the same as in the unique Nash equilibrium in which player 1 plays HEAD with probability 1/2 and player 2 with probability 2/5. In fact, every strategy of player 1 in the interval [1/2, 3/5] yields the same maxmin payoff, which makes this strategy a candidate for an Ellsberg equilibrium strategy.

For solving an Ellsberg game it is helpful to plot the (minimal expected) payoff function as in Figure 2.2. Nevertheless, to understand the general properties of Ellsberg games and their relation to the standard analysis of normal form games, we also have a look at the expected payoff function of the players.

In Figure 2.4 we show player 1's expected payoff function for Modified Matching Pennies I in two different perspectives. Studying the plots, one can see that the payoff is indeed independent of Q at P = 1/3, and, furthermore, P = 1/3 yields the maximal minimal expected payoff for player 1. So P = 1/3 is player 1's maximin strategy. We say more about the general relation between immunization strategies and maximin strategies in Section 2.4.

(P, Q) = 6P Q -2P -2Q + 1, in two different perspectives.
Because of the, in some sense, inverse relation to Nash equilibrium strategies, the necessary conditions for the existence of completely mixed immunization strategies are similar to the existence of completely mixed Nash equilibrium. However, for immunization strategies these are also sufficient. We call a strategy completely mixed if it puts positive probability on every pure strategy. Denote with A (B) the payoff matrix of player 1 (2). Then player 1(2) has an immunization strategy if and only if no convex combination of columns (rows)

of A (B) dominates convex combinations of other columns (rows). However, the following example shows that existence of completely mixed Nash equilibrium strategies does not imply existence of completely mixed immunization strategies.

Example 2.2. Consider the game with the following payoff matrix for player 1:

Player 1 U 2 -1 D 1 1
As Nash equilibrium strategy player 2 would choose a convex combination of columns such that player 1 is indifferent between U and D, that is, 2/3,1/3). But to immunize himself, player 1 plays U with zero probability and D with probability one, which is not a completely mixed strategy.

(Q * , 1 -Q * ) = (

Immunization Strategies of General 2 × 2 Games

We calculate the immunization strategies of a general two-person 2×2 game. Immunization strategies ( P , 1-P ) and ( Q, 1-Q) for players 1 and 2 are fully described by the probability P to play U and Q to play L. We use this abbreviated notation also for Nash equilibrium

(P * , Q * ) := ((P * , 1 -P * ), (Q * , 1 -Q * )).
We have the following proposition.

Proposition 2.3. Let G = �{1, 2}, (S i ), (u i )� be a normal form game with payoff matrix Player 1 Player 2 L R U a, e b, f D c, g d, h
Then the immunization strategy P of player 1 is

P = d -c a -b -c + d ,
and the immunization strategy Q of player 2 is

Q = h -f e -f -g + h , if 0 ≤ P ≤ 1 and 0 ≤ Q ≤ 1. That P and Q lie between 0 and 1 is fulfilled when d -c ≥ 0 > b -a or d -c ≤ 0 < b -a and, respectively, h -f ≥ 0 > g -e or h -f ≤ 0 < g -e.
Proof. We show this only for player 1, the calculation for player 2 is analog. Let P ∈

[P 0 , P 1 ] denote the probability with which player 1 plays U , and

Q ∈ [Q 0 , Q 1 ]
U 1 (P, [Q 0 , Q 1 ]) = min Q∈[Q 0 ,Q 1 ] aP Q + bP (1 -Q) + c(1 -P )Q + d(1 -P )(1 -Q) = min Q∈[Q 0 ,Q 1 ] (a -b -c + d)P Q + (b -d)P + (c -d)Q + d = min Q∈[Q 0 ,Q 1 ] Q((a -b -c + d)P + c -d) + (b -d)P + d =      Q 0 ((a -b -c + d)P + c -d) + (b -d)P + d if P > d-c a-b-c+d , (b-d)(d-c) a-b-c+d + d if P = d-c a-b-c+d , Q 1 ((a -b -c + d)P + c -d) + (b -d)P + d if P < d-c a-b-c+d .
Under the assumption that 0

≤ P ≤ 1, the utility function is constant for all [Q 0 , Q 1 ] if and only if P = d-c a-b-c+d .
The completely mixed Nash equilibrium of the above general game, if it exists, is

(P * , Q * ) = � h -g e -f -g + h , d -b a -b -c + d � .
It is noticeable that the immunization strategies are similar to the opponent's Nash equilibrium strategy. In fact, some of the standard two-person 2 × 2 games have the property that the immunization strategies are exactly the Nash equilibrium strategies of the opponent. This is the case when c = b for player 1 and f = g for player 2. This is what we will assume in Proposition 3.7 on the Ellsberg equilibria of general competitive games in order to make the exposition of the result more comprehensive.

Immunization as Maximin Strategy

An immunization strategy does not only make a player i indifferent about any ambiguity used by the opponents, it makes him also indifferent to any pure or mixed strategies the opponents might play. Playing the immunization strategy a player can assure himself some payoff v i that is independent of the opponents' behavior. If this payoff v i is not

Immunization as Maximin Strategy

less than the maximal Nash equilibrium payoff v * i , it can be argued that playing the immunization strategy is smarter than playing the Nash equilibrium strategy. The Nash equilibrium strategy bears the risk of yielding a smaller payoff, if, e.g., some opponent makes a mistake.

A similar phenomenon had been discussed in [START_REF] Aumann | Some thoughts on the minimax principle[END_REF]. The authors notice that in some (and not only in zero-sum!) games the maximin payoff is the same as the Nash equilibrium payoff. In a very instructive paper, [START_REF] Pruzhansky | Some interesting properties of maximin strategies[END_REF] analyses this observation further and finds, that this is the case if and only if there exists a completely mixed Nash equilibrium and the maximin strategy is an equalizer. What the author calls an equalizer is the same as an immunization strategy, only for pure and mixed strategies and not for Ellsberg strategies, that is a strategy Pi for which the expected payoff is constant for all strategies P -i ∈ ∆S -i . It is immediate that every equalizer strategy is an immunization strategy. If the expected payoff u i ( Pi , P -i ) is constant for all P -i ∈ ∆S -i , then min P -i ∈P -i u i ( Pi , P -i ) is also constant for all P -i ⊂ ∆S -i . The converse is also true, since if min P -i ∈P -i u i ( Pi , P -i ) is constant for some fixed Pi and all P -i ⊂ ∆S -i , then it is also constant for all P -i ∈ ∆S -i .

We now state the definition of maximin strategies and discuss existence and uniqueness.

Definition 2.4. Let G = �N, (S i ), (u i )� be a normal form game. A mixed strategy P i ∈ ∆S i is a maximin strategy for player i ∈ N if it maximizes his minimal expected payoff, that is P i ∈ arg max

P � i ∈∆S i min P -i ∈∆S -i u i (P � i , P -i ) .
We know that maximin strategies always exist, since u i is continuous and ∆S i compact and convex for all i ∈ N . But maximin strategies are in general not unique as the following example shows. We look at two different 2 × 2 games, one with many maximin strategies, the other with a unique one. The non-uniqueness of the maximin strategy of u 1 in the following example is due to the fact that the Nash equilibrium strategy Q * of player 2 lies on the boundary, that is, Q * is not completely mixed.

Whenever we speak about 2 × 2 games, mixed strategies are represented by one variable per player, typically in our notation (P, 1 -P ) for player 1 and (Q, 1 -Q) for player 2.

The general payoff function u 1 (P, Q) of player 1 with payoff matrix

A = a b c d is then u 1 (P, Q) = aP Q + bP (1 -Q) + c(1 -P )Q + d(1 -P )(1 -Q) = (a -b -c + d)P Q + (b -d)P + (c -d)Q .
Example 2.5 (Unique and non-unique maximin strategies). We look at two payoff functions for player 1,

u 1 (P, Q) = 2P Q -Q and u � 1 (P, Q) = 2P Q -Q -P ,
which are for example the payoff functions of the following payoff matrices.

u 1 : L R U 1 0 D -1 0 u � 1 : L R U 0 -1 D -1 0 Figure 2
.5: Two games with unique and non-unique maximin strategies.

The functions are plotted in Figure 2.6 and 2.7, respectively, each time shown in the two perspectives P and Q. We see that in Figure 2.6 the indifference strategy of player 2 is

Figure 2.6: u 1 (P, Q) = 2P Q -Q.
zero, thus not completely mixed. The minimum over Q is 0 whenever P ≥ 1/2 and else 2P -1, thus the maximin strategy is not unique. In Figure 2.7 the indifference strategy is

Q * = 1/2.
The minimum over Q is -P when P ≥ 1/2 and P -1 else. This function is maximized over P when P = 1/2. This is the unique maximin strategy of player 1.

Finally, note that immunization strategies are not always maximin strategies as one can

0 0.25 0.5 0.75 1 -1 -0.5 0 P Q 0 0.25 0.5 0.75 1 -1 -0.5 0 P Q Figure 2.7: u � 1 (P, Q) = 2P Q -Q -P .
see immediately in the following example. The game does not have a completely mixed Nash equilibrium. We prove in the next section that this is a necessary condition for a game to have immunization strategies that are maximin.

Example 2.6. In the game in Figure 2.8 below, player 1's maximin strategy is U , and his only immunization strategy is D.

Player 1 Player 2 L R U 2, 1 1, 0 D 0, 0 0, 1
Figure 2.8: Game in which the immunization strategy is not the maximin strategy.

Immunizing is Playing Against Your Zero-Sum-Self

Theorem 2.8 in this section relates maximin strategies and immunization strategies for twoplayer games with n strategies for each player. We show that if a two-player game with such square payoff matrices has a completely mixed Nash equilibrium, then an immunization strategy is a maximin strategy.

We use the following notation. A two-person normal form game G is described by the payoff matrices A and B of player 1 and player 2, we write G = (A, B). We call G square when A and B are square matrices; in this section we only consider square games. The row vectors of A (B) are denoted by subscripts, a i (b i ) ∈ R n , and the column vectors by superscripts, a j (b j ) ∈ R n . In the following it will be convenient to write the expected utility of a mixed strategy P of player 1 with payoff matrix A as P AQ when player 2 plays mixed strategy Q. Transpose signs are suppressed.

A necessary condition for G to have a completely mixed Nash equilibrium is that no player has weakly or strictly dominated strategies. That is, no row (column) of A (B) is dominated by another row (column) or a convex combination of rows (columns). This condition can be expressed as follows. Let à be the (n + 1) × n-matrix consisting of the matrix A with an additional last column (1, . . . , 1), and B the n × (n + 1)-matrix with an additional last row (1, . . . , 1). Furthermore let k = (k, . . . , k, 1) and l = (l, . . . , l, 1) with k, l ∈ R. Then G has a completely mixed Nash equilibrium (P, Q) = (P * , Q * ) when it is a nonnegative solution to the two systems of linear equations

P B = k , ÃQ = l .
The existence of immunization strategies can be expressed analogously: no column (row) of matrix A (B) is dominated by another column (row) or a convex combination of columns (rows). Let ũ = (u, . . . , u, 1) and ṽ = (v, . . . , v, 1) with u, v ∈ R. Player 1 has an immunization strategy P = P in G when the vector is a nonnegative solution to the system P Ã = ũ , and player 2 has an immunization strategy Q = Q in G when the vector is a nonnegative solution to the system BQ = ṽ .

Note that for the existence of a completely mixed Nash equilibrium both solutions P * and Q * have to exist, whereas the immunization strategy is defined for a single player.

To a square game G we define associated zero-sum games G 1 and G 2 . G 1 is the game with payoff matrices (A, -A), G 2 the game with payoff matrices (-B, B). We first prove the following lemma.

Lemma 2.7. Let G be a square two-person normal form game, G 1 and G 2 its associated zero-sum games. If G has a completely mixed Nash equilibrium (P * , Q * ) and player 1

(2) has an immunization strategy

P ( Q), then G 1 (G 2 ) has a Nash equilibrium (P * 1 , Q * 1 ) ((P * 2 , Q * 2 )) where P * 1 = P and Q * 1 = Q * (P * 2 = P * and Q * 2 = Q).
Proof. We show this only for player 1, the case for player 2 follows analogously. If (P * , Q * ) is a completely mixed Nash equilibrium of G, then Q * solves the system ÃQ = l for some l ∈ R. Furthermore, if P is an immunization strategy of player 1, then P solves the system P Ã = ũ for some u ∈ R. Then P also solves P ( -A) = -u and is therefore a Nash equilibrium strategy for player 1 of the game G 1 . Now, we present and prove the main result of the section which relates immunization and maximin strategies of a square two-person normal form game.

Theorem 2.8. Let G be a square two-person normal form game with a completely mixed Nash equilibrium (P * , Q * ). If player 1 (2) has an immunization strategy P ( Q) in G, then P ( Q) is a maximin strategy of player 1 (2).

Proof. We prove the result only for player 1. P is an immunization strategy of player 1 in the game G if and only if it is a nonnegative solution to the system P Ã = ũ. Now by assumption there exists a completely mixed Nash equilbrium (P * , Q * ) of G, thus with Lemma 2.7 there exists a completely mixed Nash equilibrium (P * 1 , Q * 1 ) in G 1 . P * 1 is therefore a nonnegative solution to the system P ( -A) = k for some k ∈ R. Then P * 1 also solves the system P Ã = ( -k) and we see that P * 1 must be the immunization strategy P of player 1. Now, since G 1 is a zero-sum game, P is by the classic Minimax Theorem for zero-sum games (which we state in Theorem 3.15) also the maximin strategy in G 1 , that means P ∈ arg max

P min Q P AQ .
This condition only depends on the payoff matrix A of player 1, thus P (= P * 1 ) is the maximin strategy in any game where player 1 has the payoff matrix A, in particular in the game G.

To show the analog statement for player 2 we use the associated zero-sum game G 2 . Lemma 1 in [START_REF] Pruzhansky | Some interesting properties of maximin strategies[END_REF] proves a result very close to our Theorem 2.8, but with different means by making use of geometric properties of the various types of strategies.

The interpretation of the result gets an interesting twist when we use the associated zero-sum games as in our proof. A player who is immunizing himself against strategic ambiguity blindfolds himself and drops the strategic element of the game. Instead of playing the original game, he plays the associated zero-sum game which results from his own payoff matrix. Thus, he plays his optimal strategy P , the immunization strategy, against an imaginary self that receives the negative of his own payoff.

The assumption of a completely mixed Nash equilibrium in Theorem 2.8 is necessary.

In the following example we construct a game with a unique Nash equilibrium in pure strategies (which is also the unique Ellsberg equilibrium) that has completely mixed immunization strategies which are not maximin.

Example 2.9. Consider the game with payoff matrix Player 1

Player 2 L R U 1, 1 0, 2 D 2, 0 3, 3
It has the unique Nash equilibrium in pure strategies (D, R) which is also the unique

Concealing Nash Equilibrium and Maximin Behavior

We now look at the former case in detail, when the immunization strategy is not the Nash equilibrium strategy.

As long as the Nash equilibrium strategy of the opponent is the distribution used by the player to evaluate his maxmin expected utility, the player is indifferent between any mixed strategy, as the classical Principle of Indifference states. The Nash equilibrium strategy is only used when it is at the boundary of the Ellsberg strategy of the opponent, and the player is only indifferent between the strategies that lie between his immunization strategy and the boundary. Any strategy outside that set makes him change his utility evaluation to the other boundary of the opponents' Ellsberg strategy. Therefore the Ellsberg equilibrium strategy of the player cannot properly contain the immunization strategy in a similar way as it cannot properly contain the Nash equilibrium strategy. The only candidates are thus always between the immunization and the Nash equilibrium strategy, or between the Nash equilibrium strategy and the boundary of the unit interval, if the immunization strategy does not lie in between. In particular, in 2 × 2 coordination games, there is always an Ellsberg equilibrium in which both players use the set of probabilities between the Nash equilibrium and the maximin strategy, that is

([P * , M 1 ] , [Q * , M 2 ]) (compare Proposition 3.2).
In these games, the strategic use of ambiguity therefore implies concealment of Nash equilibrium and maximin behavior.

3 Strategic Use of Ambiguity in Two-Player Normal Form Games

) := ((P * , 1 -P * ), (Q * , 1 -Q * )) as a
short hand for the mixed Nash equilibrium of the game (if it exists). P is the probability that player 1 plays U (or his upper strategy with a sometimes different name), and Q is the probability that player 2 plays L (or his left strategy wit a sometimes different name), throughout the chapter.

Following Proposition 1.11, in order to determine the Ellsberg equilibria of a two-person 2 × 2 game, we have to find the arg max over P � ∈ ∆S 1 of player 1's minimal expected utility when P � is played against the equilibrium strategy Q of the opponent, and likewise for player 2. Thus, we start by calculating the minimal expected utility for both players, where we fix their respective mixtures P ∈ ∆S 1 , Q ∈ ∆S 2 and minimize over the possible Ellsberg strategies of their opponent,

U 1 (P, [Q 0 , Q 1 ]) = min{u 1 (P, Q 0 ), u 1 (P, Q 1 )} , U 2 ([P 0 , P 1 ] , Q) = min{u 2 (P 0 , Q), u 2 (P 1 , Q)} .
Typically this yields a piecewise linear function for each of the players, depending on the values of P and Q, respectively. Subsequently, we find the best response correspondences Finally a remark on notation. We often express relations of the value of several variables at once. In oder to keep things as short as possible we introduce some abbreviated notation.

B 1 ([Q 0 , Q 1 ])
For example, 0 ≤ P ≤ 1/2 and 0 ≤ Q ≤ 1/2 is abbreviated by 0 ≤ P, Q ≤ 1/2. In the same way, when a ≥ b and a ≥ c and d ≥ b and d ≥ c, we simply write a, d ≥ b, c.

Strategic Use of Ambiguity in Two-Person Coordination

How can ambiguity be used in standard coordination games? In this section we analyze 2 × 2 games in which both players want to match each other. We start with an example of a coordination game where both players get the same payoff when they coordinate successfully. In what follows, we present a proposition which characterizes the Ellsberg equilibria of general coordination games.

The section closes with further interesting examples for two-person coordination. The first example is a non-symmetric coordination game; we see that this change to asymmetric payoffs has important implications for the Ellsberg equilibria of the game. The second example is of the type where the players would like to coordinate on different equilibria, that is, games of the type Battle of the Sexes. Thirdly, we discuss the electronic mail game by [START_REF] Rubinstein | The electronic mail game: strategic behavior under "almost common knowledge[END_REF] and see that coordination on the good equilibrium is more likely when players can use Ellsberg strategies.

Example of a Symmetric Coordination Game

To start, we calculate the Ellsberg equilibria of a symmetric coordination game with the payoffs given by the matrix in Figure 3.1.

Player 1 Player 2 L R U 1, 1 0, 0 D 0, 0 3, 3 Figure 3.1: Symmetric coordination game.
Note that the Nash equilibria of the game are two pure strategy equilibria (U, L), and (D, R), and a mixed strategy equilibrium

(P * , Q * ) = ((3/4, 1/4) , (3/4, 1/4)) .
Let [P 0 , P 1 ] denote the set of probabilities with which player 1 plays U , and [Q 0 , Q 1 ] the set of probabilities with which player 2 plays L. [P 0 , P 1 ] and [Q 0 , Q 1 ] are thus Ellsberg strategies of player 1 and player 2, respectively. We first fix a distribution P ∈ [P 0 , P 1 ] for player 1 and consider his payoff function

U 1 (P, [Q 0 , Q 1 ]). Player 1 evaluates his minimal expected payoff given player 2 plays Q ∈ [Q 0 , Q 1 ]: U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 P Q + 3(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(4P -3) + 3(1 -P ) =          Q 0 (4P -3) + 3(1 -P ) if P > 3/4 , 3/4 if P = 3/4 , Q 1 (4P -3) + 3(1 -P ) if P < 3/4 . (3.1)
The maximum of (3.1) over P ∈ [P 0 , P 1 ] depends on the interval [Q 0 , Q 1 ]. Observe that at the boundaries the function reduces to

U 1 (0 , [Q 0 , Q 1 ]) = 3(1 -Q 1 ) ,
and

U 1 (1 , [Q 0 , Q 1 ]) = Q 0 .
We find in particular that

U 1 (P, [Q 0 , Q 1 ]) is constant for all P , when Q 0 = 3/4 = Q 1 .
We distinguish six cases where Q 0 and Q 1 can be situated within the interval [0, 1], all these yield different arg max of U 1 (P, [Q 0 , Q 1 ]). Figure 3.2 pictures these cases. The distinction depends on the mixed strategy Q 0 = 3/4 = Q 1 that makes player 1 indifferent between any choice of [P 0 , P 1 ] ⊆ ∆S 1 . This is the Nash equilibrium strategy of player 2.

We plot the payoff function in Figure 3.2. The numbering (1)-( 6) of the functions refer to (one possible) payoff function when Q 0 and Q 1 take different values. The values are listed below. In the same list we present player 1's best response analysis to the different

strategies [Q 0 , Q 1 ]: Q 0 > 3/4 : B 1 ([Q 0 , Q 1 ]) = 1 (1) Q 0 = 3/4 < Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [3/4, 1]} (2) Q 0 < 3/4 < Q 1 : B 1 ([Q 0 , Q 1 ]) = 3/4 (3) Q 0 < 3/4 = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 3/4]} (4) Q 1 < 3/4 : B 1 ([Q 0 , Q 1 ]) = 0 (5) Q 0 = 3/4 = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 1]} (6)
As this is a symmetric game, we get the same behavior for player 2. We fix the probability Q ∈ ∆S 2 and derive the payoff functions U 2 ([P 0 , P 1 ] , Q) of player 2. Player 2 evaluates his minimal expected payoff given player 1 plays P ∈ [P 0 , P 1 ].

U 2 ([P 0 , P 1 ] , Q) = min P 0 ≤P ≤P 1 P Q + 3(1 -P )(1 -Q) = min P 0 ≤P ≤P 1 P (4Q -3) + 3(1 -Q) =          P 0 (4Q -3) + 3(1 -Q) if Q > 3/4 , 3/4 if Q = 3/4 , P 1 (4Q -3) + 3(1 -Q) if Q < 3/4 . (3.2)
Analogue to the case considered before, the maximum of (3.2) over Q ∈ [0, 1] depends on the interval [P 0 , P 1 ]. The plot of U 2 ([P 0 , P 1 ] , Q) is identical to the plot in Figure 3.2. We thus get the following best responses of player 2:

P 0 > 3/4 : B 2 ([P 0 , P 1 ]) = 1 P 0 = 3/4 < P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [3/4, 1]} P 0 < 3/4 < P 1 : B 2 ([P 0 , P 1 ]) = 3/4 P 0 < 3/4 = P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, 3/4]} P 1 < 3/4 : B 2 ([P 0 , P 1 ]) = 0 P 0 = 3/4 = P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, 1]}
To find the Ellsberg equilibria, we check which of the six cases lead to intersections of the best response correspondences. We explain each of the six cases in turn.

(1) If player 2 plays Q 0 > 3/4, player 1 maximizes his payoff function

U 1 ([P 0 , P 1 ] , [Q 0 , Q 1 ])
by playing P 0 = P 1 = 1 and then again the best response of player 2 is

Q 0 = Q 1 =
1. This is an Ellsberg equilibrium. It coincides with the Nash equilibrium in pure strategies (U, L).

(2) If player 2 plays

Q 0 = 3/4 < Q 1 , player 1's best response is [P 0 , P 1 ] ⊆ [3/4, 1]. If
he chooses the left border P 0 > 3/4, player 2 can improve by playing Q 0 = Q 1 = 1, so this would not be an equilibrium. But when player 1 chooses [P 0 , P 1 ] = [3/4, P 1 ] with 3/4 ≤ P 1 ≤ 1 and player 2 chooses

[Q 0 , Q 1 ] = [3/4, Q 1 ] with 3/4 ≤ Q 1 ≤ 1
out of his best response set, either of the players can improve by deviating. Thus,

([3/4, P 1 ] , [3/4, Q 1 ]
) where 3/4 ≤ P 1 ≤ 1 and 3/4 ≤ Q ≤ 1 is an Ellsberg equilibrium.

(3) If player 2 plays Q 0 < 3/4 < Q 1 , player 1's best response is the standard mixed strategy P 0 = P 1 = 3/4. Player 2 stays with his strategy because he is indifferent between all strategies. This is an Ellsberg equilibrium: (5) If player 2 plays Q 1 < 3/4, player 1's best response is P 0 = P 1 = 0. Thus, player 2 chooses Q 0 = Q 1 = 0, this is an Ellsberg equilibrium. It coincides with the Nash equilibrium in pure strategies (D, R).

(3/4, [Q 0 , Q 1 ]) , where Q 0 ≤ 3/4 ≤ Q 1 . (4) If player 2 plays Q 0 < 3/4 = Q 1 ,
(6) Lastly, if player 2 plays

Q 0 = 3/4 = Q 1 , player 1's payoff function is constant at 3/4, thus he is indifferent between all subsets [P 0 , P 1 ] ⊆ [0, 1].
If player 1 plays P 0 ≤ 3/4 ≤ P 1 , this is an Ellsberg equilibrium, ([P 0 , P 1 ] , 3/4), where P 0 ≤ 3/4 ≤ P 1 .

This equilibrium includes the possibility of player 1 choosing the Ellsberg strategy

P 0 = P 1 = 3/4 ∈ [0, 1]
. This play corresponds to the mixed strategy Nash equilibrium

P * .
Summarizing, we get the following proposition.

Proposition 3.1. In the symmetric coordination game the Ellsberg equilibria are of the form (U, L) and (D, R) ,

([3/4, P 1 ] , [3/4, Q 1 ]) , where 3/4 ≤ P 1 , Q 1 ≤ 1 , ([P 0 , 3/4] , [Q 0 , 3/4]) , where 0 ≤ P 0 , Q 0 ≤ 3/4 , (3/4, [Q 0 , Q 1 ]) , where Q 0 ≤ 3/4 ≤ Q 1 ,
and ([P 0 , P 1 ] , 3/4) , where P 0 ≤ 3/4 ≤ P 1 .

Looking at the Ellsberg equilibria, we see that player 1 and player 2 can coordinate to use ambiguity. They use the Nash equilibrium to bound their Ellsberg strategies from below or from above. Another equilibrium behavior is that one player uses ambiguity and the other uses his immunization strategy by playing U (or R, respectively) with probability 3/4. In the symmetric coordination game the mixed Nash equilibrium strategies coincide with the immunization strategies, we therefore get Ellsberg equilibria with full ambiguity.

This does not happen in general coordination games, as we will see later.

Ellsberg Equilibria in General Coordination Games

We generalize the observations of the preceding section to general 2 × 2 coordination games. We observe two different types of Ellsberg equilibria in these games: in general, when the Nash equilibrium strategies P * and Q * of player 1 and 2 are different probability distributions, there exists an Ellsberg equilibrium in which both player use ambiguity. The sets of probability distributions used in equilibrium are bounded by the Nash equilibrium distribution and the distribution that assures immunization against any ambiguity used by the other player.

In the special case where The conditions on the payoffs allow for different types of symmetric and non-symmetric coordination games. Let P * and Q * denote the Nash equilibrium strategies for player 1 and 2, respectively. The Nash equilibrium strategies can be computed using standard methods, they are

P * = Q * the
P * = h -g e -f -g + h and Q * = d -b a -b -c + d .
The immunization strategies of each player are denoted by M 1 and M 2 . Furthermore,

player 1's Ellsberg strategy is denoted [P 0 , P 1 ], player 2's [Q 0 , Q 1 ].
Then we have the following proposition.

Proposition 3.2. Let P * , Q * denote the mixed Nash equilibrium strategies, and M 1 , M 2 the immunization strategies of player 1 and 2, respectively. Then the Ellsberg equilibria of the general coordination game are of the following form.

M 1 ≤ P * M 1 ≥ P * M 2 ≤ Q * ([P * , P 1 ] , [Q * , Q 1 ]) ([P * , P 1 ] , [Q 0 , Q * ]) P 1 ≤ M 1 and M 2 ≤ Q 0 M 2 ≥ Q * ([P 0 , P * ] , [Q * , Q 1 ]) ([P 0 , P * ] , [Q 0 , Q * ]) M 1 ≤ P 0 and Q 1 ≤ M 2 If M 1 = P * or M 2 = Q * ,
then an additional type of Ellsberg equilibria arises,

(P * , [Q 0 , Q 1 ]), where Q 0 ≤ Q * ≤ Q 1 when M 1 = P * ; ([P 0 , P 1 ] , Q * ), where P 0 ≤ P * ≤ P 1 when M 2 = Q * .
In any case, the pure and mixed Nash equilibria, that is (U, L), (D, R) and (P * , Q * ), are also Ellsberg equilibria.

Proof. The Nash equilibrium strategies follow from the usual analysis. To calculate the Ellsberg equilibria of the general coordination game (Figure 3.3), we first derive the utility functions of player 1 and player 2. Due to the assumption that a, d ≥ b, c and e, h ≥ f, g, with abc + d � = 0 and efg + h � = 0, the denominators abc + d and efg + h are positive. This reflects in the payoff functions that the game is a coordination game;

player 1 uses Q 0 as a minimizer when P > d-c a-b-c+d , and, the same way, player 2 uses P 0 as a minimizer when

Q > h-f e-f -g+h . U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 aP Q + bP (1 -Q) + c(1 -P )Q + d(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q((a -b -c + d)P + c -d) + (b -d)P + d =          Q 0 ((a -b -c + d)P + c -d) + (b -d)P + d if P > d-c a-b-c+d , (b-d)(c-d) a-b-c+d + d if P = d-c a-b-c+d , Q 1 ((a -b -c + d)P + c -d) + (b -d)P + d if P < d-c a-b-c+d , U 2 ([P 0 , P 1 ] , Q) = min P 0 ≤P ≤P 1 eP Q + g(1 -P )Q + f P (1 -Q) + h(1 -P )(1 -Q) = min P 0 ≤P ≤P 1 P ((e -f -g + h)Q + f -h) + (g -h)Q + h =          P 0 ((e -f -g + h)Q + f -h) + (g -h)Q + h if Q > h-f e-f -g+h , (g-h)(h-f ) e-f -g+h + h if Q = h-f e-f -g+h , P 1 ((e -f -g + h)Q + f -h) + (g -h)Q + h if Q < h-f e-f -g+h .
We see that

M 1 = d -c a -b -c + d , respectively M 2 = h -f e -f -g + h
are the immunization strategies of player 1 and 2, respectively. At the boundaries the utility functions become

U 1 (0 , [Q 0 , Q 1 ]) = (c -d)Q 1 + d , U 2 ([P 0 , P 1 ] , 0) = (f -h)P 1 + h , U 1 (1 , [Q 0 , Q 1 ]) = (a -b)Q 0 + b , U 2 ([P 0 , P 1 ] , 1) = (e -g)P 0 + g .
The payoff function of player 1 is constant for all P , when Q 1 = Q 0 = Q * , and the payoff function of player 2 is constant for all Q, when P 1 = P 0 = P * . We calculate the best response correspondences:

Q 0 > Q * : B 1 ([Q 0 , Q 1 ]) = 1 (1) Q 0 = Q * < Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [M 1 , 1]} (2) Q 0 < Q * < Q 1 : B 1 ([Q 0 , Q 1 ]) = M 1 (3) Q 0 < Q * = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, M 1 ]} (4) Q 1 < Q * : B 1 ([Q 0 , Q 1 ]) = 0 (5) Q 0 = Q * = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 1]} (6) P 0 > P * : B 2 ([P 0 , P 1 ]) = 1 P 0 = P * < P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [M 2 , 1]} P 0 < P * < P 1 : B 2 ([P 0 , P 1 ]) = M 2 P 0 < P * = P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, M 2 ]} P 1 < P * : B 2 ([P 0 , P 1 ]) = 0 P 0 = P * = P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, 1]}
To find the Ellsberg equilibria, we look at the different cases in turn.

(1) Q 0 > Q * : player 1 responds P 0 = P 1 = 1. If also player 2 chooses

Q 0 = Q 1 = 1, this
is the Ellsberg equilibrium that is identical to the pure Nash equilibrium (U, L).

(2)

� d-b a-b-c+d , Q 1 � ⇒ [P 0 , P 1 ] ⊆ � d-c a-b-c+d , 1
� , then, depending on the size of M 1 in relation to P * , player 1 has the following choices:

• d-c a-b-c+d ≤ h-g e-f -g+h : player 1 can play either - � h-g e-f -g+h , P 1 � , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � h-f e-f -g+h , 1 � .
Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices:

* h-f e-f -g+h ≤ d-b a-b-c+d : � d-b a-b-c+d , Q 1 � , we thus have here the Ellsberg equilibrium �� h -g e -f -g + h , P 1 � , � d -b a -b -c + d , Q 1 �� , that is ([P * , P 1 ] , [Q * , Q 1 ]) . * h-f e-f -g+h ≥ d-b a-b-c+d : for all [Q 0 , Q 1 ]
the best response is P 0 = P 1 = 1, thus no Ellsberg equilibrium arises.

or

� P 0 , h-g e-f -g+h � with d-c a-b-c+d ≤ P 0 ≤ h-g e-f -g+h , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � 0, h-f e-f -g+h

�

. Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices:

* h-f e-f -g+h ≥ d-b a-b-c+d : � d-b a-b-c+d , Q 1 � with Q 1 ≤ h-f e-f -g+h , then the best response of player 1 is [P 0 , P 1 ] ⊆ � d-c a-b-c+d , 1 �
. We thus have here the Ellsberg equilibrium

�� P 0 , h -g e -f -g + h � , � d -b a -b -c + d , Q 1 �� , where d -c a -b -c + d ≤ P 0 and Q 1 ≤ h -f e -f -g + h , that is ([P 0 , P * ] , [Q * , Q 1 ]), where M 1 ≤ P 0 and Q 1 ≤ M 2 . * h-f e-f -g+h ≤ d-b a-b-c+d : for all [Q 0 , Q 1 ] the best response is P 0 = P 1 = 0, thus no Ellsberg equilibrium arises. • d-c a-b-c+d ≤ h-g
e-f -g+h : to any [P 0 , P 1 ] the best response is Q 0 = Q 1 = 1, thus no Ellsberg equilibrium arises.

(3) Q 0 < Q * < Q 1 : player 1 responds with P 0 = P 1 = M 1 . Only when M 1 = P * , player 1 sticks to his strategy and we get the equilibrium (P

* , [Q 0 , Q 1 ]), where Q 0 < Q * < Q 1 . (4) � Q 0 , d-b a-b-c+d � ⇒ [P 0 , P 1 ] ⊆ � 0, d-c a-b-c+d �
, then, depending on the size of M 1 in relation to P * , player 1 has the following choices:

• d-c a-b-c+d ≥ h-g e-f -g+h : player 1 can play either - � P 0 , h-g e-f -g+h � , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � 0, h-f e-f -g+h � .
Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices:

* h-f e-f -g+h ≥ d-b a-b-c+d : � Q 0 , d-b a-b-c+d

�

, we thus have here the Ellsberg equilibrium

�� P 0 , h -g e -f -g + h � , � Q 0 , d -b a -b -c + d �� , that is ([P 0 , P * ] , [Q 0 , Q * ]) . * h-f e-f -g+h ≤ d-b a-b-c+d : for all [Q 0 , Q 1 ]
the best response is P 0 = P 1 = 0, thus no Ellsberg equilibrium arises.

or

� h-g e-f -g+h , P 1 � with h-g e-f -g+h ≤ P 1 ≤ d-c a-b-c+d , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � h-f e-f -g+h , 1 �
. Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices:

* d-b a-b-c+d ≥ h-f e-f -g+h : � Q 0 , d-b a-b-c+d � with h-f e-f -g+h ≤ Q 0 , then the best response of player 1 is [P 0 , P 1 ] ⊆ � 0, d-c a-b-c+d � .
We thus have here the Ellsberg equilibrium

�� h -g e -f -g + h , P 1 � , � Q 0 , d -b a -b -c + d �� ,
where

P 1 ≤ d -c a -b -c + d and h -f e -f -g + h ≤ Q 0 , that is ([P * , P 1 ] , [Q 0 , Q * ]), where P 1 ≤ M 1 and M 2 ≤ Q 0 . * d-b a-b-c+d ≤ h-f e-f -g+h : for all [Q 0 , Q 1 ] the best response is P 0 = P 1 = 1, thus no Ellsberg equilibrium arises. • d-c a-b-c+d ≤ h-g
e-f -g+h : to any [P 0 , P 1 ] the best response is Q 0 = Q 1 = 1, thus no Ellsberg equilibrium arises.

(5) Q 1 < Q * : player 1 responds P 0 = P 1 = 0, if player 2 chooses Q 0 = Q 1 = 0 this is the Ellsberg equilibrium that is identical to the pure Nash equilibrium (D, R).

(6) Q 0 = Q * = Q 1 : player 1 responds with [P 0 , P 1 ] ⊆ [0, 1]. Only when Q * = M 2 , player 2
sticks to his strategy and we get the equilibrium ([P 0 , P 1 ] , Q * ), where P 0 < P * < P 1 .

Further Examples of Two-Person Coordination

We now calculate the Ellsberg equilibria of three other classes of coordination games. All these games are covered by Proposition 3.2 and can thus be simply read off the table, but to better understand the derivation of the Ellsberg equilibria and their mathematical structure, we present a detailed analysis. We calculate the Ellsberg equilibria of two nonsymmetric coordination games, of the game of Battle of the Sexes, and of Rubinstein's electronic mail game.

Two Non-Symmetric Coordination Games

We consider two examples of coordination games which are less symmetric than the game in Figure 3.1. The first non-symmetric example of a coordination game has the payoff matrix given in Figure 3.4. This analysis is interesting, because the game does not allow for the same use of ambiguity in equilibrium as the very symmetric example which we considered in the beginning. We see that as soon as the players are in some kind of opposition to each other, the use of ambiguity gets more interesting. The Nash equilibria

(P, [Q 0 , Q 1 ]) for a fixed P ∈ ∆S 1 is U 1 (P, [Q 0 , Q 1 ]) =          Q 1 (5P -2) + 2 -P if P < 2/5 , 8/5 if P = 2/5 , Q 0 (5P -2) + 2 -P if P > 2/5 . (3.3) U 1 (P, [Q 0 , Q 1 ]) is constant at 8/5 when U 1 (0 , [Q 0 , Q 1 ]) = 2 -2Q 1 = 8/5 = 1 + 3Q 0 = U 1 (1 , [Q 0 , Q 1 ]) ,
this is the case when

Q 1 = Q 0 = Q * = 1/5.
Thereby we get the following best response correspondence for player 1:

Q 0 > 1/5 : B 1 ([Q 0 , Q 1 ]) = 1 (1) Q 0 = 1/5 < Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [2/5, 1]} (2) Q 0 < 1/5 < Q 1 : B 1 ([Q 0 , Q 1 ]) = 2/5 (3) Q 0 < 1/5 = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 2/5]} (4) Q 1 < 1/5 : B 1 ([Q 0 , Q 1 ]) = 0 (5) Q 0 = 1/5 = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 1]} (6)
Proposition 3.3. The Ellsberg equilibria in the non-symmetric coordination game I are of the form (U, L) and (D, R) ,

([P 0 , 3/7] , [1/5, Q 1 ])
, where 2/5 ≤ P 0 ≤ 3/7 and 1/5

≤ Q 1 ≤ 2/7 .
This example in contrast to the symmetric coordination game, Figure 3.1, shows that when the coordination game is not symmetric in the sense that P * = Q * , the ambiguity intervals in equilibrium are not bounded by 0 and 1, but by the mixed strategy Nash equilibrium or the immunization strategies (or both). Furthermore, there are no Ellsberg equilibria with one player creating as much ambiguity as he wants and the other player playing his Nash equilibrium strategy.

To see an application of Proposition 3.2, we quickly derive the Ellsberg equilibria of the game in Figure 3.5 using the proposition. The mixed strategy Nash equilibria are P * = 3/7 and Q * = 1/5, and the immunization strategies are M 1 = 4/5 and M 2 = 4/7. We see that P * < M 1 and Q * < M 2 . Thus, all Ellsberg equilibria are of the form ([P 0 , 3/7] , [Q 0 , 1/5]), in addition to the Ellsberg equilibria that are identical to the pure Nash equilibria (U, L) and (D, R).

Battle of the Sexes

The next example is the game of Battle of the Sexes. In this classic game, players wish to coordinate, but they have different preferences on which action they want to coordinate.

We choose the payoffs as in the matrix in Figure 3.6. A popular interpretation of this setting is a man and a woman who want to spend the evening together; the man (player 2) wants to take the woman (player 1) to watch a football match (L and U ), whereas the woman wants to take him to the opera (D and R). Both prefer spending the evening together instead of spending it alone.

Player 1

Player 2 L R U 1, 2 0, 0 D 0, 0 2, 1
Figure 3.6: Battle of the Sexes.

Note that the Nash equilibria of the game are two pure equilibria (U, L) and (D, R) and a mixed strategy Nash equilibrium (P * , Q * ) = ((1/3, 2/3) , (2/3, 1/3)). We use the same notation as before. To find the Ellsberg equilibria of the Battle of the Sexes game, we first

derive the payoff function U 1 (P, [Q 0 , Q 1 ]) of player 1. U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 P Q + 2(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(3P -2) + 2(1 -P ) =          Q 0 (3P -2) + 2(1 -P ) if P > 2/3 , 2/3 if P = 2/3 , Q 1 (3P -2) + 2(1 -P ) if P < 2/3 .
Observe that at the boundaries the function reduces to

U 1 (0 , [Q 0 , Q 1 ]) = 2(1 -Q 1 ) ,
and

U 1 (1 , [Q 0 , Q 1 ]) = Q 0 . We see that U 1 (P, [Q 0 , Q 1 ]) is constant at 2/3 for all P ∈ ∆S 1 when Q 0 = 2/3 = Q 1 .
Therefore we get the following best responses for player 1:

Q 0 > 2/3 : B 1 ([Q 0 , Q 1 ]) = 1 Q 0 = 2/3 < Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [2/3, 1]} Q 0 < 2/3 < Q 1 : B 1 ([Q 0 , Q 1 ]) = 2/3 Q 0 < 2/3 = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 2/3]} Q 1 < 2/3 : B 1 ([Q 0 , Q 1 ]) = 0 Q 0 = 2/3 = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 1]} then this is an Ellsberg equilibrium, i.e., ([1/3, P 1 ] , [Q 0 , 2/3])
, where 1/3 ≤ P 1 ≤ 2/3 and 1/3 ≤ Q 0 ≤ 2/3 .

(

) P 0 < 1/3 < P 1 ⇒ Q 0 = Q 1 = 1/3 ⇒ P 0 = P 1 = 0, this is not an equilibrium. ( 3
) P 0 < 1/3 = P 1 ⇒ [Q 0 , Q 1 ] ⊆ [0, 1/3] ⇒ P 0 = P 1 = 0, 4
this is not an equilibrium.

(

) P 1 < 1/3 ⇒ Q 0 = Q 1 = 0 ⇒ P 0 = P 1 = 0, this is the analog Ellsberg equilibrium to 5
(1), which now coincides with the pure strategy Nash equilibrium (D, R).

(

) P 0 = 1/3 = P 1 ⇒ [Q 0 , Q 1 ] ⊆ [0, 1]. 6
We consider two cases: if player 2 plays Q 0 < 2/3 < Q 1 , player 1's best response is P 0 = P 1 = 2/3, so this is not an equilibrium.

But if player 2 chooses Q 0 = P 0 = 2/3 and player 1 P 0 = P 1 = 1/3, this is an Ellsberg equilibrium. It coincides with the mixed strategy Nash equilibrium P * .

Proposition 3.4. The Ellsberg equilibria in the Battle of the Sexes game are of the following form:

(U, L) and (D, R) ,

([1/3, P 1 ] , [Q 0 , 2/3])
, where 1/3 ≤ P 1 ≤ 2/3 and 1/3 ≤ Q 0 ≤ 2/3 .

In the Ellsberg equilibrium the players may use Ellsberg strategies that include the whole interval [1/3, 2/3] on U and L. Again, as in the preceding non-symmetric coordination example, the equilibrium interval is bounded by the mixed Nash equilibrium strategy and the immunization strategy.

This can be interpreted as an approximation of the two interests present in the Battle of the Sexes game. Whereas in Nash equilibrium the players stick to the different distributions over their pure strategies which reflect the conflict of interest, in Ellsberg equilibrium they play identical strategies, each of them incorporating the Nash equilibrium strategy of the opponent: ([1/3, 2/3] , [1/3, 2/3]).

Rubinstein's Electronic Mail Game

As a final example we look at the following game, taken from [START_REF] Rubinstein | The electronic mail game: strategic behavior under "almost common knowledge[END_REF]. Originally, Rubinstein considered two games, one with a payoff matrix as in Figure 3.7, and a second one, where the payoffs for (U, L) and (D, R) are interchanged. He observes that the introduction of an imperfect communication mechanism (electronic mail which fails to reach the opponent with some small probability) leads to selection of the "better" equilibrium (D, R) (as opposed to (U, L)). We only look at one of these games, namely the one in Figure 3.7, and analyze the outcome of the game when players are allowed to use Ellsberg strategies. We find that the possibility to use Ellsberg strategies also leads to equilibrium selection, if so in a weak sense. In the only type of proper Ellsberg equilibrium of Rubinstein's electronic mail game, (U, L) can still be the outcome with some probability, but chances are small compared to occurrence of the outcome (D, R).

Player 1 Player 2 L R U 0, 0 0, -2 D -2, 0 1, 1
Figure 3.7: Rubinstein's electronic mail game.

We calculate the minimal expected utility function for both players,

U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 -2(1 -P )Q + (1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(3P -3) -P + 1 =    0 if P = 1 , Q 1 (3P -3) -P + 1 if P < 1 , U 2 ([P 0 , P 1 ] , Q) =    0 if Q = 1 , P 1 (3Q -3) -Q + 1 if Q < 1 .
At the boundaries the functions reduce to

U 1 (0 , [Q 0 , Q 1 ]) = -3Q 1 + 1 , U 2 ([P 0 , P 1 ] , 0) = -3P 1 + 1 , U 1 (1 , [Q 0 , Q 1 ]) = 0 , U 2 ([P 0 , P 1 ] , 1) = 0 .
The utility functions are constant when Q 1 = 1/3 and P 1 = 1/3, respectively. Thus, this is the important conclusion, in Ellsberg equilibrium, players must use 1/3 as upper bound of their equilibrium strategy, the lower bound is variable. To have 1/3 as the lower bound

will not lead to an Ellsberg equilibrium. With this reasoning we have, for example, the equilibrium

([0, 1/3] , [0, 1/3]) ,
in which both players play (U, L) only with a probability between 0 and 1/3. Playing this Ellsberg equilibrium strategy thus helps to coordinate on the good equilibrium (D, R).

Note that there is no Ellsberg equilibrium that facilitates coordination on (U, L) in the same way, because an Ellsberg strategy with lower bound 1/3 (and upper bound not equal to 1/3) can never be a best response. These Ellsberg equilibria can of course also be read off the table in Proposition 3.2. We have the following proposition.

Proposition 3.5. The Ellsberg equilibria of Rubinstein's electronic mail game have the following form:

(U, L) and (D, R) ,

and the proper Ellsberg equilibria

([P 0 , 1/3] , [Q 0 , 1/3]) ,
where 0 ≤ P 0 , Q 0 ≤ 1/3 .

Strategic Use of Ambiguity in Two-Person Conflicts

Our approach to games has its most natural and fruitful applications to conflicts where players are at least to some degree in opposition to each other. We start this section by discussing a modified version of Matching Pennies. We then provide a general analysis of 2 × 2 conflict games, where for reasons of clarity we first calculate the equilibria of a more restricted case, before we prove the more general result. The general analysis will be followed by the derivation of Ellsberg equilibria of a standard symmetric Matching

Pennies game and a non-symmetric conflict game.

While our predictions are broader than the classical Nash equilibrium, they remain restrictive, and, at least in principle, testable. Our results allow to explain the experimental findings of [START_REF] Goeree | Ten little treasures of game theory and ten intuitive contradictions[END_REF], who show that humans tend to deviate from Matching Pennies in asymmetric modified Matching Pennies games, but tend to play Nash equilibrium in symmetric Matching Pennies. This corresponds and is consistent with our Ellsberg equilibria. We present and explain the experiment in Section 4.1 in the next chapter.

Example of a Conflict Game: Modified Matching Pennies

In the following game the standard Matching Pennies game is modified in an asymmetric way such that one player gets higher payoff in one strategy profile. All other payoffs remain the same. The game has the payoff matrix given in Figure 3.8. Observe that the modification is solely an increase in player 1's payoff in case (U, L) is played. Note that Player 1

Player

2 L R U 3, -1 -1, 1 D -1, 1 1, -1 Figure 3.8: Modified Matching Pennies.
there is a unique mixed strategy Nash equilibrium

(P * , Q * ) = ((1/2, 1/2) , (1/3, 2/3))
that yields expected utility u 1 (P * , Q * ) = 1/3 and u 2 (P * , Q * ) = 0 for player 1 and player 2.

Suppose player 1 chooses to play U with some probability P ∈ [P 0 , P 1 ], whereas player

2 plays U with some probability Q ∈ [Q 0 , Q 1 ].
To find the Ellsberg equilibria, we first

derive the payoff function U 1 (P, [Q 0 , Q 1 ]) of player 1. U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 3P Q -P (1 -Q) -(1 -P )Q + (1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(6P -2) -2P + 1 =          Q 0 (6P -2) -2P + 1, if P > 1/3 , 1/3, if P = 1/3 , Q 1 (6P -2) -2P + 1, if P < 1/3 . (3.4)
Observe that at the boundaries the utility function reduces to

U 1 (0 , [Q 0 , Q 1 ]) = 1 -2Q 1 ,
and

U 1 (1 , [Q 0 , Q 1 ]) = 4Q 0 -1 . We see that U 1 (P, [Q 0 , Q 1 ]) is constant at 1/3 for all P ∈ [0, 1] when Q 0 = 1/3 = Q 1 .
We distinguish six cases where Q 0 and Q 1 can be situated within the interval [0, 1], all these give different arg max of U 1 (P, [Q 0 , Q 1 ]). Figure 3.9 pictures these cases. The distinction depends on the mixed strategy Q 0 = 1/3 = Q 1 that makes player 1 indifferent between any choice of [P 0 , P 1 ] ⊆ ∆S 1 . This is the Nash equilibrium strategy of player 2.

payoff function U 2 ([P 0 , P 1 ] , Q) of player 2.

U 2 ([P 0 , P 1 ] , Q) = min Q 0 ≤Q≤Q 1 (1 -P )Q + (1 -Q)P = min Q 0 ≤Q≤Q 1 P (1 -2Q) + Q =          P 0 (1 -2Q) + Q, if Q < 1/2 , 1/2, if Q = 1/2 , P 1 (1 -2Q) + Q, if Q > 1/2 .
(3.5)

Observe that at the boundaries the function reduces to

U 2 ([P 0 , P 1 ] , 0) = P 0 ,
and U 2 ([P 0 , P 1 ] , 1) = 1 -P 1 .

The plot of U 2 ([P 0 , P 1 ] , Q) is given in Figure 3.10, and the best responses of player 2 are listed below the figure. P 0 > 1/2 : B 2 ([P 0 , P 1 ]) = 0 (1)

P 0 = 1/2 < P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, 1/2]} (2)

Ellsberg Equilibria in General Conflict Games

In this section we determine the Ellsberg equilibria for competitive games with more general payoffs. We will see that in the general case when P * � = Q * there exists an Ellsberg equilibrium where both players create ambiguity. The set of probability distributions which each player uses in equilibrium is bounded by his Nash equilibrium strategy and his immunization strategy. We provide two propositions. In Proposition 3.7 below we restrict to the case where the immunization strategies are equal to the opponents' Nash equilibrium strategies. It will become clear that this simplifies the notation and the comprehensiveness of the result. In Proposition 3.9 we derive a further generalization.

First consider the competitive two-person 2 × 2 game with payoff matrix in Figure 3.11.

We assume that a, c > b and d, f < e . Due to the assumptions on the payoff, our game has conflicting interests and no pure strategy Nash equilibria. In the unique mixed strategy Nash equilibrium, player 1 plays U with probability

P * = f -e d -2e + f ,
and player 2 plays L with probability

Q * = c -b a -2b + c .
Proposition 3.7. The Ellsberg equilibria of the above game are the following:

For P * > Q * all Ellsberg equilibria are of the form

([P * , P 1 ] , [Q * , Q 1 ]) for P * ≤ P 1 ≤ 1, Q * ≤ Q 1 ≤ P * ;
for P * < Q * all Ellsberg equilibria are of the form

([P 0 , P * ] , [Q 0 , Q * ]) for 0 ≤ P 0 ≤ P * , P * ≤ Q 0 ≤ Q * ;
and for P * = Q * all Ellsberg equilibria are of the form

(Q * , [Q 0 , Q 1 ]) , where Q 0 ≤ Q * ≤ Q 1 ,
and ([P 0 , P 1 ] , P * ) , where P 0 ≤ P * ≤ P 1 .

Proof. The Nash equilibrium strategies follow from the usual analysis. The conditions on the payoffs assure that the Nash equilibrium is completely mixed, i.e., 0 < P * < 1 and 0 < Q * < 1. Let now [P 0 , P 1 ] and [Q 0 , Q 1 ] be Ellsberg strategies of player 1 and 2, where P ∈ [P 0 , P 1 ] is the probability of player 1 to play U , and

Q ∈ [Q 0 , Q 1 ]
is the probability of player 2 to play L. Let us compute the minimal expected payoff. The minimal expected payoff of player 1 when he plays the mixed strategy P is min

Q 0 ≤Q≤Q 1 u 1 (P, Q) = min Q 0 ≤Q≤Q 1 aP Q + bP (1 -Q) + b(1 -P )Q + c(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(b -c + P (a -2b + c)) + bP + c -cP =          Q 1 (b -c + P (a -2b + c)) + bP + c -cP if P < c-b a-2b+c , ac-b 2 a-2b+c if P = c-b a-2b+c , Q 0 (b -c + P (a -2b + c)) + bP + c -cP if P > c-b a-2b+c .
Note that the payoff function is constant at P = c-b a-2b+c , which is player 2's Nash equilibrium strategy. Depending on Q 0 and Q 1 the minimal payoff function can have six different forms. It can be strictly increasing, strictly decreasing, have flat parts or be completely constant. To determine how player 1 maximizes his minimal payoff for different Q 0 and Q 1 , we look at the borders of the minimal payoff function, where P = 0 and P = 1. This gives us two functions min

Q 0 ≤Q≤Q 1 u 1 (0 , Q) = Q 1 (b -c) + c ,
and min

Q 0 ≤Q≤Q 1 u 1 (1 , Q) = Q 0 (a -b) + b .
Note that bc < 0 and ab > 0, that is, the minimal payoff function is decreasing with Q 1 at P = 0 and increasing with Q 0 at P = 1. When

Q 1 = c -b a -2b + c = Q * , then min Q 0 ≤Q≤Q 1 u 1 (0 , Q) = ac -b 2 a -2b + c ,
that is, the minimal payoff function is constant for 0 ≤ P ≤ Q * . The same is true for the other boundary. When

Q 0 = c -b a -2b + c = Q * , then min Q 0 ≤Q≤Q 1 u 1 (1 , Q) = ac -b 2 a -2b + c , that is, the minimal payoff function is constant for Q * ≤ P ≤ 1.
With this analysis one can see immediately that when Q 0 = Q * = Q 1 , the minimal payoff function is constant for all P ∈ [0, 1] thus any Ellsberg strategy [P 0 , P 1 ] ⊆ [0, 1] is a best response for player 1.

Assume that Q 0 > Q * , then (since ab > 0) the minimal payoff function is strictly increasing and the best response of player 1 is P 0 = P 1 = 1. The opposite is true for Q 1 < Q * and thus the best response is P 0 = P 1 = 0.

Observe that when Q 0 < Q * < Q 1 , the values of both boundary functions drop below Q * and the function takes its maximum at the kink P = Q * . Therefore player 1's best response in this case is

P 0 = P 1 = Q * .
Two cases are still missing. The minimal expected payoff function can be flat exclusively to the left or to the right of Q * . For all

P ∈ � 0, c-b a-2b+c � = [0, Q * ], Player 1's utility is constant at ac-b 2 a-2b+c when Q 1 = c-b a-2b+c = Q * ,
and it is strictly decreasing for P > Q * . Hence, all P ≤ Q * are optimal for player 1. He can thus use any Ellsberg strategy [P 0 , P 1 ] with P 1 ≤ Q * as a best reply. Similarly, the payoff is constant for all P ≥ Q * when Q 0 = Q * (and strictly increasing for P < Q * ). This means that player 1's best response to a strategy [Q 0 , Q * ] is any strategy [P 0 , P 1 ] ⊆ [0, Q * ], and player 1's best response to a

strategy [Q * , Q 1 ] with Q * ≤ Q 1 ≤ 1 is any strategy [P 0 , P 1 ] ⊆ [Q * , 1].
We repeat the same analysis for player 2. His minimal expected utility when he plays the mixed strategy Q is min

P 0 ≤P ≤P 1 u 2 (P, Q) = min P 0 ≤P ≤P 1 dP Q + eP (1 -Q) + e(1 -P )Q + f (1 -P )(1 -Q) = min P 0 ≤P ≤P 1 P (e -f + Q(d -2e + f )) + eQ + f -f Q =          P 0 (e -f + Q(d -2e + f )) + eQ + f -f Q if Q < f -e d-2e+f , df -e 2 d-2e+f if Q = f -e d-2e+f , P 1 (e -f + Q(d -2e + f )) + eQ + f -f Q if Q > f -e d-2e+f
.

Note that the payoff function has a fixed value at Q = f -e d-2e+f , which is player 1's Nash equilibrium strategy. Again, as for player 1, depending on P 0 and P 1 the minimal payoff function can have six different forms. We note the two functions that describe the minimal payoff function at the borders Q = 0 and Q = 1: min

P 0 ≤P ≤P 1 u 1 (P, 0) = P 0 (e -f ) + f ,
and min

P 0 ≤P ≤P 1 u 1 (P, 1) = P 1 (d -e) + e .
Note that ef > 0 and de < 0, that is, the minimal payoff function is increasing with P 0 in P = 0 and decreasing with P 1 in P = 1. When

P 0 = f -e d -2e + f = P * , then min P 0 ≤P ≤P 1 u 2 (P, 0) = df -e 2 d -2e + f ,
that is, the minimal payoff function is constant for 0 ≤ Q ≤ P * . The same is true for the other boundary: When

P 1 = f -e d -2e + f = P * , then min P 0 ≤P ≤P 1 u 2 (P, 1) = df -e 2 d -2e + f ,
that is, the minimal payoff function is constant for

P * ≤ Q ≤ 1.
Similar to the analysis of player 1 we now get the following best responses of player 2.

When P 0 = P * = P 1 player 2 can use any strategy [Q 0 , Q 1 ] ⊆ [0, 1], when P 0 > P * the best response is Q 0 = Q 1 = 0 and when P 1 < P * then Q 0 = Q 1 = 1. When P 0 < P * < P 1 the minimal payoff function takes its maximum at the kink Q = P * and accordingly player 2's best response is

Q 0 = Q 1 = P * .
Finally note that player 2's utility is constant at df -e 2 d-2e+f for all

Q ∈ � 0, f -e d-2e+f � = [0, P * ]
when P 0 = f -e d-2e+f = P * , and it is strictly decreasing for Q > P * . Hence, all Q ≤ P * are optimal for player 2. He can thus use any Ellsberg strategy [Q 0 , Q 1 ] where Q 1 ≤ P * as a best reply. Similarly, the payoff is constant for all Q ≥ P * when P 0 = P * (and strictly increasing for Q < P * ). This means that player 2's best response to a strategy [P * , P 1 ] is any strategy [Q 0 , Q 1 ] ⊆ [0, P * ], and player 2's best response to a strategy [P * , P 1 ] with

P * ≤ P 1 ≤ 1 is any strategy [Q 0 , Q 1 ] ⊆ [P * , 1].
In Ellsberg equilibrium no player wants to unilaterally deviate from his equilibrium strategy. We analyze in the following which Ellsberg strategies have best responses such that no player wants to deviate.

We assume first that Q * < P * . Three Ellsberg strategies can quickly be excluded to be part of an Ellsberg equilibrium. Suppose player 2 plays [Q 0 , Q 1 ] with Q 0 > Q * , then player 1's best response is P 0 = P 1 = 1 and since we are looking at a competitive game, player 2 would want to deviate from his original strategy to Q 0 = Q 1 = 0. A similar reasoning leads to the result that an Ellsberg strategy [Q 0 , Q 1 ] with Q 1 < Q * cannot be an equilibrium strategy. Thirdly, suppose player 2 plays [Q 0 , Q 1 ] with Q 0 < Q < Q 1 , then player 1 would respond with P 0 = P 1 = Q * . Since Q * < P * player 2 would deviate from his original strategy to

Q 0 = Q 1 = 1. Now suppose player 2 plays Q 0 = Q 1 = Q * ,
then player 1 can respond with any [P 0 , P 1 ] ⊆ [0, 1]. Any choice with P 0 ≥ P * , P 1 < P * or P 0 < P * < P 1 leads to contradictions similar to the cases above. The possibilities that P 0 and P 1 are such that P 0 < P 1 = P * or P 0 = P 1 = P * (which are Ellsberg equilibria), are contained in the Ellsberg equilibria that arise in the two remaining cases below.

Suppose player 2 plays

[Q * , Q 1 ] with Q * ≤ Q 1 ≤ 1, then if player 1 responds with [P 0 , P 1 ] = [P * , P 1 ] with P * ≤ P 1 ≤ 1 player 2 would play any strategy [Q 0 , Q 1 ] ⊆ [0, P * ] as a best response. Because Q * < P * , player 2 can choose [Q 0 , Q 1 ] = [Q * , Q 1 ] with Q * ≤ Q 1 ≤ P * .
These strategies are Ellsberg equilibria

([P * , P 1 ] , [Q * , Q 1 ]) , where P * ≤ P 1 ≤ 1 and Q * ≤ Q 1 ≤ P * .
In the case Q * < P * this is the only type of Ellsberg equilibrium. Note that the Nash equilibrium is contained in these equilibrium strategies.

When we assume that P * < Q * , the analysis is very similar. We skip the first four cases and only look at the cases where the minimal payoff function has flat parts. Suppose

player 2 plays [Q 0 , Q * ] with 0 ≤ Q 0 ≤ Q * , then if we let player 1 pick [P 0 , P 1 ] ⊆ [P 0 , P * ] with 0 ≤ P 0 ≤ P * , player 2's best response is any subset [Q 0 , Q 1 ] ⊆ [P * , 1]. Again, because P * < Q * , he can choose [Q 0 , Q 1 ] = [Q 0 , Q * ] with P * ≤ Q 0 ≤ Q * as
a best response. Player 1 would not want to deviate and thus these strategies are Ellsberg equilibria

([P 0 , P * ] , [Q 0 , Q * ]) , where 0 ≤ P 0 ≤ P * and P * ≤ Q 0 ≤ Q * .
As before, this is the only type of Ellsberg equilibrium in the case P * < Q * .

Finally let P * = Q * . Repeat the considerations above having in mind the equality of the Nash equilibrium strategies. Since it was precisely the difference between P * and Q * that led to the Ellsberg equilibria in the above cases, we see that no Ellsberg equilibria exist where both players create ambiguity. But, in difference to the above analysis, two types of Ellsberg equilibria with unilateral ambiguity arise that could not be sustained above.

Remember that when player 2 plays

[Q 0 , Q 1 ] with Q 0 < Q * < Q 1 then it is optimal
for player 1 to respond with P 0 = P 1 = Q * . Since P * = Q * , these strategies are in equilibrium, even for Q 0 ≤ Q * ≤ Q 1 . One observes that, as long as player 2 makes sure that the mixed Nash equilibrium strategy Q * is strictly contained in his Ellsberg strategy, player 1 responds with Q * and we have Ellsberg equilibria, in which player 1 immunizes against the ambiguity of player 2. An analogous type of Ellsberg equilibrium exists for player 2 immunizing against the ambiguity of player 1 by playing Q 0 = Q 1 = P * . Thus, we have the following Ellsberg equilibria

(Q * , [Q 0 , Q 1 ]) , where Q 0 ≤ Q * ≤ Q 1 ,
and ([P 0 , P 1 ] , P * ) , where P 0 ≤ P * ≤ P 1 . These equilibria do not exist in the non-symmetric case P * � = Q * since the immunization strategies are in general not equilibrium strategies. Due to the assumptions on the payoffs in this proposition, the immunization strategy equals the Nash equilibrium strategy of the opponent. Thereby the immunization strategy is an equilibrium strategy only when

P * = Q * .
Remarks 3.8.

1. In Proposition 3.7 we restrict to the case with (U, D) and (L, R)

giving the same payoffs (b, e) for both players. The more general competitive game yields two more types of Ellsberg equilibria as we find in Proposition 3.9. The nice feature of the above restriction is that players use the mixed Nash equilibrium strategy of their respective opponent as their immunization strategy.

2. Observe the asymmetry in the Ellsberg equilibria in the preceding proposition: no matter if P * < Q * or Q * < P * , it is always player 2 who creates ambiguity between the Nash equilibrium strategies, player 1 never does so. This is due to the assumptions on the payoffs. If we assume that a, c < b and d, f > e, player 1 plays between P * and Q * .

For a further generalization consider now the 2 × 2 conflict game with payoff matrix in Figure 3.12. We assume that a, d > b, c and e, h < f, g.

Player 1

Player 2 L R U a, e b, f D c, g d, h

Figure 3.12: General conflict game II.

As before, P * and Q * are the probabilities with which U respectively L are played. Let

P * = h -g e -f -g + h , respectively Q * = d -b a -b -c + d
denote the Nash equilibrium strategies for player 1 and 2, respectively. The immunization strategies of each player are denoted by M 1 , respectively M 2 . Proposition 3.9. Let P * , Q * denote the mixed strategy Nash equilibria and M 1 , M 2 the immunization strategies of player 1 and 2, respectively. Then the Ellsberg equilibria of the general conflict game above are of the following form.

M 1 ≤ P * M 1 ≥ P * M 2 ≤ Q * ([P 0 , P * ] , [Q * , Q 1 ]) ([P 0 , P * ] , [Q 0 , Q * ]) M 1 ≤ P 0 M 2 ≤ Q 0 M 2 ≥ Q * ([P * , P 1 ] , [Q * , Q 1 ]) ([P * , P 1 ] , [Q 0 , Q * ]) Q 1 ≤ M 2 P 1 ≤ M 1 If M 1 = P * or M 2 = Q * ,
then an additional type of Ellsberg equilibria arises,

(P * , [Q 0 , Q 1 ]), where Q 0 ≤ Q * ≤ Q 1 when M 1 = P * ; ([P 0 , P 1 ] , Q * ), where P 0 ≤ P * ≤ P 1 when M 2 = Q * .
In any case, we have the Ellsberg equilibrium which is identical to the Nash equilibrium in mixed strategies, (P * , Q * ).

Proof. The Nash equilibrium strategies follow from the usual analysis. To calculate the Ellsberg equilibria of the general conflict game II (Figure 3.12), we first derive the utility functions of player 1 and player 2. Due to the assumption that a, d > b, c and e, h < f, g, the denominator abc + d is positive, and the denominator efg + h is negative.

This reflects the competitiveness of the game in the payoff functions; player 1 uses Q 0 as a minimizer when P > d-c a-b-c+d , and on the contrary, player 2 uses P 0 as a minimizer when

Q < h-f e-f -g+h . U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 aP Q + bP (1 -Q) + c(1 -P )Q + d(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q((a -b -c + d)P + c -d) + (b -d)P + d =          Q 0 ((a -b -c + d)P + c -d) + (b -d)P + d if P > d-c a-b-c+d , (b-d)(c-d) a-b-c+d + d if P = d-c a-b-c+d , Q 1 ((a -b -c + d)P + c -d) + (b -d)P + d if P < d-c a-b-c+d , U 2 ([P 0 , P 1 ] , Q) = min P 0 ≤P ≤P 1 eP Q + g(1 -P )Q + f P (1 -Q) + h(1 -P )(1 -Q) = min P 0 ≤P ≤P 1 P ((e -f -g + h)Q + f -h) + (g -h)Q + h =          P 0 ((e -f -g + h)Q + f -h) + (g -h)Q + h if Q < h-f e-f -g+h , (g-h)(h-f ) e-f -g+h + h if Q = h-f e-f -g+h , P 1 ((e -f -g + h)Q + f -h) + (g -h)Q + h if Q > h-f e-f -g+h .
We see that

M 1 = d -c a -b -c + d , respectively M 2 = h -f e -f -g + h
are indeed the immunization strategies of player 1 and 2 respectively. At the boundaries the utility functions become

U 1 (0 , [Q 0 , Q 1 ]) = (c -d)Q 1 + d , U 2 ([P 0 , P 1 ] , 0) = (f -h)P 0 + h , U 1 (1 , [Q 0 , Q 1 ]) = (a -b)Q 0 + b , U 2 ([P 0 , P 1 ] , 1) = (e -g)P 1 + g .
The payoff function of player 1 is constant when

Q 0 = Q 1 = Q * ,
and the payoff function of player 2 is constant when P 0 = P 1 = P * . The best response correspondences for both players are listed below. We see immediately that, of course, there cannot be an Ellsberg equilibrium in pure strategies: when player 2 plays L, player 1 best responds U , and when player 1 plays U , player 2 best responds R. The intersections of the best response correspondences are discussed in detail below.

Q 0 > Q * : B 1 ([Q 0 , Q 1 ]) = 1 (1) Q 0 = Q * < Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [M 1 , 1]} (2) Q 0 < Q * < Q 1 : B 1 ([Q 0 , Q 1 ]) = M 1 (3) Q 0 < Q * = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, M 1 ]} (4) Q 1 < Q * : B 1 ([Q 0 , Q 1 ]) = 0 (5) Q 0 = Q * = Q 1 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 1]} (6) P 0 > P * : B 2 ([P 0 , P 1 ]) = 0 P 0 = P * < P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, M 2 ]} P 0 < P * < P 1 : B 2 ([P 0 , P 1 ]) = M 2 P 0 < P * = P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [M 2 , 1]} P 1 < P * : B 2 ([P 0 , P 1 ]) = 1 P 0 = P * = P 1 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, 1]}
To find the Ellsberg equilibria, we look at the different cases in turn.

(1) Q 0 > Q * : player 1 responds P 0 = P 1 = 1 and player 2 chooses Q 0 = Q 1 = 0, thus this is not an Ellsberg equilibrium.

(2)

� d-b a-b-c+d , Q 1 � ⇒ [P 0 , P 1 ] ⊆ � d-c a-b-c+d , 1
� , then, depending on the size of M 1 in relation to P * , player 1 has the following choices:

• d-c a-b-c+d ≤ h-g e-f -g+h : player 1 can play either - � h-g e-f -g+h , P 1 � , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � 0, h-f e-f -g+h � .
Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices:

* h-f e-f -g+h ≥ d-b a-b-c+d : � d-b a-b-c+d , Q 1 � with Q 1 ≤ h-f e-f -g+h , we thus have here the Ellsberg equilibrium �� h -g e -f -g + h , P 1 � , � d -b a -b -c + d , Q 1 �� ,
where

Q 1 ≤ h -f e -f -g + h , that is ([P * , P 1 ] , [Q * , Q 1 ]), where Q 1 ≤ M 2 . * h-f e-f -g+h ≤ d-b a-b-c+d : for all [Q 0 , Q 1 ]
the best response is P 0 = P 1 = 0, thus no Ellsberg equilibrium arises.

or

� P 0 , h-g e-f -g+h � with d-c a-b-c+d ≤ P 0 ≤ h-g e-f -g+h , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � h-f e-f -g+h , 1 �
. Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices:

* h-f e-f -g+h ≤ d-b a-b-c+d : � d-b a-b-c+d , Q 1 � , then the best response of player 1 is [P 0 , P 1 ] ⊆ � d-c a-b-c+d , 1
� . We thus have here the Ellsberg equilibrium

�� P 0 , h -g e -f -g + h � , � d -b a -b -c + d , Q 1 �� , where d -c a -b -c + d ≤ P 0 , that is ([P 0 , P * ] , [Q * , Q 1 ]), where M 1 ≤ P 0 . * h-f e-f -g+h ≥ d-b a-b-c+d : for all [Q 0 , Q 1 ] the best response is P 0 = P 1 = 1, thus no Ellsberg equilibrium arises. • d-c a-b-c+d ≥ h-g
e-f -g+h : to any [P 0 , P 1 ] the best response is Q 0 = Q 1 = 0, thus no Ellsberg equilibrium arises.

(3) Q 0 < Q * < Q 1 : player 1 responds with P 0 = P 1 = M 1 . Only when M 1 = P * , player 1 sticks to his strategy and we get the equilibrium (P

* , [Q 0 , Q 1 ]), where Q 0 < Q * < Q 1 . (4) � Q 0 , d-b a-b-c+d � ⇒ [P 0 , P 1 ] ⊆ � 0, d-c a-b-c+d �
, then, depending on the size of M 1 in relation to P * , player 1 has the following choices:

• d-c a-b-c+d ≥ h-g e-f -g+h : player 1 can play either - � P 0 , h-g e-f -g+h � , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � h-f e-f -g+h , 1 � .
Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices:

* h-f e-f -g+h ≤ d-b a-b-c+d : � Q 0 , d-b a-b-c+d � with h-f e-f -g+h ≤ Q 0 ,
we thus have here the Ellsberg equilibrium

�� P 0 , h -g e -f -g + h � , � Q 0 , d -b a -b -c + d �� , where h -f e -f -g + h ≤ Q 0 , that is ([P 0 , P * ] , [Q 0 , Q * ]), where M 2 ≤ Q 0 . * h-f e-f -g+h ≥ d-b a-b-c+d : for all [Q 0 , Q 1 ] the best response is P 0 = P 1 = 1, thus no Ellsberg equilibrium arises. -or � h-g e-f -g+h , P 1 � with h-g e-f -g+h ≤ P 1 ≤ d-c a-b-c+d , then the best response of player 2 is [Q 0 , Q 1 ] ⊆ � 0, h-f e-f -g+h � . Now, depending on the size of M 2 in relation to Q * , player 2 has the following choices: * d-b a-b-c+d ≤ h-f e-f -g+h : � Q 0 , d-b a-b-c+d � , then the best response of player 1 is [P 0 , P 1 ] ⊆ � 0, d-c a-b-c+d � .
We thus have here the Ellsberg equilibrium

�� h -g e -f -g + h , P 1 � , � Q 0 , d -b a -b -c + d �� ,
where

P 1 ≤ d -c a -b -c + d , that is ([P * , P 1 ] , [Q 0 , Q * ]), where P 1 ≤ M 1 . * d-b a-b-c+d ≥ h-f e-f -g+h : for all [Q 0 , Q 1 ] the best response is P 0 = P 1 = 0, thus no Ellsberg equilibrium arises. • d-c a-b-c+d ≤ h-g
e-f -g+h : to any [P 0 , P 1 ] the best response is Q 0 = Q 1 = 0, thus no Ellsberg equilibrium arises.

(5) Q 1 < Q * : player 1 responds P 0 = P 1 = 0 and player 2 chooses

Q 0 = Q 1 = 1 thereafter, thus this is not an Ellsberg equilibrium. (6) Q 0 = Q * = Q 1 : player 1 responds with [P 0 , P 1 ] ⊆ [0, 1]. Only when M 2 = Q * , player 2
sticks to his strategy and we get the equilibrium ([P 0 , P 1 ] , Q * ), where P 0 < P * < P 1 .

Further Examples of Two-Person Conflicts

Now we calculate the Ellsberg equilibria of some more examples of conflict games. We start with the classic symmetric Matching Pennies game to observe how the Ellsberg equilibria change compared to the modified case above. Then we apply Proposition 3.9 to a non-symmetric conflict game.

Matching Pennies

We look at a symmetric Matching Pennies game with the payoff matrix given in Figure 3.13. Note that the game has a unique mixed strategy Nash equilibrium in which both players mix with equal probabilities,

(P * , Q * ) = ((1/2, 1/2) , (1/2, 1/2)) . Player 1 Player 2 L R U 1, -1 -1, 1 D -1, 1 1, -1 Figure 3.13: Symmetric Matching Pennies.
Suppose player 1 chooses to play U with probability P ∈ [P 0 , P 1 ], whereas player 2 plays

U with probability Q ∈ [Q 0 , Q 1 ].
To find the Ellsberg equilibria, we first derive the payoff

function U 1 (P, [Q 0 , Q 1 ]) of player 1, U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 P Q -(1 -P )Q -P (1 -Q) + (1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(4P -2) -2P + 1 =          Q 0 (4P -2) -2P + 1 if P > 1/2 , 0 if P = 1/2 , Q 1 (4P -2) -2P + 1 if P < 1/2 .
Observe that at the boundaries the function reduces to

U 1 (0 , [Q 0 , Q 1 ]) = 1 -2Q 1 ,
and

U 1 (1 , [Q 0 , Q 1 ]) = Q 0 -1 .
Now we can solve for the Ellsberg equilibria by finding the intersections of the best response correspondences.

(1) P 0 > 1/2 ⇒ Q 0 = Q 1 = 0 ⇒ P 0 = P 1 = 0, this contradicts the original play by player 1, thus this is not an equilibrium.

(

) P 0 = 1/2 < P 1 ⇒ [Q 0 , Q 1 ] ⊆ [0, 1/2] ⇒ [P 0 , P 1 ] ⊆ [0, 1/2], 2
this again contradicts the original play by player 1, thus this is not an equilibrium.

(

) P 0 < 1/2 < P 1 ⇒ Q 0 = Q 1 = 1/2 ⇒ [P 0 , P 1 ] ⊆ [0, 1], 3
this is an Ellsberg equilibrium, ([P 0 , P 1 ] , 1/2), where P 0 < 1/2 < P 1 .

(

) P 0 < 1/2 = P 1 ⇒ [Q 0 , Q 1 ] ⊆ [1/2, 1] ⇒ P 0 = P 1 = 1, 4
thus this is not an equilibrium.

(

) P 1 < 1/2 ⇒ Q 0 = Q 1 = 1 ⇒ P 0 = P 1 = 1, thus this is not an equilibrium. ( 5
) P 0 = 1/2 = P 1 ⇒ [Q 0 , Q 1 ] ⊆ [0, 1]. We consider two cases: if player 2 chooses Q 0 < 1/2 < Q 1 , then player 1's best response is P 0 = P 1 = 1/2, thus (1/2, [Q 0 , Q 1 ]), where Q 0 < 1/2 < Q 1 , is an Ellsberg equilibrium. If player 2 decides to play Q 0 = Q 1 = 1/2, 6
no player can gain by deviating from this strategy. This is the mixed strategy Nash equilibrium (P * , Q * ).

Proposition 3.10. In Matching Pennies the Ellsberg equilibria are ([P 0 , P 1 ] , 1/2), where P 0 < 1/2 < P 1 , and

(1/2, [Q 0 , Q 1 ]), where Q 0 < 1/2 < Q 1 .
The Ellsberg equilibria can also be derived by applying Proposition 3.9. We see in this symmetric example that essentially no other behavior than in the mixed Nash equilibrium arises. The players may create complete ambiguity about their randomization, but this does not affect the opponent's best response.

Non-symmetric Conflict Game

To demonstrate the application of Proposition 3.9, we derive the Ellsberg equilibria of the game in Figure 3.14.

Player 1

Player 2 L R U 3, -1 -2, 4 D 0, 5 4, 2
Figure 3.14: Non-symmetric conflict game.

We find that P * = 3/8, Q * = 2/3 and M 1 = 4/9, M 2 = 1/4. That means, since M 1 > P * and M 2 < Q * , the Ellsberg equilibria of the above game are of the form

([P 0 , 3/8] , [Q 0 , 2/3]), where 1/4 ≤ Q 0 ≤ 2/3.

Overview of Ellsberg Equilibria in Coordination and

Conflict Games

In the preceding sections we derived the Ellsberg equilibria of different two-person 2 × 2 games. We summarize the results of Propositions 3.2 and 3.9 in Table 3 We denote by P * (Q * ) the probability that player 1 (2) play U (L) in the mixed Nash equilibrium, and by M 1 (M 2 ) the probability of U (L) in the immunization strategies of the respective players. Then the Ellsberg equilibria are as in Table 3.1.

Remark 3.11. When any of the weak inequalities between Nash equilibrium strategies and immunization strategies are in fact equalities, some of the Ellsberg equilibria in Table 3.1 become redundant but not wrong.

The results assembled in the above table cover all generic games, except when there is exactly one pure Nash equilibrium. In that case the pure Nash equilibrium is the only Ellsberg equilibrium. We showed this in Section 1.3.6.

Zero-Sum Games

Historically, much interest was paid to two-person zero-sum games which describe situations in which players are in pure opposition to each other. One player maximizing his own payoff is equivalent to him trying to minimize the payoff of his opponent: this makes these games mathematically very tractable.

Table 3.1: Ellsberg equilibria in general coordination (blue) and conflict games.

M 1 ≤ P * M 1 ≥ P * M 2 ≤ Q * ([P * , P 1 ] , [Q * , Q 1 ]) ([P * , P 1 ] , [Q 0 , Q * ]) P 1 ≤ M 1 and M 2 ≤ Q 0 ([P 0 , P * ] , [Q * , Q 1 ]) ([P 0 , P * ] , [Q * , Q 1 ]) M 1 ≤ P 0 M 2 ≤ Q 0 M 2 ≥ Q * ([P 0 , P * ] , [Q * , Q 1 ]) ([P 0 , P * ] , [Q 0 , Q * ]) M 1 ≤ P 0 and Q 1 ≤ M 2 ([P * , P 1 ] , [Q * , Q 1 ]) ([P * , P 1 ] , [Q 0 , Q * ]) Q 1 ≤ M 2 P 1 ≤ M 1 If M 1 = P * or M 2 = Q * ,
then in both types of games an additional type of Ellsberg equilibria arises,

(P * , [Q 0 , Q 1 ]) , where Q 0 ≤ Q * ≤ Q 1 when M 1 = P * ; ([P 0 , P 1 ] , Q * ) , where P 0 ≤ P * ≤ P 1 when M 2 = Q * .
In both types of games and all the above cases we have the Ellsberg equilibria which are identical to the Nash equilibria in pure (if they exist) and mixed strategies.

Naturally, it is interesting to see what the Ellsberg equilibria of these zero-sum games are. A two-person zero-sum game is a game G = �{1, 2}, (S i ), (u i )� such that

u 2 (s 1 , s 2 ) = -u 1 (s 1 , s 2 ) for all s 1 ∈ S 1 and s 2 ∈ S 2 .
We showed in Theorem 2.8 that in two-person games with a completely mixed Nash equilibrium immunization strategies are maximin strategies. In the context of two-person zero-sum games this implies that immunizing against ambiguity is an equilibrium strategy.

This observation is not true for general games as we have seen in the preceding sections, and it is crucial for the type of Ellsberg equilibria that arise in two-person zero-sum games. This is the content of the first part of this section. One player always finds it optimal to play his immunization strategy, and thus no Ellsberg equilibria in which both players create ambiguity arise. We prove this result in Proposition 3.12 and Corollary 3.13 for 2×2

games. The generalization to two-player games with n strategies per player is postponed to the next chapter, Theorem 4.2.

The second part of this section treats Minimax Theorems in Ellsberg games which assure that Ellsberg equilibria of zero-sum games are value preserving.

3.5.1 Ellsberg Equilibria in Two-Person 2 × 2 Zero-Sum Games

Note that Proposition 3.7 on the Ellsberg equilibria of general competitive games holds likewise for zero-sum games, but due to the assumptions on the payoffs the proposition restricts to zero-sum games where P * = Q * . This can be seen easily by setting d := -a, -c := f and e := -b, then

P * = f -e d-2e+f = c-b a-2b+c = Q * .
Therefore, in the Ellsberg equilibria of two-person 2 × 2 zero-sum games under the assumptions of Proposition 3.7 only one player creates ambiguity. This also holds for general two-person 2 × 2 zero-sum games, although in those games in general P * � = Q * . The result then hinges on the fact that the immunization strategy of player i is always equal to his own Nash equilibrium strategy.

We now calculate the Ellsberg equilibria of a general two-person 2 × 2 zero-sum game with the payoff matrix given in Figure 3.16. We assume without loss of generality that a, b, c, d ≥ 0. Player 1's Ellsberg strategy is as usual denoted

[P 0 , P 1 ], player 2's [Q 0 , Q 1 ]. Player 1 Player 2 L R U a, -a b, -b D c, -c d, -d
([P 0 , P 1 ] , Q * )
, where 0 ≤ P 0 ≤ P * ≤ P 1 ≤ 1 ,

and (P * , [Q 0 , Q 1 ]) , where 0 ≤ Q 0 ≤ Q * ≤ Q 1 ≤ 1 .
� d -c d -b , [Q 0 , 1] � , where 0 ≤ Q 0 ≤ 1 .

Proof. (a)

The Nash equilibrium of the game in Figure 3.16 is (P * , Q * ), where

P * = d-c a-b-c+d and Q * = d-b a-b-c+d .
We first calculate the minimal expected utility of player 1, when he plays P ∈ [P 0 , P 1 ].

U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 aP Q + bP (1 -Q) + c(1 -P )Q + d(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(P (a -b -c + d) + c -d) + (b -d)P + d =          Q 0 (P (a -b -c + d) + c -d) + (b -d)P + d if P > P * , ad-bc a-b-c+d if P = P * , Q 1 (P (a -b -c + d) + c -d) + (b -d)P + d if P < P * .
The same analysis for player 2 yields

U 2 ([P 0 , P 1 ] , Q) =          P 0 (Q(-a + b + c -d) -b + d) + (-c + d)Q -d if Q < Q * , bc-ad a-b-c+d if Q = Q * , P 1 (Q(-a + b + c -d) -b + d) + (-c + d)Q -d if Q > Q * .
As usual we look at the utility function on the borders, this gives

U 1 (0 , [Q 0 , Q 1 ]) = Q 1 (c -d) + d , U 2 ([P 0 , P 1 ] , 0) = P 0 (d -b) -d , U 1 (1 , [Q 0 , Q 1 ]) = Q 0 (a -b) + b , U 2 ([P 0 , P 1 ] , 1) = P 1 (b -a) -c .
The utility function of player 1 is constant when player 2 plays his Nash equilibrium strategy Q * , that is at the same time his immunization strategy. Player 2's utility function is constant when player 1 plays his Nash equilibrium (and immunization) strategy P * .

(b) Because the Nash and the immunization strategy coincide, we see immediately that the two types of Ellsberg equilibria stated in the proposition are indeed equilibria. Due to the competitiveness of the game we conclude that no other Ellsberg equilibria exist.

(c) Since a = c, we find the following utility functions for player 1 and player 2:

U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 Q(P (d -b) + c -d) + (b -d)P + d =          Q 0 (P (d -b) + c -d) + (b -d)P + d if P < d-c d-b , c if P = d-c d-b , Q 1 (P (d -b) + c -d) + (b -d)P + d if P > d-c d-b . U 2 ([P 0 , P 1 ] , Q) = min P 0 ≤P ≤P 1 P (Q(b -d) -b + d) + (d -c)Q -d =    -c if Q = 1 , P 1 (Q(b -d) + d -b) + (d -c)Q -d if Q < 1 .
As usual we look at the utility function on the borders, this gives

U 1 (0 , [Q 0 , Q 1 ]) = Q 0 (c -d) + d , U 2 ([P 0 , P 1 ] , 0) = P 1 (d -b) -d , U 1 (1 , [Q 0 , Q 1 ]) = Q 1 (c -b) + b , U 2 ([P 0 , P 1 ] , 1) = -c .
We see in the best response correspondences that player 2 finds it optimal to play Q 0 = Q 1 = 0 when player 1 plays

P 1 > d-c d-b and Q 0 = Q 1 = 1 when P 1 < d-c d-b . When player 1 plays exactly P 1 = d-c d-b , player 2 is indifferent between all distributions in [0, 1]. When Q 0 = Q 1 = 1 player 1 is indifferent between all distributions in [0, 1]. For all Q 0 < 1 = Q 1 player 1 responds optimally with [P 0 , P 1 ] ⊆ � d-c d-b , 1 �
and finally when Q 1 < 1 player 1 wants to play P 0 = P 1 = 1.

From this analysis we find that the Ellsberg equilibria are

([P 0 , P 1 ] , 1) , where P 1 > d -c d -b , and � d -c d -b , [Q 0 , 1] � , where 0 ≤ Q 0 ≤ 1 .
The cases treated in parts (b) and (c) of Proposition 3.12 are exhaustive in the sense that any other constellation of payoffs would have strictly dominant strategies and we know from Proposition 1.16 that those games do not have proper Ellsberg equilibria. Thus, we can state the following corollary.

Corollary 3.13. In two-person 2 × 2 zero-sum games, no proper Ellsberg equilibria exist in which both players create ambiguity.

To apply Proposition 3.12 we consider the following example.

Example 3.14. We derive the Ellsberg equilibria of the zero-sum game in Figure 3.17 by applying Proposition 3.12 (c).

Player 1 Player 2 L R U 2, -2 3, -3 D 2, -2 1, -1
Figure 3.17: Example of a zero-sum game.

The Ellsberg equilibria of the game are of the form

([P 0 , P 1 ] , 1), where P 1 > 1/2 and (1/2, [Q 0 , 1]) .

Minimax Theorems in Ellsberg Games

We let G = �{1, 2}, (S i ), (u i )� be a zero-sum game throughout the whole section, that is u 2 (s 1 , s 2 ) = -u 1 (s 1 , s 2 ) for all s 1 ∈ S 1 and s 2 ∈ S 2 . The following proposition collects the classic Minimax results for zero-sum games, see, e.g., [START_REF] Myerson | Game Theory: Analysis of Conflict[END_REF] p. 123, for a proof.

Proposition 3.15. Let G be a zero-sum game with value v.

(i) (P * , Q * ) is a Nash equilibrium of G if and only if

P * ∈ arg max P ∈∆S 1 min Q∈∆S 2 u 1 (P, Q) and Q * ∈ arg min Q∈∆S 2 max P ∈∆S 1 u 1 (P, Q) . (3.6) (ii) If (P * , Q * ) is a Nash equilibrium of G, then u 1 (P * , Q * ) = max P ∈∆S 1 min Q∈∆S 2 u 1 (P, Q) = min Q∈∆S 2 max P ∈∆S 1 u 1 (P, Q) = v . (3.7)
The proposition says that player 1 wants to maximize his minimal expected payoff, and player 2 likewise. This interpretation of the result is possible, since for any function f (x), max x (-f (x)) =min x (f (x)) and arg min x (-f (x)) = arg max x f (x), and therefore expression (3.6) reads

Q * ∈ arg max Q∈∆S 2 min P ∈∆S 1 u 2 (P, Q) . (3.8)
If we want to state a similar relation for Ellsberg games, we must be careful, since it is not true in general that U 2 (P, Q) = -U 1 (P, Q). Therefore the equivalence between (3.6) and (3.8) does not carry over to Ellsberg games. Anyhow, we can show an equivalent of the classical minimax theorem (3.7) and thus find that in two-person zero-sum games the Ellsberg equilibria all yield the same payoff, and this payoff is the value v of the game for player 1, and -v for player 2. We prove two lemmas that yield the result, which is, thereafter, presented in Theorem 3.18.

Lemma 3.16. Let G be a zero-sum game with two players. Then the following holds.

min

Q⊆∆S 2 max P⊆∆S 1 U 1 (P, Q) (1) = max P⊆∆S 1 min Q⊆∆S 2 U 1 (P, Q) (2) = u 1 (P * , Q * ) = v , (3.9)
and min

P⊆∆S 1 max Q⊆∆S 2 U 2 (P, Q) (1) = max Q⊆∆S 2 min P⊆∆S 1 U 2 (P, Q) (2) = -u 1 (P * , Q * ) = -v . (3.10)
Proof. We start by showing equality (2) of equation (3.9), followed by equality (1). (3.10) is shown below. For all equalities we use the fact that for all linear functions f (x, y), min x min y f (x, y) = min y min x f (x, y), and Lemma 1.12.

max

P⊆∆S 1 min Q⊆∆S 2 U 1 (P, Q) = max P⊆∆S 1 min Q⊆∆S 2 min P ∈P min Q∈Q u 1 (P, Q) = max P⊆∆S 1 min Q⊆∆S 2 min Q∈Q min P ∈P u 1 (P, Q) Lemma 1.12 = max P⊆∆S 1 min Q∈∆S 2 min P ∈P u 1 (P, Q) = max P⊆∆S 1 min P ∈P min Q∈∆S 2 u 1 (P, Q) Lemma 1.12 = max P ∈∆S 1 min Q∈∆S 2 u 1 (P, Q) = u 1 (P * , Q * ) = v .
To proof equality (1) of equation (3.9) we need the Minimax Theorem 1 which we presented in Theorem 1.15. Then we have min

Q⊆∆S 2 max P⊆∆S 1 U 1 (P, Q) = min Q⊆∆S 2 max P⊆∆S 1 min P ∈P min Q∈Q u 1 (P, Q) = min Q⊆∆S 2 max P ∈∆S 1 min Q∈Q u 1 (P, Q) Thm. 1.15 = min Q⊆∆S 2 min Q∈Q max P ∈∆S 1 u 1 (P, Q) = min Q∈∆S 2 max P ∈∆S 1 u 1 (P, Q) = u 1 (P * , Q * ) = max P⊆∆S 1 min Q⊆∆S 2 U 1 (P, Q) .
We now come to equality (2) of equation (3.10).

max

Q⊆∆S 2 min P⊆∆S 1 U 2 (P, Q) = max Q⊆∆S 2 min P⊆∆S 1 min Q∈Q min P ∈P u 2 (P, Q) = max Q⊆∆S 2 min P⊆∆S 1 min P ∈P min Q∈Q u 2 (P, Q) Lemma 1.12 = max Q⊆∆S 2 min P ∈∆S 1 min Q∈Q u 2 (P, Q) = max Q⊆∆S 2 min Q∈Q min P ∈∆S 1 u 2 (P, Q) Lemma 1.12 = max P ∈∆S 1 min Q∈∆S 2 u 2 (P, Q) = max P ∈∆S 1 min Q∈∆S 2 -u 1 (P, Q) = -u 1 (P * , Q * ) = v .
Lastly, we show equality (1) of equation (3.10).

min

P⊆∆S 1 max Q⊆∆S 2 U 2 (P, Q) = min P⊆∆S 1 max Q⊆∆S 2 min Q∈Q min P ∈P u 2 (P, Q) = min P⊆∆S 1 max Q∈∆S 2 min P ∈P u 2 (P, Q) Thm. 1.15 = min P⊆∆S 1 min P ∈P max Q∈∆S 2 -u 1 (P, Q) = min P ∈∆S 1 max Q∈∆S 2 -u 1 (P, Q) = -u 1 (P * , Q * ) = max Q⊆∆S 2 min P⊆∆S 1 U 2 (P, Q) .
Lemma 3.17. Let (P * , Q * ) be an Ellsberg equilibrium of the two-person zero-sum game G.

Then

U 1 (P * , Q * ) = u 1 (P * , Q * ) = v , (3.11) and U 2 (P * , Q * ) = -u 1 (P * , Q * ) = -v . (3.12)
Proof. We start by showing equation (3.11). Equation (3.12) follows analogously.

U 2 (P * , Q * ) ≥ U 2 (P * , Q) for all Q ⊆ ∆S 2 ⇒ U 2 (P * , Q * ) = max Q⊆∆S 2 U 2 (P * , Q) ⇒ min P ∈P * min Q∈Q * u 2 (P, Q) = max Q⊆∆S 2 min P ∈P * min Q∈Q * u 2 (P, Q) ⇒ min P ∈P * min Q∈Q * u 2 (P, Q) = max Q∈∆S 2 min P ∈P * u 2 (P, Q) ⇒ max P ∈P * max Q∈Q * u 1 (P, Q) = min Q∈∆S 2 max P ∈P * u 1 (P, Q) ⇒ max P ∈P * max Q∈Q * u 1 (P, Q) ≤ min Q∈∆S 2 max P ∈∆S 1 u 1 (P, Q) .
Furthermore,

U 1 (P * , Q * ) ≥ U 1 (P, Q * ) for all P ⊆ ∆S 1 ⇒ U 1 (P * , Q * ) = max P⊆∆S 1 U 1 (P, Q * ) ⇒ min P ∈P * min Q∈Q * u 1 (P, Q) = max P⊆∆S 1 min P ∈P min Q∈Q * u 1 (P, Q) ⇒ min P ∈P * min Q∈Q * u 1 (P, Q) = max P ∈∆S 1 min Q∈Q * u 1 (P, Q) ⇒ min P ∈P * min Q∈Q * u 1 (P, Q) ≥ max P ∈∆S 1 min Q∈∆S 2 u 1 (P, Q) .
From the above relations and Proposition 3.15 follows that max

P ∈P * max Q∈Q * u 1 (P, Q) ≤ min Q∈∆S 2 max P ∈∆S 1 u 1 (P, Q) = max P ∈∆S 1 min Q∈∆S 2 u 1 (P, Q) ≤ min P ∈P * min Q∈Q * u 1 (P, Q)
and we finally have

U 1 (P * , Q * ) = u 1 (P * , Q * ) = v .
From the two preceding lemmas we get the following minimax theorem for Ellsberg games.

Since, as explained above, in general U 2 (P, Q) � = -U 1 (P, Q), the equation is shown and stated for both players 1 and 2. In the classic mixed strategy case it suffices to state the minimax relation for one of the players.

Theorem 3.18 (Minimax Theorem 2). Let G be a two-person zero-sum game. Then for all i ∈ {1, 2} and every Ellsberg equilibrium (P * , Q * ) we have max

P i ⊆∆S i min P -i ⊆∆S -i U i (P, Q) = min P -i ⊆∆S -i max P i ⊆∆S i U i (P, Q) = U i (P * , Q * ) .

Ellsberg Equilibria of Some Classic 2 × 2 Games

We have already seen the Ellsberg equilibria of some important classes of games: of symmetric and asymmetric coordination games, competitive games and zero-sum games.

In an undergraduate game-theory course one analyzes usually three more types of games that became sort of classic, the Hawk and Dove game, the Prisoners' Dilemma and the Stag Hunt. We derive the Ellsberg equilibria of these games and find some special properties which we present in this section. These games have in common that the payoff functions only depend on the lower bound of the probability interval of the opponent. Thus they are linear, opposed to the piecewise linearity observed in the preceding examples. We then present a fourth game with linear payoff functions which is an example by [START_REF] Myerson | Game Theory: Analysis of Conflict[END_REF]. We demonstrate that despite the well-behaved (i.e., linear) payoff functions, in some of these games proper Ellsberg equilibria arise.

Hawk and Dove

We start with the two-player game which is known as Hawk and Dove with the payoff matrix in Figure 3.

(P, [Q 0 , Q 1 ]) of player 1. U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 3(P Q + P (1 -Q)) + 4(1 -P )Q = min Q 0 ≤Q≤Q 1 Q(4 -2P ) + P = Q 0 (4 -2P ) + P for all P ∈ [P 0 , P 1 ] .
The payoff function is linear and depends only on the lower bound of [Q 0 , Q 1 ]. The best response of player 1 is P 0 = P 1 = 0 when Q 0 < 1/2, and

P 0 = P 1 = 1 when Q 0 > 1/2,
these lead to the pure Nash equilibria (H, D) and (D, H). When player 2 chooses Q 0 = 1/2 the payoff function is constant at 2, thus the best response of player 1 is [P 0

, P 1 ] ⊆ [0, 1].
This is an Ellsberg equilibrium, as long as P 0 < 1/2 < P 1 . Furthermore, player 1 may choose P 0 = P 1 = 1/2 which results in the mixed strategy Nash equilibrium P * . Since the game is symmetric, the analysis for player 2 yields the same results. Hence we get the following proposition.

Proposition 3.19. The Ellsberg equilibria in the Hawk and Dove game are

([P 0 , P 1 ] , 1/2) , where P 0 < 1/2 < P 1 ,
and

(1/2, [Q 0 , Q 1 ]) , where Q 0 < 1/2 < Q 1 .
It is not surprising that in this completely symmetric game only Ellsberg equilibria exist which do not add interesting behavior to the analysis. When a player creates ambiguity about his Nash equilibrium strategy, the optimal response of his opponent does not change compared to the usual analysis.

Prisoners' Dilemma

Let us now look at the Prisoners' Dilemma with the payoff matrix in Figure 3.19.

Player 1

Player 2

C D C 3, 3 0, 4 D 4, 0 1, 1 Figure 3.19: Prisoners' Dilemma.
This game has a unique Nash equilibrium in pure strategies (D, D). To find the Ellsberg equilibrium, we first derive the payoff function

U 1 (P, [Q 0 , Q 1 ]) of player 1. U 1 (P, [Q 0 , Q 1 ]) = min Q∈[Q 0 ,Q 1 ] 3P Q + 4(1 -P )Q + (1 -Q)(1 -P ) = min Q∈[Q 0 ,Q 1 ] 3Q -P + 1 = 3Q 0 -P + 1 for all P ∈ [P 0 , P 1 ] .
The payoff function is again linear and depends only on the lower bound of [Q 0 , Q 1 ]. The best response of player 1 is always P 0 = P 1 = 0, no matter where Q 0 lies in the interval [0, 1]. The analysis for player 2 yields the same result, since the game is symmetric. This results in the Nash equilibrium in pure strategies (D, D).

Proposition 3.20. The Ellsberg equilibrium of the Prisoners' Dilemma is (D, D).

Because the only Nash equilibrium of the game is in strictly dominant strategies, the possibility of using ambiguity is not seized. However, when we add a mediator to the Prisoners' Dilemma, this mediator causes the prisoners to cooperate. We analyze the three-player mediated Prisoners' Dilemma with non-Nash outcome in Section 5.1.

Stag Hunt

Next we calculate the Ellsberg equilibria of the Stag Hunt game. We choose the following payoff matrix.

Player 1

Player 2 S H S 2, 2 0, 1 H 1, 0 1, 1 Figure 3.20: Stag Hunt.
The game has two pure Nash equilibria (S, S) and (H, H), and one Nash equilibrium in mixed strategies (P * , Q * ) = ((1/2, 1/2), (1/2, 1/2)). We calculate the minimal payoff functions of the two players.

U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 2P Q + (1 -P )Q + (1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 2P Q -P + 1 = 2P Q 0 -P + 1 for all P ∈ [0, 1] ,
and

U 2 ([P 0 , P 1 ] , Q) = 2P 0 Q -P + 1 for all Q ∈ [0, 1] .
The payoff function of player 1 is constant at 1 for Q 0 = 1/2, and the payoff function of player 2 likewise for P 0 = 1/2. They are increasing for Q 0 ≥ 1/2 (P 0 ≥ 1/2) and decreasing for Q 0 ≤ 1/2 (P 0 ≤ 1/2). Thus, we can immediately see that ambiguity is only used when P 0 = Q 0 = 1/2, and then the best responses are the whole interval [0, 1] for both players. Therefore, we have the following proposition.

Proposition 3.21. The Ellsberg equilibria of the Stag Hunt game are (S, S), (H, H) and the proper Ellsberg equilibria

([1/2, P 1 ] , [1/2, Q 1 ]) , where 1/2 ≤ P 1 ≤ 1 and 1/2 ≤ Q 1 ≤ 1 .
This proposition shows that in the Stag Hunt game the payoff dominant equilibrium (S, S)

is preferred over the risk dominant equilibrium (H, H), when ambiguity as a strategy is allowed.

Example by Myerson

The last example of this section is taken from [START_REF] Myerson | Game Theory: Analysis of Conflict[END_REF]. He chose this example to

show the power and limits in the interpretation of Nash equilibrium. We present it here,

because it has linear payoff functions also for ambiguity-averse players, and nevertheless proper Ellsberg equilibria. The game has the payoff matrix depicted in Figure 3.21. It has the unique Nash equilibrium in mixed strategies (P * , Q * ) = ((3/4, 1/4) , (1/2, 1/2)) with expected payoff (0 , 0), but it can be argued that it would be more reasonable for players to coordinate on (U, L) and thus guarantee an equilibrium payoff of 0. Of course, this is not a Nash equilibrium since player 1 would want to deviate.

Player 1

Player 2 L R U 0, 0 0, -1 D 1, 0 -1, 3 Figure 3.21: Myerson's game.
We calculate the payoff functions of the two players.

U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 (1 -P )Q -(1 -P )(1 -Q) = min Q 0 ≤Q≤Q 1 Q(2 -2P ) + P -1 =    Q 0 (2 -2P ) + P -1, if P < 1 , 0, if P = 1 .
(3.13)

Note that for P equal to 1, the utility function of player 1 is 0, and for P = 0 it is equal to 2Q 0 -1. In Figure 3.22 one can easily derive the best responses of player 1 which are listed below. We do the same analysis for player 2.

Q 0 > 1/2 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ]|P 0 = P 1 = 0} Q 0 = 1/2 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ] ⊆ [0, 1]} Q 0 < 1/2 : B 1 ([Q 0 , Q 1 ]) = {[P 0 , P 1 ]|P 0 = P 1 = 1}
U 2 ([P 0 , P 1 ] , Q) = min

P 0 ≤P ≤P 1 -P (1 -Q) + 3(1 -P )(1 -Q) = min P 0 ≤P ≤P 1 P (4Q -4) -3Q + 3 =    0, if Q = 1 , P 1 (4Q -4) -3Q + 3, if Q < 1 .
Similarly to case U 1 , utility function of player 2 is 0 when Q is equal to 1, and for Q = 0 it is 3 -4P 1 . The plot of U 2 looks very much like the payoff function for player 1. The best responses of player 2 are

P 1 < 3/4 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ]|Q 0 = Q 1 = 0} P 1 = 3/4 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ] ⊆ [0, 1]} P 1 > 3/4 : B 2 ([P 0 , P 1 ]) = {[Q 0 , Q 1 ]|Q 0 = Q 1 = 1} .
Hence, we obtain Proposition 3.22. The Ellsberg equilibria in Myerson's game are

([P 0 , 3/4] , [1/2, Q 1 ]) , where 0 ≤ P 0 ≤ 3/4 and 1/2 ≤ Q 1 ≤ 1 .
Thus, both players may use ambiguity in equilibrium, although the payoff functions of both players are linear. In a sense, one of the Ellsberg equilibria, that is (3/4, [1/2, 1]), makes coordination on the outcome (U, L) more likely.

We make an interesting observation studying these examples: even when both payoff functions are linear (and not, as often in Ellsberg games, piecewise linear) there exist Ellsberg equilibria in which ambiguity is used. But the existence of a unique strict Nash equilibrium in pure strategies as in the Prisoners' Dilemma causes, of course, that no ambiguity is used even when we allow for it. We proved this in Section 1.3.6.

Ellsberg Equilibria of 3 × 3 Games and Their Geometry

We can apply the insights on the Ellsberg equilibria of 2×2 games in the preceding sections to investigate the Ellsberg equilibria of larger games. We discuss the Ellsberg equilibria of a special class of modified 3 × 3 zero-sum games, that is, modified circulant games.

Circulant games have square payoff matrices in which the payoffs are circularly permuted in every row (column). A general circulant 3 × 3 zero-sum game has the following payoff matrix for some a, b, c ∈ R.

Player 1 Player 2 L M R T a, -a b, -b c, -c C c, -c a, -a b, -b B b, -b c, -c a, -a Figure 3.23: General circulant game.
For example, the classic game Rock Scissors Paper (RSP) is a circulant game, this is the game we analyze in this section. Our RSP has the payoff matrix given in Figure 3.24, the strategies are denoted R for Rock, S for Scissors, P for Paper.

We fix some new notation for Ellsberg strategies when we deal with games with more than two pure strategies. An Ellsberg strategy for player 1 is, as before, denoted by P, for player 2 by Q. They are convex sets of probability distributions (P 1 , P 2 , P 3 ) and

(Q 1 , Q 2 , Q 3 ) in ∆S 1 and ∆S 2 , respectively, that is P ⊆ ∆S 1 , Q ⊆ ∆S 2 . An Ellsberg Player 1 Player 2 R S P R 0, 0 1, -1 -1, 1 S -1, 1 0, 0 1, -1 P 1, -1 -1, 1 0, 0
Figure 3.24: Rock Scissors Paper.

strategy P is described as follows. Fix two vectors x, y ∈ [0, 1] 3 in the unit cube with components (x 1 , x 2 , x 3 ) and (y 1 , y 2 , y 3 ). Then

P = � (P 1 , P 2 , P 3 ) ∈ R 3 � � � 3 � i=1 P i = 1, P i ≥ 0, x i ≤ P i ≤ y i , i = 1, 2, 3 � . (3.14)
Note that P is completely described by giving the possible range of probabilities P 1 and P 2 . The range of P 3 (given by x 3 and y 3 ) then follows directly from the assumption that every triple (P 1 , P 2 , P 3 ) ∈ P is a probability distribution and some careful calculations.

We use the following abbreviation for (3.14):

P = ({x i ≤ P i ≤ y i }) i=1,2,3 = ({x 1 ≤ P 1 ≤ y 1 }, {x 2 ≤ P 2 ≤ y 2 }, {x 3 ≤ P 3 ≤ y 3 }) .
We write {x i ≤ P i } when y i = 1, and {P i ≤ y i } when x i = 0. We drop the subscript i = 1, 2, 3 when no confusion can arise. To describe an Ellsberg strategy Q for player 2, we use the vectors w, z ∈ [0, 1] 3 . An Ellsberg strategy for player 1 is then for example

({1/3 ≤ P 1 ≤ 1/2}, {P 2 ≤ 2/3}, {P 3 ≤ 2/3}) .
When we calculate Ellsberg equilibria, we typically find a large number of equilibria which are very similar but differ in that one or more boundaries of the set of probability distributions are variable. Then we use the following notation. Fix x, ŷ ∈ [0, 1] 3 , then

({x i ≤ x i ≤ P i ≤ y i ≤ ŷi }) := {({x i ≤ P i ≤ y i }) | for all x, y ∈ [0, 1] 3 such that xi ≤ x i ≤ y i ≤ ŷi } .
As before, we suppress xi and ŷi when they are 0 or 1, respectively. Such a set of Ellsberg strategies is then for example

({1/3 ≤ P 1 ≤ y 1 ≤ 1/2}, {x 2 ≤ P 2 ≤ 2/3}, {P 3 ≤ y 3 ≤ 2/3}) .
Note that this describes the set of Ellsberg strategies which necessarily include the probability distribution (1/3, 2/3, 0) at the boundary of each Ellsberg strategy contained in the set. This is the type of set of Ellsberg strategies which we encounter frequently in Ellsberg equilibrium analysis.

Now we present the Ellsberg equilibria of RSP. We make use of Theorem 4.2 which we prove in Section 4.3 and which holds for all two-player games in which at least one player's Nash equilibrium strategy is equal to his immunization strategy. This assumption is in particular true for all two-player zero-sum games, hence also for RSP. We show in Theorem 4.2 that the only type of Ellsberg equilibria these games have are of the type

(P * , Q * ) ((P * , Q * ))
, where P * (Q * ) is the Nash equilibrium strategy of player 1 (2), and Q * ⊆ ∆S 2 (P * ⊆ ∆S 1 ) can be any convex set of probability distributions in which Q * (P * ) is contained. This is the type of Ellsberg equilibrium that occurs in circulant games such as RSP.

We use the following notation. We denote by P 1 (Q 1 ) the probability with which player 1(2) plays R, P 2 (Q 2 ) the probability with which player 1(2) plays S, P 3 (Q 3 ) the probability with which player 1(2) plays P . The only Nash equilibrium of the game is (P * , Q * ) = ((1/3, 1/3, 1/3), (1/3, 1/3, 1/3)). Therefore, using the above considerations, we obtain the following proposition.

Proposition 3.23. The Ellsberg equilibria of Rock Scissors Paper are

(({P i = 1/3}), ({w i ≤ Q i ≤ z i })) , where 0 ≤ w i ≤ 1/3 ≤ z i ≤ 1 for all i = 1, 2, 3 , and
(({x i ≤ P i ≤ y i }), ({Q i = 1/3})) , where 0 ≤ x i ≤ 1/3 ≤ y i ≤ 1 for all i = 1, 2, 3 .
As in the matching pennies game, the situation changes when we slightly modify the original zero-sum game. We consider the following modification (Figure 3.25), where player 1 gets a payoff of 2, instead of 1, when (R, S) is played. Now the Nash equilibrium is

(P * , Q * ) = ((1/3, 1/3, 1/3), (1/3, 1/4, 5/12)) .
As in 2 × 2 games, the immunization strategies play an important role in the Ellsberg Player 1

Player 2 equilibria of the modified RSP. In our modified game, they are (M 1 , M 2 ) = ((1/4, 1/3, 5/12), (1/3, 1/3, 1/3)) .

R S P R 0, 0 2, -1 -1, 1 S -1, 1 0, 0 1, -1 P 1, -1 -1, 1 0, 0
We have the following proposition.

Proposition 3.24. The Ellsberg equilibria of the modified Rock Scissors Paper are

(({1/3 ≤ P 1 ≤ y 1 ≤ 2/3}, {P 2 = 1/3}, {x 3 ≤ P 3 ≤ 1/3}), ({w 1 ≤ Q 1 ≤ z 1 }, {1/4 ≤ Q 2 ≤ z 2 ≤ 1/3}, {5/12 ≤ Q 3 ≤ z 3 ≤ 3/4}))
,

where 0 ≤ w 1 ≤ 1/3 = z 1 .
This can be read as player 1 using the strategy: "I will play Rock at least with probability 1/3 but not with probability higher than 2/3. I will play Paper with probability less than 1/3 and Scissors with probability 1/3." Before we prove this, we discuss the intuition of the result. We know from the analysis of 2 × 2 games that ambiguity can only be a best response if the opponent uses his Nash equilibrium probability at the boundary of his Ellsberg strategy, and additionally this is the worst case measure which the player uses in his utility evaluation. In this case, the utility function has flat parts and the player best responds with ambiguity. When the opponent plays exactly his Nash equilibrium distribution (without using ambiguity himself), the whole utility function of the player is flat and he can play any Ellsberg strategy he likes, it will always be a best response.

However, recall that this can only be an Ellsberg equilibrium if the opponent's Nash equilibrium strategy is exactly his maximin strategy.

Consider the Ellsberg equilibrium in Proposition 3.24. We can draw the Ellsberg strategies as a projection into the 2-simplex (see, e.g., [START_REF] Ritzberger | Foundations of Non-Cooperative Game Theory[END_REF] p. 36 for explanation)

to understand how the characteristics reviewed above are apparent in 3 × 3 games. To this end we use the 'largest' Ellsberg equilibrium in the set, that is

(({1/3 ≤ P 1 ≤ 2/3}, {P 2 = 1/3}, {P 3 ≤ 1/3}), ({Q 1 ≤ 1/3}, {1/4 ≤ Q 2 ≤ 1/3}, {5/12 ≤ Q 3 ≤ 3/4})) .
In the equilateral triangle in Figures 3.26 and 3.27, for every point on the edge opposite the vertex R, the probability that R is played is zero. On the other hand, at the vertex R, R is played with probability one and S and P with probability zero. Sets of probability distributions, as we encounter in Ellsberg equilibria, are drawn as gray areas. The gray areas are the possible probabilities for each component of (P 1 , P 2 , P 3 ) ∈ P (in Figure 3.26) and Proof of Proposition 3.24. We start by calculating the minimal expected utility functions of player 1 and 2. Player 1 minimizes over

(Q 1 , Q 2 , Q 3 ) ∈ Q (in
Q 1 and Q 2 . Q 3 is expressed as 1 -Q 1 -Q 2 .
or {1/3 ≤ P 1 }. In Ellsberg equilibrium, player 2 uses 1/3 as his worst case measure, thus in the first case we only consider those elements of U 2 which use y 1 as worst case measure, in the second case only those which use x 1 . y 1 is only used when Q 2 ≥ 1/3. Since the range of Q 2 must contain 1/4 at the boundary, which is impossible when Q 2 ≥ 1/3, {1/4 ≤ x 1 ≤ P 1 ≤ 1/3} cannot be an Ellsberg equilibrium strategy. We proceed with the second case: x 1 is only used for Q 2 ≤ 1/3, this is compatible with the Nash equilibrium strategy and yields

{1/4 ≤ Q 2 ≤ z 2 ≤ 1/3}. Finally we check if {1/3 ≤ P 1 } is a best response to {1/4 ≤ Q 2 ≤ z 2 ≤ 1/3}. Player 1 must use w 2 in his utility evaluation U 1 ,
this is true whenever P 1 ≥ 1/4 which is compatible with the strategy {1/3 ≤ P 1 }. Thence we have found a best response pair.

In the second component, player 1 can play {P 2 = 1/3}, which is his Nash equilibrium probability and therefore makes player 2 indifferent between all {w 1 ≤ Q 1 ≤ z 1 }, and at the same time his best response to player 2 playing any probability

{w 1 ≤ Q 1 ≤ z 1 } with 0 ≤ w 1 ≤ 1/3 ≤ z 1 ≤ 1.
Or, conversely, player 2 can play {Q 1 = 1/3} and player

1 {x 2 ≤ P 2 ≤ y 2 } with 0 ≤ x 2 ≤ 1/3 ≤ y 2 ≤ 1.
The latter case collapses to Nash equilibrium. The restrictions on w 1 , z 1 , which are 0 ≤ w 1 ≤ 1/3 = z 1 , follow from the third component and the fact that each element of the Ellsberg equilibrium strategy must be a probability distribution.

We cannot determine the range for P 3 and Q 3 from the two utility functions, since P 3 and Q 3 are only implicitly given. Thus, we derive the utility functions again, now using

(P 1 , 1 -P 1 -P 3 , P 3 ) and (1 -Q 2 -Q 3 , Q 2 , Q 3 ) as probabilities.
Since we already know that only the boundaries x 1 and w 2 are relevant for the equilibrium, we get

U 1 (P 1 , P 3 , Q 2 , Q 3 ) = min Q 2 ,Q 3 2P 1 Q 2 -P 1 Q 3 -(1 -P 1 -P 3 )(1 -Q 2 -Q 3 ) + (1 -P 1 -P 3 )Q 3 + P 3 (1 -Q 2 -Q 3 ) -P 3 Q 2 =          w 3 (2 -3P 1 -3P 3 ) + w 2 (P 1 -3P 3 + 1) + 2P 3 + P 1 -1 if P 3 < 2/3 -P 1 , w 2 (5/3 -4P 3 ) + P 3 -1/3 if P 3 = 2/3 -P 1 , z 3 (2 -3P 1 -3P 3 ) + w 2 (P 1 -3P 3 + 1) + 2P 3 + P 1 -1 if P 3 > 2/3 -P 1 , when P 1 > 3P 3 -1.

Ellsberg Games, Human Behavior and Observational Implications

In this chapter we present some thoughts on a general interpretation of Ellsberg equilibria and a classification of when ambiguity is an option in two-player Ellsberg equilibria.1

We are on the one hand interested in human behavior in two-person games and its connection to Ellsberg equilibria. In an experiment with a modified version of Matching Pennies, [START_REF] Goeree | Ten little treasures of game theory and ten intuitive contradictions[END_REF] observed that Nash equilibrium prediction was frequently violated; interestingly, our Ellsberg equilibria can explain this behavior. In Section 4.2 on the other hand, we classify Ellsberg equilibria that are supported by strategies which are not in the support of any Nash equilibrium of the game. These Ellsberg equilibria arise frequently in games with weakly dominated strategies and have an interesting behavioral interpretation. Finally in Section 4.3, we characterize in which circumstances and how ambiguity is used in equilibrium in two-person games. This analysis provides an answer to the question in which two-person games proper Ellsberg equilibria exist.

Human Behavior in Matching Pennies Games and Ellsberg Equilibria

We have now seen a number of different two-player games and have calculated their Ellsberg equilibria. Recall the modified Matching Pennies example in Section 3.3.1. Whereas the support of the Ellsberg and Nash equilibria of the game is obviously the same, we think that the Ellsberg equilibria reveal a new class of behavior not encountered in game theory before. It might be very difficult for humans to play exactly a randomizing strategy with equal probabilities. Indeed, the ability to do so has been a source of debate since the early days of game theory, and some claim that humans cannot randomize, see [START_REF] Dang | Gaming or guessing: mixing and best-responding in matching pennies[END_REF] for a recent account and references therein. Our result shows that it is not necessary to randomize exactly to support a similar equilibrium outcome (with the same expected payoff). It is just enough that your opponent knows that you are randomizing with some probability, and that it could be that this probability is one half, but not less. It is thus sufficient that the player is able to control the lower bound of his device. This might be easier to implement than the perfectly random behavior required in classical game theory.

In fact, there are experimental findings which suggest that the Ellsberg equilibrium strategy in the modified Matching Pennies game is closer to real behavior than the Nash equilibrium prediction. To illustrate this, let us consider the interesting results by [START_REF] Goeree | Ten little treasures of game theory and ten intuitive contradictions[END_REF] who ran experiments on three different versions of Matching Pennies; the three payoff matrices can be seen in Table 4.1.

In the first game, we have a typical symmetric conflict game with a unique mixed Nash equilibrium in which both players randomize uniformly over both pure strategies. The aggregate play of humans in the experiment is closely consistent with the Nash equilibrium prediction, 48% of players choosing "Top" or "Left", respectively. Remember that the probabilities in a mixed strategy equilibrium are chosen in such a way as to render the opponent indifferent between his two pure strategies. As a consequence, if we change the payoffs of player 1 only (while keeping the ordering of payoffs), his Nash equilibrium strategy does not change because he has to make player 2 indifferent between his two pure actions, and player 2's payoffs have not been modified.

In the second game, called the asymmetric Matching Pennies game, player 1 gets 320 instead of 80 in the upper left outcome. All other payoffs remain the same. Many humans now deviate from Nash, as is reported in brackets, 96% of the players taking the action "Top". Interestingly, also the humans playing the role of player 2 change their behavior, and most of them play "Right", the best reply to "Top".

In the third case, player 1's payoff in the upper left outcome is decreased to a lowly 44.

Then only 8% of players choose "Top"; 80% of humans in the role of player 2 choose "Left".

While aggregate behavior by humans is certainly inconsistent with the predictions of Nash equilibrium, it is consistent with Ellsberg equilibria. We summarize the results in Table 4.2.

In the symmetric game, our Proposition 3.7 essentially predicts only Nash equilibrium behavior, and this is what we observe in the experiment as well. In the asymmetric Matching Pennies game, the Nash equilibrium strategies are P * = 1/2 for player 1 and Q * = 1/8 for player 2. According to our proposition, the Ellsberg equilibria allow for probabilities in the interval [1/2, 1] for player 1 choosing "Top", and for the interval [1/8, 1/2] for player 2 choosing "Left". The observed percentages of 96% and 16% do lie in these intervals.

And in the "reversed" version of the game, the Nash equilibrium strategies are P * = 1/2

Observational Implications of Ellsberg Equilibria

Game Theory studies equilibrium outcomes of social conflicts when rational agents interact. Human beings are quite different from rational agents in general, so one can only expect to see a consistency with Nash equilibrium predictions and human behavior when the situation is controlled in such a way as to bring out the rational part of humans.

Nevertheless, it does make sense to ask what the observational implications of our theory are. For three player games, this is quite clear, as our theory predicts new equilibria outside the support of Nash equilibria. This is a testable implication, and we shall proceed one day to carry out such a test.

For two player games, the situation is more subtle. Both the Nash equilibrium and the Ellsberg equilibria have full support, so the only thing that we can learn from our theory seems to be that either action is fine in a one shot game. This is indeed the stance of [START_REF] Bade | Ambiguous act equilibria[END_REF], in line with a number of predecessors.

There is, however, a way to distinguish the predictions of Ellsberg equilibria and Nash equilibria even in two player games. To understand this, we first need to explain what the law of large numbers looks like under ambiguity. The classical law states that the frequency of HEAD in an infinite sequence of independent coin tosses will converge to the probability of HEAD. Now let us look at a typical Ellsberg urn that contains 100 balls, red and black, and we only know that the number of red balls is between 30 and 60. What can we say about the average frequency drawn from independent repetitions of the Ellsberg experiment? The natural guess would be that the average lies in the interval between 30% and 60% in the long run. This is indeed correct, and mathematical versions of that theorem have recently been proven, see [START_REF] Maccheroni | A strong law of large numbers for capacities[END_REF] and Epstein and Schneider (2003a), e.g., [START_REF] Peng | Law of large numbers and central limit theorem under nonlinear expectations[END_REF] has obtained the result that the average frequency will indeed fluctuate between both bounds, and every point in the interval [0.3, 0.6] is an accumulation point of the sequence.

What is then the empirical content of such laws of large numbers? If we adhere to the point of view that our observed humans play independently one shot games, and that they should play equilibrium strategies, then the average frequency will converge to the Nash equilibrium strategy according to the classical theory, and will fluctuate between two bounds according to the new Ellsberg theory.

We thus do get observational differences between the two theories, and we interpret the Goeree-Holt results as a first evidence that our theory can accommodate deviations from Nash equilibrium observed in laboratories.

Two-Player Games: Strategies Outside the Nash Support

We now present a two-player example in which a strategy is supported in Ellsberg equilibrium which is not in the support of a Nash equilibrium of the game. When this happens, an Ellsberg equilibrium is observationally different to any Nash equilibrium of the game:

if an outside observer watched two players play a one-shot strategic game, he would be able to distinguish if the players used ambiguity in equilibrium or not. [START_REF] Bade | Ambiguous act equilibria[END_REF] introduces the notion of observational equivalence to strategic two-person games with subjective randomization devices. She shows that in two-person games with fairly general preference structure observational equivalence is always satisfied. To make the theorem hold she uses one assumption on the preferences which is not common to the mostly used preference representations, the property of strict monotonicity.

The following example is taken from [START_REF] Bade | Ambiguous act equilibria[END_REF]. She chooses it to show that without the condition of strict monotonicity the conditions of the theorem are not fulfilled and thus we have observational differences. In opposition to her, we think that the introduction of ambiguity offers an interesting outcome of the game. In addition, our analysis allows to give a full description of all possible Ellsberg equilibria of this game. The payoff matrix of the game is given in We first explain why strict monotonicity is not fulfilled in this game. Strict monotonicity means that if there is one non-null state2 in which an Ellsberg strategy is strictly preferred to another, this should be so in all non-null states. Player 1 strictly prefers D to U when in some state player 2 plays L with positive probability. Whenever player 2 plays L with some

Ellsberg strategy [0, P 1 ], P 1 > 0, player 1 discounts in his utility evaluation completely the possibility of L to occur, although it is not a null-state. Thus, D is not for all states strictly preferred to U .

The game has a pure Nash equilibrium (D, R). Notice that the game does not have a Nash equilibrium with full support, since L is never played in equilibrium. One must admit that although U is a weakly dominated strategy, it would be appealing for player 1 if he could get player 2 to play L. Player 1 could assure himself a payoff of 10 as opposed to a payoff of 0. Nash equilibrium does not allow this "coordination" on (U, L). We will now see that this is different when we look at the Ellsberg equilibria of the game.

We calculate the Ellsberg equilibria. Suppose player 1 chooses to play U with the set of probabilities [P 0 , P 1 ], whereas player 2 plays L with the set of probabilities [Q 0 , Q 1 ]. To find the Ellsberg equilibria, we first derive the payoff function U 1 (P, [Q 0 , Q 1 ]) of player 1.

U 1 (P, [Q 0 , Q 1 ]) = min Q 0 ≤Q≤Q 1 10P Q + 11(1 -P )Q = min Q 0 ≤Q≤Q 1 Q(11 -P ) = Q 0 (11 -P ) for all P ∈ [P 0 , P 1 ] . (4.1)
The payoff function is linear and depends only on the lower bound of [Q 0 , Q 1 ]. On the boundaries we have

U 1 (0, [Q 0 , Q 1 ]) = 11Q 0 , U 1 (1, [Q 0 , Q 1 ]) = 10Q 0 .
The best response of player 1 is P 0 = P 1 = 0 when Q 0 > 0, only when Q 0 = 0 the payoff function is constant equal to zero and thus player 1 is indifferent between any subsets of [0, 1]. Thus, his best response in the latter case is [P 0 , P 1 ] ⊆ [0, 1]. This can be easily observed in the plot of (4.1) in Figure 4.2. We plot

U 1 (P, [Q 0 , Q 1 ]) for (1) Q 0 = 0, (2) Q 0 = 1/2 and (3) Q 0 = 1.
Q 0 = 0, (2) Q 0 = 1/2 and (3) Q 0 = 1.
The case is different for player 2, here we get a dependence on the lower and upper bound

P 0 = P 1 = 0. But when player 2 chooses [Q 0 , Q 1 ] ⊆ [0, Q 1 ] with 0 ≤ Q 1 ≤ 1/2, then
player 1 is indifferent on the whole interval of P and may choose [P 0 , P 1 ] ⊆ [0, 1]. If in the latter case player 1 chooses to play [P 0 , P 1 ] = [P 0 , 1/2] with 0 ≤ P 0 ≤ 1/2, then this is an Ellsberg equilibrium, i.e.,

([P 0 , 1/2] , [0, Q 1 ])
, where 0 ≤ P 0 ≤ 1/2 and 0 ≤ Q 1 ≤ 1/2 .

(5

) P 1 < 1/2 ⇒ Q 0 = Q 1 = 0 ⇒ [P 0 , P 1 ] ⊆ [0, 1]
, thus if player 1 chooses [P 0 , P 1 ] with 0 ≤ P 0 ≤ P 1 ≤ 1/2 this is an Ellsberg equilibrium. Note that the pure strategy Nash equilibrium (D, R) is contained in this Ellsberg equilibrium.

(

) P 0 = 1/2 = P 1 ⇒ [Q 0 , Q 1 ] ⊆ [0, 1]. 6
We again consider two cases: if player 2 chooses

[Q 0 , Q 1 ] = [Q 0 , 1] with 0 ≤ Q 0 ≤ 1, then player 1's best response is P 0 = P 1 = 0, thus this is not an equilibrium. But if player 2 decides to play [Q 0 , Q 1 ] = [0, Q 1 ] with 0 ≤ Q 1 ≤ 1 and player 1 plays P 0 = P 1 = 1/2 ∈ [0, 1],
no player can gain by deviating from this strategy. We get the Ellsberg equilibrium

(1/2, [0, Q 1 ]) , where 0 ≤ Q 1 ≤ 1 .
Proposition 4.1. The Ellsberg equilibria of the game in Figure 4.1 are of the form

([P 0 , 1/2] , [0, Q 1 ])
, where 0 ≤ P 0 ≤ 1/2 and 0 ≤ Q 1 ≤ 1/2 , ([P 0 , P 1 ] , 0) , where 0 ≤ P 0 ≤ P 1 ≤ 1/2 ,

and (1/2, [0, Q 1 ]) , where 0 ≤ Q 1 ≤ 1 .
In the first and third of the Ellsberg equilibria there is some chance that (U, L) is played.

Player 2 creates ambiguity and makes sure that the probability that L is not played at all is contained in his Ellsberg strategy. Then player 1 finds it optimal to play U with positive probability, either by playing a random strategy as in the third equilibrium, or by creating ambiguity himself.

Of course, in all these equilibria the minimal expected utility of player 1 is 0. Therefore player 1, being ambiguity-averse, in his own estimation does not gain any utility by using an Ellsberg strategy instead of the pure Nash equilibrium strategy. Even more, player 2 can assure himself at most a utility of 1/2 which is worse than his (strict) Nash equilibrium payoff. Nevertheless, these Ellsberg equilibria are in some sense pareto dominating, because with some positive probability the outcome of the game is (U, L).

The preceding example (Figure 4.1) suggests that if we allow for not strictly monotonous payoff functions in two-person games (which is the case in most ambiguity averse preference representations) we can obtain Ellsberg equilibria which are supported by pure strategies that do not appear in the support of any Nash equilibrium. These equilibria pareto dominate the Nash equilibria of these games.

Weakly Dominated Strategies and Observational Differences

Different papers have found that the existence of weakly dominated strategies plays an important role in the classification of games with ambiguity averse players. We comment on some of the existing results.

Theorem 5 in [START_REF] Klibanoff | Stochastically independent randomization and uncertainty aversion[END_REF] shows that Weak Admissibility (WA) of a preference relation �:3 for all acts f, g, if for all s ∈ S f (s) � g(s) , then f � g and [f � g if and only if for some non-null event E , f (s) � g(s) for all s ∈ E] implies agreement on null events (AGR), let P i be a closed and convex subset of ∆S i P (E) = 0 if and only if for all P ∈ P i , P (E) = 0 in the maxmin expected utility representation by [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]. The same holds for the objective ambiguity representation we use in Ellsberg games. Agreement on null events has the following interpretation in the context of games. Each event (that is, e.g., each strategy of the opponent) is given either zero probability by all distributions in P i , or positive probability by all distributions in P i (i.e., the distributions in P i are mutually absolutely continuous, see, e.g., [START_REF] Epstein | Mutual absolute continuity of multiple priors[END_REF]).

Practically, this excludes proper Ellsberg strategies which touch the boundary of the simplex. In 2 × 2 games that would be strategies of the type [0, P 1 ] and [P 0 , 1]. When we exclude these types of strategies, those Ellsberg strategies in Proposition 4.1 that have a support outside the Nash support, collapse.

And really, two-player games with weakly dominated strategies (where the indifference is at the minimal payoff) have to be excluded if we want to assure no observational differences between Ellsberg equilibria and Nash equilibria in two-person normal form games. This is proved in Theorem 1 on observational equivalence in [START_REF] Bade | Ambiguous act equilibria[END_REF] for a large class of ambiguity averse preferences which also includes Ellsberg games. She finds that (WA) is a necessary condition for the theorem to hold.

In a similar spirit, [START_REF] Eichberger | Non-additive beliefs and strategic equilibria[END_REF] prove in Proposition 5.1, that in nperson normal form games where the minimal payoffs for each strategy are different, in an "equilibrium under uncertainty" no weakly dominated strategies are used. But note that in their concept of equilibrium under uncertainty players are not allowed to use mixed strategies, thus weak domination in the proposition is only weak domination in pure strategies.

Existence of Proper Ellsberg Equilibria: When is

Ambiguity an Option?

In the preceding examples one could observe that in some games ambiguity is played in equilibrium, either by both or at least by one of the players, and in some games no proper Ellsberg equilibria exist. In, e.g., the Prisoners' Dilemma which has a unique Nash equilibrium in pure strategies, no ambiguity is observed in Ellsberg equilibrium: in fact, the only Ellsberg equilibrium is the unique Nash equilibrium. In the modified Matching Pennies game exist Ellsberg equilibria where both players create ambiguity, and in zerosum games no more than one player uses ambiguity in Ellsberg equilibrium. We now have a closer look at the conditions under which proper Ellsberg equilibria (see Definition 1.8) are possible.

Intuitively, when thinking about ambiguity in normal form games, we imagine players using some kind of ambiguity "around" their mixed Nash equilibrium strategies. If we take P * to be the mixed Nash equilibrium strategy, this would be some Ellsberg strategy [P * -�, P * + �] with � > 0. Surprisingly, these kind of equilibria, where both players create ambiguity around their Nash strategy, never exist. When one of the players uses this type of strategy in Ellsberg equilibrium we speak of unilateral full ambiguity. "Full", because the player can then use as much ambiguity as he wants (also the whole simplex), as long as his mixed Nash equilibrium strategy is contained in the set.

There is only one situation in (generic) 2 × 2 games where one of the players uses a strategy like the one above: when the immunization strategy of one player is exactly his Nash equilibrium strategy. Then the opponent can use any set of probability measures, as long as his Nash equilibrium is an element of that Ellsberg strategy. We show that if and only if the immunization strategies are identical with the Nash equilibrium strategies there exists an Ellsberg equilibrium with unilateral full ambiguity. distributions (provided this set contains his Nash equilibrium strategy). When we change the game just a little by adding � to some payoff of player 1, this Ellsberg equilibrium type collapses and player 2 is now much more constrained in his equilibrium play (compare table 3.1). Thus, the equilibrium type in Theorem 4.2 is not robust to small changes in the payoff matrix. On the other hand, starting from a game where the Nash equilibrium strategies are not identical to the immunization strategies, the Ellsberg equilibria change continuously with changes in the payoff matrix.

5 A Third Player Can Cause Non-Nash Behavior

In the preceding chapters we presented the basic properties of Ellsberg equilibria and how players behave when we allow them to use objective ambiguity in two-player games.

Naturally, we want to characterize what can happen in games with more than two players.

To tackle this problem, we first investigate the relation of Ellsberg equilibrium to the concept of subjective equilibrium. Since there exists no other solution concept that lets players use sets of probabilities as their strategy, the question of comparison has to be answered with regard to the attainability of certain outcomes. The objective is to find out under which conditions subjective equilibria can (or cannot) have the same support as Ellsberg equilibria.

We start with an example of a three-player game, where an Ellsberg equilibrium exists that attains an outcome that is not in the support of any Nash equilibria of the game. This suggests that in very simple games with more than two players, the prediction of Ellsberg equilibrium can be quite different from the Nash equilibrium prediction. Subsequently, we define subjective equilibrium in the spirit of [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] and [START_REF] Hallin | Subjectively mixed strategies[END_REF] and subjective beliefs equilibrium like [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF]. We show that every Ellsberg equilibrium contains a subjective equilibrium and a subjective beliefs equilibrium. In two player games, subjective beliefs equilibria have the same support as Nash equilibria, but this changes for three players. This explains that in Ellsberg games with more than two players outcomes outside the Nash equilibrium support can be attained.

Three-Person Games with Non-Nash Outcomes

We want to give a simple example of a game with more than two players, where an Ellsberg equilibrium predicts very different behavior than the Nash equilibrium. To this end, we consider a classic example from [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF]. He presents a three-person game where one player has some mediation power to influence his opponents' choice. The original game is given by the payoff matrix in Figure 5.1, where we let player 1 choose rows, player 2 choose columns and player 3 choose matrices. Player 3 is indifferent between his strategies l and r, since he gets the same payoffs for both. As long as player 3 chooses l with a probability higher than 3/8, L is an optimal strategy for player 2 regardless what player 1 does; and player 1 would subsequently play D. By the same reasoning, as long as player 3 plays r with a probability higher than 3/8, D is optimal for player 1, and player 2 plays L then. Thus, the Nash equilibria of this game are all of the form (D, L, P * ), where P * is any classical mixed strategy.

This example has also been analyzed in other literature on ambiguity in games, see [START_REF] Eichberger | Ambiguity and social interaction[END_REF], Lo (2009) and [START_REF] Bade | Ambiguous act equilibria[END_REF]. Bade chooses one ambiguous act equilibrium with maxmin expected utility preferences to show that non-Nash outcomes can be sustained in games with more than two players. We characterize all Ellsberg equilibria1 and provide an interpretation of the example which highlights the strategic use of ambiguity as a mediation tool.

Let us now explain how Aumann's example can be interpreted to illustrate the strategic use of ambiguity. Suppose players 1 and 2 are prisoners, and player 3 the police officer. Let us rearrange the matrix game and put it in the form displayed in Figure 5.2. We swap the strategies of player 2 and rename the strategies of player 1 and 2 to C = "cooperate" and D = "defect" as in the classical Prisoners' Dilemma. We can merge the two matrices into one, because the strategy choice of the police officer is simply the choice of a probability that influences the payoffs of prisoners 1 and 2 in case of unilateral defection from (C, C).

If he chooses P = 1, this corresponds to strategy l in the original game (i.e., prisoner 2 gets all the reward), P = 0 would be strategy r (i.e., prisoner 1 gets all the reward).

The choice of the objectively mixed strategy P = 1/2 leads to the classical symmetric Prisoners' Dilemma with a payoff of 4 in case of unilateral defection.

In this interpretation, players 1 and 2 are facing a sort of Prisoners' Dilemma situation mediated by a player 3, the police officer. Given the payoffs, the police officer is most interested in cooperation between the prisoners. The police officer can influence how high the reward would be for unilateral defection by using an objective randomizing device.

Nevertheless, in every Nash equilibrium of the game, the players obtain the inefficient outcome of 1. Let us exhibit Ellsberg strategies that support this behavior. If prisoner 2 expects P to be lower than 3/8 and prisoner 1 expects P to be higher than 5/8, they would prefer to cooperate. This behavior corresponds to the police officer playing an Ellsberg strategy [P 0 , P 1 ] with 0 ≤ P 0 < 3/8 and 5/8 < P 1 ≤ 1. The ambiguity averse prisoners 1 and 2 evaluate their utility with P = P 0 and P = P 1 , respectively. Consequently they would prefer to play (C, C). This gives an Ellsberg equilibrium in which the prisoners cooperate.

Proposition 5.1. In the mediated Prisoners' Dilemma, the Ellsberg strategy profiles (C, C, [P 0 , P 1 ]) with P 0 < 3/8 and P 1 > 5/8

are Ellsberg equilibria that achieve the efficient outcome (3,3,3).

Again, as in the example by Greenberg (see Figure 1.3 in Section 1.4), it is important to see that both prisoners use different worst-case probabilities to compute their expected payoffs. [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] has already commented on this behavior. He observes that (C, C) can be a "subjective equilibrium point" if players 1 and 2 have non-common beliefs about the objectively mixed strategy player 3 is going to use. In his analysis player 1 believes P = 3/4 and player 2 believes P = 1/4. Note that in Ellsberg equilibrium the players have the common belief P ∈ [P 0 , P 1 ].

Subjective Equilibria and Ellsberg Equilibria

In a subjective equilibrium the players may base their actions on events of which the probability of realization is not commonly agreed on. These events are thus not the outcome of the roll of a die, but something more subjective such as the outcome of a horse race, certain stock exchange fluctuations or the outcome of a soccer match. For a game this implies that a player may best respond to a belief he has about his opponents' play, but this belief does not have to coincide with the actual strategy played.

The notion of subjective equilibrium as a static concept has been first introduced by [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF]. The only authors that to our knowledge subsequently worked with this static notion were [START_REF] Hallin | Subjectively mixed strategies[END_REF] and [START_REF] Brandenburger | Rationalizability and correlated equilibria[END_REF]. [START_REF] Hallin | Subjectively mixed strategies[END_REF] shows under which conditions subjective equilibrium payoffs can be attained by an objective equilibrium, and when, on the other hand, subjective equilibrium payoffs can strictly dominate objective ones. In both results he allows for correlated beliefs and correlated strategies. [START_REF] Brandenburger | Rationalizability and correlated equilibria[END_REF] show that correlated rationalizability is equivalent to a special case of subjective equilibria, that is subjective correlated equilibria that are also optimal after the state of the world is revealed [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] calls these "a posteriori equilibria"). Subjectivity in extensive form or repeated games has been defined and examined in another branch of literature not treated in this section.

We give an example of a subjective equilibrium before stating the definition. Consider the standard matching pennies game in Figure 5.3. This game has only one Nash equilibrium (in Aumann (1974)'s language: equilibrium in objective mixed strategies, objective, because all players have the same probability distribution on the state space, and mixed, because the strategies cannot be correlated), that is when both players mix with probabilities (1/2, 1/2). The expected payoff is then zero for both players. Now we relax the assumption that the beliefs in equilibrium have to be correct. Then (U, L) can be a subjective equilibrium when player 1 beliefs player 2 will play L (which is true), but player 2 thinks player 1 will play D and thus correctly best responds to his belief by playing L. Obviously this equilibrium seems unsatisfying, since one of the players beliefs something wrong. Nevertheless, this is a subjective equilibrium of the matching pennies game, with subjective expected utility (1, 1).

Player 1 Player 2 L R U 1, -1 -1, 1 D -1, 1 1, -1
Aumann (1974) considers a similar equilibrium of the matching pennies game in example (2.2) of that article. He supposes that there is an event to which players 1 and 2 ascribe subjective probabilities 1 and 0, respectively. Player 2 plays left in any case (he does not observe D), player 1 plays U if D occurs, else he plays D. Aumann claims, p. 76, that these strategies form an equilibrium in subjective mixed strategies, but I believe it only works when the strategies are correlated. At least player 1 has to believe that player 2 plays a strategy that is correlated to his. Else player 1's strategy cannot be a best response to any belief player 1 might have.

Definition of Subjective Equilibrium

As we said above, a subjective equilibrium relaxes the common belief assumption in Nash equilibrium by letting players base their actions on events that players may assign different probabilities to.

Let N = 1, . . . , n denote the set of players, S i the set of pure strategies of each player. A normal form game G = �N, (S i ), (u i )� is defined by the players' strategies and their utility functions. We equip each player i with a subjective randomizing device Ω i that is the set of future states of the world. Each player i has a subjective probability distribution

π i = (π i 1 , . . . , π i n ) on the profile Ω = (Ω 1 , . . . , Ω n ).
We assume that all the state spaces are stochastically independent. A subjectively mixed strategy for player i is a measurable function f i : Ω → ∆S i , where ∆S i is the set of lotteries over pure strategies s i ∈ S i . The set of measurable functions f i is denoted (∆S i ) Ω , lotteries over S i , i.e., mixed strategies, are denoted by p i . The utility for player i of a profile f of subjectively mixed strategies is evaluated as subjective payoff expectation,

Ūi (f 1 , . . . , f n ) := � Ω u i (f (ω)) dπ i .
Definition 5.2. A subjective equilibrium of the strategic game G = �N, (S i ), (u i )� is a profile (f * 1 , . . . , f * n ) of subjectively mixed strategies, such that for all i ∈ N there exists a profile of probability distributions π i on Ω, such that

f * i ∈ arg max f i ∈(∆S i ) Ω � Ω i � Ω -i u i (f i (ω i ), f * -i (ω -i )) dπ i -i dπ i i .
Or alternatively, if for all i ∈ N there exists a profile of probability distributions π i on Ω, such that for all f

i : Ω → ∆S i , Ūi (f * i , f * -i ) ≥ Ūi (f i , f * -i ) .
Note that we gave a definition for the private event case. Every player bases his action on a private event, and the state spaces of all the players are stochastically independent. [START_REF] Hallin | Subjectively mixed strategies[END_REF] gave the analog definition for the public event case and thus leaves the possibility for players to correlate their actions and have correlated beliefs.

The above definition can, the same as correlated and Ellsberg equilibrium, be reduced to the case where the states of the world are the pure strategies of each player. To this end, let µ i -i = (µ i 1 , . . . , µ i i-1 , µ i i+1 , . . . , µ i n ) denote a profile of probability distributions over the sets ∆S 1 , . . . , ∆S i-1 , ∆S i+1 , . . . , ∆S n , respectively, this means µ i -i ∈ ∆(∆S -i ). µ i -i is the belief of player i about which mixed strategy players N \{i} will play, therefore it has to be a probability distribution over the set of mixed strategies. We state the definition below.

The subjective expected payoff Ūi of the profile of acts f now reduces to the subjective expected payoff of the beliefs which are represented by mixed strategies. Therefore we denote the subjective expected payoff in the reduced case by u i .

Definition 5.3. A reduced form subjective equilibrium of a strategic game G = �N, (S i ), (u i )� is a profile (p * 1 , . . . , p * n ) of mixed strategies, if for each i ∈ N there exists a profile of probability distributions µ i -i on the set of lotteries ∆S -i , such that

p * i ∈ arg max p i ∈∆S i u i (p i , µ i -i ) = arg max p i ∈∆S i � S i � S -i u i (s i , s -i ) dµ i -i dp i .
Or alternatively, if for all i ∈ N there exists a profile of probability distributions µ i -i on the set of lotteries ∆S -i , such that for all p i ∈ ∆S i ,

u i (p * i , µ i -i ) ≥ u i (p i , µ i -i ) .
Proposition 5.4. The definitions 5.2 and 5.3 are equivalent.

Proof. The proof is analog to the proof of Proposition 1.3 on the reduced form Ellsberg equilibrium.

In the reduced form subjective equilibria players formulate their beliefs directly about which strategy their opponents will play, in the non-reduced form the belief is formed about the probability distribution governing the state space that each opponent uses to subjectively mix their strategy.

Whichever definition the reader prefers, since the beliefs are inconsistent (that is, they can differ between the players), [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] argues that in subjective equilibria someone might want to renegotiate after the state of the world is revealed. This can be observed very well in the example of a subjective equilibrium of the matching pennies game presented at the beginning of the chapter. In that equilibrium player 2 best responds to a completely false belief and thus is happy with his choice only before the state D is revealed.

A posteriori he would obviously like to change his action.

These problems can be avoided by making further assumptions on the equilibrium.

Aumann (1974) calls them "a posteriori equilibria", when he assumes that no player wants to unilaterally deviate even after the state of the world is revealed. In his definition of a posteriori equilibria he allows for correlated randomizing. He shows that the set of payoffs to a posteriori equilibrium points coincides with the set of payoffs to subjective correlated equilibrium points, when the π i are mutually continuous with respect to each other. That is, if π i (ω) = 0 for one i, then π i (ω) = 0 for all i ∈ N . This property is also called agreement on null-events. [START_REF] Brandenburger | Rationalizability and correlated equilibria[END_REF] go even further and show that the set of payoffs to a posteriori equilibria coincides with the set of correlated rationalizable payoffs.

Relation with Ellsberg Equilibrium

We now analyze how Ellsberg equilibria are connected with subjective equilibria. With his definition of a subjectively mixed strategy, [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] was the first to model a strategy that is based on an uncertain event in the Knightian sense. Of course, at the time, no ambiguity averse preference representations were available and Aumann assumes that an uncertain event is characterized by the fact that players may have different priors on its realization. Thus, he assumes that players have a probability distribution on the set of states of the world Ω according to [START_REF] Savage | Foundations of Statistics[END_REF] subjective expected utility. With this assumption the use of subjectively mixed strategies boils down to the fact that the beliefs of the players may differ in equilibrium. When we get rid of the subjective expected utility assumption and instead assume ambiguity averse preferences, these lead to ambiguous acts as defined by [START_REF] Bade | Ambiguous act equilibria[END_REF]. In contrast, by playing Ellsberg strategies players may pick any kind of ambiguous randomizing device that they find reasonable, that is, which is utility maximizing. Preferences are imposed upon the existence of objective ambiguity.

In spite of the different preference representations in [START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF]

i -i ∈ ∆P * -i .
Proof. When a Nash equilibrium (q 1 , . . . , q n ) is element of the Ellsberg equilibrium (P * 1 , . . . , P * n ), that is q i ∈ P * i for all i ∈ N , then we define p * i := q i , and µ i -i := q -i , and the proposition holds since every Nash equilibrium is a subjective equilibrium in which the subjective probability distributions coincide. Now let there not be a Nash equilibrium like above. Take p * i ∈ P * i to be any element of P * i and

µ i -i ∈ � P � -i � � � P � -i = arg min P -i ∈P * -i � S i � S -i u i (s i , s -i ) dP -i dp * i � (
p i ∈∆S i � S i � S -i u i (s i , s -i ) dµ i -i dp i for all i ∈ N .
Thus, (p * 1 , . . . , p * n ) with the beliefs µ i -i as in (5.1) for each i ∈ N is a subjective equilibrium.

Remarks 5.6. 1. To find a subjective equilibrium contained in an Ellsberg equilibrium, not every candidate for the beliefs µ i -i can be chosen within (P * 1 , . . . , P * n ). To see this, consider the Modified Matching Pennies example, Figure 3.8: if player 1 played his Nash equilibrium strategy 1/2 and his belief about player 2's strategy was not the minimizer 1/4, he would not stay with his mixed strategy.

2. On the other hand, for the beliefs µ i -i of every player i ∈ N , we need not necessarily choose the minimizer as constructed in (5.1) in the proof. To see this, consider the peace negotiation example in Figure 1.3: every belief of countries A and B about the play of the superpower C that fulfills µ A C (punish A) > 5/9 and µ B C (punish A) < 4/9 is a candidate for a subjective equilibrium.

This result is of course not very surprising, because due to the inconsistency of beliefs one can obtain a large set of outcomes with the subjective equilibrium notion.

The converse of the statement does not hold, that is, it is not possible to find for every subjective equilibrium of a normal form game G an Ellsberg equilibrium that contains the subjective equilibrium in the way prescribed by the proposition. We give an example to show this. Consider the matching pennies game in Figure 5.3. We explained that the pair (U, L) with beliefs µ 1 2 (D) = 1 and µ 2 1 (L) = 1 is a subjective equilibrium of the game. In contrast, the only Ellsberg equilibria are (

(1/2, 1/2), [Q 0 , Q 1 ]) with Q 0 < 1/2 < Q 1 and ([P 0 , P 1 ] , (1/2, 1/2)) with P 0 < 1/2 < P 1 , where P ∈ [P 0 , P 1 ], Q ∈ [Q 0 , Q 1 ]
are the probabilities of player 1 playing U and player 2 playing L, respectively. It is obvious that the subjective equilibrium (U, L) cannot be contained in any Ellsberg equilibrium.

Subjective Equilibria Explain Non-Nash Behavior in Three-Person Ellsberg Games

We now give an intuitive explanation for the observation of strategies with non-Nash support in Ellsberg games with more than two players. To this end, we define one more subjective equilibrium notion. The subjective beliefs equilibrium is a profile of possible non-common beliefs about the opponents' play, where the belief µ j i of player j about the play of player i must be a best response of player i to his belief µ i -i about the others' play.

Definition 5.7. A subjective beliefs equilibrium is a profile (µ 1 -1 , . . . , µ n -n ), where µ i -i ∈ ∆S -i , such that for all i ∈ N , for every j � = i, µ j i ∈ arg max

P i ∈∆S i u i (P i , µ i -i ) .
This is what [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF] (1996) generalizes the subjective beliefs equilibrium.

We show in Theorem 5.8 that every Ellsberg equilibrium contains a subjective beliefs equilibrium. In two-player games, every subjective beliefs equilibrium is a Nash equilibrium. When there is only one opponent, there is no room for disagreeing beliefs. This That is, arg max

P i ∈∆S i U i (P i , P * -i ) ⊆ arg max P i ∈∆S i u i (P i , µ i -i ) .
Since subjective beliefs equilibria and Nash equilibria are equivalent in two-person games, Theorem 5.8 has the following corollary. The reasoning of Theorem 5.8 and the corollary are based on [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF], Proposition 3. The corollary gives an intuitive explanation, why in two-player games the Ellsberg equilibrium behavior cannot be "too far" from the Nash equilibrium behavior. We discuss this in Remark 5.10.

Corollary 5.9. When G is a two-player game, then for every Ellsberg equilibrium (P * , Q * ) there exists P * ∈ P * and Q * ∈ Q * such that (P * , Q * ) is a Nash equilibrium and arg max

P ∈∆S i U i (P, Q * ) ⊆ arg max P ∈∆S i u i (P, Q * ) .
Remark 5.10. Note that the statement of Corollary 5.9 has an interesting consequence for the interpretation of Bade (2011b)'s theorem on observational indifference. When we relax Bade's assumption on strictly monotone payoff functions and allow for weakly dominated strategies (like in the example on observational differences in two-player games, Figure 4.1), we are likely to observe Ellsberg equilibria with a support greater than the Nash equilibrium support. But, and this follows from Corollary 5.9 and the proof of Theorem 5.8, the minimizers used by the players to derive the utility in Ellsberg equilibrium remain the Nash equilibrium strategies.

Strategic Use of Ambiguity in Dynamic Games

It is straightforward that Ellsberg urns can also be used as strategies in dynamic games. In this chapter we develop the theoretical framework to analyze such extensive form Ellsberg games. As in classic extensive form theory, players can use an urn to create ambiguity over their set of pure strategies, or they can place an Ellsberg urn at every decision point in the process of the game. In the first section of the chapter we thus extend the notions of mixed strategy and behavioral strategy to extensive form Ellsberg games.

Considering dynamic games leads directly to the question of dynamic consistency of the players' preferences. We discuss this in Section 6.2 and develop a formalism to translate rectangularity of [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] to extensive form Ellsberg games. This property of Ellsberg strategies is then used to prove a version of Kuhn's Theorem [START_REF] Kuhn | Extensive games and the problem of information[END_REF]) for extensive form Ellsberg games. We present an example for intuition. Thereafter we define Ellsberg equilibrium in extensive form Ellsberg games and finally, in Section 6.5, we compare Ellsberg equilibria to other extensive form solution concepts.

Extensive Form Ellsberg Games

We focus on dynamic games that can be represented in finite game trees. This implies a finite number n of players and a finite number of moves for each player. For simplicity, we do not allow any chance moves. To define a game tree and address its elements we use the formalism developed in chapter 3 of [START_REF] Ritzberger | Foundations of Non-Cooperative Game Theory[END_REF]. We state the definitions as briefly as possible, for further details please consult [START_REF] Ritzberger | Foundations of Non-Cooperative Game Theory[END_REF]. For further reference, the concept is also developed and explained in Alós-Ferrer and [START_REF] Alós-Ferrer | Trees and extensive forms[END_REF] and Alós-Ferrer and [START_REF] Alós-Ferrer | Trees and extensive forms[END_REF].

A tree T = (W, N ) is defined as a set of plays W and a collection of nodes N which are subsets of W such that {w} ∈ N for all w ∈ W . The tree is rooted if W ∈ N , then W is called the root. In a rooted tree the elements w ∈ W are called plays. Sets x ∈ N of plays are nodes, singleton sets of plays (x = {w} ∈ N , for some w ∈ W ) are terminal nodes, and all nodes that contain more than one play, X := N \ {{w}} w∈W , are called moves. We

(c) ∩ T (c � ) � = ∅, then T (c) = T (c � ) for all c, c � ∈ C i . An element c ∈ C i is called a choice of player i.
For an extensive form F , the system H i of disjoint subsets of moves defined by H i := {T (c)} c∈C i is player i's information partition. An information set of player i is denoted h ∈ H i . Furthermore, the set X i := � h∈H i h is the set of decision points of some player i.

To turn an extensive form into a game, we first need to specify players' preferences over plays. These are represented by a Bernoulli utility function u : W → R n that maps each play w ∈ W into a payoff vector u(w) = (u 1 (w), . . . , u n (w)). Preferences on lotteries over plays are represented by the expected value of u. We assume that players have preferences on lotteries over plays that conform to the Von Neumann and Morgenstern axioms.

In order to define an extensive form Ellsberg game in which players can use Ellsberg urns, we specify the strategies that players may use and how the utility of these strategies is evaluated. We assume that players can use Ellsberg urns to create ambiguity over their pure strategies and over the choices at each information set. Therefore, in addition to classical pure, mixed and behavioral strategies, we define Ellsberg strategies and Ellsberg behavioral strategies and specify players' preferences over these strategies.

To start with, a pure strategy of player i consists of a prescription c ∈ C i at every move x ∈ X. Definition 6.1. A pure strategy of player i is a function s

i : X → C i such that s -1 i (c) = T (c) for all c ∈ s i (X) = ∪ x∈X s i (x) . (6.1)
Condition (6.1) guarantees that s i maps each move into choices available at that move and that s i is measurable with respect to the information partition H i , i.e., that to each move x in an information set h = T (c) the same choice gets assigned. The utility of a pure strategy profile s ∈ S 1 × . . . × S n is simply u i (w), where w is the unique play induced by the profile s (remember that we do not allow chance moves in our definition of an extensive form). Sometimes we also write u i (s) directly. An Ellsberg strategy for player i is now a convex and compact set of probability distributions over the pure strategies S i ; more precisely: Definition 6.2. An Ellsberg strategy of player i is a convex and compact set P i of probability distributions P i :

S i → R + , such that � s i ∈S i P i (s i ) = 1.
We denote the set of mixed strategies P i of player i as usual with ∆S i . We assume a product structure on ∆S := ∆S 1 × . . . × ∆S n and thus have stochastic independence of mixed strategies. Thus, if P = (P 1 , . . . , P n ) ∈ ∆S, s ∈ S, then

P (s) = P (s 1 , . . . , s n ) = n � i=1 P i (s i ) for all s ∈ S .
Then the utility u = (u 1 , . . . , u n ) : S → R n of a mixed strategy profile P (we use the same notation as for the utility of a pure strategy) is

u i (P ) = � s∈S P (s)u i (s) .
We assume that the players are ambiguity-averse according to [START_REF] Gajdos | Attitude toward imprecise information[END_REF] in the special case φ = id. The utility of an Ellsberg strategy profile P is then evaluated with a maxmin rule using the worst case probability distribution. We have stochastic independence of the Ellsberg strategies by assuming the above product structure for every mixed strategy profile P contained in P = (P 1 , . . . , P n ). For details and discussion of this construction see (1.7) in Section 1.3.3. Thus, the utility of an Additionally, we define Ellsberg behavior strategies. An Ellsberg behavior strategy of player i specifies an Ellsberg urn over the available choices at every information set of player i. This is an analog of behavior strategies, the same analogy as the one between

Ellsberg strategies and mixed strategies in classic theory of extensive form games.

Definition 6.4. An Ellsberg behavior strategy Θ i of player i is a collection of convex and compact sets Θ α i of probability distributions θ α i over choices C α i at every information set h α ∈ H i , α = 0, .., a i , where a i + 1 is the number of information sets of player i.

An Ellsberg behavior strategy can thus be thought of as a collection Θ i = (Θ α i (c)) c∈C α i of possible ranges of probabilities for each c, where

� c∈C α i θ α i (c) = 1 for each θ α i ∈ Θ α i .
The collection θ i consisting of a probability distribution θ α i at every information set h α ∈ H i is thus a classic behavior strategy for player i. The collection Θ i consists of collections θ i . A profile of Ellsberg behavior strategies is denoted Θ = (Θ 1 , . . . , Θ n ), and Θ is constituted of profiles of behavior strategies θ = (θ 1 , . . . , θ n ).

Finally, we specify the utility for player i of an Ellsberg behavior strategy profile Θ. We have to define what set of probability distributions an Ellsberg behavior strategy induces on plays. Given an Ellsberg behavior strategy profile Θ, the set of realization probabilities that a singleton1 set of play w ∈ W will be reached is

Π Θ (w) = � π θ (w) = n � i=1 θ i (c i ) � � � θ ∈ Θ, (c i ) n i=1 ∈ × n i=1 C i s.t. w ∈ ∩ n i=1 c i � .
Note that (c i ) n i=1 ∈ × n i=1 C i is the unique choice combination such that w is reached. The utility of an Ellsberg behavior strategy profile Θ for an ambiguity-averse player i is then

U i (Θ) = U i (Π Θ ) = min π θ ∈Π Θ � w∈W π θ (w)u i (w) = min θ∈Θ � w∈W � n � i=1 θ i (c i ) � u i (w) .
We also fix notation for the set of realization probabilities Π P (w) of a singleton set of play The set Π P (w) collects all probabilities of pure strategies in the support of any P ∈ P which share the same play w, that is, for which s ∈ w.

Rectangular Ellsberg Strategies and Dynamic Consistency

Dynamic consistency requires that preferences over outcomes at some point in the game will not be reversed or contradicted at a later point. A compelling feature of classic extensive form games lies in the fact that for a rational player it does not matter whether he plans his strategy in advance or executes it whenever one of his information sets is reached; this means that additional information arriving at some point in the game is irrelevant for the choice of the strategy. This equivalence is captured by Kuhn's Theorem which we discuss and extend to Ellsberg games in the next section. When players are expected utility maximizers, Kuhn's Theorem establishes a connection between conditional preferences at different points in the game tree via Bayes' updating rule: when players update in that way, their choices following the expected utility model are dynamically consistent.

Updating of ambiguity-averse preference representations and its relation to dynamic consistency has been extensively analyzed. [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF] propose two updating rules for multiple-prior expected utilities, full Bayesian updating and maximum likelihood updating, which are in general not dynamically consistent. [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] characterize a particular set of priors which they call rectangular, for which full Bayesian updating is dynamically consistent. They show, more precisely, that multipleprior utilities like those used in Ellsberg games are dynamically consistent if and only if each set of priors is rectangular and it is updated by Bayes' Rule applied prior by prior. [START_REF] Riedel | Optimal stopping with multiple priors[END_REF] uses rectangularity to prove dynamic consistency of dynamic coherent risk measures. Later [START_REF] Hanany | Updating preferences with multiple priors[END_REF] use a weaker notion of dynamic consistency to obtain a larger set of dynamically consistent updating rules for maxmin expected utilities.

We translate the property of rectangularity in [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] to Ellsberg games. First we describe the information structure of an extensive form Ellsberg game (F, U ) with a filtration {W t } T 0 on a state space W . This is only possible, when the extensive form F satisfies perfect recall, otherwise for some t, W t � W t+1 . Thus, we assume that F satisfies perfect recall.2 Furthermore, we assume finite trees, T < ∞, throughout. The information structure of (F, U ) is captured in the extensive form F in form of a rooted tree T = (W, N ). The root W corresponds to time 0, and the sequence of decisions taken by the players defines the information available at time t. Let the set W of plays be the state space. Then, staying with [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF], we assume that W 0 is trivial (that is, consists only of W and ∅) and that for each t, W t is generated by the finite information partition at time t. Then W t (w) denotes the partition component containing the play w. W t (w) corresponds to some information set h, at which w is possible, along with a number of other plays. At t + 1, the number of possible plays is narrowed down again. At time T , W T (w) is the singleton set of play {w} that has materialized. [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] consider lotteries over adapted consumption processes, and preferences over W t -measurable acts from W into such lotteries. Then, for simplicity, they assume full support (Axiom 5 therein): every non-empty event in W T is considered possible at time 0. To apply the result by [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] we assume the same condition. For the game (F, U ) this implies that we only consider strategies with full support. This is without loss of generality. To allow for strategies without full support, we apply the construction to a suitable subset of plays. It is up to further research to determine whether this procedure has implications for the strategic analysis of an extensive form Ellsberg game.

In the description of the information structure {W t } T 0 , it is irrelevant, who takes the decision at some time t. The filtration only represents what is known to all players at some time t, therefore it does not define a game, but only an event tree. When we define a rectangular Ellsberg strategy profile P, we therefore only impose restrictions on the set of induced distributions over plays: as long as the set of induced realization probabilities Π P (see (6.3)) is rectangular, any Ellsberg strategy profile P which induces Π P is admissible.

We specify this in Definition 6.5. For the extension of Kuhn's theorem to extensive form Ellsberg games this leaves some degrees of freedom for the choice of an Ellsberg strategy profile P given some set of realization probabilities Π P .

We have already found a way to describe Ellsberg strategy profiles, that is, "the big urn" over all pure strategies, in the setting of [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF]: they are represented by the set of realization probabilities Π P . Now we see how Ellsberg behavior strategies are described in this setting. Define the set of W t -conditionals of measures π on (W, W T ) as

Π t P (w)(•) = � π t (w)(•) := π(•|W t )(w) � � π ∈ Π P � ,
this is the set of Bayesian updates at a time t. The set of conditional one-step-ahead measures is defined by

Π t P,+1 (w) = � π t +1 (w) � � π ∈ Π P � ,
where π t +1 is the restriction of π t to W t+1 . The sets Π t P and Π t P,+1 can be viewed as realizations of W t -measurable correspondences into ∆(W, W T ) and ∆(W, W t+1 ), respectively. The name "conditional one-step-ahead measures" has the following intuition. Each measure in Π t P,+1 (w) is a measure on W t+1 , thus one can think of Π t P,+1 (w) as the set of measures describing beliefs about the "next step": the belief at time t about what will happen at time t + 1. Think of it as cutting the tree at time t and time t + 1, then one is only left with distributions over the choices at the information sets at time t, and the distributions are induced by Π P . From this explanation it is clear that the conditional onestep-ahead measures are the induced Ellsberg behavior strategies. [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] explain further that from any set of one-step-ahead conditionals, e.g. some set Π t +1 , a rectangular set Π can be constructed with backward construction via

Π = {π ∈ ∆(W, W t ) | π t +1 ∈ Π t +1 (
w) for all t and w} .

We see that Π is the set of all measures π whose one-step ahead conditionals conform with the Π t +1 .

When is an Ellsberg strategy dynamically consistent (rectangular)? The Bayesian theory says, a probability distribution π on (W, W T ) can be for every t decomposed into its conditionals and marginals in the form

π t (w) = � w � ∈W π t+1 (w � ) • π t +1 (w) . (6.4)
The set Π P of probability distributions (and thus P) on (W, W T ) is rectangular if it admits a corresponding decomposition. Details are explained in the proof of Theorem 6.7 and in [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF]. First, we define rectangularity precisely.

Definition 6.5. An Ellsberg strategy profile P in an extensive form Ellsberg game (F, U ) is rectangular, if the set of realization probabilities Π P is rectangular in the following sense.

Let the information structure of the game (F, U ) be described by the filtration {W t } T 0 .

Then Π P is {W t }-rectangular, if for all w and all t

Π t P (w) = � � w � ∈W π t+1 (w � ) • π t +1 (w) � � �
π t+1 (w � ) ∈ Π t+1 P (w � ) for all w � ∈ W, π t +1 (w) ∈ Π t P,+1 (w) � . (6.5)

An Ellsberg strategy P i for player i is rectangular if it is part of a rectangular Ellsberg strategy profile P. When necessary, we denote the restriction of a set X to its rectangular subset with X R .

Remark 6.6. Definition 6.5 also applies to the set of realization probabilities Π Θ . Π Θ is rectangular, if it has an analog decomposition as in (6.5). In difference to Definition 6.5 this is not a property of the Ellsberg behavior strategy profile Θ.

Observe that the inclusion ⊂ is always satisfied by applying (6.4) for every π ∈ Π P . In order to assure the inclusion in the other direction, Π P is required to have the special "rectangular" form. Combinations of conditionals and marginals that arise from different π, π � ∈ Π P have to lie in Π t P . In Example 6.9 in the following section, we explain the geometric representation of a rectangular Ellsberg strategy. [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] point out some features of rectangularity. One of these is particularly important in the setting of extensive form Ellsberg games: rectangularity imposes no restrictions on onestep-ahead conditionals. This means that players can use any Ellsberg behavior strategy they wish. Only when they want to represent this Ellsberg behavior strategy by an outcome-equivalent Ellsberg strategy they have to use the induced rectangular Ellsberg strategy. We show this in detail in the following section.

6.3 Kuhn's Theorem for Extensive Form Ellsberg Games [START_REF] Kuhn | Extensive games and the problem of information[END_REF] proved that an extensive form satisfies perfect recall if and only if, for every probability distribution on plays that is induced by some mixed strategy profile, there is a behavior strategy profile that induces the same distribution, and for every probability distribution on plays that is induced by some behavior strategy profile, there is a mixed strategy profile that induces the same distribution. Under expected utility this implies that the strategies yield the same utility.

In this section we assume, as it is assumed in most applications, that the extensive form F = (T , C) satisfies perfect recall. We then show that under this assumption and the condition that the Ellsberg strategies are rectangular we get an equivalent result of Kuhn's theorem for extensive form Ellsberg games, namely: Theorem 6.7. In an extensive form Ellsberg game (F, U ) with F = (T , C) satisfying perfect recall, every rectangular Ellsberg strategy profile P induces an Ellsberg behavior strategy profile Θ P via prior-by-prior updating; every Ellsberg behavior strategy profile Θ induces a rectangular Ellsberg strategy profile P Θ such that prior-by-prior updating of P Θ yields Θ. The induced strategy profiles are payoff-equivalent, i.e., U i (P) = U i (Θ P ) and U i (Θ) = U i (P Θ ) .

Proof. We construct the induced strategies as follows. Note that we only consider strategies with full support according to Axiom 5 in [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF]. This is without loss of generality. To allow for strategies without full support, we apply the construction to a suitable subset of plays.

A rectangular Ellsberg strategy profile P has by definition a rectangular set of realization probabilities Π P . Due to rectangularity, Π P possesses at any point in time t a decomposition into marginals and one-step-ahead conditionals according to (6.5). The set of conditional one-step-ahead measures at t, Π t P,+1 , defines the induced Ellsberg behavior strategy profile Θ P by setting It remains to show that P and Θ P , as well as Θ and P Θ , yield the same minimal expected utility.

Recall that U i (P) = min

P ∈P P (s)u i (s) .
This is equal to the minimal expected utility of the realization probabilities of P, U i (P) = U i (Π P ) = min From (6.8) to (6.9) we get by definition of θ i in the construction of the proof, see (6.6). We can replace � n i=1 θ i (c i ) by � T t=0 π t +1 (w), where (c i ) n i=1 is the unique sequence of choices which yields w. The equality of (6.9) and (6.10) results from the rectangularity of P.

Because of rectangularity,

� T t=1 π t +1 (w) is an element of Π P , that is, � T t=0 π t +1 (w) = π P (w). Likewise, we show the equality of U i (Θ) and U i (P Θ ). We use (6.7) for the equality of (6.11) and (6.12), and rectangularity in the equality of (6.12) and (6.13). Remark 6.8. Note that the induced Ellsberg behavior strategy profile Θ P is unique, but the induced Ellsberg strategy profile P Θ is not. This is because the construction of P Θ from the set of realization probabilities Π R P is in general not unique.

Theorem 6.7 has the following interpretation. For every Ellsberg behavior strategy profile there exists an Ellsberg strategy profile that yields the same utility for all players. This induced Ellsberg strategy profile has to be chosen rectangular if we want it to be dynamically consistent and equivalent to the original Ellsberg behavior strategy. Note that this does not impose any restriction on the choice of the Ellsberg behavior strategy profile (this is also pointed out in [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF], p. 10), but only allows a restricted set of Ellsberg strategy profiles. Conversely, for every rectangular Ellsberg strategy, there exists an equivalent Ellsberg behavior strategy.

To gain some intuition about the nature of rectangular Ellsberg strategies and the inductions explained in the proof of Theorem 6.7, we now discuss a most simple example.

Example 6.9. Consider the following example taken from Osborne and Rubinstein (1994), p. 93. We believe this to be the simplest two-player information structure with which we can illustrate the equivalence of rectangular Ellsberg strategies and Ellsberg behavior strategies.

The game is presented in Figure 6.3. First, we start with Ellsberg behavior strategies Θ 1 = (Θ 0 1 , Θ 1 1 ) and Θ 2 that specify a set of probability distributions at every information set of player 1 and 2, respectively. does not lie in the set Θ 1 1 (E). Hence we have to restrict Π Θ to its rectangular subset Π R Θ , this is exactly the set that yields the correct conditionals.

To apply the construction of [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] we represent the game in Π R Θ ⊂ Π Θ is the set of distributions on (W, W 2 ) that for all t ∈ {0, 1, 2} and all w ∈ W admits a decomposition

Π R,t Θ (w) = � � w � ∈W π t+1 (w � ) • π t +1 (w) � � � π t+1 (w � ) ∈ Π t+1 Θ (w � ) for all w � ∈ W, π t +1 (w) ∈ Π t Θ,+1 (w)
� .

(6.15)

The set Π R Θ excludes exactly all those distributions in Π Θ that yield conditionals outside Θ 1 1 . The rectangular set Π R Θ is depicted as the blue rectangle in Figure 6.4.

From the construction of Π R,t Θ , it follows that

Π R,0 Θ,+1 = Θ 0 1 , Π R,1 Θ,+1 = Θ 1 1 ,
that is, the induced conditional one-step-ahead measures are exactly the components of the Ellsberg behavior strategy which we started with. The set Π R Θ can also be constructed recursively as the set of all distributions π ∈ ∆(W, W T ) for which π t +1 ∈ Π R,t Θ,+1 (w) for all t and all w, see [START_REF] Epstein | IID: independently and indistinguishably distributed[END_REF] as in Figure 6.2. Then the rectangular set of realization probabilities Π R P is Now, finally, we construct P Θ from the rectangular set of realization probabilities Π R P . We find

Π R P = � π = (π 1 , π 2 , π 3 ) ∈ ∆(W, W T ) � � π 1 /(1 -π 3 ) ∈ [1/2, 2/3] = Π R,1 P,+1 (
P Θ 1 = � P ∈ ∆S 1 � � P (A, E) = π 1 , P (A, F ) = π 2 , P (B, E) = π 1 π 1 + π 2 π 3 , P (B, F ) = π 2 π 1 + π 2 π 3 , for all (π 1 , π 2 , π 3 ) ∈ Π R P � .
We calculate the maxmin expected utility of player 1 of the Ellsberg strategy profile P Θ .

U 1 (P Θ ) = min (6.17)

Player 1 uses P = (1/8, 1/8, 3/8, 3/8) as his worst case probability distribution, because it puts the greatest available probability to the worst outcome d. We thus find that the Ellsberg strategy profile P Θ induced by Θ yields the same maxmin expected utility as Θ:

U 1 (P Θ ) (6.17 = U 1 (Θ) .

Furthermore, doing the reasoning backwards, one sees directly that with the rectangular construction of Π R P = Π R Θ in (6.15), the Ellsberg strategy yields the correct one-step-ahead conditionals for A, B and E, F . Now, we start out from the rectangular Ellsberg strategy profile P (the blue rectangle in Figure 6.4) and construct the induced Ellsberg behavior strategy profile Θ P . We proceed as sketched in the diagram in Figure 6.1. The Ellsberg strategy profile has a set of realization probabilities Π P which are by definition also rectangular. Then from Definition 6.5, Π P possesses a decomposition of Π t P for all t into marginals and one-step-ahead conditionals. The rectangular set Π P is constructed in the way to get exactly Π 0 P,+1 = Θ P,0 1 and Π 1 P,+1 = play off the equilibrium path can lead to non-Nash outcomes. Subsequently this "horse"like game3 has been used as an example in a number of papers [START_REF] Battigalli | Learning and convergence to equilibrium in repeated strategic interactions: an introductory survey[END_REF], [START_REF] Fudenberg | Self-confirming equilibrium[END_REF], [START_REF] Rubinstein | Rationalizable conjectural equilibrium: between Nash and rationalizability[END_REF], [START_REF] Groes | Nash equilibrium with lower probabilities[END_REF], [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF]). The information structure is the same as in Greenberg (2000)'s peace negotiation example, and also the incentive structure is very similar. We are in a situation, where ambiguity can be used as a threat against deviation. Proposition 6.12. The proper Ellsberg equilibria of Fudenberg and Kreps' Three-Player game are of the form (A 1 , A 2 , [R 0 , R 1 ]) where R ∈ [R 0 , R 1 ] is the probability that player 3 plays L, and R 0 < 1/3 and R 1 > 2/3.

Proof. Let P, Q denote the probability that player 1 plays A 1 , player 2 plays A 2 , respectively. Then we can calculate the minimal expected utility for each player.

U 1 (P, Q, [R 0 , R 1 ]) = min R 0 ≤R≤R 1 P QR + 3P (1 -Q)R + 3(1 -P )QR + 3(1 -P )(1 -Q)R + P Q(1 -R) = min R 0 ≤R≤R 1 R(-3P Q + 3) + P Q = R 0 (-3P Q + 3) + P Q for all P, Q ∈ [0, 1] .
Nash equilibrium by [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF]. Lo allows beliefs to be represented by multiple priors and demands every distribution in the sets of beliefs to be a best response to the beliefs on the other players. Players only have pure strategies at their disposition. Lo (1999) also looks at the example in Figure 6.5 and shows that the path (D 1 , D 2 ) can be supported by a Multiple Priors Nash equilibrium, that is, that there exist sets of beliefs such that playing (D 1 , D 2 ) is optimal. However, this equilibrium in beliefs is entirely subjective and in that aspect differs fundamentally from Ellsberg equilibrium. In the same way, Nash equilibrium with lower probabilities in [START_REF] Groes | Nash equilibrium with lower probabilities[END_REF] is distinguished from Ellsberg equilibrium. The Ellsberg equilibrium outcomes in Fudenberg and Kreps' game can also be achieved with subjective equilibrium by [START_REF] Kalai | Subjective games and equilibria[END_REF], but, again, the fundamentals of the two concepts differ.

σ * -equilibrium by [START_REF] Ma | Uncertainty aversion and rationality in games of perfect information[END_REF] is closest to our approach. He lets players use ambiguous plans modeled by sets of probability distributions over acts and mentions the possibility to create Knightian uncertainty. In difference to our approach he relies entirely on subjective preference representations and focuses on belief systems and thus on equilibrium in beliefs.

An interesting aspect of his paper is that it models the possibility to create ambiguity with pre-play communication.

3. 1

 1 Ellsberg equilibria in general coordination (blue) and conflict games. . . . . 4.1 The Goeree-Holt results on three different versions of Matching Pennies. . . 4.2 Comparison of Nash and Ellsberg predictions with the experimental observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

  How does the possibility to use ambiguity as a strategy change the game? I propose in this thesis a model of Ellsberg games that differs from the classical Von Neumann and Morgenstern model merely in the one aspect that players may use (objectively ambiguous)

Figure 1

 1 Figure 1.3: Peace negotiation.

Figure 1 . 4 :

 14 Figure 1.4: Peace negotiation in normal form.

  introduces the concept of equilibrium in beliefs under uncertainty where the beliefs are represented by multiple priors over other players' mixed strategies. Each player i has a set of beliefs B i over what the other players do, so over ∆S -i . The profile (B i ) then forms a beliefs equilibrium if player j puts positive weight only on strategies of player i that maximize i's minimal expected payoff given the belief set B i . This concept allows for disagreement of players' beliefs, and for correlation. Lo therefore introduces the refinement of a beliefs equilibrium with agreement in which player j and k agree about player i's actions and the beliefs of i over j and k are independent. Lo proves the nice result that every beliefs equilibrium contains a Bayesian beliefs equilibrium (where the belief sets are singletons). As a corollary, he obtains a precursor of Bade (2011b)'s main theorem (which we discuss in a later paragraph): in two player games, every beliefs equilibrium contains a Nash equilibrium.

Figure 2 . 2 :

 22 Figure 2.2: Player 1's (minimal expected) payoff (2.1) as a function of the probability P of playing HEAD when player 2 uses the Ellsberg strategy [1/4, 1/2].

Figure 2

 2 Figure 2.4: Player 1's expected payoff function of the modified Matching Pennies I game, u 1 (P, Q) = 6P Q -2P -2Q + 1, in two different perspectives.

  the probability with which player 2 plays L. Assume first that P = d-c a-b-c+d . To assure that 0 ≤ d-c a-b-c-d ≤ 1 we have either dc ≥ 0 and abc + d > 0 and abc + d ≥ dc , or dc ≤ 0 and abc + d < 0 and abc + d ≤ dc . From the first case follows dc ≥ 0 > ba, from the latter case dc ≤ 0 < ba. Now assume without loss of generality that dc ≥ 0 > ba. We derive the minimal expected utility of player 1.

Figure 3 . 2 :

 32 Figure 3.2: Coordination game: payoff function (3.1) of player 1.

  player 1 maximizes his payoff function by playing any interval [P 0 , P 1 ] ⊆ [0, 3/4]. This leads to an analog Ellsberg equilibrium to case (2), ([P 0 , 3/4] , [Q 0 , 3/4]), where 0 ≤ P 0 ≤ 3/4 and 0 ≤ Q 0 ≤ 3/4.

  Figure 3.3: General coordination game.

Figure 3

 3 Figure 3.4: Non-symmetric coordination game I.

Figure 3

 3 Figure 3.5: Non-symmetric coordination game II.

Figure 3 .

 3 Figure 3.10: Modified Matching Pennies: payoff function (3.5) of player 2 for different strategies [P 0 , P 1 ] ⊆ ∆S 2 .

  Figure 3.11: General conflict game I.

Figure 3

 3 Figure 3.15: General game.

Figure 3 .

 3 Figure 3.16: General zero-sum game.

( c )

 c Assume that a = c and b > d, i.e., U is a weakly dominant strategy for player 1. We further assume without loss of generality d < c < b. The Ellsberg equilibria in this case are ([P 0 , P 1 ] , 1) , where P 1 > dc db and 0 ≤ P 0 ≤ P 1 , and

Figure 3

 3 Figure 3.18: Hawk and Dove.

Figure 3 .

 3 Figure 3.22: Payoff function (3.13) of player 1 in Myerson's game.

Figure 3 .

 3 Figure 3.25: Modified Rock Scissors Paper.

Figure 3

 3 Figure 3.26: Ellsberg equilibrium strategy of player 1 in the modified RSP.

FigureFigure 4 . 1 :

 41 Figure 4.1: Example of a 2 × 2 game which has Ellsberg equilibria with support outside the Nash equilibrium support.

Figure 4 . 2 :

 42 Figure 4.2: Payoff function (4.1) of player 1 for the cases (1)Q 0 = 0, (2) Q 0 = 1/2 and(3) Q 0 = 1.

  Figure 5.1: Aumann's example.

  Figure 5.2: Mediated Prisoners' Dilemma.

Figure 5 . 3 :

 53 Figure 5.3: Matching Pennies game.

  define an immediate predecessor function T : N → X that maps nodes in N into moves by assigning to each node its immediate predecessor (we set T (W ) = W ). Accordingly, T -1 (x) = {y ∈ N | T (y) = x} are the immediate successors of node x. The function T and the correspondence T -1 are straightforwardly extended to subsets of N . With these preparations we can define an extensive form F = (T , C), where T is a rooted tree and C = (C 1 , . . . , C n ) is the set of choices consisting of n partitions C i of Y := N \ {W } (so X is the set of nodes without the terminal nodes, and Y is the set of nodes without the root). C satisfies the condition that, if two choices c, c � ∈ C i of the same player i have one common immediate predecessor, then they have all predecessors in common. That is, if T

w

  ∈ W induced by an Ellsberg strategy profile P,

Figure 6

 6 Figure 6.1: Rectangular P induces Ellsberg behavior strategy Θ P .

=

  w)u i (w) , where π P (w) = � s∈S,w∈s P (s). Now consider the minimal expected utility of Θ P , withπ θ (w) = � n i=1 θ i (c i ), where (c i ) n i=1 is the unique sequence of choices that yield w.U i (Θ P ) = U i (Π Θ P ) = min π θ ∈Π Θ P U i (Π P ) = U i (P) .

U=

  i (Θ) = U i (Π Θ ) = min π θ ∈Π Θ U i (Π P Θ ) = U i (P Θ ) .

  Figure 6.3: Game where player 1 moves before and after player 2.

Figure 6 .

 6 Figure 6.9 by a state space W = {E, F, B} (the set of plays) and a filtration {W t } 2 0 where

Figure 6 . 4 :

 64 Figure 6.4: Set of realization probabilities Π Θ (green hexagon), rectangular set of probabilities Π R Θ (blue rectangle). The grey numbers indicate the set of probabilities for B, the blue numbers the set of conditional one-step-ahead probabilities for E.

  The notation in (6.16) is derived as follows. Every point in the simplex is fully described by probabilities for B and E. The two lines passing through B = 1 and E = 2/3, B = 1 and E = 1/2 are functions of B and E with equationsπ 1 = π 3 /2 + 1/2 , π 1 = 2π 3 /3 + 2/3 .

  E) • a + P (A, F ) • b + (P (B, E) + P (B, F )) • d = 1/8 • a + 1/8 • b + (3/8 + 3/8) • d = 1/8 • a + 1/8 • b + 3/4 • d .

  ) = 1/8 • a + 1/8 • b + 3/4 • d (6.14)

  4

  Figure 6.5: Fudenberg and Kreps' three-player game.

  The set of these probabilities over S i is denoted ∆S i . The players evaluate such devices according to expected utility, as in Von Neumann and Morgenstern's formulation of game theory. Moreover, and this is the new part, players can use Ellsberg urns. As we said above, we imagine that the players can credibly commit to base their actions on ambiguous outcomes. Technically, we model the Ellsberg urn of player i as a triple (Ω i , F i , P i ) as explained above. Player i acts in the game by choosing a measurable function (or Anscombe-Aumann act) 4

  With this chapter I wish to provide a sort of compendium of the Ellsberg equilibria in the most important normal form games and highlight some structure of Ellsberg equilibria in 2 × 2 games. The reader should be able to only read certain sections and fully understand how the Ellsberg equilibria are derived. For this reason the impression of repetition of certain steps in the calculations is unavoidable.3.1 Computing Ellsberg Equilibria in Two-Player 2 × 2GamesWe calculate the Ellsberg equilibria of two-player 2 × 2 games. Here Ellsberg strategies are completely described by the interval of probabilities [P 0 , P 1 ] with which one of the strategies is played. The other strategy is then played with some probability in the interval [1 -P 1 , 1 -P 0 ]. In this section, we thus use the notation [P 0 , P 1 ] to denote the Ellsberg strategy of player 1, and [Q 0 , Q 1 ] to denote the Ellsberg strategy of player 2. When it is useful, we also use the notation P and Q for the Ellsberg strategies of player 1 and 2, sometimes P * and Q * is used to indicate Ellsberg equilibrium strategies. In all the 2 × 2 games we analyze in this chapter, we write (P

In this chapter we present a number of examples and properties of Ellsberg equilibria in normal form games with two players.

1

We start with an explanation on how Ellsberg equilibria of two-person 2 × 2 games are computed. Subsequently, we look at different classes of games: coordination games (Section 3.2), conflict games (Section 3.3) and zerosum games (Section 3.5). We analyze examples and compute the Ellsberg equilibria of these classes of normal form games. The Ellsberg equilibria of general coordination and conflict games are summarized in Table

3

.1 in Section 3.4. For zero-sum games we also show that Ellsberg equilibria are value preserving. In Section 3.6 we calculate the Ellsberg equilibria of some classic 2 × 2 games such as, e.g., Hawk and Dove and the Prisoners' Dilemma. We explain with an example by Myerson, that also games with linear (as opposed to in Ellsberg games often observed piecewise linear) payoff functions can have proper Ellsberg equilibria.

We close the chapter with an analysis of a special class of 3 × 3 games. We consider a slightly modified version of Rock Scissors Paper, and with the help of our experience with 2 × 2 games we succeed in calculating the Ellsberg equilibria of these games. For a better understanding of how the equilibria are determined, we provide a geometric analysis of the equilibrium strategies in the two-dimensional probability simplex. * , Q *

Table 4 .

 4 1: The Goeree-Holt results on three different versions of Matching Pennies.

			Left (48) Right (52)
	Symmetric	Top (48)	80,40	40,80
		Bottom (52)	40,80	80,40
			Left (16) Right(84)
	Asymmetric	Top (96)	320, 40	40,80
		Bottom (4)	40,80	80,40
			Left (80) Right (20)
	Reversed	Top (8)	44,40	40,80
		Bottom (92)	40,80	80, 40

Table 4 .

 4 2: Comparison of Nash and Ellsberg predictions with the experimental observations. We record the probabilities (or intervals of probabilities) for each player to play the first pure strategy ("Top", respectively "Left") and the observed aggregate frequency of these actions in the Goeree-Holt experiments.

	Game	Nash Equilibrium	Ellsberg Equilibrium	Observations
		Player 1 Player 2 Player 1 Player 2	Player 1 Player 2
	symmetric	0.5	0.5	0.5	0.5	0.48	0.48
	asymmetric	0.5	0.125	[0.5,1]	[0.125,0.5]	0.96	0.16
	reversed	0.5	0.90	[0,0.5]	[0.5,0.90 ]	0.08	0.8
	and Q allow for probabilities for "Top" in the interval [0, 1/2] for player 1, and for probabilities
	in [1/2, 10/11] for player 2. The aggregate observed quantities of 8% and 80% do lie in
	these intervals.

* = 10/11. So we have the reversed relation Q * > P * . The Ellsberg equilibria

  and our setup, both concepts rely on the fact that each player may base his action on a set of states of the world for which different probability distributions exist. In[START_REF] Aumann | Correlated equilibrium as an expression of Bayesian rationality[END_REF] each player may have a different one, but every player has only one distribution, in Ellsberg games all players have the same prior, but the prior is a set of probability distributions on each of the state spaces. How do the sets of equilibria in subjective games and Ellsberg �N, (S i ), u i �, i ∈ N , there exists a subjective equilibrium (p * 1 , . . . , p * n ) with beliefs µ i -i ∈ ∆S -i of G, such that

	p * i ∈ P * i ,
	and µ
	games relate to each other? We start by proving that every Ellsberg equilibrium contains
	a subjective equilibrium.

Proposition 5.5. For every Ellsberg equilibrium (P * 1 , . . . , P * n ) of a normal form game G =

  calls a Bayesian Beliefs Equilibrium, with the difference that we do not allow for correlated beliefs. The difference of subjective beliefs equilibrium to subjective equilibrium lies in the fact that players do no longer best respond to some subjective belief, but the (possibly non-common) subjective beliefs of the other players must be a best response to the subjective beliefs. One can see clearly now that[START_REF] Klibanoff | Stochastically independent randomization and uncertainty aversion[END_REF]'s equilibrium under uncertainty is a generalization of subjective equilibrium, whereas Lo

  An extensive form Ellsberg game with n players is a pair (F, U ) where F = (T , C) is an extensive form with n players and U = (U 1 , . . . , U n ) is the ambiguityaverse preference representation in (6.2). The definition of an extensive form Ellsberg game is somehow implicit: by specifying in the definition how players evaluate Ellsberg strategies we allow players to use Ellsberg urns as devices to create ambiguity over their pure strategies. Of course, it is still possible to play classic mixed strategies. Then U i reduces to the Von Neumann and Morgenstern expected utility u i .

	P ∈P	s∈S �	P (s)u i (s) .	(6.2)
	With the ambiguity-averse preference representation at hand, we can now give the defini-
	tion of an extensive form Ellsberg game.
	Definition 6.3.

Ellsberg strategy profile P for player i is

U i (P) = min

In this sense, the assumption of ambiguity-aversion is parsimonious as it makes it harder for players to introduce Knightian uncertainty. They do not use Ellsberg urns for love of uncertainty.

I am grateful to a seminar participant at WZB in October 2012 for suggesting the first example and to a seminar participant at Exeter University in April 2012 for suggesting the second example.

We are always going to work with convex sets of probability measures. In this case, this means that we would allow for any P2, P3 ≥ 0 with P2 + P3 = 2/3 here. In our framework, this is without loss of generality, of course.

1.3 Ellsberg Games

We assume that all such acts in this thesis are measurable, also if not explicitly stated at each appearance.

Instead of Ellsberg urns they speak of "non-unique probability spaces" (Ω, F, P),Gilboa and Schmeidler (1989), p. 150.

Given an event E ⊆ Ω and acts f and h, let fEh be the act such that (fEh)(s) = f (s) if s ∈ E, and (fEh)(s) = h(s) otherwise. An event E is null if fEh ∼ f � E h for all acts f and f � , otherwise it is non-null.

Strategic Use of Ambiguity in Normal Form Games

Italic in the original text.

We take the payoffs as in Greenberg's paper. In case the reader is puzzled by the slight asymmetry between country A and B in payoffs: it does not play a role for our argument. One could replace the payoffs 3 and 6 for country A by 0 and 9.

1.4 Non-Nash Outcomes: Strategic Use of Ambiguity in Negotiation Games

See p. 802 therein, in particular.

[START_REF] Lo | Extensive form games with uncertainty averse players[END_REF] requires every probability distribution in the belief sets to be a best response, therefore every beliefs equilibrium with agreement is a refinement of equilibrium with uncertainty aversion and rationalizable beliefs (this is shown in[START_REF] Lo | Extensive form games with uncertainty averse players[END_REF], Proposition 9).

Section

2.1 was published in the IMW working paper[START_REF] Riedel | The strategic use of ambiguity[END_REF].

Immunization Against Strategic Ambiguity

Some parts of this chapter, namely Section 3.3.1 and the first half of Section 3.3.2, were published in the IMW working paper[START_REF] Riedel | The strategic use of ambiguity[END_REF].

Some parts of this chapter, namely Section 4.1 and the first part of Section 4.2, were published in the IMW working paper[START_REF] Riedel | The strategic use of ambiguity[END_REF].

[START_REF] Bade | Ambiguous act equilibria[END_REF] also defines non-null events in the sense of[START_REF] Savage | Foundations of Statistics[END_REF].

This is the same property as strict monotonicity of preferences, in[START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF] (axiom vi) and Bade (2011b) (MON).

Ellsberg Games, Human Behavior and Observational Implications

A Third Player Can Cause Non-Nash Behavior

In this special game and the preference representation chosen by Bade, the ambiguous act equilibrium coincides with one of the Ellsberg equilibria. Recall that in difference to ambiguous act equilibria, Ellsberg equilibria use ambiguity objectively.

We only consider singleton sets of plays when we speak about realization probabilities. We should use the bracket notation {w} for singleton sets of plays, but suppress the brackets for notational clarity.

For a precise definition of this notion in the notation introduced here see Ritzberger(2002), p. 124.

Strategic Use of Ambiguity in Dynamic Games

[START_REF] Selten | Reexamination of the perfectness concept for equilibrium points in extensive games[END_REF] is first to analyze this "horse"-like information structure. In his numerical example, however, the possibility to use Ellsberg strategies does not lead to Ellsberg equilibria outside the Nash equilibrium support, because the incentive structure is such that players 1 and 2 are not in opposition to each other.

[START_REF] Kelsey | Ambiguity in partnerships[END_REF] observe that "Ambiguity can make threats more effective." In the incentive structure of the games considered here, this observation is confirmed.

Acknowledgments

In the first place I would like to express my gratitude to my supervisor Prof. Frank Riedel.

His continuous support and his enthusiasm were essential for the development and completion of my thesis. I want to thank him for pointing out the topic to me, and for giving me a picture what economic research is about. I learned a great deal during the last few years, and I hope I could adapt some of his inspiration and intuition. I also want to express my gratitude to my supervisor Prof. Jean-Marc Tallon whose support and interest were of great help, especially during my stay in Paris. Likewise, I wish to thank Prof. Pierpaolo Battigalli and Prof. Wolfgang Leininger for agreeing to be external referees for my cotutelle. I want to thank the members of the Institute for Mathematical Economics at Bielefeld University for their support, interest and inspiring discussions. In particular, I would like to thank my fellow doctoral students in EBIM and BiGSEM for their friendly company in the last years. We spent many nice moments together, and their interest and our fruitful discussions were of great help for completing my Ph.D. I am also grateful to Paris School of Economics and its members, and to the participants of the Paris Game Theory seminar. In particular, I want to thank Frédéric Koessler whose interest in my research greatly motivated me during my stay in Paris. I am grateful to the DFG about financial support in the first three years of my Ph.D. via the International Graduate School EBIM. I also thank the IMW for financial support

scholarship to finance my cotutelle between Bielefeld University and Paris 1. The scholarship gave me the possibility to visit interesting conferences and workshops, travel between Bielefeld and Paris, and it helps me finance my defense at Bielefeld University. Finally, I wish to thank the rectorate of Bielefeld Uni-

List of Figures

The same analysis for player 2 yields the best response correspondence P 0 > 3/7 :

B 2 ([P 0 , P 1 ]) = 1

, 1]} P 0 < 3/7 < P 1 : B 2 ([P 0 , P 1 ]) = 2/7

B 2 ([P 0 , P 1 ]) = 0

We look at the intersections of the best response correspondences to find the Ellsberg equilibria:

(1) Q 0 > 1/5 ⇒ P 0 = P 1 = 1, when player 2 chooses Q 0 = Q 1 = 1. This is the Ellsberg equilibrium that coincides with the pure Nash equilibrium (U, L).

(2) Q 0 = 1/5 < Q 1 ⇒ [P 0 , P 1 ] ⊆ [2/5, 1], if player 1 chooses [P 0 , 3/7] with 2/5 ≤ P 0 < 3/7 then player 2's best response is [Q 0 , Q 1 ] ⊆ [0, 2/7]. Thus, we obtain the Ellsberg equilibrium ([P 0 , 3/7] , [1/5, Q 1 ]) , where 2/5 ≤ P 0 < 3/7 and 1/5 < Q 1 ≤ 2/7 .

(3) Q 0 < 1/5 < Q 1 ⇒ P 0 = P 1 = 2/5 ⇒ Q 0 = Q 1 = 0. This contradicts that Q 0 < 1/5 < Q 1 , thus this is not an Ellsberg equilibrium.

(4) Q 0 < 1/5 = Q 1 ⇒ [P 0 , P 1 ] ⊆ [0, 2/5]. We observe that 2/5 < 3/7, thus player 2 responds optimally with Q 0 = Q 1 = 0. Thus, this does not lead to an Ellsberg equilibrium.

(5) The case Q 1 < 1/5 results, analogously to (1), in the Ellsberg equilibrium that coincides with the other pure Nash equilibrium (D, R).

(6) In the last case, players find it optimal to deviate from the immunization strategy, since it does not coincide with their Nash equilibrium strategy: Q 1 = 1/5 = Q 0 ⇒

[P 0 , P 1 ] ⊆ [0, 1], then player 2 responds optimally with Q 0 = Q 1 = 2/7 and player 1 would, since 2/7 > 1/5, respond with his pure strategy U . But what player 1 can do is respond with his respective Nash equilibrium strategy 3/7, then we have the equilibrium (3/7, 1/5).

Now we fix the probability Q ∈ [Q 0 , Q 1 ] and consider the payoff function U 2 ([P 0 , P 1 ] , Q) of player 2. Player 2 evaluates his minimal expected payoff given player 1 plays P ∈ [P 0 , P 1 ]:

U 2 ([P 0 , P 1 ] , Q) = min

Observe that at the boundaries the function reduces to U 2 ([P 0 , P 1 ] , 0) = 1 -P 1 , and U 2 ([P 0 , P 1 ] , 1) = 2P 0 .

We see that U 2 ([P 0 , P 1 ] , Q) is constant at 2/3 when P 0 = 1/3 = P 1 . Therefore we get the following best responses for player 2:

(1)

Now we can calculate the Ellsberg equilibria:

(1)

. This is the Ellsberg equilibrium that coincides with the pure strategy Nash equilibrium (U, L).

(

. Player 2 has the possibility to play different subsets of [1/3, 1], but only one leads to a best response that is consistent with the original play of player 1,

If in the last case player 1 chooses to play [P 0 , P 1 ] = [1/3, P 1 ] with 1/3 ≤ P 1 ≤ 2/3, We plot the payoff function in Figure 3.9. The numbering (1)-( 6) of the functions refer to (one possible) payoff function when Q 0 and Q 1 take different values. The values are listed below Figure 3.9. In the same list we present player 1's best response analysis to the different strategies [Q 0 , Q 1 ]. We see that the payoff function is constant when 6)), then player 1 has any set of distributions in the whole interval [0, 1] as a best response. In cases ( 2) and ( 4), only one boundary of player 2 is fixed at 1/3, hence only subintervals of [1/3, 1] or [0, 1/3] are possible best responses for player 1.

When player 2 plays an interval around his Nash equilibrium strategy (case (3)), player 1's best response is the classic mixed strategy 1/3.

Next we look at the behavior of player 2. We fix the probability Q ∈ ∆S 2 and derive the

Now we can calculate the Ellsberg equilibria by finding the intersections of the best response correspondences.

(1) P 0 > 1/2 ⇒ Q 0 = Q 1 = 0 ⇒ P 0 = P 1 = 0, thus this is not an equilibrium.

(

Player 2 can choose different subsets of the interval [0, 1/2], but only one leads to a best response of player 1 that is consistent with his original play:

If player 1 chooses to play [P 0 , P 1 ] = [1/2, P 1 ] with 1/2 ≤ P 1 ≤ 1, then this is an Ellsberg equilibrium,

, where 1/2 ≤ P 1 ≤ 1 and 1/3 ≤ Q 1 ≤ 1/2 .

(3) P 0 < 1/2 < P 1 ⇒ [Q 0 , Q 1 ] = 1/2 ⇒ P 0 = P 1 = 1, thus this is not an equilibrium.

(

, 1] ⇒ P 0 = P 1 = 1, thus this is not an equilibrium.

(

1's best response is P 0 = P 1 = 1/3, thus this is not an equilibrium. But if player 2 decides to play Q 0 = Q 1 = 1/3 and player 1 plays P 0 = P 1 = 1/2, no player can gain by deviating from this strategy. This is the mixed strategy Nash equilibrium (P * , Q * ).

Proposition 3.6. In modified Matching Pennies the Ellsberg equilibria are of the form

In addition to the Nash equilibrium in mixed strategies, in asymmetric modified Matching Pennies are Ellsberg equilibria in which both players create ambiguity. In the game analyzed above, in particular, player 2 uses a interval between his Nash equilibrium strategy and his immunization strategy. We generalize this observation in the next section.

We see that

Therefore the best responses of player 1 are as follows:

We get an analog behavior for player 2, just for the opposite strategies. We fix the probability Q ∈ ∆S 2 and derive the payoff function U 2 ([P 0 , P 1 ] , Q) of player 2. Player 2 evaluates his minimal expected payoff given player 1 plays P ∈ [P 0 , P 1 ].

Observe that at the boundaries the function reduces to

The best response correspondence of player 2 is thus the following:

.27: Ellsberg equilibrium strategy of player 2 in the modified RSP.

The lower and upper bounds of Q 1 and Q 2 are w 1 , z 1 and w 2 , z 2 , respectively.

Player 2 minimizes over P 1 and P 2 , P 3 is expressed as 1 -P 1 -P 2 . The lower and upper bounds of P 1 and P 2 are x 1 , y 1 and x 2 , y 2 , respectively.

Now we proceed as follows to derive the Ellsberg equilibria. Recall the Nash equilibria and immunization strategies of the game:

Player 1 has to use his Nash equilibrium strategies at the boundary of his Ellsberg equilibrium strategy, except for the component P 2 , where the Nash equilibrium probability is the same as the immunization strategy. Furthermore, the set of probability distributions in the Ellsberg equilibrium may not extend across the immunization strategy.

Hence, for the first component for player 1, we consider either {1/4 ≤ x 1 ≤ P 1 ≤ 1/3}, On the other hand,

Now we use the following reasoning: we see from U 2 that Q 3 must be greater than (or in fact equal to, this is suppressed in the shortened statement of U 2 ) 1/3. Since 5/12 is the Nash equilibrium probability of Q 3 , we get the two following possible probability sets for

In the first case, player 1 must use z 3 in his utility evaluation, according to U 1 this is the case only when P 3 ≥ 2/3 -P 1 . We know that P 1 ≥ 1/3, therefore P 3 ≥ 1/3. This leads to the set of probabilities {1/3 ≤ P 3 ≤ y 3 ≤ 5/12} for P 3 , since 1/3 as the Nash equilibrium probability must be part of the set. Here x 3 is used by player 2, hence from U 2 we see this is only the case if Q 2 ≥ 2/3 -Q 3 . Q 3 ≥ 1/3 and therefore Q 2 must be greater than 1/3. This is not possible, since we have already found Q 2 to be played with the set of

Consider the second possible set for Q 2 . Here w 3 is used by player 1, and hence P 2 ≤ 1/3. This leads to {x 3 ≤ P 3 ≤ 1/3} for P 3 . Player 2 uses y 3 and we conclude that Q 2 must be less than or equal to 1/3. This is exactly what we found to be true for Q 2 before.

Therefore we have found the Ellsberg equilibrium of RSP to be

of [P 0 , P 1 ], and thus kinks in the payoff function.

U 2 ([P 0 , P 1 ] , Q) = min

The maximum of (4.2) over Q ∈ [Q 0 , Q 1 ] depends on the interval [P 0 , P 1 ]. Observe that at the boundaries player 2's payoff function is the following.

Player 2 maximizes (4.2) over [Q 0 , Q 1 ], his best responses are

(1)

Now we can solve for the Ellsberg equilibria by finding the intersections of the best response correspondences.

(

thus this is not an equilibrium.

(

, 0] ⇒ P 0 = P 1 = 0, thus this is not an equilibrium.

(3) P 0 < 1/2 < P 1 ⇒ Q 0 = Q 1 = 1/2 ⇒ P 0 = P 1 = 0, thus this is not an equilibrium.

(

. We look at different subsets of the interval [0, 1/2]. When player 2 chooses Q 0 > 0, then player 1 always finds it optimal to play Theorem 4.2. Let G = �{1, 2}, (S i ), (u i )� be a normal form game with a completely mixed Nash equilibrium (P * , Q * ). Let M 1 and M 2 denote the immunization strategies of player 1 and 2, respectively. Then there exist Ellsberg equilibria with unilateral full ambiguity if and only if P * = M 1 or Q * = M 2 . The Ellsberg equilibria are of the following form:

and

Proof. We show the result only for the first type of equilibrium, the other type follows

Now let M 1 = P * . When player 1 plays M 1 = P * , player 2 is indifferent between all pure and mixed strategies in ∆S 2 according to the Principle of Indifference for Nash equilibrium. Thus, player 2 is free to play and

with Q 0 < Q * < Q 1 the minimal expected utility function of player 1 is maximized when he plays the maximin strategy M 1 . Thus, (M 1 ,

In an unpublished working paper, [START_REF] Ryan | The maximin criterion and randomized behavior reconsidered[END_REF] proves a result for 2 × 2 normal form games using the Beliefs Equilibrium concept by [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF]. Beliefs Equilibrium is a wider (it allows correlation of beliefs) and entirely subjective concept (see Section 1.5 on the related literature for details), however, the result applies to 2 × 2 Ellsberg games. He

shows that if Q is a closed and convex subset of ∆S 2 and player 1's set of best responses B 1 (Q) does not contain pure strategies, then player 1 has a unique maximin strategy M 1 and B 1 (Q) = M 1 . The property of no pure strategies in the set of best responses has the geometric representation of a minimal expected utility function with a unique optimal point in the interior of [0, 1] (see, e.g., graph (3) in Figure 3.2). Theorem 4.2 adds to this result. The Ellsberg strategy Q with the property above is part of an Ellsberg equilibrium, if and only if B 1 (Q) = M 1 and M 1 = P * is the Nash equilibrium strategy of player 1.

The result of Theorem 4.2 also provokes reflection on the robustness of Ellsberg equilibria. From what we have seen, when player 1's Nash equilibrium strategy is equal to his immunization strategy, Ellsberg equilibrium has the particular form that player 1 plays his Nash equilibrium strategy and player 2 can play an arbitrarily large set of probability changes when we allow for more players, as [START_REF] Lo | Extensive form games with uncertainty averse players[END_REF] points out, because now players j and k may disagree about what player i will play. The consequence of this is immediate: we have an intuitive explanation, why in games with more than two players Ellsberg equilibria arise that are not obtainable by a Nash equilibrium.

Theorem 5.8. If (P * 1 , . . . , P * n ) is an Ellsberg equilibrium, then there exist

) is a subjective beliefs equilibrium and arg max

Proof. It is sufficient to show that there exists µ i -i ∈ P * -i such that arg max

The latter inclusion means that the set of best responses to P * -i is contained in the set of best responses to µ i -i . This and the assumption that (P * 1 , . . . , P * n ) is an Ellsberg equilibrium imply µ j i ∈ P * i ⊆ arg max

and therefore (µ 1 -1 , . . . , µ n -n ) is a subjective beliefs equilibrium. We know from the Minimax Theorem 1 (Theorem 1.15) and the Principle of Indifference in Distributions (Theorem 1.14) that max

) if and only if (by the Principle of Indifference in Distributions) min P -i ∈P * -i u i (P * i , P -i ) = c * . From this follows that

for all P -i ∈ P * -i , for all P * i ∈ arg max

Then we have

(5.3)

When we combine equations (5.2) and (5.3), we have

Suppose player 1 plays

with Θ 0 1 the set of distributions over A and B, and Θ 1 1 the set of distributions over E and F . For the purpose of the example, let player 2 have an Ellsberg behavior strategy Θ 2 that chooses C with probability 1, without creating any ambiguity.

The set of plays W is then given by W = {E, F, B} with respect to the outcomes a, b and d. We derive the set of realization probabilities Π Θ corresponding to the Ellsberg behavior strategy Θ,

Now we can calculate the maxmin expected utility for player 1 of the Ellsberg behavior strategy Θ.

What is the rectangular Ellsberg strategy profile P Θ induced by the Ellsberg behavior strategy profile Θ? We follow the construction in Figure 6.2.

The set of induced realization probabilities Π Θ is a hexagon (colored green in Figure 6.4),

given by the surface that lies in the intersection of

It can be easily seen that Π Θ is too large in the sense that it induces conditional probabilities over E and F which lie outside the set Θ 1 1 of original Ellsberg behavior strategies. Take for example the distribution (1/8, 3/8, 1/2) (probability distribution over E, F, B) at the lower right corner of the green hexagon. This distribution yields a conditional one-step-ahead probability for E which is equal to 1/8 1/2 = 1/4, and 1/4

6.4 Ellsberg Equilibrium in Extensive Form Ellsberg Games Θ P,1 1 as one-step-ahead conditionals, and hence

� .

Obviously we have

6 Every Nash equilibrium is an Ellsberg equilibrium. Conversely, every Ellsberg equilibrium in which P is a single probability distribution is a Nash equilibrium.

Relation to Other Extensive Form Solution Concepts

We discuss the relation of Ellsberg equilibrium to other extensive form solution concepts with an example which [START_REF] Fudenberg | Learning, experimentation, and equilibrium in games[END_REF] used to show that mistakes about

Relation to Other Extensive Form Solution Concepts

U

Player 1 gets a payoff less than 1 if player 1 or 2 defect, when

Player 2 gets a payoff less than 1 if player 1 or 2 defect, when

This yields the equilibrium (A

The information structure of the game has been characterized by [START_REF] Fudenberg | Self-confirming equilibrium[END_REF] in the analysis of their concept of Self-confirming equilibrium. They define a property of unobserved deviators which captures the fact that a player does not observe who of his opponents leaves the equilibrium path at some point in the game. These games have a natural property to allow for "wrong" beliefs on plays off the equilibrium path. Therefore games with unobserved deviators can have Self-confirming or Conjectural [START_REF] Battigalli | Comportamento razionale ed equilibrio nei giochi e nelle situazioni sociali[END_REF], [START_REF] Battigalli | Conjectural equilibria and rationalizability in a macroeconomic game with incomplete information[END_REF], [START_REF] Battigalli | Conjectural equilibria and rationalizability in a macroeconomic game with incomplete information[END_REF]) equilibria which are not Nash equilibria. [START_REF] Fudenberg | Self-confirming equilibrium[END_REF] show that games that have only observed deviators do not have non-Nash equilibria. We suppose that a similar characterization holds for Ellsberg equilibria as well. This is an interesting question for further research.