René Van Den Brink

Agnieszka Rusinowska
email: agnieszka.rusinowska@univ-paris1.fr.

Frank Steffen
email: steffen2@liverpool.ac.uk.

Marlies Ahlert

Shanti Chakravarty

Anke Gerber

Manfred Holler

David Hojman

Hartmut Kliemt

László Kóczy

Annick Laruelle

Garance Marechal

Jack Stecher

Jens Tiedemann

Measuring Power and Satisfaction in Societies with Opinion Leaders: An Axiomatization 1

Keywords: Collective choice, follower, opinion leader, power, satisfaction, axiomatization choix collectif, disciple, leader d'opinion, pouvoir, satisfaction, axiomatisation JEL Classification: D71, D85

A well-known model in sociology and marketing is that of opinion leadership. Opinion leaders are actors who are able to affect the behavior of their followers. Hence, opinion leaders have some power over their followers, and they can exercise this power by influencing their followers choice of action. We study a two-action model for a society with opinion leaders. We assume that each member of the society has an inclination to choose one of these actions and that the collective choice is made by simple majority of the actions chosen by each member. For this model we axiomatize satisfaction and power scores, which allow us to investigate the effects of different opinion leader-follower structures.

Résumé

Un modèle bien connu en sociologie et gestion est celui du leadership d'opinion. Les leaders d'opinion sont des acteurs qui peuvent influer sur les comportements de leurs disciples. En conséquence, les leaders d'opinion ont un certain pouvoir sur leurs disciples et ils peuvent exercer ce pouvoir en influençant le choix d'action de leurs disciples. Nous étudions un modèle de deux actions pour une société avec des leaders d'opinion. Nous supposons que chaque membre de la société a une inclination de choisir une des actions et que le choix collectif est fait par la majorité simple des actions choisies par chaque membre. Pour ce modèle nous axiomatisons les scores de satisfaction et de pouvoir, ce qui nous permet d'examiner les effets des différentes structures de leader d'opinion -disciples.

Introduction

The concept of opinion leadership received considerable attention in sociology and marketing. It rose out of the two-step flow of communication theory introduced by the 'Lazarsfeld group' (see, e.g., [START_REF] Katz | Personal influence: the part played by people in the flow of mass communication[END_REF], and [START_REF] Lazarsfeld | The peoples choice -How the voter makes up his mind in a presidential campaign[END_REF]). In its most rudimentary form it claims that 'ideas often flow from radio and print to the opinion leaders and from them to the less active sections of the population', see [START_REF] Lazarsfeld | The peoples choice -How the voter makes up his mind in a presidential campaign[END_REF]. They investigated the influence of mass communication on the 1940 presidential election campaign in the US and found that the voters' choices were more influenced by actors which they called opinion leaders than by mass communication. Based on this observation they arrived at the conclusion that the communication process is not a one-but a two-step process. According to their model information distributed by mass media first reaches the so-called opinion leaders. These are actors who are specified as highly self-confident with strong opinions. In [START_REF] Lazarsfeld | The peoples choice -How the voter makes up his mind in a presidential campaign[END_REF] they act as intermediaries between the mass media and the recipients. In general, the latter actors are called followers. They feel attracted by the opinion leaders holding them in high esteem and are prepared to accept their opinion for their own behavior. Hence, a major characteristic of opinion leaders is their exercised power over their followers. After critiques of the model by the 'Lazarsfeld group' (see, e.g., [START_REF] Bostian | The two-step flow theory: cross-cultural implications[END_REF]), [START_REF] Troldahl | A field test of a modified two-step flow of communication model[END_REF] introduced a modified version of their model called the two-cycle flow of communication model which corresponded to other results in the field (see, e.g., [START_REF] Deutschmann | Diffusion of knowledge of the major new story[END_REF]). Troldahl's model distinguishes between two phases in the communication process. Phase one is a flow of information from the mass media to the members of the society which is assumed to be a one-step process, i.e., the information goes directly to all members of the society. Phase two is the flow of influence on beliefs and behavior which is assumed to be a two-step process. In a first step opinion leaders form their own opinion based on additional information provided by experts, such as academics, while in a subsequent second step they try to influence the behavior of their followers. Since Troldahl's contribution the literature on opinion leadership has provided a strong body of knowledge of how and why opinion leaders influence followers choices (see [START_REF] Hoyer | A comparison of antecedents and consequences of market mavens and opinion leaders[END_REF]).

Opinion leaders form an attractive group for marketing and policy purposes (see, e.g., Hoyer and Stockburger-Sauer (2007)) as the existence (or non-existence) of opinion leaders in a society and their relations to their followers may have a considerable impact on market behavior (such as consumer or financial markets), and other social agglomerations being made up of individual actors choosing among a number of alternatives (open to them at a given time). Hence, it appears to be interesting to investigate the effect of different opinion leader-follower structures in markets or other collective decision-making 1 Documents de Travail du Centre d'Economie de la Sorbonne -2011. [START_REF] Deutschmann | Diffusion of knowledge of the major new story[END_REF] situations in a society. This includes questions such as whether it would be worthwhile to establish a new opinion leader in a society or whether a change in an existing opinion leader-follower structure can be expected to make a difference to the society. However, to our best knowledge apart from our own recent work there exists no study which addresses such issues on bare theoretical grounds. In van den Brink, Rusinowska, and Steffen (2011), we laid the foundation to fill out this lacunae by introducing novel power and satisfaction scores for societies with opinion leaders and discussing some properties of both scores. 1 The power scores inform us about the power distribution among the members of the society with respect to their ability to affect the state of the society concerning a specific outcome, while satisfaction scores tell us to which degree members of the society can be expected to end up with an outcome that they like. Based upon these results, in this paper we provide a full axiomatization of the power and satisfaction scores for a specific opinion leader-follower collective choice situation.

For our analysis we consider the example of binary choice as it can also be found in [START_REF] Sinha | Emergence of two-phase behavior in markets through interaction and learning in agents with bounded rationality[END_REF] who study the effect of opinion leaders on market outcomes. It is assumed that an actor can choose among two alternatives. For instance, this can be a market in which the actors have to decide whether they should buy or not buy a joint product, or a voting situation in which the members of the society have to choose to vote either yes or no on a specific proposal. From now onwards we will refer to a voting situation only. However, all results presented in this paper also apply to markets. We assume that the actors in a society have to decide whether they would like to remain with the status quo or whether a specific exogenous proposal leading to a new state of the society should be adopted. We assume that the proposal has been distributed among all actors. Each actor has to form its own opinion on the proposal, i.e., without being influenced by any other actor. We will call this the actor's action inclination. The society is partitioned into opinion leaders, followers, and independent actors. In line with the inherent idea of opinion leadership we suppose that via informal discussions of the proposal the action inclinations of the opinion leaders are becoming public information prior to the real decision. Only after these discussions, all actors will choose their action which coincide with the actors action inclination if it is an opinion leader or independent actor. Concerning the followers we assume that for their choice of action they -independently of their own action inclination -adopt the action inclination of their opinion leaders if all of these have the same action inclination. Finally, based on the individual choices of all actors, a decision-making mechanism determines the collective choice, i.e., whether the proposal is adopted or not. We assume that the collective choice is made by simple majority of the actions chosen by the actors.

In the literature we can find several scores and measures being introduced for analyzing collective decision-making situations with a possible influence between the actors. 2For instance, some measures for arbitrary digraphs have been studied in van den Brink and Borm (2002) and van den Brink and Gilles (2000). Our analysis in the present paper and our earlier study (see van den Brink, Rusinowska, and Steffen (2011)) is related to these contributions as we represent opinion leader-follower structures by bipartite digraphs. Coming from a slightly different direction are the works presented in the voting power literature. One of the traditional measures is the Rae index (Rae (1969)) which measures the success of an actor in a voting situation. An actor is said to be successful if its vote coincides with the voting outcome. Such a successful actor can be additionally powerful. For the calculation of the voting power of an actor a number of measures have been suggested. The most prominent measures are the Banzhaf measure [START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF], see also [START_REF] Dubey | Mathematical properties of the Banzhaf power index[END_REF], and Owen (1975)), and the Shapley-Shubik index [START_REF] Shapley | A method for evaluating the distribution of power in a committee system[END_REF]). They ascribe power to an actor if its vote coincides with the voting outcome, but this outcome would have been different if the actor changed its vote. 3As we are concerned with measuring satisfaction and power distributions, our research is also related to the work on measurement of voting power. We use the notion of satisfaction in order to distinguish our approach from those in the standard voting power literature. We are aware of the fact that at least in [START_REF] Straffin | Probability models for power indices[END_REF] and [START_REF] Straffin | Power and satisfaction in an ideologically divided voting body[END_REF] the notion of satisfaction is used as a synonym of success. However, in our opinion in their framework referring to the relation between votes and the voting outcome, success appears to be the more appropriate notion, while in our context where we refer to the relation between action inclinations and the social outcome, the notion of satisfaction appears to be more natural.

In van den Brink, Rusinowska, and Steffen (2011) we measured satisfaction by the number of times the collective choice is the same as the action inclination of an actor. 4We measured the power of an actor in a bipartite digraph by the number of times the actor has a swing, where a swing is defined as a situation where an actor by changing its action inclination, given the action inclinations of the others, enforces a change in the collective choice via a change in its action. In that paper we demonstrated that the power and satisfaction scores we introduced have some dictator and opinion leader properties in common in case the followers, for their choice of action, adopt the action inclination of their opinion leaders independently of their own inclination if a certain fraction of these have the same action inclination. In the present paper we consider the specific, but quite usual, case that this fraction is equal to one, i.e., unanimity. In the current context this assumption appears to be in line with other findings in the literature. For instance, [START_REF] Asch | Effects of group pressure upon the modification and distortion of judgments[END_REF][START_REF] Asch | Social psychology[END_REF][START_REF] Asch | Studies of independence and conformity: a minority of one against a unanimous majority[END_REF] results imply that when a group takes a unanimous position, people may feel more pressure to conform. A very recent study underpinning this view comes from an experiment conducted by [START_REF] Verhulst | Conformity in groups: the effects of groups on expressed attitudes[END_REF]. They found that participants were more likely to conform to the attitudes expressed by a unanimous group than by a non-unanimous group.

Assuming that a unanimity of opinion leaders is required for a follower to adopt their action inclination for its own choice of action implies that a follower only chooses an action which is not in line with its individual action inclination if all its opinion leaders have the (same) opposing action inclination. Based upon this assumption, in the present paper we show that the power and satisfaction scores satisfy even stronger opinion leader properties than those studied in van den Brink, Rusinowska, and Steffen (2011). Moreover, we introduce two different normalizations (i.e., units of measurement) and obtain full axiomatizations of both scores which differ in the normalization only.

The paper is structured as follows. In Sect. 2 we describe the model, and in Sect. 3 we define and illustrate the satisfaction and power scores for actors in societies with opinion leaders. In Sect. 4 we provide axiomatizations of the satisfaction and power scores differing only in the normalization that is applied. Finally, in Sect. 5 we draw some conclusions and discuss some possible extensions and applications of the model.

The model

Let N = {1, ..., n} denote a society containing n actors which is partitioned into opinion leaders, followers, and independent actors. Adopting a distinction applied by [START_REF] Vanberg | Rational choice and moral order[END_REF] and [START_REF] Heckathorn | Collective incentives and the creation of Prisoner's Dilemma norms[END_REF] we assume that each actor k ∈ N has two types of inclinations: constitutional and action inclinations, where constitutional inclinations can be regarded to be on a 'higher level' than action inclinations as action inclinations have to be formed within the framework given by the constitutional inclinations. In our context constitutional inclinations are related to the organization of the society. They determine (i) whether an actor k is an opinion leader, follower, or independent actor, (ii) which opinion leader an actor k chooses if k is a follower, and (iii) the procedure for followers to follow their opinion leaders. Instead, action inclinations are related to the outcome of the collective choice to be made by the society, i.e., they state which actions an actor k would choose being 'on its own' and not being influenced by others. 5In this paper we assume that constitutional inclinations are exogenous. This implies that (i) we have a given partition of our society into opinion leaders, followers, and independent actors, and that (ii) our opinion leader-follower relationships are already fixed, i.e., it is given which actors might influence the choice of action of certain other actors by exercising some power over them. 6 Moreover, (iii) we assume a unanimity requirement applies for followers to follow their opinion leaders, i.e., a follower will only choose an action against its own action inclination if all its opinion leaders have an inclination different from its own. In this case the follower will adopt the action inclination of its opinion leaders. Action inclinations are left 'unspecified'. As we assumed the constitutional inclinations to be exogenous, from now onwards we will just refer to inclinations instead of action inclinations.

Formally, we represent the structure of our society and the 'opinion leader-follower' relations by a bipartite directed graph (or bipartite digraph) (N, D) with a finite set of nodes N representing the actors, and D ⊂ N × N a binary relation on N such that each actor is either an opinion leader, a follower, or an independent actor. Since we take the set of actors N fixed, we represent a digraph (N, D) just by its binary relation D. Let S D (k) and P D (k) denote the set of successors and predecessors of actor k in digraph D, respectively, i.e., for each k ∈ N ,

S D (k) = {j ∈ N : (k, j) ∈ D} and P D (k) = {j ∈ N : (j, k) ∈ D}.
As we assume that each actor is either an opinion leader, follower or independent actor, we consider digraphs D such that Regarding the flow of information among the actors and their inclination formation we assume that (via the mass media) an exogenous proposal will be distributed among all actors k ∈ N . Having been informed about this proposal, each actor k ∈ N will form its inclination on it. Concerning the nature of the proposal, following [START_REF] Sinha | Emergence of two-phase behavior in markets through interaction and learning in agents with bounded rationality[END_REF] we assume that an actor has a binary choice: actor k can have the inclination either to support the proposal in order to obtain a new state of the society (inclination to choose the yes-action denoted by 1), or to reject it in order to remain with the status quo (inclination to choose the no-action denoted by 0). The inclinations chosen by the members of our society are represented by an inclination vector I = (I 1 , ..., I n ) ∈ {0, 1} n . This is a vector which k th component, I k , is 1 if actor k has the inclination to support the proposal, and 0 if it is inclined to reject it. Now we can define an opinion leader-follower collective choice situation as a pair (I, D) with I ∈ {0, 1} n and D ∈ D N being a bipartite graph as described above.

|S D (k)| • |P D (k)| = 0 for each k ∈ N, (2.1
In line with the inherent idea of an opinion leader, we assume that via informal (public) discussions of the proposal the inclinations of opinion leaders are becoming public information prior to the formal decision on the proposal, i.e., each follower is aware of the inclination of its opinion leader(s). 7 Only after these informal discussion actors will secretly (or simultaneously) choose their action. We assume that the actors in k ∈ OL(D)∪IND(D) make their simultaneous choice of action according to their own inclinations. For each follower we assume that, independently from its own inclination, it will choose the action which corresponds to the inclination of its opinion leaders if all of them have the same inclination. Otherwise, if the inclinations of its opinion leaders are not all the same, the follower will choose the action which corresponds to its own inclination.

Let V = V (I, D) ∈ {0, 1} n denote the choice vector, that is, a vector which k th component, V k , is 1 if actor k has chosen to support the proposal, and 0 if k has chosen to reject it. Thus, the choice vector V = V (I, D) ∈ {0, 1} n is given by:

V k = I k if k ∈ OL(D) ∪ IND(D),
and for k ∈ FOL(D) :

V k = x if I j = x for all j ∈ P D (k) I k otherwise, (2.2)
where x ∈ {0, 1}.

After all actors have chosen their actions, a collective choice is resulting according to the decision-making mechanism in use. The decision-making mechanism is given by the collective decision function C: {0, 1} n × D N → {0, 1} which assigns an outcome to every pair (I, D) ∈ {0, 1} n × D N , that is, the value 0 if the collective decision is no, and the value 1 if the collective decision is yes. Usually, one only considers collective decision functions C that are neutral8 and anonymous 9 . In this paper, we define the collective decision function by simple majority voting. Let for an action x ∈ {0, 1} and choice vector V = V (I, D) ∈ {0, 1} n the number of actors choosing the action x be denoted by

n x (V (I, D)) = |{k ∈ N : V k = x}|.
Restricting our analysis to situations in which the number of actors is odd, the collective decision function is defined, for each (I, D), as follows:

C(I, D) = 1 if n 1 (V (I, D)) > n 0 (V (I, D)) 0 if n 0 (V (I, D)) > n 1 (V (I, D)). (2.3)
Before we present our satisfaction and power scores in the next section, it appears to be appropriate to add a remark on the common measurement of success and power in voting games. Both usually relate the vote(s) of each actor to the voting outcome. An actor is said to be successful in a voting game if the actor's vote coincides with the voting outcome, i.e., the collective choice. In addition, if an actor is successful then power is ascribed to such an actor if, given the votes of the others, by changing its own vote the actor changes the voting outcome. It is said that the actor has a swing. Hence, roughly speaking, power in this context refers to an ability of an actor, i.e., what the actor is able to do (by changing its vote) against some resistance of others (represented by those given votes of the others which are not in line with the 'new' vote of the actor in question) irrespective of the actual occurrence of this resistance (see van den Brink and Steffen (2008) referring to [START_REF] Braham | Social power and social causation: towards a formal synthesis[END_REF]). These definitions of success and power are sufficient for many applications. However, being applied to the measurement of success and power in voting games they come along with a number of implicit simplifying assumptions (see Morriss 1987Morriss /2002:154-156:154-156). Among others they assume (i) that actors vote in line with their inclination, and (ii) that each actor's choice of a vote is not influenced by another actor. The implication of (i) is that for measurement purposes it is sufficient to consider the votes of the actors and the resulting voting outcome, and to ignore the fact that inclinations are usually part of any definition of power defined as an ability. Following Morriss (1987Morriss ( /2002:26):26), 'abilities are things that we can do when we want'. As our opinion leader-follower collective choice situations allow for situations under which both assumptions above are violated which, in fact, is the usual case in our context, we have to relax both. This implies that for our purposes we have to include the actors' inclinations into our measurement.

Measuring satisfaction and power

In van den Brink, Rusinowska, and Steffen (2011) satisfaction and power scores for bipartite digraphs which represent opinion leader-follower collective choice situations as described in the previous section, are introduced. In general, a score for bipartite digraphs is a function f : D N → IR n which assigns an n-dimensional real vector to every bipartite digraph on N .

Since in our model actors will actually choose either the yes-or no-action if they have the corresponding inclination to do so, or if they are influenced by their opinion leaders to make that choice, we propose to ascribe satisfaction to an actor if the actor's inclination prior to its actual choice coincides with the collective choice and to measure the satisfaction of an actor in an opinion leader-follower collective choice situation by the actor's likelihood to be satisfied.

Formally, first, in order to ascribe satisfaction to an actor we define a satisfaction score of an actor under a given inclination vector, i.e., for each (I,

D) ∈ {0, 1} n × D N and k ∈ N SAT k (I, D) = 1 if C(I, D) = I k 0 otherwise.
Next, based on the satisfaction of an actor under each inclination vector, we define the satisfaction score in a bipartite digraph, SAT : D N → IR n , given by

SAT k (D) = I∈{0,1} n SAT k (I, D) for each k ∈ N. (3.4)
In a similar way we ascribe power to an actor if the actor, by changing its inclination, is able to alter the collective choice and measure the power of the actor in an opinion leaderfollower collective choice situation by the actor's likelihood to be powerful.

Hence, formally, actor k ∈ N has a swing in (I, D) according to collective decision function C if C(I, D) = C(I ′ , D) with I ′ k = I k and I ′ j = I j for all j ∈ N \ {k}. In order to ascribe power to actor k we define a power score of actor k under a given inclination vector:

P OW k (I, D) = 1 if k has a swing in (I, D) 0 otherwise.
Then, the power score P OW : D N → IR n is given by

P OW k (D) = I∈{0,1} n P OW k (I, D) for each k ∈ N. (3.5)
To illustrate the satisfaction and power scores we present two examples, one with three actors and one with five actors.

Example 1 Let N = {1, 2, 3}. For this set of actors the first column in Table 1 contains all feasible inclination vectors. As a bench mark case let us begin with the empty digraph D ∅ = ∅, i.e., all actors are independent. Hence, the choice vector, defined in (2.2), is the same as the inclination vector, i.e., V (I, D ∅ ) = I. Having defined the collective decision function C(I, D) by simple majority voting, the resulting outcomes of the choices given by V (I, D ∅ ) = I are displayed in the second column. This results in SAT (D ∅ ) = [START_REF] Bolger | A value for games with n players and r alternatives[END_REF][START_REF] Bolger | A value for games with n players and r alternatives[END_REF][START_REF] Bolger | A value for games with n players and r alternatives[END_REF] and POW (D ∅ ) = (4, 4, 4).

Table 1: Three actors

I C(I, D ∅ ) V (I, D 1 ) C(I, V (I, D 1 )) V (I, D 2 ) C(I, V (I, D 2 )) V (I, D 3 ) C(I, V (I, D 3 )) (0,0,0) 0 - - - - - - (1,0,0) 0 (1,1,0) 1 (1,1,1) 1 - - (0,1,0) 0 (0,0,0) - (0,0,0) - (0,0,0) - (0,0,1) 0 - - (0,0,0) - - - (1,1,0) 1 - - (1,1,1) 1 - (1,0,1) 1 (1,1,1) - (1,1,1) - (1,1,1) - (0,1,1) 1 (0,0,1) 0 (0,0,0) 0 - - (1,1,1) 1 - - - - - -
Next, let us consider the digraph D 1 = {(1, 2)}, i.e., the case of actor 1 being the opinion leader of actor 2 (implying that actor 2 is the follower of actor 1), while actor 3 remains independent. Column 3 of Table 1 displays the choice vectors V (I, D 1 ) and column 4 the resulting outcomes of the choices for this case under simple majority voting. By a '-' in column 3 we indicate that the choice is identical to the inclination. By a '-' in column 4 we indicate that the outcome is the same as with the empty digraph, i.e., when all actors choose according to their own inclination. Obviously, when the choices do not change then also the outcome does not change, but the choices may change without affecting the outcome.

Based on Table 1 we can now compute the satisfaction and power scores for digraph D 1 = {(1, 2)}: SAT (D 1 ) = [START_REF] Bolger | Characterizations of two power indices for voting games with r alternatives[END_REF][START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF][START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF] and POW (D 1 ) = (8, 0, 0). Note that these scores are the same in case actor 1 is a 'real' dictator, i.e., if we consider the digraph D 2 = {(1, 2), (1, 3)}, although the choices are different for all inclinations except (0, 0, 0) and (1, 1, 1). Note also that the scores for D 3 = {(1, 2), (3, 2)} are the same as for the empty digraph, although the choices for inclinations (0, 1, 0) and (1, 0, 1) are different. So, in this three actor example satisfaction and power are distributed either according to the situation that there are only independent actors, or there is a dictator.

2

Some more interesting cases are obtained when we consider five actors. We take the following opinion leader-follower structures from van den Brink, Rusinowska, and Steffen (2011), but discuss the unanimity model of the underlying paper.

Example 2 Consider N = {1, 2, 3, 4, 5}. Like in Table 1 of the previous example, also the first two columns of Table 2 display all feasible inclinations and the resulting outcomes under simple majority voting for the empty digraph, i.e., for the case that all actors behave independently. By the following columns we provide the corresponding choice vectors and the outcomes for the following five digraphs as also represented by Figure 1: Next, we consider two dictator properties. Clearly, a dictator, i.e., a unique opinion leader who is followed by all other actors, has the power to change the outcome for any voting profile by changing its own inclination and, if the dictator votes according to its inclination, then the outcome will be the inclination of the dictator. Therefore the dictator property states that, if there is a dictator, then the score of the dictator is equal to the total number of possible inclination vectors. Note, that since we assume that no actor can be at the same time a follower and an opinion leader, the dictator as defined above cannot be a follower.

D ∅ = ∅ D 1 = {(1, 2)} D 2 = {(1, 2), (3, 2)

Dictator property

If D ∈ D N and h ∈ N is such that S D (h) = N \{h}, then f h (D) = 2 n .
Secondly, since a follower who has only one opinion leader has always to follow this opinion leader, dictated independence states that the score of a follower with one opinion leader does not change as long as this follower is dictated by a sole opinion leader.

Dictated independence If D, D ′ ∈ D N and k ∈ N are such that |P D (k)| = |P D ′ (k)| = 1, then f k (D) = f k (D ′ ).
Next we present two opinion leader properties saying something about the changes in score for different actors when the opinion leader-follower structure changes, in particular when an actor gets a new opinion leader. The properties that we consider are inspired by similar properties for solutions in cooperative game theory, where studying these kinds of properties has a longer history.

In the context of cooperative TU-games, [START_REF] Lehrer | An axiomatization of the Banzhaf value[END_REF] and [START_REF] Haller | Collusion properties of values[END_REF] introduced properties that consider collusion of players. In particular, [START_REF] Haller | Collusion properties of values[END_REF] considers different types of collusion neutrality properties requiring that the sum of the payoffs of two colluding players does not change, see also [START_REF] Malawski | Procedural values for cooperative games[END_REF]. In van den Brink (2010) these properties are stated in terms of games in which the players belong to some hierarchical structure, the so-called games with a permission structure. There deleting a domination link between a successor and a predecessor is interpreted as a collusion between this predecessor and another predecessor with respect to the influence over the successor. In that context power neutrality states that the sum of payoffs of the two colluding predecessors should not change. We are now going to apply this idea to define two axioms for opinion leaderfollower collective choice situations.

Suppose that an independent actor gets an opinion leader. The opposite gain property states that, if an actor becomes a sole opinion leader of another actor who was previously independent, then the sum of the scores of these two actors does not change. This implies that in case the opinion leader gains this goes fully at the cost of the follower.

Opposite gain property

Let D, D ′ ∈ D N , j ∈ IND(D), h ∈ OL(D) ∪ IND(D), and D ′ = D ∪ {(h, j)}. Then f h (D ′ ) -f h (D) = f j (D) -f j (D ′ ).
Horizontal neutrality states that, if a follower gets one more opinion leader, then the sum of scores of the 'new' and an 'old' opinion leader does not change. In other words, the change for the new opinion leader is opposite but in absolute value equal to the change for an 'old' opinion leader. 10Horizontal neutrality Let D,

D ′ ∈ D N , j ∈ FOL(D), g ∈ P D (j), h ∈ OL(D) ∪ IND(D), and
D ′ = D ∪ {(h, j)}. Then f h (D ′ ) -f h (D) = f g (D) -f g (D ′ ).
One of the most widely studied and applied properties in the context of cooperative games in which the players belong to some binary communication structure, is fairness introduced by [START_REF] Myerson | Graphs and cooperation in games[END_REF]. It states that deleting a communication link between two players changes their individual payoffs by the same amount. In van den Brink (1997) this type of equal gain/loss property is stated in terms of the above mentioned games with a hierarchical permission structure. For the opinion leader-follower collective choice situation of the underlying paper, suppose that a follower gets one more opinion leader. In van den Brink, Rusinowska, and Steffen (2011) it is shown that in case a follower follows a qualified majority of its opinion leaders, both their satisfaction and power score satisfy the equal absolute change property stating that the changes in scores of this follower and of its new opinion leader are either the same or are opposite, but the same in absolute values. It turns out that in the unanimity case considered in this paper (i.e., a follower chooses against its own inclination only if all its opinion leaders have the same inclination that is different from its own) both the satisfaction score SAT and power score POW even satisfy the stronger equal gain property stating that the changes in scores of this follower and of its new opinion leader are the same.

Equal gain property

Let D, D ′ ∈ D N , j ∈ FOL(D), h ∈ OL(D) ∪ IND(D), and D ′ = D ∪ {(h, j)}. Then f h (D ′ ) -f h (D) = f j (D ′ ) -f j (D).
We illustrate these axioms with the digraphs presented in Figure 1.

Example 3 Consider the five actor opinion leader-follower collective choice situation of Example 2. We see that the satisfaction and power scores of a dictated actor (i.e., an actor that is subordinate to exactly one opinion leader) stay the same as long as the actor is dictated by this opinion leader (see, for example, the scores for actor 2 in D 1 and D 4 ), illustrating dictated independence. Further, if we let an independent actor become the follower of one opinion leader, then both the gain in power and the gain in satisfaction of the opinion leader go at the cost of the follower. For example, by going from the empty digraph D ∅ to D 1 actor 1 becomes opinion leader for actor 2, and the sum of their satisfaction and power (44, respectively, 24) does not change, illustrating the opposite gain property. If also actor 3 becomes an opinion leader for actor 2 (i.e., we go from D 1 to D 2 ), then the gain in satisfaction and power of actors 2 and 3 change the same (an increase of 4, respectively, 8), illustrating the equal gain property, and is in absolute value equal to the loss of actor 1, illustrating horizontal neutrality.

2

As we will prove below, both, the satisfaction score SAT and the power score POW satisfy the above six properties. Obviously, since satisfaction and power are related but different concepts (see, e.g., [START_REF] Dowding | Power[END_REF]) each will satisfy a different normalization. As normalization of satisfaction we take that the sum of all scores is equal to the total number of individual satisfactions, i.e., for each inclination vector we count how many actors have an inclination that coincides with the social outcome, and we add all these satisfactions over all inclination vectors.

Satisfaction normalization For every D ∈ D N it holds that

k∈N f k (D) = I∈{0,1} n |{k ∈ N : I k = C(I, D)}|.
As normalization of power we take that the sum of all scores is equal to the total number of individual swings, i.e., for each inclination vector we count how many actors have a swing, and we add all these swings over all inclination vectors.

Power normalization For every D ∈ D N it holds that

k∈N f k (D) = I∈{0,1} n |{k ∈ N : k has a swing in (I, D)}|.
It turns out that adding satisfaction normalization to the first six axioms characterizes the satisfaction score.

Theorem 1 Let the choice vector V be defined by (2.2). A score f : D N → IR n is the satisfaction score SAT : D N → IR n if and only if it satisfies symmetry, the dictator property, dictated independence, the opposite gain property, horizontal neutrality, the equal gain property, and satisfaction normalization.

Proof SAT satisfying symmetry, the dictator property, dictated independence, and the opposite gain property is shown in van den Brink, Rusinowska, and Steffen (2011). 11 It is obvious that SAT satisfies satisfaction normalization.

To show the equal gain property and horizontal neutrality, let D, D ′ ∈ D N , j ∈ FOL(D), h ∈ OL(D) ∪ IND(D), and D ′ = D ∪ {(h, j)}. If C(I, D) = C(I, D ′ ) then it must hold that actor j initially had to vote against its inclination and now can vote according to its inclination because its new opinion leader h has the same inclination. Moreover, all 'other' opinion leaders of j must have the opposite inclination. Thus, for g ∈ P D (j) we have

C(I, D) = I g = I j = I h and C(I, D ′ ) = I j = I h = I g . So, SAT j (I, D ′ ) -SAT j (I, D) = SAT h (I, D ′ ) -SAT h (I, D) = SAT g (I, D) -SAT g (I, D ′ ) = 1. Obviously, if C(I, D) = C(I, D ′ ), then SAT j (I, D ′ ) -SAT j (I, D) = SAT h (I, D ′ ) -SAT h (I, D) = SAT g (I, D) -SAT g (I, D ′ ) = 0. Thus, with (3.4) we have SAT j (D) -SAT j (D ′ ) = SAT h (D) -SAT h (D ′ ) = SAT g (D ′ ) -SAT g (D)
, showing that SAT satisfies the equal gain property and horizontal neutrality.

To prove uniqueness, suppose that the score f : D N → IR n satisfies the seven axioms, and let D ∈ D N . We prove that f must be equal to SAT in several steps. We first prove uniqueness in case there is at most one actor that is an opinion leader by induction on its number of followers.

First, suppose that the opinion leader is a dictator, i.e., there is an h ∈ N such that S D (h) = N \ {h}. Then the dictator property implies that f h (D) = 2 n . Since each actor k ∈ N \ {h} has the same inclination as h in half of the inclination vectors in {0, 1} n , satisfaction normalization implies that k∈N \{h} f k (D) = k∈N \{h} 2 n-1 = (n -1)2 n-1 . Symmetry then implies that f k (D) = 2 n-1 is determined for all k ∈ N \ {h}.

Proceeding by induction, suppose that f ( D) is uniquely determined whenever |S D (h)| > |S D (h)|. Take a j ∈ N \ ({h} ∪ S D (h)). Note that j is an independent actor since h is the only actor with successors. Consider D ′ ∈ D N given by D ′ = D ∪ {(h, j)}. Dictated independence and the induction hypothesis imply that f k (D) is uniquely determined for all k ∈ S D (h). Symmetry implies that there is a constant c ∈ IR such that

f k (D) = c for all k ∈ N \ ({h} ∪ S D (h)). (4.6)
The opposite gain property implies that

f h (D ′ ) -f h (D) = f j (D) -f j (D ′ ), (4.7)
where f h (D ′ ) and f j (D ′ ) are given by the induction hypothesis. Then, with satisfaction normalization, (4.6) and (4. From above it follows that f (D) is uniquely determined if D = ∅. 12Proceeding by induction, assume that f ( D) is uniquely determined whenever | D| < |D|.

We distinguish the following cases with respect to actor k ∈ N (of which at least one must occur):

1. If |P D (k)| = 1 then dictated independence and the case with a dictator considered before imply that f k (D) = 2 n-1 is uniquely determined.

2. If there is a j ∈ S D (k) with |P D (j)| = 1 then the opposite gain property implies that

f k (D) + f j (D) = f k (D \ {(k, j)}) + f j (D \ {(k, j)}). (4.8)
Since actor j is as in case 1, we determined f j (D). With the induction hypothesis f k (D \ {(k, j)}) and f j (D \ {(k, j)}) are determined. Thus, with (4.8),

f k (D) = f k (D \ {(k, j)}) + f j (D \ {(k, j)}) -f j (D) is uniquely determined.
3. If there is a j ∈ S D (k) with |P D (j)| ≥ 2 then take h ∈ P D (j) \ {k}. The equal gain property implies that

f k (D) -f k (D \ {(k, j)}) = f j (D) -f j (D \ {(k, j)}) (4.9)
and

f h (D) -f h (D \ {(h, j)}) = f j (D) -f j (D \ {(h, j)}). (4.10)
Horizontal neutrality implies that which lies in the normalization being applied. 15 This expresses the basic difference in measuring satisfaction or power. Furthermore, note that satisfaction normalization and the dictator property imply that the satisfaction score SAT also satisfies the property that in case there is a dictator, the satisfaction score of a dictated actor is half the satisfaction score of the dictator. On the other hand, power normalization and the dictator property imply that the power score POW satisfies the property that the power score of actors that are subordinate to a dictator is equal to zero. 16 This expresses another difference between measuring satisfaction and power. If there is a dictator then the other actors cannot influence the outcome of the voting process since they have to follow the dictator. Therefore their power is equal to zero. However, also an actor that is subordinate to a dictator might end up with a social outcome that coincides with its inclination. Ex ante, a subordinate of a dictator will have its inclination coincide with that of the dictator in half of the cases. Since a dictator always dictates the outcome we arrive at a satisfaction score of the subordinate that is half the satisfaction score of the dictator.

f k (D) -f k (D \ {(k, j)}) = f h (D \ {(k, j)}) -f h (D). ( 4

Future Research

The existence of opinion leaders and their influence over other actors can be seen in every day life situations: in small as well as in large societies, be it in politics or business. Both satisfaction and power are the very natural measures of actors' strength or status in such situations. Since both are different concepts, it is worth to analyze what the common and different properties of the scores for both concepts display. Although, as mentioned in the introduction, there exist several related theoretical studies in the literature on voting models and on networks, the approach which we use in this paper, i.e., the analysis of opinion leader-follower structures, and the properties of the scores in question has brought 15 A similar difference is shown by van den Brink and Gilles (2000) for the outdegree measure and the βmeasure as scoring methods for directed graphs, highlighting that a normalization is not always so innocent as it might appear. 16 The power score POW even satisfies a stronger property which states that if a follower has a unique opinion leader, then its power is equal to zero, i.e., if D ∈ D N , j ∈ FOL(D) and h ∈ OL(D) such that P D (j) = {h}, then f j (D) = 0.

up several innovative elements and can also be regarded to contribute to knowledge, in particular, in marketing. Some generalizations that we will consider in the future are the following. First, one could allow for an actor to be an opinion leader and follower at the same time, i.e., to have a society with more than two 'layers'. In terms of [START_REF] Troldahl | A field test of a modified two-step flow of communication model[END_REF] two-cycle flow of communication model this would allow us to include the experts as an additional group of actors into our analysis acting as opinion leaders of the opinion leaders. Secondly, one could enlarge the set of possible actions, i.e., instead of allowing for just two actions one could follow some works on abstention (see, e.g., [START_REF] Braham | Voting power in games with abstentions[END_REF], [START_REF] Felsenthal | Ternary voting games[END_REF][START_REF] Felsenthal | The Measurement of voting power: theory and practice, problems and paradoxes[END_REF][START_REF] Felsenthal | Models and reality: the curious case of the absent abstention[END_REF], Tchantcho, Diffo Lambo, Pongou, and Mbama Ebgoulou (2008)), and multi-choice games (see, e.g., [START_REF] Grabisch | A model of influence with an ordered set of possible actions[END_REF], and Hsiao and Raghavan (1993)). Related models are also games with r alternatives, where the alternatives are not ordered; see [START_REF] Bolger | Power indices for multicandidate voting games[END_REF][START_REF] Bolger | A value for games with n players and r alternatives[END_REF][START_REF] Bolger | A consistent value for games with n players and r alternatives[END_REF][START_REF] Bolger | Characterizations of two power indices for voting games with r alternatives[END_REF]. Also in Freixas (2005aFreixas ( , 2005b) ) and [START_REF] Freixas | Weighted voting, abstention, and multiple levels of approval[END_REF], the authors consider decision-making situations, i.e., voting systems, with several levels of approval in the input and output, where those levels are qualitatively ordered. They introduce (j, k) simple games, in which each actor expresses one of j possible levels of input support, and the output consists of one of k possible levels of collective support. Thirdly, one could endogenize the 'higher level' constitutional inclinations.

In this paper the choice vector is determined by unanimity. A further item for future research is to axiomatize the satisfaction and power score for other methods determining the choice vector, such as those defined by qualified majority. Moreover, one could apply the formal approach in the present paper to related problems in organizational hierarchies where an organizational choice is to be made, see, e.g., [START_REF] Hammond | Invisible decisive coalitions in large hierarchies[END_REF]. Finally, one could apply our framework to concrete real life situations. Here, in particular, in the light of the financial crisis of 2007-2010 the herding behavior in financial markets (see, e.g., [START_REF] Devenow | Rational herding in financial economics[END_REF]) appears to be a promissing field for an application.

The dictator property follows straightforward since a dictator is followed in all 2 n inclination vectors in {0, 1} n , i.e., if S D (h) = N \ {h}, then C(I, D) = I h for all I ∈ {0, 1} n .

To show dictated independence, note that actor k always chooses an action according to j's inclination if P D (k) = {j}. That means that the collective choice is independent of actor k's inclination, i.

2

Proposition A.2 Let the choice vector V be defined by (2.2). The power score POW satisfies symmetry, the dictator property, dictated independence, and the opposite gain property.

Proof

It is straightforward that P OW satisfies symmetry.

Since a dictator has a swing in every inclination vector, P OW satisfies the dictator property.

Since an actor with a unique opinion leader never has a swing, P OW satisfies dictated independence.

  ) where |X| denotes the cardinality of set X. Let OL(D), FOL(D), and IND(D) denote the sets of all opinion leaders, followers, and independent actors in digraph D, respectively, i.e., OL(D) = {k ∈ N : S D (k) = ∅} FOL(D) = {k ∈ N : P D (k) = ∅} and IND(D) = N \ (OL(D) ∪ FOL(D)). Therefore, by assumption (2.1) we have that OL(D) ∩ FOL(D) = ∅, and thus the sets OL(D), FOL(D) and IND(D) form a partition of the set N . We denote the collection of all bipartite digraphs on N , represented by their binary relation, by D N .

} D 3 =Fig. 1 :10

 31 Fig. 1: Digraphs of Example 2

  7) yield (n -1 -|S D (h)|) + 1 + 1 = n -|S D (h)| + 1 linearly independent equations with the n -|S D (h)| + 1 unknowns, c and f i (D), k ∈ N \ S D (h). Thus f (D) is uniquely determined. Next, we prove that f (D) is uniquely determined for all D ∈ D N by induction on |D|.

. 11 )

 11 With the induction hypothesis the values in graphs D \ {(k, j)} and D \ {(h, j)} are uniquely determined. Thus, with the three linearly independent equations (4.9), (4.10) and (4.11), the payoffs f k (D), f j (D) and f h (D) are uniquely determined.4. If |PD (k)| ≥ 2 then f k (D)is uniquely determined as in the previous case (with the roles for k and j reversed).

Proposition 3

 3 (i) (Dictator domination satisfaction property) If D ∈ D N , OL(D) = {h}, and FOL(D) = N \ {h}, then SAT k (D) = 1 2 SAT h (D) for all k ∈ N \ {h}. (ii) (Dictator domination power property) If D ∈ D N , OL(D) = {h}, and FOL(D) = N \ {h}, then POW k (D) = 0 for all k ∈ N \ {h}.

  e., C(I, D) = C(I ′ , D) if I h = I ′ h for all h ∈ N \ {k}. Hence, in half of the inclination vectors C(I, D) = I k and in the other half C(I, D) = I k . So, SAT satisfies dictated independence. To show the opposite gain property, let D, D ′ ∈ D N , j ∈ IND(D), h ∈ OL(D) ∪ IND(D), and D ′ = D ∪ {(h, j)}. Suppose that C(I, D) = C(I, D ′ ). Then it must hold that actor j had to deviate from its inclination and follow h, and this must result in a change of collective choice from I j to I h , with I j = I h . So, C(I, D) = I j = I h and C(I, D ′ ) = I h = I j . Then, SAT j (I, D ′ ) = SAT j (I, D) -1 and SAT h (I, D ′ ) = SAT h (I, D) + 1. So, SAT h (I, D ′ ) -SAT h (I, D) = SAT j (I, D) -SAT j (I, D ′ ). Obviously, this last equality also holds if C(I, D) = C(I, D ′ ). Thus, we have SAT h (D ′ ) -SAT h (D) = SAT j (D) -SAT j (D ′ ), showing that SAT satisfies the opposite gain property.

2 25

 2 To show the opposite gain property, let D, D ′ ∈ D N , j ∈ IND(D), h ∈ OL(D) ∪ IND(D), and D ′ = D ∪ {(h, j)}. Since in D ′ voter j has to choose an action according to its unique opinion leader h, j never has a swing in D ′ , i.e., POW j (D ′ ) = 0. So, we have to show that POW h (D ′ ) -POW h (D) = POW j (D). We distinguish the following three cases. (i) If h does not have a swing in (I, D) but j has a swing in (I, D) then h has a swing in (I, D ′ ), i.e., if POW h (I, D) = 0 and POW j (I, D) = 1 then POW h (I, D ′ ) = 1. (ii) If h has a swing in (I, D) then h has a swing in (I, D ′ ), i.e., if POW h (I, D) = 1 then POW h (I, D ′ ) = 1. If, moreover, also j has a swing in (I, D) then h also has a swing in (I ′ , D ′ ) with I ′ j = I ′ h = I h = I j , i.e., if POW h (I, D) = 1 and POW j (I, D) = 1 then POW h (I ′ , D ′ ) = 1. (iii) Finally, if h does not have a swing in (I, D) and j has a swing in (I, D) then the only possibility for h to have a swing in (I, D ′ ) is as described in the last case before. So, POW h (D ′ ) = I∈{0,1} n POW h (I, D ′ ) = I∈{0,1} n (POW h (I, D) + POW j (I, D)), showing that POW satisfies the opposite gain property. Documents de Travail du Centre d'Economie de la Sorbonne -2011.18

Note that this research is in some respects also related to work on opinion leaders and the Condorcet Jury Theorem (see, e.g.,[START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF]), threshold models of collective behavior (see, e.g.,[START_REF] Granovetter | Threshold models of collective behavior[END_REF], and Granovetter and Soong (1986)) and, in more general terms, to the literature on network externalities.

Documents de Travail du Centre d'Economie de la Sorbonne -2011.18

For the distinction between scores, measures, and indices, see[START_REF] Felsenthal | The Measurement of voting power: theory and practice, problems and paradoxes[END_REF].

Both measures can also be derived from a probabilistic framework, see, e.g.,[START_REF] Straffin | Homogeneity, independence, and power indices[END_REF][START_REF] Straffin | Probability models for power indices[END_REF] and, more recently,[START_REF] Laruelle | Assessing success and decisiveness in voting situations[END_REF].

Note that in van den Brink, Rusinowska, and Steffen (2011) we have just used the term inclination instead of action inclination.

Documents de Travail du Centre d'Economie de la Sorbonne -2011.18

Documents de Travail du Centre d'Economie de la Sorbonne -2011.18

Note that action inclinations are the type of inclinations which were introduced by Hoede and Bakker (1982) for their power analysis of organizational structures.

As a result of this influence, the ability of the followers to determine the outcome of the collective choice, i.e., their power to do something (with respect to the outcome of the collective choice) might be affected. 5 Documents de Travail du Centre d'Economie de la Sorbonne -2011.18

Note that while our model allows that the public discussions may change the inclinations of some actors, we suppose that when choosing their action, and after that, inclinations do not alter. 6 Documents de Travail du Centre d'Economie de la Sorbonne -2011.18

A collective decision function C is neutral if [C(I, D) = 1 if and only if C(I c , D) = 0], where I c k = 1 if and only if I k = 0.

A collective decision function C is anonymous if for every permutation π: N → N , C(I, D) = C(π(I), π(D)) with π(I) k = I π(k) and (π(k), π(j)) ∈ π(D) if and only if (k, j) ∈ D.

Note that this is a stronger version of the power neutrality for two opinion leaders used in van den Brink, Rusinowska, and Steffen (2011).

To make the paper self-contained we also put these proofs in the appendix of this paper.

Note that this also follows from symmetry and satisfaction normalization.

(0,0,0,0,0) 0 --------(1,0,0,0,0) 0 (1,1,0,0,0) -----(1,1,0,0,0) -(0,1,0,0,0) 0 (0,0,0,0,0) -(0,0,0,0,0) -(0,0,0,0,0) -(0,0,0,0,0) -(0,0,1,0,0) 0 ------(0,0,1,1,0) -(0,0,0,1,0) 0 ------(0,0,0,0,0) -(0,0,0,0,1) 0

For the empty digraph we have SAT (D ∅ ) = [START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF] and POW (D ∅ ) = [START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF]. In case actor 1 is a dictator, i.e., D = {(1, 2), [START_REF] Asch | Effects of group pressure upon the modification and distortion of judgments[END_REF][START_REF] Asch | Studies of independence and conformity: a minority of one against a unanimous majority[END_REF], [START_REF] Asch | Effects of group pressure upon the modification and distortion of judgments[END_REF][START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF], (1, 5)} we obtain the scores SAT (D) = [START_REF] Haller | Collusion properties of values[END_REF][START_REF] Brink R Van Den | Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties[END_REF][START_REF] Brink R Van Den | Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties[END_REF][START_REF] Brink R Van Den | Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties[END_REF][START_REF] Brink R Van Den | Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties[END_REF] and POW (D) = (32, 0, 0, 0, 0). (The same scores we obtain if one actor is the unique opinion leader of at least half of the other actors, although the choices may be different.) For the other digraphs we obtain the following scores: SAT (D 1 ) = [START_REF] Freixas | Weighted voting, abstention, and multiple levels of approval[END_REF][START_REF] Brink R Van Den | Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties[END_REF][START_REF] Dowding | Power[END_REF][START_REF] Dowding | Power[END_REF][START_REF] Dowding | Power[END_REF], POW (D 1 ) = (24, 0, 8, 8, 8) SAT (D 2 ) = [START_REF] Felsenthal | The Measurement of voting power: theory and practice, problems and paradoxes[END_REF][START_REF] Dowding | Power[END_REF][START_REF] Felsenthal | The Measurement of voting power: theory and practice, problems and paradoxes[END_REF][START_REF] Dowding | Power[END_REF][START_REF] Dowding | Power[END_REF], POW (D 2 ) = (16, 8, 16, 8, 8) SAT (D 3 ) = [START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF][START_REF] Estlund | Opinion leaders, independence, and Condorcets jury theorem[END_REF], POW (D 3 ) = [START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF][START_REF] Brink R Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF] SAT (D 4 ) = [START_REF] Felsenthal | The Measurement of voting power: theory and practice, problems and paradoxes[END_REF][START_REF] Brink R Van Den | Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties[END_REF][START_REF] Felsenthal | The Measurement of voting power: theory and practice, problems and paradoxes[END_REF][START_REF] Brink R Van Den | Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties[END_REF][START_REF] Felsenthal | The Measurement of voting power: theory and practice, problems and paradoxes[END_REF], POW (D 4 ) = (16, 0, 16, 0, 16).

4 Axiomatizations

In this section we provide full axiomatizations of the satisfaction score SAT and power score POW . In van den Brink, Rusinowska, and Steffen (2011) it is shown that both scores satisfy the first four properties discussed below. Symmetry states that the value of a score for actors with a symmetric position in the bipartite digraph is the same.

5. Finally, symmetry implies that there is a constant c ∈ IR such that f k (D) = c for all k ∈ IND(D). Since above we determined all f j (D) for j ∈ OL(D) ∪ FOL(D), satisfaction normalization determines c.

Thus, all f k (D), k ∈ N , are uniquely determined. 2

As mentioned earlier, the power score POW satisfies all properties except satisfaction normalization. Replacing satisfaction normalization by power normalization characterizes POW .

Theorem 2 Let the choice vector V be defined by (2.2). A score f : D N → IR n is the power score POW : D N → IR n if and only if it satisfies symmetry, the dictator property, dictated independence, the opposite gain property, horizontal neutrality, the equal gain property, and power normalization.

Proof POW satisfying symmetry, the dictator property, dictated independence, and the opposite gain property is shown in van den Brink, Rusinowska, and Steffen (2011). 13 It is obvious that POW satisfies power normalization.

To show the equal gain property and horizontal neutrality 14 , let Uniqueness follows similar as the uniqueness part of the proof of Theorem 1, but using the alternative power normalization.

2

Note that Theorems 1 and 2 characterize the satisfaction score SAT and the power score POW by the same axioms except the normalization axiom. Thus, we have two comparable axiomatizations which exactly illustrate the difference between satisfaction and power 13 Again, to make the paper self-contained we put these proofs in the appendix of this paper. 14 In van den Brink, Rusinowska, and Steffen (2011) it is already shown that horizontal neutrality holds if the follower has exactly one opinion leader.

Appendix

To make the paper self-contained, this appendix contains proofs of parts of Theorems 1 and 2 that are also given in van den Brink, Rusinowska, and Steffen (2011).

Proposition A.1 Let the choice vector V be defined by (2.2). The satisfaction score SAT satisfies symmetry, the dictator property, dictated independence, and the opposite gain property.

Proof

It is straightforward that SAT satisfies symmetry.