Vladimir Ivanovitch Danilov
email: danilov@cemi.rssi.ru

Ariane Lambert-Mogiliansky

Non-classical expected utility theory with application to type indeterminacy

Keywords: non-classical, uncertainty, decision-making

In this paper we extend Savage's theory of decision-making under uncertainty from a classical environment into a non-classical one. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expected utility. We also propose an application in simple game context in the spirit of Harsanyi.

Introduction

In this paper we propose an extension of the standard approach to decisionmaking under uncertainty in Savage's style from the classical model into the more general model of non-classical measurement theory. Formally, this means that we substitute the Boolean algebra model with a more general ortholattice structure (see [6]).

In order to provide a first line of motivation for our approach we turn back to Savage's theory in a very simplified version. In Savage [13], the issue is about the valuation of "acts" with uncertain consequences or results. For simplicity we shall assume that the results can be evaluated (cardinally) in utils. Acts lead to results (measurable in utils), but the results are uncertain (they depend on a state of Nature). How can one formalize acts with uncertain outcomes?

The classical approach amounts to the following. There exists a set X of states of nature, which may in principle occur. For simplicity, we assume that the set X is finite. An act is a function f : X → R. If the state s ∈ X is realized, our agent receives a utility of f (s) utils. But before hand it is not possible to say which state s is going to be realized. To put it differently, the agent has to choose among acts before he learns about the state s . This is the heart of the problem.

Among possible acts there are "constant" acts, i.e. acts with a result that is known before hand, independently of the state of nature s. The constant act is described by a (real) number c ∈ R. It is therefore natural to link an arbitrary act f with its "certainty equivalent" CE(f ) ∈ R (such that our decision-maker is indifferent between the act f and the constant act which gives utility CE(f )). The first postulate of our simplified Savage model asserts the existence of the certainty equivalent:

• S1. There exists a certainty equivalent CE : R X → R and for the constant act 1 X we have CE(1 X ) = 1.

It is rather natural to require monotonicity of the mapping CE:

• S2. If f ≤ g then CE(f ) ≤ CE(g).
The main property we impose on CE is linearity:

• S3. CE(f + g) = CE(f ) + CE(g) for any f and g ∈ R X .

The requirement S3 looks like a very strong condition indeed. Savage himself and his followers preferred to appeal to the so-called "sure thing principle" so that the linearity is derived from some other axioms. But the related considerations are not relevant to the point we make in this paper.

In fact axiom S3 should be understood as a condition of additivity rather linearity. But together with monotonicity axiom S3 implies true linearity, that is CE(αf + βg) = αCE(f ) + βCE(g) for any α, β ∈ R. As a linear functional on the vector space R X , CE can be written in a form CE(f ) = P x f (x)µ(x). By axiom S2, µ ≥ 0; by CE(1 X ) = 1 we have P x µ(x) = 1. Therefore µ(x) can be interpreted as the " probability" [START_REF] Atmanspacher | Quantum Zeno features of bistable perception[END_REF] for the realization of state x. With such an interpretation, CE(f ) becomes the "expected" utility of the act f . Of course, we may assign probabilities not only to single states but also to any subset of states (or to any event) A ⊂ X. µ(A) can be understood either as the sum P x∈A µ(x), or as CE(1 A ) , where 1 A is the characteristic function of the subset A. The interpretation in the second approach is clear: the act 1 A is a bet on event A such that we receive 1 util if event A is realized and 0 util otherwise (if the opposite or complementary event A occurs). The decisionmaker can compare such bets on events and thereby compare events with respect to their likelihood. So this implies a notion of qualitative probability measure.

In this paper we propose to substitute the Boolean lattice of events with a more general ortholattice. The move in that direction was initiated long ago, in fact with the creation of Quantum Mechanics (QM). The Hilbert space entered into the theory immediately, beginning with von Neumann [14] who proposes to use a lattice of projectors in the Hilbert space as the suitable model for QM instead of the classical (Boolean) logic. In their seminal paper from 1936 [2], Birkhoff and von Neumann investigate the necessary properties of such a non-distributive logic. The necessity to use more general ortholattices than the Boolean one, arises as soon as the measurements (in this paper we understand acts as measurements) affect the measured system and change its state. [START_REF] Birkhoff | The logic of quantum mechanics[END_REF] This is particularly important when one does not limit attention to a single measurement, but is interested in a sequence of measurements or decision problems. If our measurements do not change the state of the object, one can use Savage's classical paradigm.

Recently a few decision-theoretical papers appeared (see for example, [11, 7, 9, 8]) in which the standard expected utility theory was transposed into Hilbert space model. Lehrer and Shmaya write "We adopt a similar approach and apply it to the quantum framework... While classical probability is defined over subsets (events) of a state space, quantum probability is defined over subspaces of a Hilbert space." Gyntelberg and Hansen (2004) apply a general event-lattice theory (with axioms that resemble those of von Neumann and Morgenstern) to a similar framework. One could expect that Gyntelberg and 1 Sometimes this probability is called subjective or personal, because it only expresses the likelihood that a specific decision-maker assigns to event x.

2 For a formal exposition of this argument see our article in Mathematical Social Sciences (2007).

Hansen truly would be working with general ortholattices. But no, they too work with subspaces of a Hilbert space. Our first aim is to show that there is no need for a Hilbert space, the Savage approach can just as well (and even easier) be developed within the frame of more general ortholattices. Beside making this formal argument, a motivation for this research is that a more general description of the world allows to explain some behavioral anomalies e.g., the Eldsberg paradox (see [7]). In an illustration we show that the results in this paper are relevant to modelling interaction in simple games when a decisionmaker faces a type indeterminate opponent i.e., an agent whose type changes under the impact of decision-making as proposed by Lambert-Mogilliansky, S. Zamir and H. Zwirn [10].

For the sake of comparison with the Savage setup, we develop the theory in a static context. But non-classical measurement theory was developed to deal with situations where the measurements impact on the states of the measured system . Therefore, a genuine theory of non-classical expected utility should apply to sequences of acts or measurements. In the Discussion we mention some dynamic implications.

Ortholattices

A lattice is an ordered set such that any of its subsets (including the empty subset) has a greatest lower bound (∨ or sup) and a lowest higher bound (∧ or inf), which guarantees the existence of a maximal element 1 and a minimal element 03 . An ortholattice is a lattice L equipped with an operation of orthocomplementation ⊥: L → L. This operation is assumed to be an involution (a ⊥⊥ = a), to reverse the order (a ≤ b if and only if b ⊥ ≤ a ⊥ ) and to satisfy the following property a ∨ a ⊥ = 1 (or, equivalently, a ∧ a ⊥ = 0). Example 1. Let X be a set and L = 2 X be the set of all subsets of X. The order is defined by set-theoretical inclusion. For A ⊂ X, A ⊥ = X -A, is the set-theoretical complement. It is the classical situation.

Example 2.Take some finite dimensional Hilbert space H (over the field of real or complex numbers). Let L be the lattice of vector subspaces of H and ⊥ be the usual orthogonal complementation. This example is standard in Quantum Mechanics as well as in the articles mentioned in the Introduction. But it was early understood that the lattice L(H) (sometimes it is called the lattice of projectors) is endowed with a number of special properties [START_REF] Busemeyer | Quantum Dynamics of Human Decision Making[END_REF] . We discuss a significantly more general case in the next example.

Example 3. Let (X, ⊥) be an orthospace that is a set X equipped with an irreflexive and symmetric binary relation (orthogonality) ⊥. For A ⊂ X A ⊥ = {x ∈ X, x ⊥ a for all a ∈ A}.

The sets of the form A ⊥ are called orthoclosed subsets or flats. When equipped with the relation ⊂ and the operation ⊥, flats form an ortholattice F(X, ⊥) (for details see [6]). Moreover almost any (at least any finite) ortholattice has the form of F(X, ⊥) for a suitable orthospace (X, ⊥).

In order to get better acquainted with this subject, let us consider a few concrete examples. 3a) Assume that all distinct points of X are pairwise orthogonal. Then any subset of X is orthoclosed and the ortho-complementation coincides with the usual set-theoretical complementation. That is we obtain the Boolean model of Example 1. 3b) Let X consist of four points r, l, r 0 , l 0 . The orthogonality relation is represented by the graph below, where we connect points with a plain line when they are NON-ORTHOGONAL (so that orthogonal points as "far from each other" are unconnected).

d d d d r l 0 r 0 l
The point r is orthoclosed since r = {r 0 , l} ⊥ ; similarly the point l is orthoclosed. There are two other (nontrivial) flats: the set {r 0 , l} = r ⊥ and {l 0 , r} = l ⊥ . The corresponding ortholattice is represented below

d d d d d d @ @ ¡ ¡ @ @ ¡ ¡ 0 1 l r r ⊥ l ⊥
3c) Let us consider the orthospace represented by the following graph

R L B F ¡ ¡ @ @ ¡ ¡ @ @ d d d d
The corresponding ortholattice is

d d d d d d ¡ ¡ ¡ @ @ @ H H H H H H @ @ @ ¡ ¡ ¡ © © © © © © 0 1 L R = L ⊥ F B = F ⊥ 3d)
On the left side below we depicted another orthospace and on the right side the corresponding ortholattice.

d d d d d F L M R B ¡ ¡ ¡ @ @ @ @ @ @ ¡ ¡ ¡ d d d d d d d d d d b b b b b b " " " " " " " " " " " " b b b b b b @ @ ¡ ¡ ¡ ¡ @ @ ¡ ¡ © © © © © @ @ H H H H H @ @ ¡ ¡ F B = F ⊥ L M R 0 1
We want to defend the thesis that an ortholattice is a natural structure for applying all the concepts that are used in the classical theory of decisionmaking under uncertainty. As in the Boolean model we may speak of the intersection (∧) and union (∨), as well as of the complementation (or as the negation, and understand it as ortho-complementation). All the usual relations between these operations are preserved with one exception: the law of distributivity is not satisfied in the general case. But how often is it used? In the proofs of some theorems and propositions, perhaps. But hardly in the formulation of the concepts.

A central point is that it is possible to speak about probabilities which can be considered as a quantified saturation of the ortholattice skeleton.

Non-classical probability

The theory of probability starts with the definition of a set X of elementary events. Thereafter it moves over to general events. In our language events (or properties) are elements of an ortholattice L. The next key concept is a "collection of mutually exclusive events". In the classical model this is simply a partition of the set X, that is a decomposition X = A 1 q ... q A n . In the general case the notion of a collection of mutually exclusive events should be replaced by the notion of an Orthogonal Decomposition of the Unit (ODU).

Definition 1 An Orthogonal Decomposition of the Unit in an ortholattice L is a (finite) family of α = (a(i), i ∈ I(α) ) of elements of L satisfying the following condition: for any i ∈ I(α)

a(i) ⊥ = _ j6 =i a(j).
The justification for this formulation is provided by that a(i) ⊥ a(j) for i 6 = j and ∨ i a(i) = 1. The proof is obvious.

For instance, the single-element family 1 is a (trivial) ODU. For any a ∈ L, the two-element family (a, a ⊥ ) is an ODU. We call this kind of family the question about property a.

Intuitively, the family α is to be understood as a measurement (or a source of information) with a set of possible outcomes I(α). If such a measurement yields an outcome i ∈ I(α), we conclude that our system is endowed with property a(i) (or that the event a(i) occurs). Assume that we can "prepare" our system in some state and repeatedly measure the system (each time prepared in that same state). The measurement outcomes can differ from one trial to another. Imagine that we performed n such measurements (for n relatively large) and that outcome i was obtained n i times. Then we can assign each outcome i a "frequency" p i = n i /n. In fact we have that p i ≥ 0 and

P p i = 1.

This leads us to

Definition 2 An evaluation on an ortholattice L is a mapping ν :

L → R. An evaluation ν is called 1) nonnegative if ν(a) ≥ 0 for any a ∈ L; 2) monotone if ν(a) ≤ ν(b) when a ≤ b; 3) normed if ν(1) = 1; 4) additive (or a measure) if ν(a ∨ b) = ν(a) + ν(b)
for orthogonal events a and b. We write a ⊕ b instead of a ∨ b to emphasize that a ⊥ b.

5) probabilistic (or a probability) if it is nonnegative and

P i ν(a(i)) = 1 for any ODU (a(i), i ∈ I).
We make a few simple remarks on the links between these concepts. From 4) or 5) it follows easily that ν(0) = 0; clearly then 2) =⇒ 1). It is also clear that 5) =⇒ 3), and 1), 3) and 4) together imply 5). In the classical (Boolean) case 5) implies 1) -4), but that is not true in the general case. Indeed, let us consider Example 3b, where (excluding the trivial events 1 and 0) we have four events r, l, r ⊥ , l ⊥ and where r ≤ l ⊥ and l ≤ r ⊥ . To assign a probability is equivalent to giving two numbers ν(r) and ν(l) both between 0 and 1 but otherwise arbitrary. Such a probability is monotone if

ν(r) + ν(l) ≤ 1 and is additive if ν(r) + ν(l) = 1.
There exists an important case when everything simplifies and approaches the classical case. It is the case of orthomodular lattices. So are called the lattices that satisfy the property of orthomodularity

(if a ≤ b then b = a ∨ (b ∧ a ⊥ )).
It is clear that any Boolean lattice is orthomodular and so are the lattices from Examples 2, 3c, and 3d. In contrast, the lattice from Example 3b is not orthomodular. We assert that for orthomodular lattices, property 5) implies 3) and 4).

Lemma 1 If L is orthomodular ortholattice, then any probability on L is additive and monotonic.

Proof. Let ν be a probability on L. We first establish additivity. Suppose

a ⊥ b and pose c = (a ⊕ b) ⊥ . Since (c, c ⊥ ) is an ODU, ν(c) + ν(c ⊥ ) = 1.
We assert that (a, b, c) is an ODU as well. To prove that we need to show that a ⊥ = b ⊕ c. Since a, b and c are pairwise orthogonal, b ⊕ c ≤ a ⊥ . By force of the property of orthomodularity we have that a

⊥ = (b ⊕ c) ⊕ (a ⊥ ∧ (b ⊕ c) ⊥ ). But a ⊥ ∧ (b ⊕ c) ⊥ = (a ∨ b ∨ c) ⊥ = (a ⊕ b) ⊥. ∧ c ⊥ = c ∧ c ⊥ = 0. Hence a ⊥ = b ⊕ c. Similarly b ⊥ = a ⊕ c. The equality c ⊥ = a ⊕ b is satisfied by definition. Thus, the triplet (a, b, c) is an ODU.
Therefore we have the equality ν(a)

+ ν(b) + ν(c) = 1. Hence ν(a ⊕ b) = ν(c ⊥ ) = 1 -ν(c) = ν(a) + ν(b)
, which yields the additivity of ν.

Monotonicity follows trivially from the formula b = a ⊕ (b ∧ a ⊥ ), the additivity and the nonnegativity of the number ν(b ∧ a ⊥ ). QED Thus, for the case of orthomodular lattices, a probability may also be defined as a nonnegative normed measure.

Qualitative Measures

As was already explained we model uncertainty by an ortholattice of properties or events. If we understand the elements of the lattice as events, we may talk of smaller or larger probability for the realization of these events. Further, we focus on the qualitative relation corresponding to the "more (or less) likely than" relation between events .

Definition 3 A qualitative measure on an ortholattice L is a binary relation (of "likelihood") ¹ on L satisfying the following two axioms: QM1. ¹ is complete and transitive. QM2. Let a ¹ b and a 0 ¹ b 0 . Then a ⊕ a 0 ¹ b ⊕ b 0 (recall that it means that a ⊥ a 0 and b ⊥ b 0 ). The last inequality is strict if at least one of the first inequalities is strict. [START_REF] Busemeyer | A Quantum Game theory Explanation of the Disjunction Effect[END_REF] A qualitative measure ¹ is generated by a (quantitative) measure µ when a ¹ b if and only if µ(a) ≤ µ(b). In this section we are interested by the question as to when a qualitative measure can be generated by a quantitative measure (or when there exists a probabilistic sophistication). For simplicity we shall assume that the ortholattice L is finite. But even in the classical context the answer is generally negative (Kraft, Pratt, Seidenberg, 1959). Therefore, in order to obtain a positive answer we have to impose some additional conditions which strengthen QM2. We shall here consider a condition generalizing the classical "cancellation condition". We prefer to call it "hyperacyclicity".

Definition 4 A binary relation on L is said to be hyperacyclic if the following condition holds:

Assume that we have a finite collection of pairs (a i , b i ) and that a i ¹ b i for all i and for some i the inequality is strict. Then P µ(a i ) 6 = P µ(b i ) for some measure µ on L.

It is obvious that hyperacyclicity implies acyclicity as well as . Clearly, if the qualitative relation ¹ is generated by a measure µ then it is hyperacyclic. The main result of this section (and the analog of Theorem 1 in [9]) asserts that for finite ortholattice the reverse is true.

Theorem 2 Let ¹ be a hyperacyclic qualitative measure on a finite ortholattice L. Then ¹ is generated by some measure on L.

A complete proof of Theorem 1 can be found in the Appendix. Here we confine ourselves with describing the logic of the proof: We first embed the ortholattice L into a vector space V and identify linear functionals on V with measures on L. With the qualitative measure ¹ we construct a subset P ⊂ V and show that 0 does not belong to the convex hull of P . The separability theorem then guarantees the existence of a linear functional on V (that is of a measure on L) which is strictly positive on P . It is easy to show that this measure generates the relation ¹.

Clearly, if the relation ¹ is monotonic (that is a ¹ b for a ≤ b), then any measure µ generating ¹ is also monotonic. If, in addition, 0 ≺ 1 then µ(1) > 0; dividing the measure µ by µ(1) we can assume that µ is a normalized measure. Thus, the measure µ is a monotonic probability.

Non-classical utility theory

First of all we need to formulate a suitable generalization of the Savagian concept of act. Roughly speaking an act is a bet on the result of some measurement.

Definition 5 An act is a pair (α, f ), where α = (a(i), i ∈ I(α)) is some ODU (or a measurement), and f :

I(α) → R is a function.
We call the measurement α the basis of our act. Intuitively, if an outcome i ∈ I(α) is realized as a result of measurement α, then our agent receives f (i) utils.

In such a way the set of acts with basis α can be identified with the set (vector space, indeed) F (α) = R I(α) . The set of all acts F is the disjoint union of F (α) taken over all ODU α.

We are concerned with the comparison of acts with respect to their attractiveness to our decision-maker. We start with a implicit formula for such a comparison. Assume that the agent knows (more precisely, he thinks he knows) the state of the system, that is he has in his mind a (subjective) probability measure µ on the ortholattice L. Then, for any act f on the basis α = (a(i), i ∈ I(α)), he can compute the following number (expected value of the act f )

CE µ (f ) = X i µ(a(i))f (i).
Using those numbers our agent can compare different acts. We now shall (following Savage) go the other way around. We begin with a relation ¹ representing preferences over the set of all acts F , thereafter we formulate axioms, impose conditions and arrive at the conclusion that the preferences are explained by some probability measure µ on L.

More precisely, instead of a preference relation ¹ on the set F of acts, we at once assume the existence of a certainty equivalent CE(f ) for every act f ∈ F . (Of course this does simplify our task. But this step is unrelated to the issue of classicality or non-classicality of the "world"; it is only an assertion about the existence of a utility on the set of acts. It would have been possible to obtain the existence of CE from yet other axioms. We chose this more direct and shorter way).

Given that we shall only impose three requirements on CE. The first two relate to acts defined on a fixed basis. Such acts are identified with elements of the vector space F (α) = R α .

Monotonicity axiom. The restriction of CE on each F (α) is a monotone functional.

Linearity axiom. For any measurement α the restriction of CE on F (α) is a linear functional.

The third axiom links acts between different but in some sense comparable basis. For this we need to be able to compare at least roughly two different measurements. Consider two ODU α = (a(i), i ∈ I(α)) and β = (b(j), j ∈ I(β)). We say the measurement α is finer than β if there exists a mapping ϕ : I(α) → I(β) such that a(i) ≤ b(ϕ(i)) for any i ∈ I(α). Simply stated it means that as we know a result i of the first measurement, we know the result of the second measurement without performing it, it is j = ϕ(i). We note also that the transformation mapping ϕ is uniquely defined. In fact assume that ϕ(i) simultaneously belongs to b(j) and b(k). Then a(i) belongs to b(j) ∧ b(k). But since b(j) and b(k) are orthogonal b(j) ∧ b(k) = 0, so a(i) = 0. But this type of events does only formally enter in the decomposition of the unit and it can be neglected.

In any case, any such mapping ϕ : I(α) → I(β) defines a mapping

ϕ * : F (β) → F (α).
For a function g on I(β) the function ϕ * (g) in a point i has the value g(ϕ(i)).

Intuitively, the payoffs from both functions (acts) g and f = ϕ * (g) are identical in all situations. Therefore our agent should consider them as equivalent and assign them the same certainty equivalent. This is the idea of the following axiom.

Agreement axiom. Suppose that a measurement α is finer than β and ϕ : I(α) → I(β) is the corresponding mapping. Then CE(g) = CE(ϕ * (g)) for each g ∈ F (β).

Take for instance f to be the constant function in I(α) with value 1. The agreement axiom says that the agent is indifferent between two acts. The first is to receive one util without performing any measurement. The second is to perform the measurement α and (independently of the outcome) to receive a unit of utils.

The last requirement which cannot really be called an axiom says that the utility of the trivial act with payoff 1 is equal to 1. That is CE(1) = 1.

Theorem 3 Suppose that a certainty equivalent CE satisfies the monotonicity, linearity and agreement axioms. Then there exists a probabilistic valuation µ on L such that CE(f ) = P i µ(a(i))f (i) for any act f on the basis of measurement α = (a(i), i ∈ I(α)). Moreover this valuation µ is uniquely defined.

Proof. For a ∈ L we denote 1 a the bet on property a. It gives 1 util if we receive the answer YES on the question (a, a ⊥ ) and 0 for NO. Let µ(a) = CE(1 a ). Since 1 a ≥ 0 we have µ(a) ≥ 0 for any a ∈ L.

Let now α = (a(i), i ∈ I(α)) be an arbitrary ODU, and f : I(α) → R be an act on the basis α. We denote with the symbol 1 i the act on the basis of α which yields 1 on i and 0 on F (α) -{i}. By the agreement axiom we have that CE(

1 i ) = µ(a(i)). Since f = P i f (i)1 i we conclude that CE(f ) = X i µ(a(i))f (i)
In particular, if f = 1 we obtain that 1 = CE(1) = P i µ(a(i)). Therefore µ is a probabilistic valuation. QED We do not assert that the valuation µ is monotone. In the next section we substitute the agreement axiom with a stronger "dominance" axiom and we obtain the monotonicity of µ.

The Dominance axiom

Let α = (a(i), i ∈ I(α)) be a measurement (or an ODU). And let b ∈ L be an event (or a property). We say that an outcome i ∈ I(α) is impossible under condition b (or in presence of the property b), if a(i) ⊥ b. All other outcomes are in principle possible, and we denote the set of possible outcomes as I(α|b).

Clearly b ≤ _ i∈I(a|b) a(i) = a(I(α|b)),
and I(α|b) is the smallest subset of I(α) with that property. In fact if b ≤ a(J) then a(J) ⊥ ≤ b ⊥ . But a(J) ⊥ = a(I(α) -J), therefore for any i, not belonging to J, we have

a(i) ≤ b ⊥ , that is a(i) ⊥ b.
Consider for instance a situation when we have two measurements α = (a(i), i ∈ I(α)) and β = (b(j), j ∈ I(β)). Suppose that the measurement α is finer than β and ϕ : I(α) → I(β) is the corresponding mapping. Since b(j) = a(ϕ -1 (j)), it is easily seen that I(α|b(j)) = ϕ -1 (j) and I(β|a(i)) = {ϕ(i)}.

We go back to acts. Let f : I(α) → R and g : I(β) → R be acts on the α and β basis respectively. We say the g dominates f (and write f ≤ g) if for any i ∈ I(α) and any j ∈ I(β|a(i)) (that is j is possible at the event a(i)) the inequality f (i) ≤ g(j) is true. Intuitively, this means that the act g always gives no less than the act f . With such an interpretation it is natural to assume that our rational decision-maker must assign to g no less utility than to f. We formulate this as

Axiom of dominance. If f ≤ g then CE(f ) ≤ CE(g).
It is clear that the dominance implies monotonicity. We assert that the dominance axiom also implies the axiom of agreement. In fact let β be a measurement coarser than α and let f = ϕ * (g) for some act g on the β basis.

From the description above it is clear that f ≤ g and g ≤ f such that CE(f ) = CE(g).

Theorem 4 Assume that the axiom of linearity and dominance are satisfied. Then CE is an expected utility for some monotonic probability measure µ on L.

Proof. The first statement follows from earlier remarks and theorems. Therefore we should prove the monotonicity of the measure µ. Let a ≤ b. Consider two measurement-questions α = (a, a ⊥ ) and β = (b, b ⊥ ). Let f = 1 a, that is f is a bet on the event (property) a : the agent receives one util if measurement α reveals (actualizes) property a, and receives nothing in the opposite case. We define 1 b similarly on the β basis. Clearly 1 a ≤ 1 b . In fact if the first measurement reveals (actualizes) property a, then b is true for sure since a ≤ b. Therefore 1 b gives the agent one utils when a occurs, and ≥ 0 utils when a ⊥ occurs, which is not worth less than 1 a . By force of the axiom of dominance CE(α) ≤ CE(β). The first term is equal to µ(a) and the second to µ(b). QED

Illustration: Non-classical type uncertainty

In this section we want to address the question as to whether the general approach developed in this paper can be relevant in decision theory. Isn't it possible to find a suitable classical representation? Indeed, consider the case when the states of Nature relevant to an act is whether the egg you are about to add to the omelet is rotten or not (Savage's own example). There, it is clear that the ortholattice representing Nature is Boolean. One may really wonder if real life ever offers examples of decision situations where uncertainty needs to be modelled with a non-Boolean ortholattice.

Non-classical versus classical representation

In this section we propose a method for constructing a classical model of a system starting from a description in terms of the general framework developped in this paper. We recall that an act corresponds to a measurement α and a payoff function f : I (α) → R, defined on the set of outcomes of the measurement. In order to evaluate an act, the decision-maker must guess the state of Nature. When we describe Nature with an the ortholattice L, a state is a probability measure on L. Formally, everything is the analogue of Savage. The only difference is that Savage assumes that the ortholattice is Boolean. Therefore, the question regarding the comparison with a classical model boils down to the question as to whether one can represent Nature i.e., our ortholattice L with an equivalent or suitable Boolean ortholattice L 0 . And whether for each measurement α in L we can define a corresponding ODU α 0 = (a 0 (i), i ∈ I(α)) in the new ortholattice L 0 . Note that the set of outcomes of measurements α and α 0 is the same.

A most natural way of doing this is to look for an homomorphism ϕ : L → L 0 of ortholattices which commutes with operations ∨, ∧ and ⊥ . And since we are interested in a classical explanation we have to choose L 0 as a Boolean ortholattice, we call it B, and a suitable homomorphism ϕ so we can represent the measure σ with some measure σ 0 on B.

Let our system (here Nature) be represented by a (finite) orthospace (X, ⊥) . For simplicity we shall assume that our orthospace is orthoseparable. [START_REF] Danilov | Measurable Systems and Behavioral Sciences[END_REF] We call orthobasis any subset B ⊂ X endowed with the following two properties:

1. Distinct elements of B are orthogonal to each other; 2. B is maximal with property 1 , i.e., for all x ∈ X, x / ∈ B, you can find b ∈ B such that x and b are non-orthogonal.

We understand an orthobasis as a complete measurement; B is the set of outcomes of measurement B.

Definition 6

We call a state (or a probability measure) in X the correspondance σ : X → R + such that for any orthobasis the following equality holds:

σ (B) := P b∈B σ (b) = 1.
With this definition, the numbers σ (b) can be viewed as probabilities for the outcomes of measurement B in state σ.

A state is called dispersion-free or classical if the outcome of any measurement is deterministic, that is if σ (x) ∈ {0, 1} . In other words a dispersion-free state can be identified with a subset S ⊂ X such that its intersection with any orthobasis is exactly one element. We may call such a subset an antibasis because it only contains elements that are non-orthogonal to each other. We denote the set of all antibases A (X, ⊥) .

We view the set A (X, ⊥) as a natural set of classical states of our system. Indeed, we see that, by construction, every such states gives a deterministic response to any measurement. In such a way we have defined a natural Boolean counterpart of our system. The set A (X, ⊥) is not the only possible classical model of the system (X, ⊥) however. In particular, one may construct complex models with hidden variables. [START_REF] Gyntelberg | Expected utility theory with "small worlds[END_REF] Our view is that for the limited purpose of this section (and because there is a lack of concensus on the meaning of hidden variables), we can confine attention to simple classical models constructed whithout additional structure as described above.

Playing with a Non-classical opponent

We shall consider a situation where our decision-maker faces uncertainty about the type (or preferences) of the agent that he is interacting with. That is what we earlier called "Nature" is another decision-maker. The idea that agents (represented by their preferences and beliefs) may be viewed as non-classical systems was first proposed in Lambert-Mogiliansky, S. Zamir and H. Zwirn (2003) and further developped in e.g. Busemeyer et al (2006a, 2006b) and Danilov and Lambert-Mogiliansky (2007). The motivation for this approach is that a variety of empirical phenomena, so-called behavioral anomalies, can be explained when representing uncertainty about the type (preferences) of a decision-maker with a non-boolean ortholattice. In that context the term type is equivalent to the term "state" when talking about arbitrary systems. A decision situation or DS is an ODU that measures a type characteristics. [START_REF] Mura | Decision Theory in the Presence of Risk and Uncertainty[END_REF] We next formulate a simple game situation in terms of the general theory exposed above. We immediately wish to emphasize that this is only an exploratory first step of games with indeterminate players. In the next section we illustrate the distinction between classical and non-classical models in a few examples.

The closest formal equivalence with the non-classical Savage model obtains when dealing with a sequential move game where our decision-maker (we call him player 1) moves first and the opponent (player 2) moves second which ends the game. We consider the problem from the point of view of player 1, a classical decision-maker who knows his type but is uncertain about the type of his opponent.

The opponent is represented by his type i.e., an element θ ∈ (X, ⊥) . Or equivalently a type is described as a probability measure µ over L the corresponding lattice of type characteristics. The type t or the measure µ captures player 1's (subjective) beliefs about player 2's type. An act is a pair (α, f ) , where α = (a (i) , i ∈ I (α)) is some ODU (measurement), and f : I (α) → R is a payoff function. The interpretation is that α is a move that implies a bet on the uncertain behavior of the opponent and a payoff function depending on the behavior of the opponent. Only the move by player 1 qualifies as an act. It induces a decision node for the player 2 which is understood as measurement α of player 2 0 s type. The parallel with the general model is straightforward. When player 2 chooses i, i.e., when the type a (i) ∈ I (α) is actualized, player 1 receives a payoff of f (i) (cardinal) utils. [START_REF] Lehrer | A Subjective Approach to Quantum probability[END_REF] Assume that Player 1 knows (or thinks that he knows) the type t ∈ X of player 2 is or equivalently player 1 knows the probability distribution µ on L. That is a probability distribution over decisions in various nodes α, β ∈ F . Then, for f : I (α) → R on the basis α = (a (i) , i ∈ I (α)) and g : I (β) → R on basis β = (b (i) , i ∈ I (β)), player 1 can compute and compare the following numbers (expected values)10

CE (f ; µ) = X i µ t (a (i)) f (i) and CE (g; µ) = X i µ t (b (i)) g (i) .
It follows from Theorem 4 that if the axioms of linearity and dominance are satisfied, we can define a CE that is an expected utility for any initial type t of player 2. Our rational decision-maker (player 1) selects among the possible acts the one associated with the largest expected payoff. So we see that the probability measures on the ortholattice of type characteristics play a role similar to beliefs about the (Harsanyi) type of the opponent in the analysis of a classical player 1's decision-making.

Examples

Our first example involves two games the Prisoner's Dilemma (PD) [START_REF] Pitowsky | Betting on the outcomes of measurements[END_REF] and the Ultimatum Game(UG) [START_REF] Rockafeller | Convex Analysis[END_REF] .

Example 1 Player 1 is confronted with a choice between playing the PD or UG with player 2. From his point of view the PD is an act: (α, f ) with I (α) = {C, D} and f is player 1's payoff. [START_REF] Savage | The Foundations of Statistics[END_REF] Similarly, the UG is an act: (β, g) with I (β) = {G, E} corresponding to two options for sharing the pie: one generous(G) and the other egoist(E). Measurements α and β are incompatible ODU so the corresponding set of pure types is X = {C, D, G, E} .

Any type t of player 2 is described by four non-negative numbers corresponding to player 1's subjective probability for the realization of the different type characteristics in each of the two acts:

prob

(C| t) = µ, prob (D| t ) = (1 -µ) , prob (E| t) = λ, prob (G| t) = (1 -λ) .
Following the method exposed in section 7, we build a classical model of player 2. It is given by the set of antibases (or classical pure states)A (X, ⊥) = {CG, CE, DG, DE}. It is easy to see that the agent of type t can be represented as a mixture of the four classical states with prob (CG| t) = λµ and similarly for the probabilities of the other states. Such a classical model gives exactly the same predictions and makes the same recommendation as the non-classical one. So in this example there is no reason to embrasse the more general approach.

Example 2

The model of example 1 is enriched as follows. First, the PD and the UG are augmented with one common option, that of "not playing" N. Second, we add a third game, the Trust Game(TG) [START_REF] Von Neumann | Mathematische Grunlagen der Quantummechanik[END_REF] which is constructed so as to share an option in common with the two first games. More precisely, we consider the act (δ, h ) corresponding to second move in a TG with three possible levels of effort: low L, high H and medium M. The orthospace is given in fig. 1. [START_REF] Wright | The state of the Pentagon[END_REF] The representation entails that the L-type in TG is the same as the D-type in the PD and H = G. Our orthospace has six pure states: C, D, M, G, E, N. The classical counter part is given by the set of antibases A (X, ⊥) = {CEM, CG, NM, DE} .

Consider now type t 0 of player 2, defined by following probabilities: prob (N| t 0 ) = .5, prob (G| t 0 ) = .5, prob (D| t 0 ) = .5, implying that all other types have probability zero.

We now show that we cannot reproduce the predictions of the non-classical model by assigning probabilities to the four classical types. Indeed, since C has probability zero both CEM, CG must be assigned probability zero. The same holds for NM and DE because M and E also have probability zero. So none of the classical types can be assigned positive probability! So here we see that there is no way to represent state t 0 in the classical model. 16 equal probability. If our decision-maker can only do a single measurement, the two models calls for selecting the P D.

Suppose now that our decision-maker can make a series of 3 measurements. In both models the optimal strategy includes selecting the P D as a first measurement and, if the outcome is C, to repeat the choice of the P D two more times. If the outcome is D then again both models call for selecting UG as a second measurement. But from here the recommendations of the two models differ. Whatever the outcome of UG, it is optimal in the classical model to select UG as the third act. This is because if he is lucky he may earn 20 euros and if unlucky i.e., the result of UG is E, he believes that the type of the opponent is DE and so he has nothing to hope for anymore. In contrast, the non-classical model calls for selecting PD as the last act whatever the outcome of UG. This is because the agent knows that whether the type is G or E there is (again) 1/2 probability for obtaining outcome C in the PD in which case he earns 100. On average after an initial draw of D, the agent with the nonclassical model in mind earns 55 while the agent with the classical model in mind earns 10. So in a dynamic context, and even when two acts have no outcome in common, the recommendations of the classical and the non-classical models differ and that they are associated with different expected payoffs.

Discussion

In this paper we show that Savage's theory of decision-making under uncertainty can be formulated in terms of a very general algebraic structure called an ortholattice instead of the more restrictive Boolean algebra. Our results shed new light on the generality of the Savage's approach. They also extends it so as to allow considering decision situations where the payoff relevant uncertainty pertains to non-classical objects. In this section we want to discuss some limitations of our approach.

Savage argument is formulated in a static context. In a static context a classical state space can represent uncertainty even when measurements are incompatible provided they have disjoint outcome sets. This is what we illustrated with our first example in section 7. For the case the outcome sets intersect as example 2 of section 7, one can construct states that cannot be represented in a classical state space. These states are quite peculiar however. Indeed, returning to our t 0 state in the triangle example, it is easy to see that such a state cannot be obtained as the result of any of the three measurements. So if the system started from that state, as we performed measurements on it, the system never returns to it. It is an "ephemeris" state. This seems to be a rather general characteristics of non-classical states that cannot be expressed in a classical state space. It is therefore legitimate to question the practical value of the proposed generalization of Savage. Our response is that the non-classical representation of uncertainty becomes truly valuable when we consider a dynamic situations i.e., a situation when a series of decisions under uncertainty is to be made. In a classical world, the state pre-exists the measurement, it is only revealed by it. As the decision-maker proceeds in the series of decisions, properties of the world (type characteristics of the agent) become known to him. The decision-maker, with a classical representation in mind, makes his next decisions on the basis of updated beliefs according to the Bayes' rule. But if the system is non-classical performing measurements on it alters its state. Bayes' rule which assumes that the state remains unchanged is not longer appropriate. In example 3 of section 7, we show that in the simplest case the classical and the non-classical representation of uncertainty yield distinct recommendations for decision-making.

Here we prove Theorem 1.

1. Construction of the vector space V . Denote R⊗L the vector space generated by L. It consists of (finite) formal expressions of the form P i r i a i , where r i ∈ R and a i ∈ L. Denote K the vector subspace in R⊗L generated by expressions a ⊕ bab (recall that a ⊕ b means that a ⊕ b = a ∨ b and a ⊥ b.) Finally, V = V (L) is the quotient space R⊗L by the subspace K, V = (R⊗L)/K.

The ortholattice L naturally maps into V ; the image 1 • a of an element a ∈ L we denote simply as a. Any linear functional l on V restricted to L gives a valuation on L. Since l(a⊕b-a-b) = l(a⊕b)-l(a)-l(b) = 0, the valuation l is additive, that is a measure on the ortholattice L. Conversely, let l be a measure on L. We extend it by linearity to R⊗L assuming l( P r i a i ) = P r i l(a i ). By force of additivity, l yields 0 for elements of the form a ⊕ bab, that is l vanishes on the subspace K. Therefore l factors through V and is obtained from a linear functional defined on V . We just proved Proposition 1. The vector space of measures on L is identified with the space V * of linear functional on V .

Remark. The canonical mapping L → V (L) can be considered as the universal measure on the ortholattice L. It is injective if and only if the ortholattice L is orthomodular.

2. Construction of the set of "strictly positive" P . Let ¹ be a binary relation on L; as usual, ≺ denote the strict part of ¹. By definition, P = P (¹) consists of (finite) expressions of the form P i (a ib i ), where b i ¹ a i for all i and b i ≺ a i for some i. (P is empty if the relation ≺ is empty, that is if all elements in L are equivalent relatively to ¹.) We note also that P is stable with respect to the addition.

3. Suppose now that a relation ¹ is hyperacyclic. Note that the hyperacyclicity of ¹ means precisely that 0 does not belongs to P . Proposition 2. If the relation ¹ is hyperacyclic then 0 does not belong to the convex hull of P .

Proof. Assume that 0 is a convex combination of elements of P , 0 = P i r i p i , where p i ∈ P , r i ≥ 0, and P i r i = 1. By Caratheodory's theorem we can assume that the p i are affinely independent (and therefore the coefficients r i are uniquely defined). We assert that in this case the coefficients are rational numbers.

It would be simplest to say that the set P is defined over the field of rational numbers. But it is not so easy to provide a precise meaning to it. For that purpose we choose and fix some subset L ⊂ L, such that its image in V is a basis of that vector space. We also choose a subset M of expressions of the form a ⊕ bab, which constitute a basis of the subspace K. The union of L and M is a basis of the vector space R⊗L. On the other side, L is a basis of R⊗L as well. Since elements of L ∪ M are rational combinations of elements of the L,basis elements of L, in turn, can be rationally expressed in terms of L ∪ M. In particular, the images of elements of L in V are rational combinations of elements of the L basis. All the more, the elements p i ∈ P can be rationally expressed in terms of L. It follows (see, for example, Proposition 6 in [3], Chap. 2, § 6) that 0 can be expressed rationally through p i . Since the coefficients r i are defined uniquely, they are rational numbers. Now the proof can be easily completed. We have an equality 0 = P i r i p i , where p i ∈ P and r i are rational numbers (not all equal to zero). Multiplying with a suitable integer we may consider r i themselves as integers. Since P is stable with respect to addition, we obtain that 0 ∈ P , in contradiction with hyperacyclicity of the relation ¹.

4. Together with Separation theorem of convex sets (see [12]) the results above imply existence of a (non-trivial) linear functional µ on V , non-negative on P . But we need strict positivity on P . To obtain it we show that (in the case of a finite ortholattice L) the convex hull of P is a polyhedron.

Let us introduce some notations. where a i , b i , c j , d j ∈ L (more precisely, belong to their image in V ), a i  b i for any i, c j º d j for any j, α i , β i are nonnegative, and P i α i = 1. Proposition 3. The convex hull of P coincides with Q.

Proof. It is clear from the definitions that any element of P belongs to Q. By the convexity of Q, the convex hull of P is also contained in Q.

It remains to show the converse, that any element q of Q belongs to the convex hull of P . For that (appealing to the convexity of co(P )) we can assume that q has the form in ( * ) with n and m equal to 1, that is

q = (a -b) + β(c -d),
where a  b, c º d and β ≥ 0. If β is an integer, it is clear that q ∈ P. In general case β is a convex combination of two nonnegative integers β 1 and β 2 ; then q is the corresponding convex combination of two points (a-b)+β 1 (c-d) and (ab) + β 2 (cd) both belonging to P .

Corollary. Assume that an ortholattice L is finite. Then the convex hull of P is a polyhedron.

In fact, in this case the sets A and B are finite. Therefore (see [12], theorem 19.1) Q is a polyhedra.

Thus, if 0 does not belong to the convex hull of P (see Proposition 2) then there exists a linear functional µ on V which is strictly positive on P . As we shall see, this immediately provides us with a proof of Theorem 1.

5.

Proof of Theorem 1. The assertion in the theorem is trivially true if all elements of L are equivalent to each other. Therefore we can assume that there exists at least one pair (a, b) such that a  b. Let µ be a linear functional on V (we may consider µ as a measure on the ortholattice L) strictly positive on P . We assert that this measure generates the relation ¹ .

  A denotes the set of expression ab, where a  b. B denotes the set of rays of the form R + (ab), where a º b. Finally, Q is the convex hull of A∪B in V . By definition, Q consists of elements of the formq = α 1 (a 1b 1 ) + ... + α n (a nb n ) + β 1 (c 1d 1 ) + ... + β m (c md m ), ( * )

  Let us suppose c º d. Since for any integer positive number n the element (ab) + n(cd) belongs to P , we have µ(a)µ(b) > n (µ(d)µ(c)) for any n. This implies µ(d) ≤ µ(c). Conversely, let us suppose µ(c) ≥ µ(d) for some c, d ∈ L. We have to show that c º d. If this is not the case then, by completeness of the relation º, we have d  c. But then dc belongs to P and µ(dc) = µ(d)µ(c) > 0, which contradicts to our first assumption. This completes the proof of Theorem 1.

It is more natural to call what we just defined, a complete ortholattice. Usually one only requires the existence of finite bounds. However we shall not interest us much for the general case, assuming finiteness of L.

Maybe we should give examples here?

The special case of QM2 when a 0 = b 0 is referred to in[START_REF] Lehrer | A Subjective Approach to Quantum probability[END_REF] as De Finetti axiom.

Orthoseparability means that all singletons sets are flats.

It is well-known that it is possible with unobservable hidden variables to construct a model that replicates the quantum predictions in most cases including what concerns the position and momentum of a particle.

The classical approach to uncertainty in interactive decision situations i.e., in "games" is due to Harsanyi's. All uncertainty about physical outcome functions, utility functions and strategy spaces is captured in uncertainty about the type of the players. Uncertainty about the type is represented by a Boolean ortholattice of type characteristics. We next consider an example of a classical player interacting with a "type indeterminate" player that is a player represented by a general ortholattice of type characteristics.

The type a (i) is defined as the type with property that the choice of i maximizes his utility.

As in the general model we assume the existence of a certainty equivalent.

The Prisoner's Dilemma is an achetypical game of two players where each player chooses between cooperation and defection. We consider a simultaneous move version of this game.

The ultimatum game is a game of two players concerned with the division of a pie. The first player (our player 2) moves to propose a division of the pie. The second player (our player 1) either accepts in which case the pie is allocated according to the proposal. Or he refuses in which case both players get zero payoff.

We do not explicite player 1's action in the games. Player 1's payoffs should be understood as the best reply payoffs. That is the payoffs associated with the best reply to player 2's expected play.

The trust or investment game is a two-players game where player 1 invests a fixed amount which pays back depending on player 2's effort.

An orthospace is a graph that connects all orthogonal elements.

It is true that this result relies on an (implicit) assumption there does not exist a

The financial support of the grant #NSh-6417.2006.6, School Support, is gratefully acknowledged.

In order to better understand the implications of the distinction between the two models consider the following payoffs (for player 1): in the PD: f (D) = f (N)) = 1 and f (C) = 10; in the UG: g (N) = g (G) = 2 and g (E) = 0; and in the TG, h (L) = 0 and h (M) = h (H) = 3. When the player has the classical representation in mind we can express the expected utility of any act as follows

where the vector of probability (p 1 , ..., p 4 ) represent (the belief of player 1 about) player 2's type. We compute the expected utility of the 3 acts:

) . It appears clearly that for any type of player 2, UG is dominated by T G which in turn dominates P D. So UG is never recommended. Consider now, a decisionmaker who has the correct non-classical model in mind and knows that his opponent is of type t 0 with the probabilities given above. We compute the expected utility of each act given type t 0 :

EU ncl (P D; t 0 ) = 1, EU ncl (UG; t 0 ) = 2 and EU ncl (T G; t 0 ) = 1.5. We see that UG dominates both T G and P D so the recommendation is play UG which was a dominated option in the classical model.

Example 3

We conclude this section with an example that involves a series of decisions. We return to the first example with the two acts: PD with outcome (C,D) and UG with outcomes (E,G). The set of possible types of the opponent is X = {C, D, G, E} . We know that C ⊥ D and G ⊥ E. As for the other correlations we assume the following: If the agent is of type D (or C) then UG will give outcome E (or G) with probability 1/2. And when player 2 is of type E (or G), the PD results in type C or D with equal probability. As we already saw this system has a classical counter-part with types CG, CE, DG, DE.

Consider the case when the outcome C yields 100 euros i.e., f (C) = 100, f (D) = g (E) = 0 and g (G) = 10. Suppose all types have initially measurement with outcome set {D, N, G} . In the opposite case the ortholattice would boil down to a Boolean algebra to which we associate a classical probability space. But we could had taken any similar example with 5 or more (an impair number) elements common to two games. The orthospace with 5 such elements depicts a pentagone with each segment being the basis of a triangle. The same impossibility result obtains with no need of an assumption on the non-existence of a measurement. We chose the three acts case for the ease of presentation.


Jean-Pascal Bénassy
email: benassy@pse.ens.fr

Interest rate rules and global determinacy: An alternative to the Taylor principle

Keywords: E43, E52, E58, E62, E63 Taylor principle, global determinacy, interest rate rules, Taylor rules, fiscal theory of the price level

A most wellknown determinacy condition on interest rate rules is the "Taylor principle", which says that nominal interest rates should respond more than hundred percent to inflation. Unfortunately, notably because interest rates must be positive, the Taylor principle cannot be satisfied for all interest rates, and as a consequence global determinacy may not prevail even though there exists a locally determinate equilibrium.

We propose here a simple alternative to the Taylor principle, which takes the form of a new condition on interest rate rules that ensures global determinacy. An important feature of the policy package is that it does not rely at all on any of the fiscal policies associated with the "fiscal theory of the price level", which was so far the main alternative for determinacy.

Introduction

In the recent years an important concern of macroeconomists has been to find out which combinations of monetary and fiscal policies lead to determinacy of the dynamic equilibrium. Notably the early seminal work of Leeper (1991) shows how either monetary or fiscal policies can lead to local determinacy.

On the monetary side the focus has been recently on interest rate rules. A central result is the wellknown "Taylor principle", which says that the nominal interest rate should respond to inflation with an elasticity greater than one [START_REF] Barro | Are government bonds net wealth?[END_REF] (Taylor, 1993, 1998).

Unfortunately, as is wellknown, the Taylor principle is a condition for local, not global determinacy. Notably if one takes into account the fact that nominal interest rates must be positive, then the Taylor principle cannot be satisfied for all values of the interest rates, and as a consequence global determinacy may not prevail even though there exists a locally determinate equilibrium (Benhabib, Schmitt-Grohe and Uribe, 2001).

If the Taylor principle cannot deliver global determinacy, there is another wellknown determinacy condition (Sims, 1994, Woodford, 1994, 1995), which has been known under the name of "fiscal theory of the price level" (FTPL). The basic idea behind the FTPL is that the government pursues fiscal policies such that, in off-equilibrium paths, it will not satisfy its intertemporal budget constraint, and run an explosive debt policy. This leaves only one feasible equilibrium path [START_REF] Bénassy | Interest rate rules, price determinacy and the value of money in a non Ricardian world[END_REF] . Now although such off-equilibrium paths are not observed in the model's equilibrium, it would be extremely optimistic to assume that in real life situations the economy would follow at every instant the equilibrium path while the government pursues such policies. As a result many people would be reluctant to advise such policies to a real life government.

In view of the above qualifications the purpose of this article is to propose a simple condition on interest rate rules that will ensure global determinacy, even when no FTPL-type policies are implemented.

A key to the results is that we will make the realistic assumption that the economies we work in are non-Ricardian. By non-Ricardian economies we mean economies where, as in OLG models, new agents enter in time, so that in particular Ricardian equivalence fails (Barro, 1974). We shall see that in such a framework global determinacy can be obtained under reasonable fiscal and monetary policies.

The model

We want to have a non Ricardian monetary model that "nests" the traditional infinitely lived agent model, so we shall use a monetary model due to Weil (1987, 1991), and assume that new "generations" of households are born each period, but nobody dies. Denote as N t the number of households alive at time t. So N t -N t-1 ≥ 0 households are born in period t. We will mainly work below with a constant rate of growth of the population n ≥ 0, so that N t = (1 + n) t . The Ricardian case is obtained by taking the limit case n = 0.

Consider a household born in period j. We denote by c jt , y jt and m jt its consumption, endowment and money holdings at time t ≥ j. This household maximizes the following utility function:

U jt = ∞ X s=t β s-t Log c js (1)
and is submitted in period t to a "cash in advance" constraint:

P t c jt ≤ m jt (2)
Household j begins period t with a financial wealth ω jt . First the bond market opens, and the household lends an amount b jt at the nominal interest rate i t . The rest is kept under the form of money m jt , so that:

ω jt = m jt + b jt (3)
Then the goods market opens, and the household sells his endowment y jt , pays taxes τ jt in real terms and consumes c jt , subject to the cash constraint (2). Consequently, the budget constraint for household j is:

ω jt+1 = (1 + i t ) ω jt -i t m jt + P t y jt -P t τ jt -P t c jt (4)
Aggregate quantities are obtained by summing the various individual variables. There are N j -N j-1 agents in generation j, so for example aggregate assets Ω t and taxes T t are equal to:

Ω t = X j≤t (N j -N j-1 ) ω jt T t = X j≤t (N j -N j-1 ) τ jt (5)
Similar formulas apply to output Y t , consumption C t , money M t and bonds B t . We now must describe how endowments and taxes are distributed among households. We assume that all households have the same income and taxes, so that:

y jt = y t = Y t /N t τ jt = τ t = T t /N t (6)
and that real income per head grows at the rate ζ:

y t+1 /y t = ζ Y t+1 /Y t = (1 + n) ζ (7)
Let us now consider government. Households' aggregate financial wealth Ω t has as a counterpart an identical amount Ω t of financial liabilities of the government. These are decomposed into money and bonds:

Ω t = M t + B t (8)
The evolution of these liabilities is described by the government's budget constraint:

Ω t+1 = (1 + i t ) Ω t -i t M t -P t T t (9)
Note that, to simplify the exposition and to concentrate on tax policy, we assume that governement spending is zero. If not, the results would be essentially the same, but the formulas would be more clumsy.

Monetary policy

We shall consider "Taylor rules" where the nominal interest rate responds to inflation:

1 + i t = Φ (Π t ) Φ (Π t ) ≥ 1 (10)
with Π t = P t /P t-1 . We shall also define the elasticity of this function:

φ (Π t ) = ∂LogΦ (Π t ) ∂LogΠ t = Π t Φ 0 (Π t ) Φ (Π t ) (11)
One says that the Taylor principle is satisfied at

Π t if φ (Π t ) > 1.

Fiscal policy

If the budget was balanced, taxes would be equal to interest payments on bonds i t B t , so that one would have:

P t T t = i t B t (12)
We shall actually assume that the government can run a deficit or a surplus, so taxes have the form:

P t T t = i t B t -D (Ω t , P t Y t ) (13)
where the function D (Ω t , P t Y t ), which represents the fiscal deficit in nominal terms, is homogeneous of degree 1 in its two arguments. The argument P t Y t represents some type of income tax, while the argument Ω t reflects the fact that the government may want to raise taxes in order to diminish its financial liabilities Ω t .

Putting together equations ( 8), ( 9) and ( 13), we find:

Ω t+1 = Ω t + D (Ω t , P t Y t ) (14)
3 Dynamics

Equation ( 14) is our first dynamic equation. Turning now to nominal income P t Y t , it is shown in the appendix that, assuming N t+1 /N t = 1 + n, the dynamics of nominal income P t Y t is given by:

P t+1 Y t+1 = β (1 + n) (1 + i t ) P t Y t -(1 -β) nΩ t+1 (15)
Combining with equation ( 10) we obtain:

P t+1 Y t+1 = β (1 + n) Φ (Π t ) P t Y t -(1 -β) nΩ t+1 (16)
We shall actually not use the two dynamic equations ( 14) and ( 16) as such, but rather use as working variables inflation Π t and the predetermined variable X t defined as3 :

X t = Ω t P t-1 Y t-1 (17)
Then dividing both sides of equation ( 16) by ζ (1 + n) P t Y t it becomes:

Π t+1 = Φ (Π t ) ξ -νX t+1 (18)
where ξ = ζ/β is the autarkic interest rate [START_REF] Blanchard | The solution of linear difference models under rational expectations[END_REF] and:

ν = (1 -β) n (1 + n) ζ (19)
Let us now consider equation ( 14) and divide both sides by P t Y t . We find, in view of the homogeneity properties of the function D:

X t+1 = Ω t + D (Ω t , P t Y t ) P t Y t = Ω t P t Y t + D µ Ω t P t Y t , 1 ¶ (20)
Let us define the "fiscal function" F :

F (Z t ) = Z t + D (Z t , 1) (21)
Then equation ( 20) becomes, taking Z t = Ω t /P t Y t :

X t+1 = F (Z t ) = F µ Ω t P t Y t ¶ (22)
Now we have:

Z t = Ω t P t Y t = Ω t P t-1 Y t-1 P t-1 Y t-1 P t Y t = X t ζ (1 + n) Π t (23)
so that finally:

X t+1 = F (Z t ) = F • X t ζ (1 + n) Π t ¸(24)
The fiscal function F will play an important role in what follows. We shall assume F 0 (Z t ) > 0 and denote as f (Z t ) its elasticity:

f = f (Z t ) = ∂Log [F (Z t )] ∂LogZ t (25)

Ricardian economies and local determinacy

We begin our investigation with the traditional Ricardian version of the model. For that it is enough to take n = 0. The dynamic system (18), (24) simplifies as:

Π t+1 = Φ (Π t ) ξ (26) X t+1 = F µ X t ζΠ t ¶ (27)
Steady states (Π, X) of this system (when they exist) are characterized by:

Π = Φ (Π) ξ (28) X = F µ X ζΠ ¶ (29)
Linearizing ( 26), ( 27) around a steady state (Π, X) we find:

Π t+1 -Π = φ (Π t -Π) (30) X t+1 -X = f (X t -X) - Xf Π (Π t -Π) (31)
or in matrix form:

• Π t+1 -Π X t+1 -X ¸= • φ 0 -Xf/Π f ¸• Π t -Π X t -X ¸(32)
with:

φ = φ (Π) = ΠΦ 0 (Π) Φ (Π) f = f (Z) = ZF 0 (Z) F (Z) (33)
The two roots are thus φ and f . Since there is one predetermined variable, X t , and one non predetermined, Π t , there will be local determinacy if one of the roots has modulus smaller than 1, the other bigger.

So we have two possibilities for local determinacy [START_REF] Buiter | The fiscal theory of the price level: a critique[END_REF] . The first is:

φ > 1 f < 1 (34)
We recognize with φ > 1 the Taylor principle. But we see that with the tax function appears a new possibility for local determinacy, i.e.:

φ < 1 f > 1 (35)
The condition φ < 1 says that the Taylor principle is not satisfied. But f > 1 means that the ratio of government's financial liabilities to nominal income will have divergent dynamics along off-equilibrium paths (equation 31). As we indicated above, we want to avoid such policies, whose potentially destabilizing effects in real life situations certainly contributed making the FTPL controversial [START_REF] Christiano | Understanding the fiscal theory of the price level[END_REF] .

Global determinacy in the non-Ricardian case

We just saw that in the Ricardian framework there are two alternative conditions for local determinacy, corresponding to the Taylor principle and the FTPL, and expressed respectively as φ (Π t ) > 1 and f (Z t ) > 1.

What we want to show is that in a non-Ricardian world it is possible to find some interest rate rules Φ (Π t ) such that (a) global determinacy obtains, and (b) no FTPL-type fiscal policy is used:

f (Z t ) ≤ 1 (36)
That such a combination is indeed possible is demonstrated in the following proposition: Proposition 1: Let us assume:

Φ 0 (Π t ) > ξ ∀Π t (37) f (Z t ) ≤ 1 ∀Z t ( 38
)
Then there is a unique globally determinate equilibrium.

Proof: The proof will proceed in several steps. As a first step we shall show that the equilibrium is unique. From ( 18) and (24) the equations of the curves X t+1 = X t and Π t+1 = Π t are respectively:

X t = F • X t ζ (1 + n) Π t ¸(39) Φ (Π t ) ξ -Π t = νF • X t ζ (1 + n) Π t ¸(40)
Let us differentiate logarithmically the two equations. We obtain for the locus X t+1 = X t (equation 39):

dX t X t = f µ dX t X t - dΠ t Π t ¶ (41)
Assumption (38) implies that the curve X t+1 = X t is downward sloping.

Let us now differentiate the equation of the locus Π t+1 = Π t (equation 40). We find:

Π t [Φ 0 (Π t ) -ξ] Φ (Π t ) -ξΠ t dΠ t Π t = f µ dX t X t - dΠ t Π t ¶ (42)
We may note that:

Φ (Π t ) -ξΠ t = Π t Z 0 [Φ 0 (Π s ) -ξ] dΠ s (43)
so that assumption (37) implies that:

Φ (Π t ) -ξΠ t > 0 (44)
So using ( 42) and (44) we see that assumptions (37) and (38) imply that the curve Π t+1 = Π t is upward sloping.

Since the curve X t+1 = X t is downward sloping and the curve Π t+1 = Π t is upward sloping, their intersection, if it exists, is unique.

We can now move to the study of global determinacy of this equilibrium. The dynamics of the system is given by:

X t+1 > X t if X t < F • X t ζ (1 + n) Π t ¸(45) Π t+1 > Π t if νF • X t ζ (1 + n) Π t ¸< Φ (Π t ) ξ -Π t (46)
This is represented in figure 1 where it appears that the unique equilibrium has saddle path dynamics and global determinacy.

Q.E.D.

Figure 1

There remains now to check that there exist functions Φ (Π t ) such that condition (37), which replaces the Taylor principle, is satisfied.

Linear interest rate rules

We shall now show that there are interest rate rules that satisfy condition (37). Let us consider indeed here simple linear interest rate rules:

Φ (Π t ) = AΠ t + B A > 0 B > 1 (47)
Condition (37) will be satisfied if:

A > ξ (48)
We can further compute the elasticity of these policy rules:

φ (Π t ) = ∂LogΦ (Π t ) ∂LogΠ t = ∂Log (AΠ t + B) ∂LogΠ t = AΠ t AΠ t + B < 1 (49)
We note that this elasticity is always below 1. So, whatever the inflation rate, these interest rate rules do not satisfy the Taylor principle.

Fiscal policy

To make the example complete, we have to add to this monetary rule an explicit fiscal policy. We can take the following particularly simple policy:

F (Z t ) = γZ t ( 50
)
which corresponds to the following evolution of government liabilities:

Ω t+1 = γΩ t (51)
For this policy f = 1, so condition (38) is satisfied. So the set of policies characterized by (47), ( 48) and (50) does satisfy the conditions of proposition 1, and therefore leads to global determinacy

Conclusions

A recent lively subject of research has been the search for conditions on monetary and fiscal policies that ensure global determinacy of the economic system. Two classic conditions, the Taylor principle and the fiscal theory of the price level (FTPL) turned out to be somewhat unsatisfactory: The Taylor principle delivers only local, but not global determinacy. The FTPL is implicitly based on fiscal policies that can make government's liabilities explosive in off-equilibrium paths, and this could be dangerous in real life situations.

So our purpose was to find a new interest rate rule that ensures global determinacy even when associated with "reasonable" fiscal policies (in the sense of condition 36). Proposition 1 shows that such a combination is actually feasible, and that condition (37) is an operational alternative to the Taylor principle.

A key to our results is the use of a "non Ricardian" framework where new agents enter the economy every period and Ricardian equivalence does not hold. This assumption seems more realistic than the traditional "Ricardian" framework with no birth. It has already proved quite useful in studying local determinacy (Bénassy, 2005), and appears here to be central for global determinacy as well.

It may finally be noted that the conditions given in proposition 1 can be extended to different frameworks. For example in appendix 2 the assumption of rational expectations is replaced by a very simple learning process, and it is shown that the very same policies assumed in proposition 1 lead to stable dynamics around the same unique equilibrium.

= νf 2 (X t-1 -X) + £ 1 -f 2 (µ -1) ¤ (Π t-1 -Π) (72)
or in matrix form:

• Π t -Π X t -X ¸= M • Π t-1 -Π X t-1 -X ¸(73) M = • [1 -f 2 (µ -1)] / [χ + f (µ -1)] f 2 ν/ [χ + f (µ -1)] -f (µ -1) /ν f ¸(74)
The characteristic polynomial is:

Ψ (λ) = λ 2 - 1 + fχ χ + f (µ -1) λ + f χ + f (µ -1) (75)
We can compute:

Ψ (0) = f χ + f (µ -1)
> 0 (76)

Ψ (1) = (1 -f ) (χ -1) + f (µ -1) χ + f (µ -1) > 0 (77)
So both roots are on the same side of 1. The decisive number is the product of the roots which is f/ [χ + f (µ -1)]. The system will be locally stable if this product is smaller than one, i.e. if:

f < χ + f (µ -1) (78)
If χ (Π t ) > 1, then µ (Π t ) > 1, and condition (78) is satisfied for all fiscal rules such that f (Z t ) < 1. Note that the roots can be real or complex depending on the values of the parameters.

Q.E.D.

The original Taylor rule (1993) had actually both inflation and output as arguments, but the debate has mostly centered around inflation.

Appraisals (with different degrees of criticism) of the FTPL appear in[START_REF] Buiter | The fiscal theory of the price level: a critique[END_REF],[START_REF] Christiano | Understanding the fiscal theory of the price level[END_REF],[START_REF] Cochrane | Money as stock[END_REF], Kocherlakota and Phelan (1999),Mc- Callum (2001),[START_REF] Niepelt | The fiscal myth of the price level[END_REF] and[START_REF] Weil | Reflections on the fiscal theory of the price level[END_REF].

This representation is borrowed from[START_REF] Guillard | Politique monétaire et fiscale dans un monde non-Ricardien: une théorie fiscale de l'inflation[END_REF].

The autarkic interest rate is the one which would prevail if each generation lived in complete autarky. Alternatively it is the real interest rate that would prevail in an economy with a single dynasty, and the utility function[START_REF] Barro | Are government bonds net wealth?[END_REF].

This duality was uncovered in[START_REF] Leeper | Equilibria under "active" and "passive" monetary and fiscal policies[END_REF].

Note that the condition f > 1 should not be identified with the FTPL, since the FTPL is usually based on a global determinacy argument, whereas we give here a local determinacy condition. But the condition f > 1 is anyway a necessary one, and captures in a simple and elegant manner the controversial feature of the FTPL.

Appendix 1

In this appendix we shall derive the fundamental dynamic equation (15). Consider the household's budget equation (4), and assume that i t is strictly positive. The household will thus satisfy the "cash in advance" equation exactly, so that m jt = P t c jt and the budget constraint is written:

Let us define the following discount factors:

Maximizing the utility function (1) subject to the sequence of budget constraints (52) from time t to infinity yields household j's consumption function:

Summing this across the N t agents alive in period t, and using the equilibrium condition C t = Y t we obtain the equilibrium equation:

Let us divide both sides by N t , subtract from it the corresponding equation for t + 1, and then divide by R t+1 . We obtain:

Divide the government's budget equation ( 9) by N t and insert it into equation (56). This yields:

Now multiply (57) by N t+1 , and assume N t+1 /N t = 1 + n. We obtain equation (15).

Appendix 2 Learning

We shall now show that the sufficient conditions of proposition 1 are not limited to the rational expectations framework, but apply as well in a very simple learning process. Let us recall the two dynamic equations in the non-Ricardian system under rational expectations (equations 18 and 24):

Now in a framework with learning Π t+1 in the first equation has to be replaced by an inflationary expectation, denoted Π e t+1 . We shall assume a particularly simple "learning scheme" by making the simple "static expectations" assumption:

so that the first equation becomes:

The dynamic system now consists of the two equations ( 59) and (61). We want to show that the very same conditions that led to determinacy under rational expectations (proposition 1) will lead to a stable system under learning. This is done through the following proposition: Proposition 2: Let us assume:

Then there is a unique dynamically stable equilibrium.

Proof: We shall first show that the equilibrium is unique. As in proposition 1 the equation of the curve X t+1 = X t is:

and from (63) it is downward sloping. Now from (61) the equation of the locus Π t = Π t-1 is:

or, forwarding one period, the equation of the locus Π t+1 = Π t is:

which, combining (62) and (63) as in proposition 1, is upward sloping. Since the locus X t+1 = X t is downward sloping, and the locus Π t+1 = Π t upward sloping, the intersection of the two loci, when it exists, is unique.

We shall now determine the eigenvalues of the dynamic system. Let us call as before:

and linearize the two equations ( 59) and (61) around the steady state (Π, X):

From (61) we know that in steady state:

So rearranging and combining (68) and (69) we obtain:

[χ + f (µ -1)] (Π t -Π) = Π t-1 -Π + νf (X t -X)