Rationalizability in Games with a Continuum of Players∗

Pedro Jara-Moroni †

Instituto Cient´ıﬁcoMilenio Sistemas Complejos de Ingenier´ıa,Fac. de Scs. F´ısicasy Matem´aticas,Universidad de Chile, Santiago, Chile. ‡ This version March 2010

Abstract The concept of Rationalizability has been used in the last fifteen years to study stability of equilibria on models with a continuum of agents such as competitive markets, macroeconomic dynamics and currency attacks. However, Rationalizability has been formally defined in a general setting only for games with a finite number of players. We propose then an exploration of Rationalizability in the context of Games with a Continuum of Players. We deal specifically with a special class of these games, in which the payoff of a player depends only on his own strategy and on an aggregate value that we call the state of the game, which is obtained from the complete action profile. In this paper we define the sets of Point-Rationalizable States and Rationalizable States. For the case of continuous payoffs and compact strategy sets we characterize these sets and prove their convexity and compactness. We provide as well results on equivalence of Point and standard Rationalizability.

Keywords: Rationalizable Strategies, Non-atomic Games, Expectational Coordination, Rational Expectations, Eductive Stability, Strong Rationality. JEL Classification: D84, C72, C62.

1 Introduction

The concept of Strong Rationality was ﬁrst introduced by Guesnerie (1992) in a model of a standard market with a continuum of producers. There, an equilibrium of the market is said to be Strongly Rational, or Eductively Stable, if it is the only Rationalizable Solution of the economic system. Inspired by the work of Muth (1961), the purpose of such an exercise was to give a rationale for the Rational Expectations Hypothesis. Strong rationality has been studied as well in macroeconomic models in terms of stability of equilibria. See for instance Evans and Guesnerie (1993), where they study Eductive Stability in a general linear model of Rational Expectations or Evans and Guesnerie (2003, 2005) for dynamics in macroeconomics. More examples of applications of Strong Rationality can be found in Chamley (2004), where the author presents models of Stag Hunts in the context of coordination in games with strategic complementarities.1

∗This article is part of my dissertation, prepared at Paris School of Economics (PSE) under the supervision of Roger Guesnerie and Alejandro Jofré. †I would like to thank Ali Khan, Thomas Boulogne, for helpful conversations; Bruno Strulovici and NicolásFigueroa for pointing me out relevant references; and the audiences at the seminars at PSE, Department of Mathematical En- gineering (DIM) of the University of Chile and Economics at Johns Hopkins University; as well as the audiences at Franco-Chilean Congress on Optimization, Chilean Economic Society Meeting, Chilean-Brazilean Workshop on Eco- nomic Theory, Third World Congress of the Game Theory Society and of the European Meeting of the Econometric Society for their valuable questions and remarks. I am especially grateful to two anonymous referees whose constructive criticism and suggestions contributed significantly to improve the presentation of the main contributions of the paper. ‡Postal address: Departamento de Ingenier´ıaCivil, Blanco Encalada No 2002, 5o piso, Santiago, Chile. Phone: +56 2 9784376. Fax +56 2 6718788 E-mail: [email protected]. 1See as well Morris and Shin (1998), Chamley (1999), Desgranges (2000), Guesnerie (2002), Gauthier (2002), Chatterji and Ghosal (2004), Desgranges and Heinemann (2005) and Ghosal (2006) although not all these models fit necessarily the framework that is presented below.

1 The Rationalizable Solution of the economic system used by Guesnerie in the definition of Strong Rationality, refers to the concept of Rationalizable Strategies as defined by Pearce (1984) in the context of games with a finite number of players and finite sets of strategies. Rationalizable Strategies were formally introduced by Bernheim (1984) and Pearce (1984) as the “adequate” solution concept under the premises that players are rational utility maximizers that take decisions independently and that rationality is common certainty. In both papers and later treatments, however, the definition and characterization of rationalizable “solutions” were developed for games with a finite number of players. 2 On the other hand, several economic models that deal with strong rationality feature a continuum of agents, including the seminal work by Guesnerie (1992) and the works that are mentioned in the first paragraph above. In those models, the definition of the Rationalizable Solution is usually intu- itive and/or context-specific, adapting the original definitions and characterizations of Rationalizable Strategies to such models. To our knowledge, there is no established definition for Rationalizable Strategies, or Rationalizability for what matters, in a general framework with a continuum of players. The present paper addresses this gap between the established theory and its economic applications exploring the ideas of rationalizability in a class of non-atomic non-cooperative games with a continuum of players. One part of the task is to provide a suitable model of such games, that allows to define and characterize Rationalizable Outcomes. We will work in a framework where payoffs depend both on players’ own actions and on the average of the actions taken by all the players. We will call this average the state of the game, and we will define the sets of Point-Rationalizable States and Rationalizable States. This last approach is not evident nor a generalization of finite player games, since in “small” games, and opposed to what we do here, players can actually affect directly and unilaterally the payoff of other players. Our approach will be that of an iterative process of elimination of non-best-responses (or non- expected-utility-maximizers). This process is proven to deliver the set of Rationalizable Strategies in games with a finite number of players, compact strategy sets and continuous payoff functions (Pearce, 1984; Bernheim, 1984).3 In a more general context, however, the Rationalizable Set may fail to be the result of the iterated elimination of non-best-responses since it should be required to be as well a fixed point of the process. Recent papers address this issue providing examples in general normal form games with finite number of players, where the set obtained as the limit of this process is not a fixed point of the process itself. 4 The authors go beyond to explore more complex iterated processes of elimination of strategies (see for instance Dufwenberg and Stegeman, 2002; Apt, 2007; Chen et al., 2007, 5). The question emerges on how should this process be defined in the context of a continuum of players? When can we claim that the result of the iterative elimination process gives a set that we may call Rationalizable? Our main results are Theorems 3.8 and 3.11 where we characterize these sets as the results of iterated elimination of states. More precisely, we extend Propositions 3.1 and 3.2 in Bernheim (1984) to (Point-)Rationalizable States in the context of games with compact strategy sets, continuous utility functions and a continuum of players. The need for these two Theorems comes from the proof of Proposition 3.2 in Bernheim (1984), where a convergent subsequence extraction argument is used, argument that is no longer valid in the context of a continuum of players. A different limit concept is needed to conclude. Moreover, certain measurability properties must be required to have a well defined process of iterated elimination. The remainder of the paper is as follows: in section 2 we introduce games with a continuum of players and some notation; in section 3 we discuss how the rationalizability arguments may fit in this context. For the particular class of games with an aggregate state, in subsections 3.1 and 3.2 we define Point-Rationalizable and Rationalizable States and we characterize them in subsection 3.3. As Corollaries we obtain their convexity and compactness. In section 4 we explore the relation between Point-Rationalizable and Rationalizable States. We close the presentation with comments and conclusions in section 6. 2See as well Tan and da Costa Werlang (1988). 3See Propositions 3.1 and 3.2 in Bernheim (1984). Proposition 3.2 proves as well, as Bernheim and Pearce claim in their articles, that their definitions are indeed equivalent. 4 The problem rises when the assumptions on utility functions and strategy sets are relaxed (namely the cases for unbounded strategy sets and/or discontinuous utility functions). 5Dufwenberg and Stegeman (2002) and Chen et al. (2007) put emphasis in Reny’s (1999) better-reply secure games.

2 2 The model

We consider games with a continuum of players. Schmeidler (1973) introduced a concept of equilibrium and provided existence results in games where a strategy profile is an equivalence class of measurable functions from the set of players into a strategy set, and the payoff function of a player depends on his own strategy and the strategy profile played. A different approach was presented later by Mas-Colell (1984) 6 and more general frameworks can be found in Khan and Papageorgiou (1987) and Khan et al. (1997) as well. For a comprehensive review of games with many players see Khan and Sun (2002). We will focus mainly in Schmeidler’s general setting and specially in games where payoffs depend on an “average” of the actions taken by all the players (Rath, 1992). In this setting, the set of players I is the unit interval in R endowed with the Lebesgue measure. We assume that all the players have the same set of available actions, denoted A, which is a compact subset of Rn. An action profile is a measurable function from I to A and the set of action profiles is denoted by AI . Our aim is to capture the relevant features of a class of models where there is a set S ⊆ RK and a variable s ∈ S that represent, respectively, the set of states and the state of an economic system. Payoff for each player depends on his action and the value of this state. To obtain the value of the state we have an aggregation operator Ag : AI → S that associates to each action profile a, the state s = Ag(a). Players are negligible in the sense that they are unable to affect the state of the system unilaterally. For each agent i, there is a payoff function defined on the product of the set of available actions A and S, u(i, · , · ): A × S → R. For simplicity, we take the aggregation operator mentioned above as the integral of the strategy profile with respect to the Lebesgue measure: Z Ag(a) ≡ a(i) di I so that AI , the set of measurable functions from I to A, is contained in dom Ag, the set of integrable functions from I to Rn, and the set of states of the game is S ≡ co {A}.7 8 We denote the set of real valued bounded continuous functions defined on a space X by Cb(X). Let UA×S denote the topological space of Cb(A × S) endowed with the sup norm topology. Throughout the document we will be using the following assumption:

HM The function u : I → UA×S that associates players with their payoff functions is measurable (Rath, 1992). This formulation leaves us in Rath’s extension of Schmeidler’s formulation of games with a continuum of players. To denote games with a continuum of players that have an aggregate state as above, we will use the notation u. We will denote then equivalently u(i) and u(i, · , · ). The definition of Nash Equilibrium is simply re-stated for a game u as a strategy profile a∗ ∈ AI such that λ-almost-everywhere in I: Z Z u i, a∗(i) , a∗(i) di ≥ u i, y, a∗(i) di ∀ y ∈ S, I I In this framework Rath shows that for every game there exists a pure strategy Nash Equilibrium.

6In Mas-Colell (1984) what matters is not strategy profiles but a distribution on the product set of payoff functions and strategies. 7If the strategy sets differed across agents, strategy profiles could be identified with measurable selections of the set R valued mapping i ⇒ A(i). Hypothesis over the correspondence i ⇒ A(i) that assure that the set S ≡ I A(i) di is well defined can be found in Aumann (1965) or in Chapter 14 of Rockafellar and Wets (1998). 8The aggregation operator can as well be the integral of the strategy profile with respect to any measure that is absolutely continuous with respect to the Lebesgue measure, or the composition of this result with a continuous function. That is, Z Ag(a) ≡ G a(i) dλ¯(i) I where λ¯ is absolutely continuous with respect to the Lebesgue measure and G : co {A} → S is a continuous function; the results in this work could well be extended to this setting. See footnote 20

3 3 Rationalizability

Our approach to Rationalizability follows from Guesnerie (1992) who is in turn inspired by Pearce (1984), where the rationalizable set is obtained as the result of the iterated elimination of unreasonable outcomes. This approach is a natural way to grasp the rationalizability argument. Nevertheless, from the characterizations of rationalizability presented by Bernheim and Pearce in their seminal papers and the literature that follows we require that the rationalizable set must not only be the the result of the iterated elimination process, but it should as well be a fixed point of this process.9 In the light of Bernheim’s Proposition 3.2, care is needed to claim that the limit of the process of iterated elimination is in fact a set that we could call of rationalizable outcomes. In particular, when we pass from the finite to the continuous player sets there are three main issues to take into account. The first one is how to address forecasts. In the finite player case it is direct to use (product) measures as forecasts and take expectation over payoff functions to make decisions. This is not evident in the context of continuum of players. A difficulty relates with the continuity or measurability properties that must be attributed to subjective beliefs, as a function of the player’s name. The second issue stems from the first one and is related to the space in which one should seek the rationalizable set. The set of action profiles may not be appealing in contexts where the set of players is a continuum. Finally, the process of iterated elimination of outcomes itself could well be undefined without proper assumptions. Consequently it is necessary to provide conditions . Following Pearce (1984) and Guesnerie (1992) then, consider the following line of reasoning. Given the strategy profile set AI , all players know that each player will only play an action that is a best response to some forecast over strategy profiles. Actual strategy profiles then must be selections of a best response correspondence constructed, for each player, as the image through their best response mapping of the complete set of forecasts over AI . This will give a new (hopefully strict) subset of AI over which it is possible to apply the same reasoning. This process may continue ad-infinitum. In the present framework it is natural to model agents as having forecasts on the set of states rather than on the set of action profiles, since the relevant information that agents need to take a decision is the value of the state s.10 Our first example, taken from Guesnerie’s (1992) paper, gives clear insight on how to treat rationalizability in the context of a continuum of players. For more details, the reader is referred to the original paper.

Example 1. Consider a group of farmers, represented by the [0, 1] ≡ I interval, that participate in a market in which production decisions are taken one period before production is sold. Each farmer i ∈ I has a cost function ci : R+ → R. The price p at which the good is sold is obtained from the (given) R inverse demand function P : R+ → [0, pmax] evaluated in total aggregate production p = P q(i) di where q(i) is farmer i’s production. Since an individual change in production does not change the value of the price, the product is sold under price-taking behavior, so each farmer i ∈ I maximizes his payoff function u(i, · , · ): R+ × [0, pmax] → R defined by u(i, q(i) , p) ≡ pq(i) − ci(q(i)). An equilibrium of this system is a price p∗ such that p∗ = P R q∗(i) di and u(i, q∗(i) , p∗) ≥ u(i, q, p∗) ∀ q ∈ R, ∀ i ∈ I. At the moment of taking the production decision, farmers do not actually know the value of the price at which their production will be sold. Consequently they have to rely on forecasts of the price or of the production decision of the other farmers. The concept in scrutiny in our work is related to how this (these) forecast(s) is (are) generated. Forecasts of farmers should be rational in the sense that no unreasonable price should be given positive probability of being achieved. It is in this setting that Guesnerie introduces the concept of strong rationality or eductive stability 11 as the uniqueness of rationalizable prices which are obtained from the elimination of the unreasonable forecasts of possible outcomes. To obtain these rationalizable prices, Guesnerie describes, in what he calls the eductive procedure, how the unreasonable prices can be eliminated using an iterative process of elimination of non-best-response actions. From the farmers problem we can obtain for each farmer his supply function s(i, · ) : [0, pmax] → R+. The structure of the payoff function implies that for a given forecast µ of a farmer i over the value of the price, his optimal production is obtained evaluating his supply function in the expectation

9This pertains to some type of best response property that the rationalizable set must satisfy. 10See as well Guesnerie (2002) for a discussion on this matter. 11An equilibrium of an economic system is said to be strongly rational or eductively stable if it is the only Rationalizable outcome of the system. We will refer equivalently to outcomes as being strongly rational or eductively stable.

4 under µ of the price, Eµ [p]: s(i, Eµ [p]). Farmers know that a price higher than pmax gives no demand and so prices higher than those are unreasonable. Since all farmers can obtain this conclusion, all farmers know that the other farmers should not have forecasts that give positive weight to prices that are greater than pmax. The expectation of each of the farmers’ forecasts then cannot be greater than pmax and so under necessary measurability hypothesis we can claim that aggregate supply can not R 1 be greater than S(pmax) = 0 s(i, pmax) di. Since all farmers know that aggregate supply can not be greater than S(pmax), they know then that the price, obtained through the inverse demand function, 1 can not be smaller than pmin = P (S(pmax)). All farmers know then that forecasts are constrained by 1 1 the interval pmin, pmax . They have discarded all the prices above pmax and below pmin. This same reasoning can be made now starting from this new interval.

q 6

@ P −1(p) @ @ © S(p) @ ? @ !! @ ! !! @ !! ∗ @ ! q !! !! @ ! @ !! !! @ @ @ @@ - p1 p∗ pmax min 6 p 2 ? pmax

Figure 1: The eductive process

In Figure 1 we can see the aggregate supply function depicted along with the demand function. We have seen that to eliminate unreasonable prices in this model we only need these two functions. The process, as described in the Figure, continues until the farmers eliminate all the prices except the unique equilibrium price p∗. We say then that this price is (globally) eductively stable. Note that the eductive process could give more than only the equilibrium point. This could happen for instance if −1 S(pmax) ≥ P (0). In this situation the rationalizable set would be the whole interval [0, pmax], since farmers would not be able to eliminate prices belonging to this interval.

The particular structure of Example 1 allows us to look at outcomes on the set of prices (or aggregate production), instead of the set of action profiles (production profiles). The eductive procedure consists in eliminating the prices that do not emerge as a consequence of farmers taking productions decisions that are best responses to the remaining action profiles, or equivalently remaining values of aggregate production or prices, and delivers the rationalizable set of prices.12 The intuition described above will allow us to rigorously formulate the processes of elimination of states of the game that are not consequence of best response to any possible (subjective probability) forecast profile over a given set of states. Following the terminology of Bernheim we will make a difference between Rationalizability and Point-Rationalizability. Rationalizability differs from Point-Rationalizability in that forecasts are

12A second characteristic of this setting is that the eductive procedure can be done by simply eliminating prices that are beyond the upper and lower bounds that are obtained in each iteration. However, this comes from the monotonicity properties of the aggregation operator (the integral) and the supply function of the farmers. It is not always the case that the eductive process works this way. This second feature of Example 1 is related to the ordered structure of the games studied in Milgrom and Roberts (1990). We give a result regarding these games in section 4 (see as well Guesnerie and Jara-Moroni, 2007).

5 probability distributions whose supports are contained in the sets of outcomes; while for Point- Rationalizability, forecasts are points or Dirac probability measures on these sets. We will continue now by giving formal deﬁnitions for Point and Standard Rationalizability for games with a continuum of players and an aggregate state.

3.1 Point-Rationalizable States

Consider the mapping B(i, · ): S ⇒ A that gives, for each player, the optimal action set given a state of the system,

B(i, s) := argmax {u(i, y, s): y ∈ A} .

A feature that distinguishes this approach from a more general assessment of the subject as that the mapping B(i, · ) goes from S ⊂ Rn, instead of AI for instance, to A ⊂ Rn. It is direct to see, however, that for a given action profile a, in the context of a game u, the best response of a player i to a is given by Bi, R a. For each i ∈ I and a set X ⊆ S, consider the image through B(i, · ) of the set X [ B(i, X) := B(i, s) . s∈X Given X ⊆ S consider the set of all the measurable selections taken from the correspondence i ⇒ B(i, X) that associates the set B(i, X) to each player i ∈ I. Then, take all the possible images through the aggregation mapping of such functions. Denoting by P(X) the family of subsets of X, we define the mapping P r : P(S) → P(S) that - to each set X ⊆ S - associates the set P r(X) ⊆ S defined by:

a is a measurable selection of the P r(X) := s ∈ S : s = Ag(a), . (3.1) correspondence i ⇒ B(i, X) Our assumptions on the aggregation operator Ag allow us to re-write deﬁnition (3.1) as the integral of a set valued mapping:13 Z P r(X) ≡ B(i, X) di. I The set P r(X) gives the set of states that are obtained as consequence of optimal behavior when forecasts are: heterogeneous, constrained to the set X and in the form of points in this set. Before continuing, we state a relevant property associated to the mapping B.

Lemma 3.1. In a game u, for a non-empty closed set X ⊆ S the correspondence i ⇒ B(i, X) is measurable and has non-empty compact values. The proof is relegated to the Appendix. Lemma 3.1 above and Theorem 2 in Aumann (1965) assure that P r(X) is non empty and closed whenever X is non empty and closed. This set to set mapping allows us to rigorously deﬁne a process of iterative elimination of states and a set of point rationalizable states. Consider the process given by iterations of P r. That is,

P r0(S) := S and P rt+1(S) := P rP rt(S) for t ≥ 1.

As noted above, the set of Point-Rationalizable States, noted PS, must then satisfy: ∞ \ t PS ⊆ P r (S) . (3.2) t=0 13 Rn The integral of a correspondence F : I ⇒ is calculated, following Aumann (1965), as the set of integrals of all the integrable selections of F . This is, Z Z F (i) di ≡ f(i) di : f is an integrable selection of F I I R R R where fdi := f1(i) di, . . . , fn(i) di .

6 The right hand side of (3.2) represents the iterative elimination of non reachable states. At each step of this process, players only keep in mind the states that could be reached following rational actions based on point expectations given by the set of the previous step. If a state is not reached by actions following forecasts constrained at a certain step of the process, then it is not rationalizable. Since t +∞ Rn the family of sets P r (S) t=0 is a nested (decreasing) family of closed subsets of , the inﬁnite intersection in expression (3.2) turns out to be the exact Painlev´e-Kuratowski limit of the sequence of sets. The second condition that the set of Point-Rationalizable States must satisfy is:

PS ⊆ P r(PS) . (3.3) T∞ t Note that if a set satisﬁes condition (3.3), then it is a subset of the set t=0 P r (S). We deﬁne the set of Point-Rationalizable States, as follows:

Deﬁnition 3.2. The set of Point-Rationalizable States, PS, is the maximal subset X ⊆ S that satisﬁes condition (3.3).

3.2 Rationalizable States

Before we enter into context we need to introduce some concepts and some notation. For a Borel set X ⊆ Rn we denote by P(X) the set of probability measures on the Borel subsets of X, B(X). Equivalently this is the set of probability measures on the Borel sets of Rn whose support is in X. ∗ We will endow the set P(X) with the weak* topology w = σ(P(X) ,Cb(X)). For a Borel subset Z of X in Rn, P(Z) can be considered to be a subset of P(X) and the weak* topology on P(Z) is the relativization of the weak* topology on P(X) to P(Z). The set X can be topologically identified with a subspace of P(X) by the embedding x 7→ δx. An important property that we will use is that X is compact if and only if P(X) is compact (and metrizable, since we use the norm in Rn).14 As we said before, in the setting of Rath there is a simple way to get through the inconvenience of defining an induced probability measure over the set of action profiles, using the presence of the state variable of the game over which players have an infinitesimal influence. We define then for each player the set valued mapping that gives the actions that maximize expected utility given a probability measure µ over the set of states S, µ ∈ P(S), B(i, · ): P(S) ⇒ A: B E (i, µ) : = argmaxy∈A µ [u(i, y, s)] Z ≡ argmaxy∈A u(i, y, s) dµ(s) . S The process of elimination of unreasonable states, taking into account that players could now use probability forecasts is described with the mapping R : B(S) → P(S):

Z a ∈ AI , a is a measurable selection R(X) := a(i) di : B I of i ⇒ (i, P(X)) Z (3.4) ≡ B(i, P(X)) di, I where B(i, P(X)) is the image through B(i, · ) of the set P(X). The set R(X) gives the set of states that are obtained as consequence of optimal behavior when forecasts are: heterogeneous, constrained to the set X and in the form of probability measures whose supports are contained in this set. As we stated earlier, the diﬀerence between P r and R is that the second process considers expected utility maximizers. For a given Borel set X ⊆ S, X can be embedded into P(X). This means that B(i, X) ⊆ B(i, P(X)) and so we have that P r(X) ⊆ R(X). Proposition 3.3. In a game u, if X ⊆ S is nonempty and closed, then R(X) is nonempty, convex and closed. For the proof we will use the following Lemma (see proof in appendix).

14See Aliprantis and Border (1999) for detailed treatments of these and other results.

7 n Lemma 3.4. Let Y and X be compact subsets of R . Given a function u ∈ Cb(Y × X), the function U : Y × P(X) → R, which to each (y, µ) ∈ Y × P(X) associates the expectation Z U(y, µ) ≡ u(y, x) dµ(x) , X is continuous when P(X) is endowed with the weak* topology. Proof of Proposition 3.3. n From our assumptions, u(i) belongs to Cb(A × S), and A and S are compact sets in R , so Lemma 3.4, along with Berges’ Theorem, imply that for each i ∈ I the correspondence B(i, · ): P(S) ⇒ A is upper semi continuous. Since X is closed, it is compact, therefore P(X) is a compact subset of P(S) and so the set B(i, P(X)) is closed for every i ∈ I. From Theorem 4 in Aumann (1965) we get that R B I (i, P(X)) di is closed. On the other hand, Lemma 3.1 states that the correspondence i ⇒ B(i, X) is measurable, which means that it has a measurable selection a. Since B(i, X) ⊆ B(i, P(X)), a is also a selection of B R B i ⇒ (i, P(X)). This implies that I (i, P(X)) di is nonempty. Convexity comes from the fact that R(X) is obtained as an integral of a set valued mapping.

As in the previous case, the set of Rationalizable States RS must satisfy

RS ⊆ R(RS) (3.5) and so we give an analogous definition. Definition 3.5. The set of Rationalizable States is the maximal subset X ⊆ S that satisfies: X ⊆ R(X) and we note it RS. Remark 3.6. Conditions (3.3) and (3.5) are related to the definition of Tight Sets Closed Under Rational Behavior (Tight CURB Sets) given in Basu and Weibull (1991). Indeed Basu and Weibull make the observation that the set of rationalizable strategy profiles in a finite game with compact strategy sets and continuous payoff functions, is in fact the maximal tight curb set, which is analogous to our definitions of (Point-)Rationalizability. Remark 3.7. At this point, it is appropriate to make the connection between the states set approach and the strategic approach and see that the latter provides a fairly convincing view of standard Ratio- nalizability in our context. Let us note that players do not care to differentiate between action profiles that give the same aggregate state. In mathematical terms, action profiles that have the same integral. Looking at the integral as a function from AI to S, we see that the measurable space structure considered in S may induce a measurable space structure on AI , AI , AI . The σ-field AI , is then the “integral induced” σ-field.15 Any probability measure, ν, defined on AI induces a probability measure in P(S) as usual: µ( · ) ≡ νAg−1( · ) . Thus, when applying the process defined in (3.4), we are in fact considering at least all probability measures that can be defined over AI . Reciprocally, for a given probability measure in P(S) the I 16 function ν : A → R+ defined by: ν(H) := µ(Ag(H)) is in fact a probability measure on AI . The question rises on whether a richer σ-field would make a difference in terms of the Rationalizable sets. The answer is given by the structure of the game. Payoff functions are defined on the values of the integrals of the action profiles. Consequently, when taking expectation, agents precisely only care about the probabilities of events of the form a ∈ AI : R a = s , for each s ∈ S which brings us back to the induced σ-field AI . We may point out, however, in light of this remark, that the relevant strategic concept associated to our approach would be that of Correlated Rationalizable Strategies (Brandenburger and Dekel, 1987).

15The σ-ﬁeld of the pre-images of the Borel subsets of S through the integral. AI ≡ Ag−1(X): X ∈ B(S) .

16By construction, if H ∈ AI then Ag(H) ∈ B(S).

8 3.3 Characterization

We now give an answer to the question of whether we can obtain, in our context, the same conclusion of Bernheim’s Propositions 3.1 and 3.2 for (Point-)Rationalizable Strategies in the case of ﬁnite player games with compact strategy sets and continuous utility functions. Our ﬁrst main result, Theorem 3.8, states that when passing to the continuous the set of Point- Rationalizable States can actually be obtained from the eductive process providing an appealing characterization of this set.

0 T∞ t Theorem 3.8. Let us write PS := t=0 P r (S). The set of Point-Rationalizable States of a game u can be calculated as

0 PS ≡ PS Proof. We will show that:

0 0 P r(PS) ≡ PS

0 0 0 Let us begin by showing that P r(PS) ⊆ PS. Indeed, if s ∈ P r(PS) then, by the deﬁnition of P r, 0 R 0 t there exists a measurable selection a : I → A of i ⇒ B(i, PS), such that s = a. Since PS ⊆ P r (S) 0 t t ∀ t ≥ 0, we have that B(i, PS) ⊆ B i, P r (S) ∀ t ≥ 0 ∀ i ∈ I. So a is a selection of i ⇒ B i, P r (S) t+1 0 and then s ∈ P r (S) ∀ t ≥ 0, which means that s ∈ PS. 0 0 t Now we show that PS ⊆ P r(PS). For this inclusion, consider the following sequence F : I ⇒ A, t ≥ 0, of set valued mappings:

F 0(i) := A ∀ i ∈ I ∀ i ∈ IF t(i) := Bi, P rt−1(S) t ≥ 1

We have that Z P rt(S) ≡ F t(i) di. I Since u(i) ∈ UA×S, then ∀ i ∈ I the mapping B(i, · ): S ⇒ A is u.s.c. and, as a consequence, the R 0 set B(i, X) is compact for any compact subset X ⊆ S (Berge, 1997). Since S ≡ I F , from Aumann (1965) we get that S is non empty and compact.17 From Lemma 3.1 we get that F 1 is measurable and compact valued and by induction over t, we get that for all t ≥ 1, P rt−1(S) is non empty, convex and compact, and F t is measurable and compact valued. Consider then the set valued mapping F : I ⇒ A deﬁned as the point-wise lim sup of the sequence t t F , noted p-lim supt F , obtained as: t t F (i) := p-lim supt F (i) ≡ lim sup F (i) t

t t t where the right hand side is the set of all cluster points of sequences {y }t∈N such that y ∈ F (i). From Rockafellar and Wets18 we get that F is measurable and compact valued. 0 R t So now let us take a point s ∈ PS. That is, s ∈ I F for all t ≥ 0. This gives a sequence of t R t measurable selections {a }t∈N, such that s = a . From the Lemma proved in Aumann (1976) we get R t that s ∈ I F , since for each i ∈ I the cluster points of {a (i)}t∈N belong to F (i) and a is the trivial limit of the constant sequence R at. 0 Now it suﬃces to check that F (i) ⊆ B(i, PS), since then we would have Z Z 0 0 s ∈ F di ⊆ B(i, PS) di ≡ P r(PS) . I I This comes from the upper semi continuity of B(i, · ). Take y ∈ F (i). From the deﬁnition of F , y is t t t a cluster point of a sequence {y }t∈N such that y ∈ F (i). That is, there is a sequence of elements

17We have already noted that in fact S ≡ co {A} and so in particular it is also convex, which is of no relevance in this proof, but is the key property in Corollary 3.9. 18See Rockafellar and Wets (1998) Ch. 4 and 5 and Theorem 14.20 .

9 k k tk−1 tk k tk of S, s k∈N such that s ∈ P r (S), y ∈ B i, s and limk y = y. Through a subsequence of k tk 0 k s k∈N we get that the limit of {y }k∈N must belong to B(i, PS), since all cluster points of s k∈N 0 are in PS, being the intersection of a nested family of compact sets. The main challenge in the proof of Theorem 3.8 is to identify the adequate convergence concept for the eductive process. In the case of finite player games there is no such a question, since in that case the technique is simply to take a convergent subsequence of points (in the finite dimensional strategy profile set) from the sequence of sets that participate in the eductive procedure and using (semi) continuity arguments of the best response mappings obtain the result. 19 However, in our setting these arguments fail to prevail. From the proof of the Theorem, we see that the set of Point-Rationalizable States is obtained as the integral of the point-wise upper limit of a sequence of set valued mappings. So the relevant improvement in the proof (besides measurability requirements) is to give the adequate limit concept. From Theorem 3.8 we get as a Corollary that in the games that we are considering, the set of Point-Rationalizable States is “well behaved”. 20 Corollary 3.9. The set of Point-Rationalizable States of a game u is well defined, non-empty, convex and compact. Proof. From Theorem 3.8, PS is the intersection of a nested family of non-empty compact convex sets. From Rath’s existence Theorem we get that there is point s∗ ∈ S such that s∗ ∈ P rt(S) ∀ t. These two facts lead us to conclude that this intersection is compact, convex and non empty. Regarding Rationalizability, Proposition 3.3 allows us to define the Eductive Process. As before, we consider the iterative elimination of non generated states, but now allowing for probability forecasts of players. The iterative process begins with the whole set of outcomes R0(S) := S. Then, on each iteration, the states that are not reached by the process R are eliminated

Rt+1(S) := RRt(S) .

Recall that since we start the process at S, what we get is a nested family of sets that, following Proposition 3.3, are nonempty convex and closed. The Eductive Process delivers then the set,

∞ 0 \ t RS := R (S) . t=0

0 Theorem 3.10. In a game u, the set RS is non empty, convex and compact. Proof. 0 RS is the intersections of closed convex sets, so it is convex and closed. Again, Rath’s existence 0 t Theorem assures that RS is nonempty, since equilibria belong to every set R (S). 0 From the deﬁnition of the set of Rationalizable States, we have RS ⊆ RS. The following Theorem, analogous to Theorem 3.8, shows that these two sets are actually the same. Theorem 3.11. The set of Rationalizable States of a game u can be calculated as

0 RS ≡ RS Although there are certain details where attention must be put, the proof of Theorem 3.11 follows the proof of Theorem 3.8 and is therefore relegated to the appendix.

19See the proof of Proposition 3.1 in Bernheim (1984). 20Theorem 3.8 holds as well for an aggregation operator deﬁned using a function G as in Footnote 8 and if G is aﬃne, Corollary 3.9 holds as well.

10 An important property stated in Corollary 3.9 and Theorem 3.10 is the convexity of the Point- Rationalizable and Rationalizable States sets. It is crucial to obtain reduction procedures from Ra- tionalizable to Point-Rationalizable Sets (see Section 4) and more simple iterative procedures as well (Guesnerie and Jara-Moroni, 2007). Moreover, we know that the convex hull of the set of equilibria is also contained in the set of Point-Rationalizable States and this inclusion may be strict. To see that the Theorems are not vacuous consider the following example.

Example 2. Consider the game where I ≡ [0, 1], A ≡ [0, 1] and u(i) ≡ u : [0, 1]2 → R for all i ∈ I, such that it generates the following best reply correspondence, depicted in Figure 2a:

s∗ if s ≤ s¯, B(s) = {0, s¯(1 − α) + sα} if s > s¯, where s∗,s ¯ and α are in ]0, 1[ and s∗ < s¯. It is clear that this game does not satisfy the hypothesis of Theorem 3.8 since no continuous utility function may give rise to such a best response correspondence.

6 6 6 1 1

s¯(1 − α) + α ¨ s1 ¨ .. ¨ s2 ¨ ...... ¨ s ¨ ... ¨ 3 ... s¯ ¨ ...... b b ... S ...... B(S) ...... ©.. ∗ 0 .. s .. PS ...... r r r ...... - ..-. ∗ 0 ss ¯ 1 z ∗ }| { s¯ 10 b {s } ≡ PS rr (a) The best reply correspondence (b) The elimination of non-generated-states with B : [0, 1] ⇒ [0, 1]. P r(X) ≡ co {B(X)} and the Point-Rationalizable State set.

0 Figure 2: Example 2: The set of Point-Rationalizable States is not the set PS.

The only equilibrium of the game is s∗. Since all the players have the same best reply correspondence, the process of elimination of non- generated-states is obtained by: P r(X) ≡ co {B(X)} The image through the best reply correspondence of the state set is

B(S) ≡ {0, s∗} ∪ ]¯s, s¯(1 − α) + α] , then P r(S) ≡ co {B(S)} ≡ [0, s¯(1 − α) + α]. We see from the form of the best reply correspondence that on each iteration of P r we get an +∞ interval of the form [0, st] where the sequence {st}t=0 satisﬁes:

st+1 =s ¯(1 − α) + stα, with s0 = 1, which gives for t ≥ 1,

t t st =s ¯ 1 − α + α ,

11 and so the sequence is decreasing and converges tos ¯ (see Figure 2b). This allows for us to see that 0 PS ≡ [0, s¯]. However, 0 0 ∗ ∗ P r(PS) ≡ co {B(PS)} ≡ co {B([0, s¯])} ≡ co {{s }} ≡ {s } 0 PS. 0 ∗ and so PS is not equal to PS, which is in fact equal to the set of equilibria: the singleton {s }.

The properties stated in Corollary 3.9 and Theorem 3.10 are not trivial. It is possible to ﬁnd examples where the outcome of the iterative elimination of non-best-replies is an empty set. We present below an example of a game with non-continuous payoﬀs:

Example 3 (Based on Dufwenberg and Stegeman (2002)). Consider the game where I ≡ [0, 1], A ≡ [0, 1] and u(i) ≡ u : [0, 1]2 → R for all i ∈ I, such that:

1 − y if 0 < s ≤ y, u(y, s) = 2 y if not.

Then, the best reply correspondence is the same for all players:

1 if s = 0, B(s) = s 2 if s > 0.

6 1 r

¨¨ ¨ r ¨¨ ¨¨ ¨¨ ¨¨ ¨ - 10 b

Figure 3: The best reply correspondence B : [0, 1] ⇒ [0, 1].

In this game there is no equilibrium. Let us study the Point-Rationalizable States set. The image through the best reply correspondence of the state set is B(S) ≡ {1} ∪ ]0, 1/2], then

P r(S) ≡ co {B(S)} ≡ ]0, 1] , the second iteration gives

P r2(S) ≡ co {B(P r(S))} ≡ co {B(]0, 1])} ≡ ]0, 1/2] and the third

P r3(S) ≡ co BP r2(S) ≡ co {B(]0, 1/2])} ≡ ]0, 1/4] , . . . , etc..

0 Then the set PS ≡ ∅ and so PS ≡ ∅, that is, in this example there is no set X ⊆ S that satisﬁes X ⊆ P r(X).

12 Although Theorem 3.8 and Corollary 3.9 assure that the eductive procedures achieve non-empty (Point-)Rationalizable sets of states, we see from Examples 2 and 3 that even in cases where the eductive procedure fails, in the sense that it does not deliver the (Point-)Rationalizable set, we may still identify a set as the correct (Point-)Rationalizable State sets following our Deﬁnitions 3.2 and 3.5 (in Example 2 the set PS is the singleton that contains the equilibrium and in Example 3 it is the empty set). Even more, the eductive procedure can (obviously) help to locate such sets even in the case of failure and emptiness. The original motivation to introduce rationalizability in economic contexts is the plausibility of the Rational Expectations Hypothesis. In consequence we allow an empty (Point-)Rationalizable set, under the deﬁnition of rationalizability, interpreting such a situation as a pessimistic answer to the possibility of the coordination of expectations.

4 Rationalizability vs Point-Rationalizability

So far we have deﬁned and characterized the sets of Rationalizable and Point Rationalizable States. In this section we give equivalence results between these two objects in fairly general contexts. It is clear that we always have

PS ⊆ RS. (4.1) The opposite inclusion is not necessarily true. We continue with an example where these two sets are not equal. Example 4. Consider the game where A = {T,M,B} and u(i) ≡ u : A × ∆ → R defined by: 21   3 if s ≥ 1/2 3 if s ≤ 1/2  B  B u(T, s) = 0 if sB ≤ 1/3 u(M, s) = 2 u(B, s) = 0 if sB ≥ 2/3 ,  1 1  1 2 18sB − 6 if 3 < sB < 2 12sB − 18 if 2 < sB < 3 R3 where S = ∆ = (sT , sM , sB) ∈ +|sT + sM + sB = 1 . Clearly, B(∆) ≡ {T,B} so P r(∆) ≡ 2 co {T,B}. Moreover B(co {T,B}) ≡ {T,B}, so P r (∆) ≡ co {T,B}. Thus, PS ≡ co {T,B}. However, if µ is such that µ({s ∈ ∆ : sB = 0}) = µ({s ∈ ∆ : sB = 1}) = 1/2 we have that,  1.5 if y = T  U(y, µ) = 2 if y = M 1.5 if y = B and so B(µ) = M, so RS ≡ ∆. Following definitions 3.2 and 3.5, the rationalizable sets are well defined and differ.

We now provide a simple condition that assures equivalence of the rationalizable sets. Proposition 4.1 states that if the utility functions are aﬃne in the state variable, then the Point-Rationalizable States set is equal to the set of Rationalizable States. Proposition 4.1. If in a game u, we have ∀ µ ∈ P(S):

Eµ [u(i, y, s)] ≡ u(i, y, Eµ [s]) then

PS ≡ RS Proof. If Eµ [u(i, y, s)] ≡ u(i, y, Eµ [s]) then

B(i, µ) ≡ B(i, Eµ [s]) ,

21I am grateful to the editor for pointing out an illustrative example that inspired the one herein presented.

13 which implies that [ B(i, P(X)) ≡ B(i, Eµ [s]) . µ∈P(X)

For a convex set X ⊆ S we have Eµ [s] ∈ X, ∀ µ ∈ P(X).

This implies that under the hypothesis of the Proposition if X is convex, [ [ B(i, Eµ [s]) ⊆ B(i, s) µ∈P(X) s∈X and consequently [ [ B(i, X) ⊆ B(i, P(X)) ≡ B(i, Eµ [s]) ⊆ B(i, s) ≡ B(i, X) . µ∈P(X) s∈X This is, B(i, P(X)) ≡ B(i, X) and so R(X) ≡ P r(X).

Noting that S is convex, we get R(S) ≡ P r(S) which are convex as well. By induction we get that P rt(S) ≡ Rt(S) ∀ t which yields (using the previous notation)

∞ ∞ 0 \ t \ t 0 RS ≡ R (S) ≡ P r (S) ≡ PS (4.2) t=0 t=0 where these intersections are closed convex sets.

Finally we get, 0 0 0 0 0 R(RS) ≡ R(PS) ≡ P r(PS) ≡ PS ≡ RS 0 0 which implies that RS ≡ RS. The first inequality comes from (4.2), the second one is true because PS 0 is convex, the third one comes from Theorem 3.8 which states that PS ≡ PS and the last one comes again from (4.2). 0 0 We conclude that RS ≡ RS ≡ PS ≡ PS. In Proposition 4.1 we obtain the equality of the Rationalizable and Point-Rationalizable sets by providing a condition that implies that the images of the sets X and P(X) through the mappings B and B (resp.) are equal. The affinity condition exploits convexity of the aggregate states set and it’s subsets obtained in the iterative processes. Convexity comes from the fact that the states are obtained through the integral of the strategy profile. However, the equality of these sets is not necessary to obtain the result. Taking further advantage of the properties of the integral, it suffices to require that B(i, P(X)) ⊆ co {B(i, X)}, since following Aumann (1965) the integrals of B(i, X) and co {B(i, X)} coincide. Nevertheless, this property is harder to obtain for every Borel subset of S. 22 This is indeed the case however in the context of Nice Games with a continuum of players.23 A game u is a Nice Game with a continuum of players if A is a compact interval in R and u(i, · , s) is strictly quasi-concave with respect to a.24

Proposition 4.2. If the game u is a Nice Game with a continuum of players, then PS ≡ RS. Proof. Since u is a Nice Game with a continuum of players, A ≡ [a, a¯] ≡ S and B(i, · ): S → A is a continuous function. We will prove the following statement:

22A local version of this argument may be found in Guesnerie and Jara-Moroni (2007) or Jara-Moroni (2008). 23Nice Games were introduced by Moulin (1984) in the context of a ﬁnite number of players, here in Proposition 4.2 we extend the notion to a continuum of players. 24Strict quasi-concavity means that: u(i, λx + (1 − λ)y, s) > min {u(i, x, s) , u(i, y, s)} , for all x, y ∈ A, s ∈ S, λ ∈ ]0, 1 [.

14 For an interval [x, x¯] ⊆ A, the inclusion B(i, P([x, x¯])) ⊆ co {B(i, [x, x¯])} holds. Then, since A is an interval we get that R(A) ≡ P r(A), which is again an interval. By induction, we get the result.

Now the proof of the statement. We know that co {B(i, [x, x¯])} is an interval. Since B(i, · ) is continuous B(i, [x, x¯]) is compact, soy ¯ := sup {x ∈ B(i, [x, x¯])} and y := inf {x ∈ B(i, [x, x¯])} are met in B(i, [x, x¯]) and consequently co {B(i, [x, x¯])} ≡ y, y¯. Suppose that B(i, P([x, x¯])) * co {B(i, [x, x¯])}. Then there existsa ˜ ∈ B(i, P([x, x¯])), such that a˜ 6∈ y, y¯. Suppose thata ˜ < y. This is, ∀ s ∈ [x, x¯],a ˜ < B(i, s). In particular, by deﬁnition of B(i, s),

∀s ∈ [x, x¯] , u(i, a,˜ s) < u(i, B(i, s) , s) .

Moreover, by strict quasi-concavity we have

∀s ∈ [x, x¯] , ∀a ∈ ]˜a, B(i, s)] , u(i, a,˜ s) < u(i, a, s) .

Since ∀ s ∈ [x, x¯], y ∈ ]˜a, B(i, s)] we get

∀s ∈ [x, x¯] , u(i, a,˜ s) < ui, y, s . Thus, if µ has support in [x, x¯], Eµ [u(i, a,˜ s)] < Eµ u i, y, s . which is a contradiction witha ˜ ∈ B(i, P([x, x¯])). A similar argument holds for the casea ˜ > y¯. Convexity due to the use of the integral allows us to introduce another family of models where general forecasts are superfluous. We will say that a game u presents increasing (decreasing) differences in a ∈ A and s ∈ S if for all i ∈ I and ∀ a, a0 ∈ A, such that a ≥ a0 and ∀ s, s0 ∈ S such that s ≥ s0:25 u(i, a, s) − u(i, a0, s) ≥ (≤)u(i, a, s0) − u(i, a0, s0) Proposition 4.3. If the game u presents increasing or decreasing differences in a ∈ A and s ∈ S and A is a compact interval in R, then PS ≡ RS. Proof. We know that both the sets PS and RS are convex. Since they are subsets of R, PS ⊆ RS must both be intervals. The hypothesis of Proposition 4.3 imply that the upper and lower bounds of RS are in fact Point-Rationalizable (Guesnerie and Jara-Moroni, 2007), which gives the converse inclusion. Restricting the action set A to R plus the hypothesis of increasing or decreasing differences, turn the game u in a game with strategic complements or substitutes (respectively). In general games with strategic complements or substitutes (this is withdrawing the one-dimensional hypothesis for the action set) it is still possible to have PS ≡ RS. This would be the case if the set PS was an interval (see Guesnerie and Jara-Moroni, 2007). 26 We close this section with a final remark on the second part of Schmeidler’s original article. In this setting the strategy sets are finite and payoffs depend on the integral of the profile of mixed strategies (instead of the profile of actions). In this context then, there is no identifiable set of states, but it is possible to interpret payoff as expected utility and mixed strategies can be in turn interpreted as probability forecasts. Consequently, it is possible to define a proper set of Rationalizable Strategy Profiles, and an equivalence result may be obtained between Point and standard Rationalizability in terms of strategy sets (Jara-Moroni, 2008).

25 n For x, y ∈ R we say that x ≥ y if and only if ∀ j ∈ {1, . . . , n}, xj ≥ yj . 26In any ordered space (E, ≤), we refer to an interval, as usual, as a set of the form {x ∈ E : s ≤ x ≤ s¯} for given s, s¯ ∈ E.

15 5 More of Example 1

To illustrate our deﬁnitions and results let us go back to the example that motivates this presentation. In this example the action set was R+ and without loss of generality we can assume it to be a 27 compact interval A ≡ [0, qmax]. As said earlier, we could choose the state set to be the set of aggregate production quantities or the set of prices. This depends on the aggregation operator that we are considering (see footnote 8).

Total production as state of the game. Let us consider ﬁrst Ag to be the operator that gives I aggregate production quantities. This is, Ag : A → R+ Z Ag(a) ≡ a(i) di. I R I R The set S is the interval S ≡ q ∈ : ∃ a ∈ A , q = I a(i) di ≡ co {[0, qmax]} ≡ [0, qmax]. The payoﬀ function u(i, · , · ) : [0, qmax] × [0, qmax] → R is then:

u(i, a(i) ,Q) ≡ P (Q) a(i) − ci(a(i)) .

If we assume P and ci to be continuous and that the measurability requirement over the function i → ci( · ) is met (for instance if all the producers have the same cost function), Theorems 3.8 and 3.11 hold and so we can compute the (Point-)Rationalizable State sets by the iterative elimination of states. We get as well the results of Corollary 3.9 and Theorem 3.10 so we know then that these sets are compact intervals. Usual assumptions on these type of models turn the game in either a Nnice game or a game with strategic substitutes. Applying then Proposition 4.2 or 4.3 we can conclude that the rationalizable sets coincide.

Price as state of the game. Now suppose that we use a variation, as in footnote 8, of the aggregation operator. In the same setting we will consider the state set to be the set of prices. This is, I Ag : A → R+ Z Ag(a) ≡ P a(i) di . I This is not the aggregation operator for which we obtained the results. However, we will see below that they still hold. The set of states S will be identiﬁed with the set:

S ≡ {p ∈ R : ∃ q ∈ [0, qmax] , p = P (q)} ≡ P ([0, qmax]) ≡ [0, pmax] .

We see that since P is a continuous function that goes from the one-dimensional aggregate-production set R+ to the set of prices [0, pmax] ⊂ R, this set turns out to be convex. The utility function is now u(i, · , · ) : [0, qmax] × [0, pmax] → R:

u(i, a(i) , p) ≡ pa(i) − ci(a(i)) .

Continuity of P implies that Theorems 3.8 and 3.11 still hold and so we can as well compute the (Point-)Rationalizable State sets by iteratively eliminating states, using Ag instead of the integral. Since the action and state sets are unidimensional we know then that these sets are compact intervals. Furthermore, in this second approach to the example we have that the payoﬀ function is aﬃne in the state variable and so Proposition 4.1 holds. Therefore, the rationalizable sets coincide.

6 Comments and Conclusions

In this work we have formally introduced the concept of Rationalizability into a class of economic models that feature a continuum of agents. We have proposed a deﬁnition for (Point-)Rationalizable States in the context of games with a continuum of players whose payoﬀs depend on other players’

27 −1 Where qmax could be the quantity that makes the price equal to 0 : qmax := inf P (0) .

16 actions through an aggregate variable that cannot be unilaterally affected by any player. Our definition relies on the original characterization of Rationalizable Strategies for games with finite set of players, compact strategy sets and continuous utility functions (Bernheim, 1984). The sets of Rationalizable and Point-Rationalizable States are defined as the maximal subsets of the state set, that are fixed points of processes of elimination of non-best response states (the eductive process). When forecasts are deterministic, we obtain the set of Point-Rationalizable States. When forecasts are in the form of probability distributions over the set of states, we obtain the set of Rationalizable States. We provide sufficient conditions to have well defined (Point-)Rationalizable sets. These conditions come down to the original conditions for existence of an equilibrium given by Rath (1992). We have characterized as well the (Point-)Rationalizable States sets as limits of the processes of elimination of states. As in the case of finite player games, continuity properties of the payoff functions are crucial to assure convergence of the processes of elimination of non-best reply states to the rationalizable set. For the continuum of players case, an additional measurability assumption must be made on the mapping that associates players to their payoff functions to be able to have existence of equilibrium. It turns out that this same assumption is sufficient to assure the integrability of the set valued mapping that is used in the eductive process and, in consequence, to obtain the constructive characterization of the two rationalizable sets introduced throughout the document. As corollaries we obtain non-emptiness, convexity and compactness of these sets. We discuss as well the equivalence between the set of Rationalizable and Point-Rationalizable States. We show that if payoffs are affine in the state variable, then standard and point Rationalizability deliver the same outcomes. We mention as well the case of strategic complementarity or substitutability as a world where probabilistic forecasts are redundant. In these last cases, the equivalence is obtained if the rationalizable sets are intervals. We have defined a key concept in a unified exploratory framework that encompasses a variety of economic settings such as models of currency attacks, stag hunts, standard markets and macroeconomic dynamics, in which Rational Expectations equilibrium and its properties (such as (local) eductive or iterative stability) have been extensively studied (see Chamley, 2004). Applications to models with continuum of agents that feature strategic complementarities or substitutes are possible as well (Cooper, 1999; Guesnerie, 2005; Guesnerie and Jara-Moroni, 2007).

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A Relegated proofs

Proof of Lemma 3.1. We show ﬁrst that the mapping G : I ⇒ S × A, that associates with each agent i ∈ I the graph of the best response mapping B(i, · ), G(i) := gph B(i, · ), is measurable.

Take a closed set C ⊆ S × A. We need to prove that the set

G−1(C) ≡ {i ∈ I : C ∩ gph B(i, · ) 6= ∅} is measurable. Consider the subset U ⊆ UA×S deﬁned by:

U := { g ∈ UA×S : ∃ (s, a) ∈ C such that g(a, s) ≥ g(y, s) ∀ y ∈ A} note that u−1(U) ≡ G−1(C) and so, from the measurability assumption over u, it suﬃces to prove ν ν ∗ that U is closed. That is, we have to show that for any sequence {g }ν∈N ⊂ U, such that g → g uniformly g∗ ∈ U. Since the functions gν are ﬁnite and continuous in A×S, from Weierstrass’ Theorem g∗ is continuous ν ∗ and so it belongs to UA×S. Moreover, g converges continuously to g , that is, for any convergent sequence (sν , aν ) with limit (s∗, a∗), the sequence gν (aν , sν ) converges to g∗(a∗, s∗). Indeed, consider ∗ any ε > 0. By the continuity of g there existsν ¯1 ∈ N such that ∀ ν > ν¯1, ε 2|g∗(aν , sν ) − g∗(a∗, s∗)| < . 2

ν From the uniform convergence of g we get that there existsν ¯2 ∈ N such that, ε |gν (a, s) − g∗(a, s)| < for all ν ≥ ν¯ and ∀ (a, s) ∈ A × S, 2 2

19 in particular this is true for all the elements of the sequence of points. We get then that ∀ ν ≥ max {ν¯1, ν¯2}, |gν (aν , sν ) − g∗(a∗, s∗)| ≤ |gν (aν , sν ) − g∗(aν , sν )| + |g∗(aν , sν ) − g∗(a∗, s∗)| < ε. We have to show then that there exists a point (s, a) ∈ C such that g∗(a, s) ≥ g∗(y, s) ∀ y ∈ A. Since gν ∈ U, we have for each ν ∈ N, points(sν , aν ) ∈ C such that gν (aν , sν ) ≥ gν (y, sν ) ∀ y ∈ A. Let ∗ ∗ ν ν (s , a ) ∈ C be the limit of a convergent subsequence of {(s , a )}ν∈N, which without loss of generality we can take to be the same sequence. We see that (s∗, a∗) is the point we are looking for since for a ﬁxed y ∈ A, continuous convergence implies that in the limit g∗(a∗, s∗) ≥ g∗(y, s∗) . We conclude then that g∗ ∈ U. Thus, U is closed and since u is a measurable mapping, u−1(U) is measurable.

With this in mind, consider a closed set X ⊆ S and the mapping i ⇒ B(i, X). Applying Theorem 14.13 in Rockafellar and Wets (1998) to the constant mapping i ⇒ X along with G above, we get that the correspondence i ⇒ B(i, X) is measurable and has closed values (hence compact since A is compact). Proof of Lemma 3.4. We write U as the composition of two functions:

(y, µ) ∈ Y × P(X) → (u(y, · ) , µ) ∈ Cb(X) × P(X) Z and (f, µ) ∈ Cb(X) × P(X) → f(x) dµ(x) X

If we endow Cb(X) with the sup norm topology and P(X) with the weak* topology, from Corollary R 15.7 in Aliprantis and Border (1999) we get that (f, µ) → fdµ is continuous on Cb(X) × P(X). Therefore, the result will follow from the continuity of the function (y, µ) → F(y, µ) =(u(y, · ) , µ) . Note ﬁrst that this function is deﬁned component to component by functions that depend each only on one variable, this is F(y, µ) = (F1(y) , F2(µ)), and second that F2 is the identity. Thus, we only need to prove that F1 : Y → Cb(X) is continuous for the sup norm topology in Cb(X).

Let yν → y and take ε > 0. Since Y ×X is compact and u is in Cb(Y × X), this function is as well uniformly continuous. Thus, ∃ δ > 0 (that depends only on ε) such that ε k (y, x) −(y0, x0)k < δ =⇒ |u(y, x) − u(y0, x0)| < 3 N Since X is compact ∃ {x1, . . . , xN } ⊂ X such that X ⊆ ∪i=1B(xi, δ). This is, for any x ∈ X there exists xi in the previous set such that x ∈ B(xi, δ). ν Finally, since y → y, there exist for each xi numbersν ¯i such that ν k (y , xi) −(y, xi)k < δ for all ν ≥ ν¯i. All together gives, for ν ≥ max {ν¯i : i ∈ {1,...,N}} and x ∈ X: ν ν ν ν |u(y , x) − u(y, x)| < |u(y , x) − u(y , xi)|+|u(y , xi) − u(y, xi)|

+|u(y, xi) − u(y, x)| ε ε ε < + + = ε. 3 3 3 We conclude that u(yν , · ) converges to u(y, · ) for the sup norm topology, which completes the proof.

20 We use this Lemma to prove Theorem 3.11: Proof of Theorem 3.11. We will show that: 0 0 R(RS) ≡ RS 0 Theorem 3.10 assures that R(RS) is correctly deﬁned.

0 Suppose that s ∈ R(RS). By the deﬁnition of R, there exists a measurable selection a : I → A B 0 R 0 0 t of i ⇒ (i, P(RS)), such that s = a. Since RS is a Borel set and RS ⊆ R (S), which are as 0 t well Borel sets ∀ t ≥ 0, we have that P(RS) ⊆ P(R (S)) are well deﬁned and, ∀ t ≥ 0, ∀ i ∈ I, B 0 B t B t t+1 (i, P(PS)) ⊆ (i, P(R (S))). So a is a selection of i ⇒ (i, P(R (S))) and then s ∈ R (S) ∀ t ≥ 0, 0 0 0 which means that s ∈ RS. This proves that R(RS) ⊆ RS.

t For the other inclusion, we consider again a sequence of set valued mappings F : I ⇒ A, t ≥ 0, whose p-lim sup limit will be again very helpful. Consider then ∀ i ∈ I,

F 0(i) := A F t(i) := Bi, PRt−1(S) t ≥ 1

We have now ∀ t ≥ 0, Z Rt(S) ≡ F t(i) di. I R 0 We know that S ≡ I F is non empty and compact. From Proposition 3.3 we get that so are the sets Rt(S) for all t ≥ 1. From Lemma 3.4, we get that ∀ i ∈ I the mapping B(i, · ): P(S) ⇒ A is u.s.c. and, as a consequence, the set B(i, P(X)) is compact for any compact subset X ⊆ S. So the correspondences F t are compact valued. From the proof of Theorem 3.8 we get that they all have a measurable selection, since for a closed set X, B(i, X) ⊆ B(i, P(X)) and from Lemma 3.1 the mappings i ⇒ B(i, X) are measurable and hence have a measurable selection. Consider then the set valued mapping F : I ⇒ A deﬁned as the point-wise lim sup of the sequence F t: t t F (i) := p-lim supt F (i) ≡ lim sup F (i) . t 0 R t So now let us take a point s ∈ RS. That is, s ∈ I F for all t ≥ 0. This gives a sequence of t R t measurable selections {a }t∈N, such that s = a . From the Lemma proved in Aumann (1976) we get R t that s ∈ I F , since for each i ∈ I the cluster points of {a (i)}t∈N belong to F (i) and a is the trivial limit of the constant sequence R at. B 0 To show that F (i) ⊆ (i, P(RS)), we use that the weak* topology in P(S) is metrizable and the upper semi continuity of B(i, · ): P(S) ⇒ A to give an argument that follows the one at the end of the proof of Theorem 3.8.