Players-Econometricians, Adverse Selection, and Equilibrium.

Transcription

1 Players-Econometricians, Adverse Selection, and Equilibrium. Jernej Čopič February 14, 2014 Abstract We study strategic environments of adverse selection, where in an outcome of strategic interaction players-econometricians observe the distribution over own payoffs and all players actions, and then form assessments regarding the underlying uncertainty. Tailored to such situations where players may face recovery problems, we define adverse equilibrium outcomes: players optimize, their assessments are consistent, and they can justify one-another s behavior. We then study the problem from the perspective of an impartial econometrician who can only observe the distribution over the actions taken by the players, and not their types. We define four conditions on the payoff structure and the strategy profile: positivity, incentive imbalance, pooling, and informational adversity. These conditions characterize the environments where the econometrician would reject equilibrium hypothesis if she didn t take into account the recovery problem faced by the players. jcopic@econ.ucla.edu. I am grateful to Andy Atkeson, Eddie Dekel, Faruk Gul, Jin Hahn, Rosa Matzkin, Romans Pancs, and Bill Zame for helpful discussions, and to my students for encouragement. I am also grateful to the UCLA econometrics, macro, and theory-proseminar audiences. 1

2 JEL codes: C10, C70, D80 Keywords: consistency, recovery, adverse selection, equilibrium, incentives, information. 1 Introduction Ever since the publications of Akerlof s (1970) Market for Lemons, Spence s (1970) Job market signaling, and Rothschild and Stiglitz s (1976) Equilibrium in Competitive Insurance Markets, various literatures have emerged documenting adverse selection arising from asymmetric information as one of the fundamental problems of information economics. In a nutshell, adverse selection describes the informational problem resulting from an equilibrium whereby given the behavior of informed agents on one side of the market, the uninformed agents on the other side cannot fully deduce private characteristics, or types, of the informed agents solely from observations of what the informed agents do. These problems can be so severe to lead to market breakdowns, or in the case of Rothschild and Stiglitz (1976), to non-existence of equilibrium. 1 One assumption underlying most of the literature on adverse selection and signaling has been that the agents are nevertheless for some reason able to know the true distribution of characteristics of the informed side of the market. To know is meant in the sense of having a well-formed and accurate prior belief consistent with the data. The problem is, however, that it is precisely the adverse selection which might lead uninformed agents to face recovery problems 2 regarding the underlying objective distribution of characteristics of the other side of the market. This, in turn, would allow for the uninformed agents to have assessments different from, 1 In a general-equilibrium setting, Zame (2007) shows that equilibrium in general exists. 2 We use the term recovery, in the sense of Mass-Colell, Whinston and Green (1995), to not confuse with the econometric term identification. The latter is more general in that it may concern also the payoff structure and the decision-making model itself, and not just the underlying uncertain parameters as is the case here. 2

3 yet entirely consistent with the truth (the objective joint distribution over agents characteristics), even when one assumes that the dataset at the uninformed agents disposal is comprised of infinitely many stationary observations with no additional noise. In other words, in environments with asymmetric information, the assumption that in an adverse-selection problem the agents assessments equal the truth may be inconsistent with the informational problem of adverse selection itself. The purpose of this paper is to address such strategic environments of adverse selection, and study the problem from the perspective of an impartial econometrician who can only observe the actions taken by the players, and not the realizations of their types. The econometrician is thus much like the players themselves, and can be imagined as one of the players in the game such a player has no actual effect on the outcome and a constant payoff function. What is meant by the econometrician s observation is an infinite dataset of observations of realizations of players actions. The econometrician can therefore recover the distribution over players actions, what we call the behavioral outcome. The question then is whether this distribution over players actions could have been generated by equilibrium play of players facing the recovery problem pertinent to adverse selection. That is, to test the equilibrium hypothesis, under what conditions will the econometrician have to take into account the recovery problem that such rational players consistent econometricians face. The focus of this paper is purely positive. We provide a tight set of conditions characterizing environments with such recovery problems. These conditions pertain only to the underlying payoff structure and the actual strategy profile that is played. Our main results then derive the equilibrium outcomes and the corresponding behavior that an econometrician could observe if these conditions are satisfied. The appropriate definition of adverse equilibrium outcomes is more complicated, as it involves an infinite regress of how players justify all the privately observed parameters as arising from play of such rational 3

4 players consistent econometricians. 3 The definition of equilibrium is that players optimization and consistency of their assessments are common belief. This definition of adverse equilibrium is precisely tailored to the problem under consideration. It is devoid of any exogenous assumptions regarding how players make assessments from the infinite datasets that they are imagined to have accummulated. It is also rigorously absent of any notion of irrationality or myopia. Not surprisingly, if the players assessments were to coincide, then such equilibrium outcomes would be supportable in Bayes Nash equilibrium the common prior over uncertainty and strategies could then be imagined as such common assessments. Of course, in that case, our econometrician would not need the present theory to test the equilibrium hypothesis. We are therefore primarily interested in situations where players assessments cannot coincide in an adverse equilibrium. Our conditions thus provide a tight characterization of outcomes where Bayes Nash equilibrium would be rejected, while adverse equilibrium would in fact be true. The conditions that we provide here are of incentive- and informational nature. On the incentive side, we provide two new notions, positivity and incentive imbalance. A strategy profile is positive if for each player s type, there is a draw of the other players types such that the player s strategy is in that state a best reply to the other players strategies. Positivity is therefore an ex-post-type condition on the payoff structure and the strategy profile. Incentive imbalance is a very simple and intuitive condition guaranteeing that there is no common assessment of uncertainty, under which a given strategy profile could simultaneously satisfy the incentive constraints for all players. These two conditions provide a tight characterization of outcomes that are not supportable in Bayes Nash equilibria, yet leave open the possibility of adverse equilibrium with differing players assessments. The informational conditions 3 Here we use the word justify rather than rationalize to not confuse such justification with the standard notion of rationalizability, e.g., Bernheim (1984). As defined here, in such a justification a player considers not only the others possible behaviors but also their probabilistic consistency with the outcomes of such behaviors. 4

5 are also pertinent to the payoff structure and the strategy profile, and are called pooling and informational adversity. These guarantee that in such outcomes players cannot recover the underlying uncertainty, so that their assessments can be different. If an outcome satisfies these four conditions then it is not supportable in Bayes Nash equilibrium, and it is supportable in adverse equilibrium. In two-player games this characterization is tight. This paper was inspired by studies in econometric theory, e.g., Manski (1993, 2004). These studies raise a concern as to how an impartial econometrician should evaluate decision makers choices when decision makers face identification problems (or recovery problems). The difference is that here we study equilibrium problems, which is quite different, and we do not impose any exogenous identifying assumptions. Instead, we treat the players estimation procedures by which they make their assessments as a black box. 4 The only requirement is that such an estimation procedure satisfies asymptotic consistency, so that each agent can potentially hold a variety of consistent assessments. Players are here standard expected-utility maximizers, who play optimally under some such consistent assessment. The definition of equilibrium here can be viewed as a version of minimal equilibrium given in Čopič (2014) tailored to players in environments with adverse selection. Adverse equilibrium can thus be viewed as a stationary steady state of any learning process with no memory loss, where players make asymptotically consistent estimates of the data they observe, players optimize, and justify the observed data as arising from such other players. This definition of equilibrium could then also be viewed as a refinement of the Self-confirming equilibrium in games with uncertainty, see Dekel et al (2004). There, players are myopic in the sense that they do not justify their observations as having arised from play by rational and consistent players. Adverse equilibrium thus imposes additional constraints. In Section 6 we provide an exam- 4 The present paper was also inspired by Anderson and Sonnenschein (1985), who consider rational expectations equilibria of agents using linear econometric models. 5

6 ple demonstrating that adverse equilibrium outcomes are different from outcomes that would require justification to any finite order rather than in an infinite regress (in particular, demonstrating the difference with self-confirming equilibrium, which requires no justification at all). 5 That example also shows that our characterization is tight: in such non-adverse-equilibrium outcomes our conditions are no longer satisfied. In Section 2 we describe the basic model and the econometric problem. In Section?? we give the definition of adverse equilibrium outcomes and some relationships to Bayes Nash equilibrium outcomes. In Section 4 we give the incentive and informational conditions and our main results, theorems 1 and 2. In Section 5 we give an example illustrating these results and an additional example illustrating how to incorporate states of fundamentals of the economy in the model. In Section 5 we discuss some generalizations of our conditions and results, and give the finite-order example, which shows tightness of the statements of our theorems. Most of the proofs are in the appendix. 2 The model and the econometric problem In a game, we define the equilibrium concept, which is suited for studying situations of adverse selection, i.e., where all players and an econometrician observe the players actions in an environment with uncertainty. Since a player s strategy is a contingent plan of action, given her private information, each player s strategy is not directly observable. What is observed by each player is the joint distribution over the player s own payoffs and actions of all players, conditional on the player s own types. What 5 Adverse equilibrium is also related to Self-confirming equilibrium, Fudenberg and Levine (1993), and especially Rationalizable Partition-Confirmed Equilibrium of Fudenberg and Kamada (2013), which is defined in extensive-form games with no uncertainty, allows for correlation in players strategies, and takes into account players trembles. Less related are various notions of conjectural equilibria, see e.g., Battigalli and Guatoli (1997), and Esponda (2013), and the notions in Jackson and Kalai (1997) and Jehiel and Koessler (2008). 6

7 is observed by the econometrician is the behavioral outcome, which is the distribution over all players actions resulting from the underlying uncertainty and players contingent strategies. In this section, we begin by defining such environment, and two benchmark cases, which are closely related to the standard model of Bayes Nash equilibrium with a common prior. A game, or the payoff structure, is Γ = {N, A, Θ, u}, where N = {1,..., n} is a finite set of players, A = i N A i is a product of finite sets of players actions, Θ = i N Θ i is the finite set of states of the world, i.e., a product of finite sets of players payoff types, and u : A Θ R N is a vector of players payoff functions. The player indexed by n can be interpreted as the state of fundamentals of the economy, in which case A n {a n }, u n const., and in the payoff structure the types of the other players are interpreted as their signals. 6 S is the set of players mixed strategies. Given an s S, s[θ] (A) is the probability distribution over players actions when the draw of types is θ, and s[θ](a) is the probability of action profile a, that is, s[θ](a) = i N s i [θ i ](a i ). To emphasize that any player can be thought of also as an econometrician (or statistician), think of the disinterested econometrician (a passive observer, who records the data) as player n + 1, in which case Θ n+1 = A n+1 = 1, and u n+1 const, where const is any constant. Γ is assumed to be 6 The argument of how this is done is from Čopič (2014), and we include it here for the sake of completeness. Interpreting the state of the fundamentals of the economy as a player allows working within the simpler normal-form representation with types alone, while preserving the generality of the model that includes the states of fundamentals apart from players types or signals. The types of players 1,..., n 1 are then players signals, i.e., θ i is the signal to player i when the state of the world is θ n, with the appropriate conditional payoff consequences to player i. Under the usual Bayesian assumption of a common prior over the states of the world and the signal structure, such signaling model is reducible to the normal-form payoff-type representation in the standard way: by computing for each player the conditional expected payoffs for each vector of signals to all players. When there is no prior (and no common prior), as is the case here, such standard procedure is no longer possible, and different assessments may induce different normalform representations. When the states of fundamentals are included as a player, that allows for such normal-form representation in the present case as well. This description therefore embeds the presumably more general model with the states of fundamentals into a reduced-form framework without having to consider the states of fundamentals in any special way. In such representation, the states of fundamentals are indexed as player n rather than 0 because index 0 is here used to describe the objective outcome. See also our second example in Section 5. 7

8 common knowledge. An outcome of Γ is given by (P, s), P (Θ), s S, and an outcome realization is a draw (θ, a) (P, s); A behavioral outcome is given by a probability distribution β (A). Denote by P r P,s (Θ A) the probability distribution over types and actions in an outcome (P, s), so that, P r P,s (θ, a) = P (θ) s[θ](a), a A. Given an outcome (P, s) (Θ) S, the behavioral outcome corresponding to (P, s) is then given by β P,s P r P,s, that is,7 A β P,s (a) = θ Θ P (θ)s[θ](a), a A. For each player i, denote by V i the set of i s possible payoffs, i.e., V i Image(u i ), and let V = i N V i. In a game Γ, fix an outcome (P, s) (Θ) S, and imagine infinitely many independent realizations of the outcome (P, s). Let ψ(.) be some statistic of the outcome (P, s), where ψ maps onto some parameter space Ψ, so that ψ : Θ A Ψ. An ideal dataset is comprised of infinitely many independent observations of statistic ψ. Let P,s;T P ˆ r ψ be the empirical distribution of T independent observations of the statistic ψ, i.e., T independent realizations ψ(θ t, a t ), t {1,..., T }, where (θ t, a t ) (P, s). Let P ˆ r P,s ψ observations tends to, be the empirical distribution of ψ in the limit, as the number of Pˆ r P,s ψ = lim T Pˆ r P,s;T ψ. 7 Throughout, the marginal distribution of some probability distribution P r over a set X is denoted by P r X, and the conditional is denoted by P r X. 8

9 By the Kolmogorov s strong law of large numbers, Pˆ r P,s ψ P r P,s ψ. Here the statistic ψ will be of the form of a projection onto the parameter space Ψ. It will then be convenient to use Ψ in the notation. In particular, if ψ(θ, a) = (θ i, a, u i (θ, a)), then Ψ = Θ i A V i, and P r P,s ψ simply the marginal probability distribution P r P,s Θ i A V i is then in a slight abuse of notation note that P r P,s is a probability distribution over Θ A, which naturally induces a probability distribution over Θ A V. As mentioned above, we imagine an econometrician simply as player n + 1. The econometrician assembles an ideal dataset of observations of statistic ψ n+1 of the outcome (P, s), and then adjudicates whether these observations could have been generated by equilibrium play. Similarly, all players assemble ideal datasets of observations ψ i, i N, and form consistent assessments. From such an assessment a player can then deliberate the optimality of play of herself as well as others. In the rest of this section we consider two benchmark cases. The first case is one where ψi BN (θ, a) = (θ, a), so that Ψ BN n Θ A, i N {n+1}. That is, all players and the econometrician assemble ideal datasets of complete observations of realizations of the outcome. In this first benchmark case, as the number of observations T, if each player uses any estimation procedure, which satisfies asymptotic consistency, then it must be that their assessments equal the objective outcome. In other words, there is no recovery problem as the observations of the realizations of the outcome are complete. In equilibrium, players must then optimize relative to the true objective outcome. Moreover, each player can a fortiori justify the other players behavior: a player i can attribute to player j the assessment over uncertainty given by (P, s), whereby player j optimizes, and i can further ascertain that j could similarly attribute to any 9

10 other player k the assessment (P, s), whereby k optimizes, and so on. That is, there is a common belief that players optimize and their assessments are fully consistent with (P, s). Since each player s assessment is (P, s), and each player optimizes, an equilibrium outcome therefore coincides with a Bayes Nash equilibrium outcome. In the second benchmark case, ψ i (θ, a) = (θ, a), for all i N, but ψ n+1 (θ, a) = a, or equivalently, ψ n+1 (θ, a) = (θ n+1, a, u n+1 (θ, a)) (recall that a disinterested econometrician only has one type and plays one action). By the same argument as before, the behavior of players is still supportable in Bayes Nash equilibrium, but the econometrician can no longer necessarily recover the true objective outcome. We can thus reinterpret Bayes Nash equilibrium to encompass the two benchmark cases. Given a (P, s) (Θ) S, denote by U P,s i (θ i, a i ) the expected utility of player i, when she is of type θ i and plays action a i, U P,s i (θ i, a i ) = θ i Θ i,a i A i u i (a i, a i, θ i, θ i )P (θ) j i s j [θ j ](a j ). Definition 1. A (P, s) (Θ) S is a Bayes-Nash equilibrium outcome, if, U P,s i (θ i, a i ) U P,s (θ i, a i), i a i, a i A i, s.t., s i [θ](a i ) > 0, θ i Θ i, s.t., P (θ i ) > 0, i N. A strategy profile s is supportable in Bayes-Nash equilibrium if there exists a P (Θ), such that (P, s) is a Bayes-Nash equilibrium outcome. A behavioral outcome β (A) is supportable in Bayes-Nash equilibrium, if there exists a (P, s) (Θ) S, such that, (P, s) is a Bayes-Nash equilibrium outcome, and β = β P,s. The subject of the next section is the case of interest, where each player also has the ideal dataset of only partial observations of the outcome, and the econometrician only observes the behavioral outcome. Then players, too, are facing a problem of recovering the true outcome (P, s). A player has a greater scope of assessments that 10

11 are consistent with her observations and she may no longer be certain regarding the strategy profile of the other players. Therefore, each player also has a greater scope of assessments that justify each other s behavior. We now make the formal construction that corresponds precisely to this situation. From now on we consider the problem from the perspective of the econometrician so that our focus will be on what outcomes and consequent behavioral outcomes are supportable in equilibrium. 3 Adverse equilibrium In the case of interest, apart from observing the other players behavior (the distribution over their actions), each player also recovers the probability distribution over her payoffs for each draw of her type, and for each action that she plays. Therefore, Ψ AE i = Θ i A V i, i N. Since a player s payoffs may vary with the other players types, a player can thus gain some additional information. Instead of focusing on the statistic Ψ AE i, we from now on focus simply on players information. Formally, a player s information is represented by the σ-algebra of events that she can discern given the actual strategy profile played a player can only obtain information from the outcome realizations that have a positive probability. Given an outcome (P, s) (Θ) s, denote by P r P,s Θ i A (θ i, a) the probability of θ i Θ i, a A, that is, P r P,s Θ i A (θ i, a) = P Θi (θ i ) s i [θ i ](a i ) j i s j [θ j ](a j ). θ j Θ j For θ i Θ i, a A, such that, P r P,s Θ i A (θ i, a) > 0, let V P,s θ i,a = {u θ i,a(θ i ) θ i Θ i, P Θi (θ i ) s i [θ i ](a i ) j i s j [θ j ](a j ) > 0}. Then, the information of i, conditional on her type θ i and action profile a A, is 11

12 given by F P,s θ i,a, the σ-algebra over Θ A generated by the events, {u 1 θ i,a (v) {(θ i, a)} v V P,s θ i,a } ((Θ i A) \ {(θ i, a)}) Θ i. For θ i Θ i, a A, such that, P r P,s Θ i A (θ i, a) = 0, define F P,s θ i,a information of player i in an outcome (P, s) is given by, = {Θ A, }. The F P,s i,a = θ i Θ i,a AF P,s θ i,a. Definition 2. Given (P, s), (P, s ) (Θ) S, (P, s ) is i A-consistent with (P, s), if, (θ,a) E P (θ) i N s i[θ i ](a i ) = (θ,a) E P (θ) i N s i [θ i ], E F P,s i,a. An assessment (P, s ) that is i A-consistent with (P, s) can be thought of as an assessment that player i made from her ideal dataset using some asymptotically consistent procedure. That is, (P, s ) is i A-consistent with (P, s), if and only if, P r P,s (E) = P r P,s Ψ AE i Ψ AE i (E), E F P,s i,a. However, we make no assumption regarding precisely what procedure a player used there is no more information that can be discerned from the ideal dataset barring exogenous assumptions. Thus, there is a scope of different assessments that are i A-consistent with (P, s). Definition 3. For a given θ i Θ i, (P, s) satisfies θ i -IC, if P Θi (θ i ) > 0 implies that, U P,s i (θ i, a i ) U P,s i (θ i, a i), a i, a i A i, s.t., s i [θ](a i ) > 0. A (P, s) (Θ) S is optimal for i, or satisfies i-ic, if (P, s) satisfies θ i -IC, θ i Θ i. The description of equilibrium behavior is now slightly more involved. Since players do not know the underlying distribution over uncertainty, nor other players 12

13 strategies, they make assessments regarding the outcome. In order to justify the other players behavior, each player also makes higher order assessments, which describe assessments that a player imputes on the other players. 8 A construct is an infinite hierarchy of such assessments. A first-order assessment of player i is denoted by (P (i), s (i) ) (Θ) S. For k N, let N k be the set of all sequences of length k with elements from N; for k = 0, define N 0 =. Let L i be the set of all finite sequences of integers from N, such that the first element is i, and no two adjacent elements are the same. That is, L i = {l N k ; k 1, l 1 = i, l m+1 l m, 1 m < k}. Also, let L = {l N k ; k 1, l m+1 l m, 1 m < k}, so that L = i N L i. For l L, define first(l) as the first element of l, last(l) as the last element of l. Let (P 0, s 0 ) (Θ) S be the objective outcome. Player i s construct is a hierarchy of i s assessments, C i = (P l, s l ) l L i [ (Θ) S] Li. A profile of constructs is given by C = (C 1, C 2,...C n ). Given l L, first(l) = i, C (l) = (P (l), s (l) ) is i s l-th order assessment. For example, when l = (i), C (i) is the assessment of player i about the outcome (P 0, s 0 ). We adopt the convention that P = P 0 and s = s 0, so that when l =, C (l) is the outcome of the game. An objective outcome (P, s) is an adverse equilibrium outcome if at (P, s) each player can hold a consistent assessment such that she optimizes, and such that she can also believe that other players hold consistent assessments under which they optimize, and so on, ad infinitum. That is, (P, s) is an adverse equilibrium outcome if it admits a common belief in consistency and optimality. 8 Here we define all assessments in the infinite hierarchy only over payoff-relevant uncertainty. This is without loss of generality, see Čopič (2014), Theorem 3. As in Čopič (2014), the equilibrium here can be thought of as a stationary steady state of any learning process with no memory loss of sophisticated rational players. 13

14 Definition 4. An outcome (P 0, s 0 ) is an adverse equilibrium outcome of Γ if there exists a supporting profile of constructs C, s.t., 1. (P l, s l ) satisfy j-ic, for j = last(l), for every l L, 2. (P (i), s (i) ) is i A-consistent with (P 0, s 0 ), for every i N, and, 3. (P (l,j), s (l,j) ) is j A-consistent with (P l, s l ), for every j N, for every l L. A strategy profile s 0 is supportable in adverse equilibrium if there exists a P (0) (Θ), s.t., (P 0, s 0 ) is an adverse equilibrium outcome. A behavioral outcome β (A) is supportable in adverse equilibrium, if there exists an adverse equilibrium outcome (P 0, s 0 ), s.t., β β P 0,s 0. We now briefly outline some relevant relationships between the outcomes that are supportable in adverse equilibrium and the two benchmark cases of the previous section. Proposition 1. Let (P 0, s 0 ) be an adverse equilibrium outcome. Then, s 0 is supportable in Bayes Nash equilibrium, if and only if, there exists a supporting profile of constructs C for (P 0, s 0 ), C = (P l, s l ) l L, such that, (P (i), s (i) ) = (P (j), s (j) ), i, j N. Proof. Let (P 0, s 0 ) be a Bayes Nash equilibrium outcome. Then let (P l, s l ) = (P 0, s 0 ), l L. This profile of constructs satisfies the requirements of Definition 4. In particular, (P (i), s (i) ) = (P (j), s (j) ), i, j N. For the converse, let (P 0, s 0 ) be an adverse equilibrium outcome, with (P (i), s (i) ) = (P (j), s (j) ), i, j N. Denote (P (i), s (i) ) = ( P, s). By the requirement 1 of Definition 4, the outcome ( P, s) is a Bayes Nash equilibrium outcome. By the requirement 2, P Θi P 0 Θ i and s i s 0 i, i N. Hence, s s 0, and the claim follows. Next, we provide a sufficient condition for the first-order assessments from Definition 4 to coincide with the objective outcome. One may call such equilibrium 14

15 outcomes fully revealing. 9 Define a conditionally-generic payoff structure as one where for each type of each player and for each profile of actions, the payoff to that player takes a different value for every profile of types of the other players. Definition 5. Let V θi,a = {u θi,a(θ i ) θ i Θ i }. A payoff structure Γ is conditionally generic if u θi,a : Θ i V θi,a, is one-to-one, θ i Θ i, a A, i N. Proposition 2. Let the payoff structure Γ be conditionally generic and let (P 0, s 0 ) be an adverse equilibrium outcome. Then: 1. Every assessment in the supporting constructs from Definition 4 coincides with the objective outcome, (P l s l ) (P 0, s 0 ), l L. 2. (P 0, s 0 ) is a Bayes Nash equilibrium outcome. Proof. To prove 1, take an i N, θ i Θ i, and a A, such that, P Θi (θ i ) s i [θ i ](a i ) j i s j [θ j ](a j ) > 0. θ j Θ j Since u θi,a : Θ i V θi,a, is one-to-one, it follows that for each v V θi,a, u 1 θ i,a (v) = {θ i }, for some θ i Θ i. Therefore, F P,s θ i,a is generated by the events {(θ i, a)} ((Θ i A) \ {(θ i, a)}) Θ i, so that F P,s i,a is generated by singletons. Hence, P,i Θ i θ i = P Θ i θ i 2 follows from 1 and Proposition 1. and s,i i s i. 9 That is in the tradition of the literature on the Rational Expectations equilibrium, see e.g., Radner (1982). However, it should be pointed out that that literature assumes a common prior over the fundamental parameters, or, in the present sense, that economic agents assessments coincide. 15

16 4 Information and Incentive conditions of adverse equilibrium outcomes In this section we parse the incentive and informational conditions that describe equilibrium behavior. Due to the recovery problem, adverse equilibrium outcomes satisfy weaker conditions than the Bayes Nash equilibrium outcomes. In this main section of the paper, we address the question under precisely what conditions an outcome is supportable in adverse equilibrium, while it is no longer supportable in Bayes Nash equilibrium. Our answer is a characterization of informationally-adverse environments, that is, environments where due to equilibrium behavior players are unable to recover the underlying distribution of uncertainty. Of course, some adverse equilibrium outcomes are still supportable in Bayes Nash equilibrium if the econometrician assumed a common prior, she could explain the players behavior even if such assumption wasn t necessarily true. Other such outcomes are no longer supportable in Bayes Nash equilibrium but are nonetheless supportable in adverse equilibrium. Our primary concern are the outcomes of this latter sort. Those are precisely the outcomes where, if the econometrician assumed Bayes Nash equilibrium, she would falsely reject the equilibrium hypothesis. Our first step is to characterize conditions under which a strategy profile s is not supportable in Bayes Nash equilibrium of Γ but which still leave open the possibility that s be supportable in adverse equilibrium of Γ. For instance, given a strategy profile of the other players, a strategy for player i which is not a best reply for any type draw will evidently not be supportable in Bayes Nash equilibrium of Γ. But such a strategy of player i will also not satisfy incentive constraints for i. We first provide a sufficient condition, which guarantees that a strategy profile is not supportable in Bayes Nash equilibrium of Γ, but which rules out strategies that are never best replies. 16

17 Let, Θ s,i = {θ i Θ i u i (θ i, θ i, s i, s i ) u i (θ i, θ i, s i, s i ) 0, s i S i }, and let Θ s,θ i = Θ s,i [{θ i } Θ i ]. Thus, Θ s,i Θ is the set of draws of types θ for which i would not have any incentive to deviate from s i to some other strategy s i, given s i ; and Θ s,θ i is the section of Θ s,i when player i s type is θ i. Definition 6. A strategy profile s is positive, if Θ s,θi, θ i Θ i, i N. Thus, a positive strategy profile might allow for player i to best-reply to other players strategies if P (Θ) put sufficient mass on Θ s,i. Proposition 3. If a strategy profile s S is supportable in Bayes Nash equilibrium then s is positive. Proof. If s is not positive, then there is an i N and a θ i Θ i, s.t., u i (θ i, θ i, s i, s i ) u i (θ i, θ i, s i, s i ) 0, s i S i. Therefore, no matter the conditional distribution P Θ i θ i, when player i is of type θ i she would have an incentive to deviate. Thus, s is not supportable in a Bayes Nash equilibrium of Γ. The reverse implication is not true: a strategy profile s may be positive and it may nonetheless not be supportable in Bayes Nash equilibrium. Proposition 4. If a strategy profile s is not supportable in Bayes Nash equilibrium of Γ, then for every θ Θ, there exists an i N, s.t., u i (θ, s i, s i ) u i (θ, s i, s i ) > 0. Proof. If there is a θ Θ, s.t., u i (θ, s i, s i ) u i (θ, s i, s i ) 0, i N, then set P (θ) = 1, and s is supportable in a Bayes Nash equilibrium of Γ. 17

18 Corollary 1. If a strategy profile s is positive and is not supportable in Bayes Nash equilibrium of Γ, then i N Θ s,i =. A sufficient condition for a strategy profile to not be supportable in Bayes Nash equilibrium must also take into account the magnitude of players incentives. To do that in a tractable way, the sufficient condition we provide here also slightly strengthens the necessary condition given by Corollary 1 to empty pair-wise intersections. For a two-player game, along with the strategy profile being positive, the next condition is then necessary and sufficient. Take a game Γ, and suppose s S is positive. For each θ i Θ i, define α θ i,s as the minimal weight that must be assigned to Θ s,θ i, such that θ i -IC may still be satisfied. Formally, α θ i,s min{p Θ i θ i (Θ s,θ i ) (P, s) satisfies θ i IC}. Definition 7. A strategy profile s S is incentive-imbalanced in Γ if, Θ s,i Θ s,j =, i, j N, and, ( i N θ i Θ i α ) θ i,s > 1. 1 Θ i Proposition 5. Let s be positive and incentive-imbalanced in Γ. Then s is not supportable in Bayes Nash equilibrium of Γ. Proof. Assume (s, P ) is a Bayes Nash equilibrium outcome of Γ. If Θ s,i Θ s,j =, i, j N, this implies that i N 1 Θ i (P, s), satisfies i-ic for all i N, which is a contradiction. ( θ i Θ i P θi (Θ s,θ i ) ) 1, since As an observation, positivity and incentive imbalance together have specific geometric implications on Γ and s. First, in order for a strategy profile to satisfy both of these properties, it must be that Θ i > 1, i N. Otherwise, it is impossible to simultaneously satisfy Θ s,i Θ s,j =, i, j N, and Θ s,θi, θ i Θ i, i N. Thus, to satisfy both, positivity and incentive imbalance, asymmetric information must be two sided. Second, for a similar reason, it must be that for some players, 18

19 Θ s,i is not a connected subset of Θ. That is most easily seen when N = 2. Our main example of Section 5 of course satisfies these geometric properties. Our next step is to provide conditions, which guarantee that a given strategy profile s can be supported in adverse equilibrium, while possibly satisfying incentive imbalance. These conditions must be of informational nature in conditionallygeneric environments, by Proposition 2 players must in equilibrium correctly recover the true underlying uncertainty. Then, such an outcome would have to be supportable in Bayes Nash equilibrium outcome, contradicting Proposition 5. The following conditions therefore guarantee that in equilibrium players can hold different assessments of uncertainty. We focus on strategy profiles satisfying strong informational requirements. In Section 6 we indicate to what extent these requirements can be relaxed. Given a strategy profile s S, define for each i N, A s, i = θi Θ i support(s i [θ i ]), and let A s, = i N A s, i. Definition 8. Take a game Γ and a player i N. A strategy profile s S is pooling for i, if, u i (θ, a i, a i ) = u i (θ, a i, a i ), a i, a i A s, i, a i A s, i, θ Θ. (1) A strategy profile s is pooling if it is pooling for all i N. A strategy profile s is pooling for i, if for any profile of actions a i that other players play under s, i is for any of her types indifferent between all the actions that she takes under s i. In particular, if one of these actions is optimal for a given θ i Θ i, then all the actions in A s, i over A s, i, when she is of type θ i. are optimal for θ i, and i could play any mixed strategy 19

20 Definition 9. Take a game Γ. A strategy profile s S is informationally adverse, if, u i (θ i, θ i, a) = u i (θ i, θ i, a), a A s,, θ i Θ i, θ i, θ i Θ i, i N. (2) A strategy profile s is informationally adverse if no player can recover any information regarding the conditional distribution over the other players types from the variation in her own payoffs. That is the case when for each of her types and actions that she takes, her payoffs do not vary with the other players types. We can now provide our main results. Theorem 1. Take a game Γ and let s S be positive, pooling, and informationally adverse. Then, (P, s) is an adverse equilibrium outcome of Γ, for any P (Θ). Combining Theorem 1 and Proposition 5 we obtain the next corollary. Corollary 2. If in addition to the requirements in Theorem 1, s is incentive imbalanced, then s is supportable in adverse equilibrium of Γ, and s is not supportable in Bayes Nash equilibrium of Γ. The next theorem describes behavioral outcomes that are supportable in adverse equilibrium but not in Bayes Nash equilibrium. Theorem 2. A behavioral outcome β (A) is supportable in adverse equilibrium, and is not supportable in Bayes Nash equilibrium of Γ, if there exists an outcome (P, s), s.t., s is pooling, informationally adverse, positive, incentive imbalanced, and β = β P,s A. Finally, we have the following proposition. 20

21 Proposition 6. Suppose that s 0 S is pooling, informationally adverse, and (s 0, P 0 ) is an adverse equilibrium outcome of Γ. Then, for every P (Θ) and s S, if, P r P 0,s 0 A P r P,s A, (3) then outcome (P, s) is supportable in adverse equilibrium of Γ. Moreover, for every P (Θ), there exists an s satisfying (3). Proposition 6 is of independent interest. Suppose a certain probability distribution over players actions can arise in equilibrium play; An econometrician only observes the marginal distribution over players actions. By the last claim in Proposition 6, under pooling and informational adversity, econometrician would have no way of pinning down the underlying objective distribution over Nature s moves. Hence, barring the possibility to somehow gather additional information about P 0, there is little scope for normative prescriptions. 5 Examples In this section we provide two examples. Our first example illustrates our main results. The example here is perhaps somewhat more complicated than would be minimally required. All of the complication is to show the full scope of the results of Section 4. Each player i has 2 types, denoted by θ i and θ i, and 3 actions, denoted by a i, a i, and a i. We will focus on the outcomes where each player puts zero probability mass on action a i, so that the payoffs associated with that action are relevant only in terms of the deviations from other actions. All payoffs that are not important when considering deviations from the other actions are chosen to be negative and constant, equal to 1. The payoff structure is given by the four payoff matrices below. 21

22 (θ 1, θ 2 ) a 2 a 2 a 2 a 1 1, 4 2, 4 1, 6 a 1 1, 0 2, 0 1, 2 a 1 0, 1 1, 1 1, 1 (θ 1, θ 2) a 2 a 2 a 2 a 1 1, 3 2, 3 1, 2 a 1 1, 1 2, 1 1, 2 a 1 3, 1 4, 1 1, 1 (θ 1, θ 2 ) a 2 a 2 a 2 a 1 2, 4 1, 4 1, 3 a 1 2, 0 1, 0 1, 2 a 1 4, 1 3, 1 1, 1 (θ 1, θ 2) a 2 a 2 a 2 a 1 2, 3 1, 3 1, 5 a 1 2, 1 1, 1 1, 3 a 1 1, 1 0, 1 1, 1 Consider the strategy profile given by, s i = (s i [.](a i ), s i [.](a i)), where s i [.](a i ) + s i [.](a i) = 1, i.e., each player i for each of her types mixes between a i and a i. We first verify that this strategy profile satisfies pooling and informational adversity. The actions that player i plays under s are A s, i = {a i, a i}. When player i is either of type θ i or θ i, for each of her actions, her payoff does not vary with the other players actions, so that s is pooling for both players, i.e., s is pooling. Next, for each profile of actions a A s,, each player s payoff does not vary with the other players types, so that s satisfies informational adversity. Note also that each player i s payoff does not vary with her own actions (from A s, i ), for each of her types, and for each action played by the other player. Due to pooling and informational adversity, this is of course a necessary condition for s to be an optimal strategy (but not necessarily sufficient). Next, we provide conditions on s i [.] under which s is positive. For player 1, positivity imposes no restricitions on s: for θ 1, player 1 has no incentives to deviate from 22

23 s 1 at θ 2 ; for θ 1, 1 has no incentive to deviate at θ 2. Hence Θ s,1 = {(θ 1, θ 2 ), (θ 1, θ 2)}, with no additional restrictions on s. on s: at θ 2 it can only be that Θ 2,θ 2 For player 2, positivity imposes restrictions = {(θ 1, θ 2 )}, and it must additionally be that s 1 [θ 1 ](a 1 ) s 1 [θ 1 ](a 1); at θ 2 it can only be that Θ 2,θ 2 = {(θ1, θ 2 ) }, and then it must be that, s 1 [θ 1](a 1 ) 2s 1 [θ 1](a 1). Hence, Θ s,2 = {(θ 1, θ 2 ), (θ 1, θ 2)}, with the additional restrictions, s 1 [θ 1 ](a 1 ) s 1 [θ 1 ](a 1) and s 1 [θ 1](a 1 ) 2s 1 [θ 1](a 1). (4) In particular, s 1 [θ 1 ](a 1 ) [ 1 2, 1] and s 1 [θ 1](a 1 ) [ 2 3, 1]. To verify incentive imbalance of s, first observe that Θ s,1 Θ s,2 =. Next, to satisfy the incentive constraints of each type, the minimal weight α θ i,s on Θ s,θ i each type given by, is for Thus, α θ 2,s = [ s 1 [θ 1 ](a 1 ) α θ 2,s = i N so that as s is incentive imbalanced. 2 3s 1 [θ 1](a 1 ) ( ) 1 α θi,s 4 Θ i 3, θ i Θ i α θ 1,s = 2 3, (5) α θ 1,s = 2 3, (6) ] 3, 1, (7) [ ] 2 3, 1. (8) Therefore, by Theorem 1 and its Corollary 2, s is supportable in adverse equilibrium and it is not supportable in Bayes Nash equilibrium. By Theorem 2, the corresponding behavioral outcomes that are supportable in adverse equilibrium are given by β, such that its marginal over A 1 satisfies β A 1 (a 1 ) 1 2 and β A 1 (a 1) = 0, its marginal over A 2 satisfies β A 2 (a 2) = 0 but is otherwise unrestricted, and β β A 1 β A 2. 23

24 Our second example is very simple. It is intended to illustrate how to include the states of the fundamentals of the economy in the game, even while that is in a way evident. 10 The example also shows the distinction between a Bayes Nash equilibrium outcome and a strategy being supportable in Bayes Nash equilibrium. In this example, there are 2 players, where player 2 represents the states of the fundamentals of the economy player 2 has one action, called O, and player 1 has 2 actions, up or down. Each player has 2 types. The types of player 2 represent the state of the fundamentals of the economy, and the types of player 1 are the corresponding signals to player 1. The states are given by the low state θ 2 and the high state θ 2, and the corresponding signals to player 1 are θ 1 and θ 1. For the sake of the example, assume that the signals are entirely uninformative of the state of the fundamentals, i.e., the objective distribution P is uniform over Θ. The payoff structure is given by the four matrices below (const. is any constant), ( θ 1, θ 2 ) O ( θ 1, θ 2 ) O up 0, const. up 3, const. down 1, const. down 1, const. (θ 1, θ 2 ) O up 0, const. down 1, const. (θ 1, θ 2 ) O up 3, const. down 1, const. This example is therefore effectively a decision problem entertained by player 1. Depending on player 1 s strategy, i.e., the action that she plays for each of her signals, she may or may not be able to recover the underlying objective distribution over the uncertainty. In fact, if s (up, O), then (s, P ) is a Bayes Nash equilibrium 10 Still, it is important to keep in mind that if signals are purely informational, so that there is no taste-shock component in the signal, then the player s preferences depend only on the state of the fundamentals. One could of course also explicitly include a taste-shock component in the signal, however, that would be unnecessary as such case could always be represented by simply adding additional states of the fundamentals with the appropriate conditional distribution over the players signals. 24

25 outcome. But if player 1 plays down regardless of her signal, and makes an assessment that her signals are informative but pessimistic, then her incentive constraints may still be satisfied. Additionally, since in that case for each of her signals, player 1 cannot recover the conditional distribution over Θ 2, such assessments can satisfy 1 A-consistency. For instance, one can set player 1 s assessment over Θ at every order to P, P θ 2 θ2 θ θ We evidently do not have to explicitly consider either the incentives or the assessments of player 2. Therefore, the outcome (P, s), where s 1 down, and s 2 O, is supportable in adverse equilibrium of this game. Here s is also supportable in Bayes Nash equilibrium, i.e., (P, s) is a Bayes Nash equilibrium outcome, while (P, s) is not. One interpretation of this example is that as a static equilibrium version of a two-armed bandit problem as in Rothschild (1974). Therefore, player 1 might not have incentives to ever play the action other than down, due to her consistent assessments, even while that might have turned out to be beneficial to player 1. 6 Discussion We now briefly discuss some of the possible extensions of theorems 1 and 2. The main assumptions in both of these results are the informational assumptions of pooling and informational adversity. These assumptions guarantee that the players can neither recover more information due to directly observable action profiles or payoffs, nor can they recover more information from strategic considerations. These assumptions are slightly stronger than necessary. For example, informational adversity could also be partial, in which case a player would not be able to recover the conditional 25

26 distribution over the other players types only for some subset of the other players types. Similarly, pooling could be partial, to the extent that there existed at least one action that were optimal for all types of a player. Then, each player might be able to partly recover the conditional distribution over the other players types from the behavior of these other players, but there would be some residual recovery problem e.g., in a two-player game, the player would be unable to determine which type of the other player played the one action that all types could play, so that the player would be unable to invert the observed probability of play of that action. We do not formally describe these extensions here as that would require a substantial amount of additional notation and definitions. 11 One important point, which we here discuss in some more detail, is that the informational conditions as given in theorems 1 and 2 must hold across all types (for all actions that are played). The reason for that can be found in the proofs of these results: in order for players to be able to justify one-another s behavior in an infinite regress, at each level of reasoning, the appropriate (minimal) amount of probability mass must be assigned to the positive sets of player under consideration. Therefore, the probability mass must be assigned and reassigned between these different positive sets in a way that satisfies i A-consistency. If informational adversity and pooling do not hold accross all types and actions that are played, then it is no longer possible to do so ad-infinitum. Thus, while the informational conditions depend only on the payoff structure and the strategy profile, the reasons why these conditions are as they are stem from the infinite regress of common belief. We illustrate this with the following example. 12 Consider the game given by the payoff structure below, and consider the strategy 11 To write down results analogous to theorems 1 and 2, one would also have to specify conditions such that on the set of types where there is no recovery problem, the resulting behavior is supportable in Bayes Nash equilibrium. The combination of these conditions is not too difficult but requires a large amount of additional notation, and the statements themselves are rather lengthy. 12 This example is similar to an example in Čopič (2012). 26

27 profile given by s 1 [θ 1 ] = s 1 [θ 1] up and s 2 [θ 2 ] = s 2 [θ 2] R, i.e., player 1 plays up regardless of her type, and player 2 plays R regardless of her type. (θ 1, θ 2 ) L R up 2, 2 1, 1 down 0, 0 0, 0 (θ 1, θ 2 ) L R up 1, 0 0, 1 down 1, 1 1, 2 (θ 1, θ 2) L R up 1, 1 0, 2 down 1, 1 2, 3 (θ 1, θ 2) L R up 3, 3 0, 2 down 2, 1 1, 0 It is easily verified that s is positive, where Θ s,1 = {(θ 1, θ 2 ), (θ 1, θ 2)} and Θ s,2 = {(θ 1, θ 2), (θ 1, θ 2 )}, and that s is incentive imbalanced. Of course, s is also pooling since each player only plays one action. However, s is not informationally adverse: it fails for type θ 1 of player 1, since u 1 (θ 1, θ 2, up, R) = 1 and u 1 (θ 1, θ 2, up, R) = 0. For the sake of the argument assume that the distribution over Nature s moves P stacks al the probability mass on type draw (θ 1, θ 2 ), i.e., P (θ 1, θ 2 ) = 1, so that P can be represented by the following matrix, P θ 2 θ 2 θ θ Now we will explicitly construct hierarchies of supporting assessments, to the highest possible order: at every order, the next order of the supporting assessments will be as close as possible to the previous order. We will show that at some order, it becomes impossible to simultaneously satisfy θ 1 -IC and i A-consistency with the previous order. Since the assessments in the hierarchy will be as close as possible to each other, that will imply that the above s cannot be supported in an infinite 27

28 hierarchy. Since s is such that each player only plays one action, in order for i A- consistency to hold, the strategy profile s must here effectively be common knowledge, i.e., s l s, l L. It is therefore enough to specify only the supporting assessments over Θ. First, (P, s) satisfies 1-IC but does not satisfy 2-IC, which fails for the type θ 2. We therefore set P (1) = P. To satisfy 2-IC along with 2 A-consistency, P (2) must assign at least equal probabilities to type draws (θ 1, θ 2 ) and (θ 1, θ 2 ), and by 2 A- consistency it must assign marginal probability 0 to type θ 2. Hence, the following P (2) is on the boundary of admissible first-order supporting assessments for Player 2, P (2) θ 2 θ 2 θ θ By this same argument, it must be that P (12) = P (2), and by a similar argument, applied to type θ 1 of player 1, the closest admissible supporting assessment P (21) to P (2) is given by, P (21) θ 2 θ 2 θ θ Similarly, at order 3, P (121) = P (21), and P (212) is given by, P (212) θ 2 θ 2 θ θ At order 4, P (1212) = P (212), and P (2121) is given by, 28