Abstract We study evolution in Bayesian supermodular population games. We define such games in the context of large populations of agents and establish the existence of a maximum and a minimum pure strategy Nash equilibria which are monotone in types. To study evolution in such games, we introduce Bayesian perturbed best response dynamics and the corresponding aggregate perturbed best response dynamic. Using the theory of cooperative dynamical systems, we show that solution trajectories of the aggregate perturbed best response dynamic converge to the set of perturbed equilibrium distributions. Using results from Ely and Sandholm (2005), we then conclude that the L1 solution trajectories of the Bayesian perturbed best response dynamic also converge to the set of Bayesian perturbed equilibria. Keywords: Supermodular Games; Bayesian Population Games; Perturbed Best Response Dynamics. JEL classification: C72; C73.

∗

I thank Bill Sandholm and two anonymous referees for their comments and suggestions on an earlier version of this paper. All errors remain my own. † Institute for Financial Management and Research, 24 Kothari Road, Nungambakkam, Chennai, 600034, India. email: [email protected]

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1

Introduction

The objective of this paper is to study evolution in Bayesian supermodular population games. Following the pioneering work in Topkis (1979), Milgrom and Roberts (1990) and Vives (1990), supermodular games have attracted wide attention from economists. A wide range of economic problems has been modeled as supermodular games.1 Supermodular games are characterized by a natural ordering of strategies. Consequently, the set of equilibria in supermodular games has certain attractive properties. For example, the set of equilibria is bounded by minimal and maximal equilibria, both in pure strategies. Moreover, the monotonicity of best responses, which is a consequence of the strategy ordering, ensures that a variety of learning schemes converge to the set of equilibria (Milgrom and Roberts, 1990, Vives, 1990). Hence, equilibrium prediction in supermodular games is very robust. The large population version of supermodular games has been studied in Hofbauer and Sandholm (2007) and Sandholm (2011). A variety of economic problems, particularly macroeconomic problems like search and consumption, can be more naturally modeled as large population supermodular games.2 Hofbauer and Sandholm (2007) studies evolution in supermodular games. They show that under perturbed best response dynamics3 , the set of equilibria in supermodular games is globally asymptotically stable. While the technical details are fairly involved, it is the same monotonicity of best responses that drives convergence to equilibria in the large population models that also drives convergence of learning schemes in the finite player supermodular games. In this paper, we extend the analysis in Hofbauer and Sandholm (2007) to supermodular games with diverse preferences. We call these games Bayesian supermodular population games. In common preference games all members of a population have the same payoffs. But in Bayesian games, payoffs within the same population vary with the type of the player. The framework for Bayesian population games has been established in Ely and Sandholm (2005). We use the same framework to define Bayesian supermodular population games. Individual players play Bayesian strategies, which are functions from the set of types to the set of mixed strategies of the player. The expectation of the Bayesian strategies with respect to the measure of types generates the aggregate social state. We define Bayesian supermodular population games as games whose payoffs satisfy two characteristics: increasing difference with respect to actions and social states for all types; and increasing difference with respect to actions and types, for every social state. Economics has benefited immensely by using the theory of Bayesian games to model of situations where uncertainty or diverse preferences are inalienable elements. Consistent with this approach, it would be natural and interesting to postulate the existence of players of diverse types while 1

Economic applications of supermodular games include: industrial organization games (various examples appear in Milgrom and Roberts, 1990); many macroeconomic models in which investment and production decisions have complementarities (Cooper and John, 1988; Murphy, Schleifer and Vishy, 1989; Cooper, 1999); games with network externalities (Sundararajan, 2004). 2 See Sandholm (2011) for such examples. 3 Perturbed best response dynamics are based on the idea that players respond by playing a best response to a noisy approximation of their true payoffs. Hence, we obtain a smoothed version of the best response dynamic. See Sandholm (2011) for a detailed discussion on these dynamics.

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modeling economic situations as supermodular population games. For example, in search models, one can allow for different agents having different search costs. Or in oligopoly games, different firms may face different costs.4 Moreover, it is straightforward to incorporate such diverse types in a way that is economically meaningful while retaining the supermodular structure of the model, albeit now in the Bayesian sense. The returns, on the other hand, that one can obtain in the form of richer conclusions from extending the model in this way are very attractive. It is this consideration that forms the basis of our analysis of Bayesian population supermodular games. For finite player games, Bayesian supermodular games have been studied by Athey (2001), McAdams (2003), Reny (2005), and Van Zandt and Vives (2007). They focus on characterizing of the set of equilibria of these games. In the Bayesian context, they recover the main conclusion of the early papers on common preference supermodular games–the existence of a minimal and maximal equilibria in pure strategies for all types. We are able to prove a similar result in the large population context. The method of proof we use, that of Cournot tatˆonnement is closest to that used in Van Zandt and Vives (2007). We show that the minimal and maximal equilibria, both in pure strategies, are monotonic in types. These results are presented in Theorems 4.4 and 4.5. In large population models, it is unrealistic to expect immediate coordination on any equilibrium given the level of knowledge and cognitive sophistication that agents would require to possess for such coordination. Such models are therefore more amenable to analysis using methods of evolutionary game theory. Under this approach, we seek to determine if from a variety of initially arbitrary distribution of strategies, the population converges to some equilibrium under some behaviorally credible dynamic process. If indeed this is the case, then we may conclude that the set of Nash equilibria in such games is behaviorally robust. However, in Bayesian games, this task is made challenging by the fact that Bayesian strategies live in an abstract L1 functional space. Hence, we cannot directly rely on conventional ODE techniques to prove evolutionary stability. Instead, we use techniques developed by Ely and Sandholm (2005) to introduce the Bayesian perturbed best response dynamic.5 Using results from Ely and Sandholm (2005), we show the equivalence of convergence under this dynamic with its finite dimensional counterpart –the aggregate perturbed best response dynamic. The special properties of Bayesian supermodular games–the monotonicity of best responses–enables us to analyze the aggregate perturbed best response dynamic using the techniques of cooperative differential equations (Smith, 1995) and show convergence under this dynamic.6 Exploiting the equivalence of convergence under the two dynamics, we arrive at our desired conclusion. The set of Bayesian equilibria in Bayesian supermodular games is globally asymptotically stable under Bayesian perturbed best response dynamic. This result is presented in the main 4

See Chapter 3, Sandholm, 2011 for examples of such search models and oligopoly games. Our use perturbed best response dynamic is motivated by certain behavioural and technical factors that we explain at the beginning of Section 5. The use of these dynamics do entail a cost–their rest points are only approximations of the Nash equilibria of the game. However, for small perturbations, the approximation is almost exact. Hence, convergence of these dynamics imply society moves towards the set of Nash equilibria. 6 The analysis of the aggregate dynamic is similar to the analysis of the perturbed best response dynamic in Hofbauer and Sandholm (2007) for common preference supermodular games. We are, however, able to go further than their conclusion and apply our result to the Bayesian games by using results in Ely and Sandholm (2005). 5

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message of the paper, Theorem 7.4. Theorem 7.4 implies that even accounting for the diversity of types, in many economic and social situations that can be modeled as supermodular games, we should realistically expect to see behaviour closely approximating Nash equilibrium. Since Bayesian supermodular games can find wide applicability in economics, we believe that in a wider context, this result can significantly enhance the applicability of evolutionary game theory in economics. The plan of the paper is as follows. In section 2, we define Bayesian games. Sections 3-4 define and characterize Bayesian supermodular games and their equilibrium set. Sections 5 and 6 develop the Bayesian perturbed best response dynamic and the aggregate perturbed best response dynamic respectively. Section 7 shows that Bayesian perturbed equilibria are globally asymptotically stable. Section 8 concludes.

2

Bayesian Population Games

In this section, we follow the framework of Ely and Sandholm (2005) to define Bayesian population games and establish the notation that we use throughout the paper. Let P = {1, 2, · · · , p} be a society consisting of p ≥ 1 populations. Agents in population p ∈ P form a continuum of mass 1.7 Agents in population p select actions from the action set S p = {1, 2, · · · , np }. The total P number of strategies in the entire society is n = p∈P np . The set of states in population p is P X p = {xp : 0 ≤ xpi ≤ 1, i∈S p xpi = 1}, with xpi being the proportion of agents in population p Q playing action i ∈ S p . We denote a social state with x ∈ X = p∈P X p . In contrast to common preference population games, we are interested in games in which players in a population may have type-dependent preferences. We denote the type space of population p by T p . In order to compare elements in a type space, we impose the partial order T p on T p . A typical type in population p is tp ∈ T p . The distribution over the type space is given by the probability measure µp . Given a set A ⊆ T p , µp (A) denotes the proportion of agents in population p having types in set A. We do not make any assumptions on the dimension of T p or whether the type space is continuous or discrete. We define Bayesian strategies of population p by the function σp : T p → X p. Thus, σ p (tp ) denotes the mixed strategy employed by an agent of type tp . The set of Bayesian Q strategies in population p is Σp = {σ p : T p → X p }. Σ = p∈P Σp is the product set of the population specific Bayesian strategy sets. The expected value of the Bayesian strategies σ p with respect to µp generates the population 7

The population mass is normalized to 1 merely for convenience. We can assume each population is of a different mass mp > 0 without affecting any of the results.

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state in population p. We denote the population state E p σ p ∈ X p and define its i-th component as E p σip

Z = Tp

σip (tp ) µp (dtp ) .

The population state describes the proportion of agents playing any particular action in that population. From the population state E p σ p , we obtain the social state Eσ ∈ X given by Eσ = (E 1 σ 1 , E 2 σ 2 , · · · , E p σ p ).

(1)

To define dynamics in Bayesian population game, we we need a means to measure the distance between two Bayesian strategies. Hence, we need to impose a metric on the set of Bayesian strategies Σ. Following Ely and Sandholm (2005), we specify the L1 metric as the appropriate metric in this space. The distance between σ and ς ∈ Σ under this metric is given by ! kσ − ςk =

XX p∈P i∈S p

E

p

|σip

−

ςip |

=

X

E

p

X

|σip

−

ςip |

=

i∈S p

p∈P

X

E p |σ p − ς p | .

(2)

p∈P

This norm satisfies the intuitive requirement that if two Bayesian strategies are identical for all but a small measure of types, then those two strategies are close together. We denote the payoff obtained from strategy i ∈ S p by Fip (x, tp ). Thus, the payoff of an agent depends upon the present social state x and the type tp of the agent. We assume that the payoff function Fip is differentiable with respect to both x and tp , for all i ∈ S p , tp ∈ T p and p ∈ P. The p

vector F p (x, tp ) ∈ Rn denotes the payoff vector of an agent with type tp . In the following sections, we use the notation F to refer to a Bayesian game as we have described here. Given a social state x, we define the best response correspondence for population p, B p (x) : T p → Σp , as B p (x)(tp ) = arg maxp y · F p (x, tp ). y∈X

A Bayesian strategy σ is a Bayesian Nash equilibrium if for all tp ∈ T p , p ∈ P, B p (Eσ)(tp ) = σ p (tp ).

3

Bayesian Supermodular Population Games

We now define Bayesian supermodular population games. We define supermodularity along two dimensions; supermodularity with respect to the action i and the social state x, and supermodularity with respect to the action i and the type tp . In order to define supermodularity, we need to order social states and types. We assume that T p defines a partial order on T p . To order social states, we use the particular partial order defined in Sandholm (2011) for common preference supermodular games. To define that order, Sandholm (2011) introduces the transformation matrices,

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p −1)×np

Γ ∈ R(n

˜ ∈ Rnp ×(np −1) , on X p where, and Γ 0 0 Γp = . ..

1

···

···

0 .. .

1 .. .

··· .. .

0 ···

···

0

−1 0 · · · 1 1 −1 . . . .. . ˜p .. , Γ = . 1 0 .. . . . . . .. .. . 1 0 ··· 0

0 .. . 0 . −1 1

We then obtain the corresponding transformation matrices on all of X as the following block ˜ ∈ Rn×(n−p) , diagonal matrices Γ ∈ R(n−p)×n and Γ ˜ = diag Γ ˜ 1, Γ ˜ 2, · · · , Γ ˜p . Γ = diag Γ1 , Γ2 , · · · , Γp , Γ The stochastic dominance ordering X on X defined in Sandholm (2011) is as follows. Definition 3.1 Let x, y ∈ X be two social states. State y is greater than state x, y X x, if Γy ≥ Γx. In this definition, Γx is the decumulative frequency of x. Thus, y is greater than x if the proportion of agents playing strategy i ∈ S p or higher, in every population p is higher under y than in x; that P P is if j≥i yjp ≥ j≥i xpj , for all i ∈ S p , p ∈ P. We use Definition 3.1 to define the following partial ordering Σ on Σ. Definition 3.2 Let σ and σ ˆ ∈ Σ. We say σ Σ σ ˆ if ΓEσ ≥ ΓE σ ˆ. We now define Bayesian supermodular population games as follows. Definition 3.3 A Bayesian population game F p (x, tp ) is supermodular if the following two increasing difference conditions are satisfied. 1. (Supermodularity with respect to (i, x)). Let y X x. F p (x, tp ) is supermodular with respect to actions and social state if p p Fi+1 (y, tp ) − Fip (y, tp ) ≥ Fi+1 (x, tp ) − Fip (x, tp ),

for all i < np , tp ∈ T p , p ∈ P. 2. (Supermodularity with respect to (i, tp )). Let tp T p sp . F p (x, tp ) is supermodular with respect to actions and types if p p Fi+1 (x, tp ) − Fip (x, tp ) ≥ Fi+1 (x, sp ) − Fip (x, sp ),

for all i < np , x ∈ X, p ∈ P.

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We now provide the following alternative characterization of supermodularity in terms of the derivative of the payoff function. The common preference version of this result can be found in Sandholm (2011). The result follows from an exercise in Taylor expansion. Theorem 3.4 The Bayesian game F is supermodular with respect to (i, x) if and only if

˜p Γ

0

˜ q ≥ 0, Dxq F p (x, tp ) Γ

for every p, q ∈ P, tp ∈ T p , and where Dxq F p (x, tp ) is the derivative matrix of F p (x, tp ) with respect to xq . Proof. In Sandholm (2011). Example: As an example of Bayesian supermodular population games, we consider the search model from Sandholm (2011). We extend that example to incorporate diverse types. A population of agents choose levels of search effort from S = {1, · · · , n}. Each search effort leads to a cost which also depends upon the agents’ type. Hence, if an agent is of type t and chooses effort i, the payoff is, Fi (x, t) = m(i)b(a(x)) + c(i, t), where a(x) =

Pn

k=1 kxk

is the aggregate search effort, b is some increasing benefit function, and c is

the cost function that depends upon strategy i and type t. Suppose the cost function is c(i, t) = ti. We assume t ∈ [−1, 0]. It is easy to check that this game is a Bayesian supermodular population game.

4

Minimal and Maximal Pure Strategy Nash equilibria in Bayesian Supermodular Population Games

In this section, we establish the structure of Nash equilibria in Bayesian supermodular population games. Consider a function f : S × A → R, where S and A are two ordered sets. Suppose the function satisfies the increasing difference condition f (s0 , a0 ) − f (s0 , a) ≥ f (s, a0 ) − f (s, a), ∀a0 > a, s0 > s, Let us define the set a∗ (s) = arg max{f (s, a) : a ∈ A}. Topkis’ (1978) monotonicity theorem says that the minimal and maximal elements of a∗ (s), a ¯(s) and a(s), are increasing functions.8 8

This special version of Topkis’ (1978) monotonicity theorem in which both S and A are subsets of the reals are taken from Amir (2003). The original theorem by Topkis applies to a much more general lattice theoretic setting.

6

Milgrom and Roberts (1990) use this theorem to establish the existence of monotone pure strategy Nash equilibria in finite player supermodular games. They define increasing differences with respect to a player’s own action and the profile of opponents’ actions. Analogous to Topkis’ monotonicity theorem, they show that the minimal and maximal best responses of a player are increasing in their opponents’ action profile. This result leads to their theorem about the monotonicity of pure Nash equilibria. Sandholm (2011) extends this result to common preference supermodular games in a large population setting. For finite player Bayesian supermodular games, variants of the result on monotonicity of pure equilibria has been established by Athey (2000), Reny (2005) and Van Zandt and Vives (2007). Here, we follow the approach in Van Zandt and Vives (2007) to prove such a result for a large population Bayesian supermodular game. First, we establish the monotonicity of maximal and minimal best responses of each type with respect to the social state. Given population state x, we denote the maximal and minimal best responses for type tp as max B p (x)(tp ) and min B p (x)(tp ) respectively. To prove this result, we only need to replicate, for each type separately, the argument in the corresponding result for supermodular games in Sandholm (2011). Hence, we dispense with the proof and merely state the following result. Proposition 4.1 Let F be a Bayesian supermodular population game, and y X x. Then, for all types tp ∈ T p and all populations p ∈ P, max B p (y)(tp ) ≥ max B p (x)(tp ) and min B p (y)(tp ) ≥ min B p (x)(tp ). Proposition 4.1 uses supermodularity in (i, x) to establish the existence of monotone best responses in social states. We now use supermodularity in (i, tp ) to establish, for every social state, the existence of maximal and minimal best responses that are monotone in types. First, we define a pure Bayesian strategy that is monotone in types. Definition 4.2 A pure Bayesian strategy σ is monotone if, for all p ∈ P, σ p (tp ) ≥ σ p (sp ), if tp ≥ sp . Supermodularity in (i, tp ) then follows from the same basic monotonicity theorem in Topkis (1978). We state the result in the following proposition. Proposition 4.3 Consider population p ∈ P. Let F be a Bayesian supermodular population game, and tp , sp ∈ T p , tp T p sp . Given a population state x, the maximal and minimal best response strategies are monotone in type, that is, max B p (x)(tp ) ≥ max B p (x)(sp ) and min B p (x)(tp ) ≥ min B p (x)(sp ). We now show the existence of a minimum and maximum Nash equilibrium in pure strategies in terms of the stochastic dominance ordering Σ introduced in Definition 3.2. We then show that the maximum and minimum Nash equilibrium are monotone in types.

7

¯ p (x) : T p → S p defined by B ¯ p (x)(tp ) = Given a state x, let us consider the function B ¯ p ∈ Σp and B ¯= B ¯ 1, · · · , B ¯ p ∈ Σ. Similarly, we define B p (x) : T p → S p max B p (x)(tp ). Clearly, B by B p (x)(tp ) = min B p (x)(tp ). We also define the maximum and minimum Bayesian strategies as, σ ¯ p (tp ) = (0, 0, · · · 1) σ p (tp ) = (1, 0, · · · 0), for all types tp ∈ T p , and all populations p ∈ P. Let σ ¯ = (¯ σ1, · · · , σ ¯ n ). Corresponding to σ ¯ , we have the highest social state x ¯ = (¯ x1 , x ¯2 , · · · , x ¯p ) = (E 1 σ ¯1, E2σ ¯2, · · · , Ep σ ¯ p ). Similarly, we define σ and the lowest social state x, with xp = E p σ p . The main idea behind the proof of the existence of a minimal and maximal Nash equilibrium is that of Cournot tatˆ onnement. Under this method, we converge to the maximal and minimal ¯ and the B mappings Nash equilibria from the highest and the lowest social states using the B respectively. Van Zandt and Vives (2007), Lemma 6, use this idea to establish a similar result for finite player Bayesian supermodular games. Sandholm (2011) uses a similar idea for common preference large population supermodular games. ¯0 (E σ ¯ σ To state and prove the theorem, let B ¯ ) = B(E ¯ ) and B 0 (Eσ) = B(Eσ). We now recursively define the following two sequences. ¯k (E σ ¯ E B ¯k−1 (E σ B ¯) = B ¯) , B k (Eσ) = B E B k−1 (Eσ) . k ∞ ¯k (E σ Theorem 4.4 Consider the sequences {B ¯ )}∞ k=0 and {B (Eσ)}k=0 .

∞ p ∞ ¯ (E σ ¯ )(tp ) k=0 is a decreasing sequence and B pk (Eσ)(tp ) k=0 1. For every p ∈ P, tp ∈ T p , B k is an increasing sequence. ∞ ¯k (E σ ¯ ) k=0 and {B k (Eσ)}∞ 2. Let σ ¯ ∞ and σ ∞ be the pointwise limits of B k=0 respectively. Both σ ¯ ∞ and σ ∞ are Nash equilibria in pure strategies for all types. 3. For any other Nash equilibrium σ, σ ¯ ∞ Σ σ Σ σ ∞ . Proof. 1. Since σ ¯ and σ are the highest and the lowest Bayesian strategies respectively, the monotonicity of the two sequences follow from the repeated application of Proposition 4.1. 2. Consider σ ¯ ∞ . Since the strategy set S p is finite for every population, the limit of the monoton p ∞ ¯ (E σ ically decreasing sequence B ¯ )(tp ) exists for every tp . Hence, we have the existence k

k=0

8

of the pointwise limit σ ¯ ∞ . We need to show that σ ¯ ∞ is a Nash equilibrium. ¯p

∞

p

¯p

B (E σ ¯ ) (t ) = B

p p ¯k (E σ ¯k (E σ ¯ lim E B ¯ ) (tp ) E lim B ¯ ) (t ) = B

¯p = lim B k→∞

k→∞

k→∞

¯p ¯ ¯k (E σ ¯ ) (tp ) = (¯ σ ∞ )p (tp ) , EB ¯ ) (tp ) = lim B k+1 E Bk (E σ k→∞

where the second equality follows from the dominated convergence theorem. A similar proof applies to σ ∞ . Clearly, both σ ¯ ∞ and σ ¯ ∞ are Nash equilibria in pure strategies for all types. 3. For any σ ∈ Σ, σ ¯ Σ σ Σ σ. Hence, the repeated application of Proposition 4.1 for every type tp gives (σ ∞ )p (tp ) = lim B pk (Eσ) (tp ) ≤ lim Bkp (Eσ) (tp ) k→∞

k→∞

¯ p (E σ ¯ ) (tp ) = (¯ ≤ lim B σ ∞ )p (tp ) , for every type tp k k→∞

For any Nash equilibrium σ ∗ , we have σ = limk→∞ B k (Eσ). So, (σ ∞ )p (tp ) ≤ (σ ∗ )p (tp ) ≤ (¯ σ ∞ )p (tp ), for every tp . This implies σ ¯ ∞ Σ σ ∗ Σ σ ∞ . Theorem 4.4 uses supermodularity in (i, x) to prove the existence of a maximum and minimum Nash equilibrium. We now use supermodularity in (i, tp ) to show that σ ¯ ∞ and σ ∞ , are monotone in types. Theorem 4.5 Let tp , sp ∈ T p , tp > sp . For all p, (¯ σ ∞ )p (tp ) ≥ (¯ σ ∞ )p (sp ), and (σ ∞ )p (tp ) ≥ (σ ∞ )p (sp ). Proof. By Proposition 4.3, ¯ p (E σ ¯ p (E σ B ¯ ) (tp ) ≥ B ¯ ) (sp ) k k B pk (Eσ) (tp ) ≥ B pk (Eσ) (sp ), ¯k (E σ for all k ∈ Z+ , for all p ∈ P. Since lim B ¯) = σ ¯ ∞ , and lim B k (Eσ) = σ ∞ , we obtain the desired result.

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k→∞

k→∞

Bayesian Perturbed Best Response Function and Bayesian Perturbed Best Response Dynamic

We now turn to the evolutionary analysis of Bayesian supermodular population games. As a first candidate, we can consider the Bayesian best response dynamic introduced in Ely and Sandholm 9

(2005) as an evolutionary model. However, on certain technical and behavioral grounds, we prefer to conduct our analysis using the Bayesian perturbed best response dynamic.9 Technically, perturbed best response dynamics allow us to exploit the monotonicity properties of Bayesian supermodular games and use the theory of cooperative dynamics (Smith, 1995). We use these technical features in Section 7 to prove the stability results in these games. From the behavioral point of view, perturbed best response dynamics do not suffer from the abrupt and discontinuous changes in strategy that characterize best responses. In an evolutionary setting where agents are myopic, this is a desirable property for a model to possess. We follow the framework in Ely and Sandholm (2005) to introduce the Bayesian perturbed best response function. We then use the function to generate the Bayesian perturbed best response dynamic. We then check the conditions of Lipschitz continuity and forward invariance that ensure that these dynamics have well defined solutions.

5.1

Bayesian Perturbed Best Response Function

Perturbed best response dynamics are based on perturbed best response functions. Formally, the perturbed best response function is derived by assuming that agents play best responses to payoffs that have been subject to some stochastic perturbations. Suppressing the dependence on the state x, let us write the payoff vector of an agent of type tp in a Bayesian population game as π p (tp ). Let p

εp ∈ Rn be the shock to π p (tp ) of an agent with type tp . Hence, the perturbed payoff corresponding to πip (tp ) is πip (tp ) + εpi . The probability of a type tp agent playing strategy i is then given by ˜ p (π p (tp )) M i

p p p = P i = arg maxp πj (t ) + εj . j∈S

For the Bayesian population game F , we define the Bayesian perturbed best response function ˜ p : T p → X p by B ˜ p (x)(tp ) = M ˜ p (F p (x, tp )). B i i

(3) p

To be an admissible stochastic perturbation, εp must admit a positive density on Rn , and this ˜ p (π p (tp )) to be continuously differentiable. The density must be smooth enough for the function M most well known perturbed best response function is the logit best response function,10 ˜ p (π p (tp )) M i

exp η −1 πip (tp ) . =P p n −1 π p (tp ) exp η j=1 j

9

(4)

Perturbed best response dynamics have been studied extensively in the context of evolution in common preference population games. Such models are based on the idea that payoffs are subject to random perturbations and and that players respond by playing a best response to the perturbed payoffs. These models first appeared in the work of Blume (1993,1997) and Young (1998) who imposed certain parametric restrictions on the form of the disturbance distributions. In Hofbauer and Sandholm (2002, 2007), these models are explored in their full generality with very minimal restrictions on the form that payoff disturbances can take. 10 The logit best response function is generated by the perturbation factor εp having the double exponential distri bution P (εpi ≤ c) = exp − exp −η −1 c − γ , where γ ≈ 0.5772 is the Euler’s constant. See Sandholm (2011) for a derivation.

10

˜ p (x)(tp ) > 0, for all i ∈ S p . A Bayesian The restrictions on the distribution of εp imply that B i strategy σ ˜ is a Bayesian perturbed equilibrium if ˜ p (E σ σ ˜p = B ˜ ), for all p ∈ P.

(5)

Typically, a Bayesian perturbed equilibrium σ ˜ is very close to a Bayesian Nash equilibrium σ ¯ if the density function of εp is concentrated around zero, for all p ∈ P. In the case of the logit choice function, limη→0 σ ˜=σ ¯. ˜ p (π p (tp )) matrix in the following We now present certain fundamental properties of the DM lemma. Sandholm (2011) lists these properties for common preference games. The same proof can be applied to obtain the following results for Bayesian games. Lemma 5.1 Let π p (tp ) be the payoff vector of an agent with type tp . Suppose that the Bayesian ˜ p (π p (tp )) is derived from admissible stochastic perturbations. perturbed best response function M ˜ p (π p (tp )) is symmetric, has strictly negative off-diagonal elements, Then the derivative matrix DM p p ˜ p (π p (tp )) · 1 = 0. is positive definite with respect to Rn × Rn and satisfies DM 0

0

We have used the method of stochastic perturbation to derive the Bayesian perturbed best response function. It has been shown in Hofbauer and Sandholm (2002) that the method of deterministic perturbation is a more general way to derive this function.11 However, for our purpose, we ˜ p (π p (tp )) need to confine ourselves to stochastic perturbations since certain properties of the DM listed in Lemma 5.1 are true only if the Bayesian perturbed best response is generated using this method. These properties will play a crucial role in the proof of convergence of our dynamics to perturbed equilibria in Bayesian supermodular games.

5.2

Bayesian Perturbed Best Response Dynamic

An evolutionary dynamic on Σ, the space of Bayesian strategies, describes the law of motion of the ˆ = {σ : T → Rn } denote strategy of every type of agents involved in the population game. Let Σ the linear space containing all directions of motion through Σ. A dynamic on Σ is therefore a law of motion σ˙ = f (σ),

(D)

ˆ where f maps Σ to Σ. The Bayesian perturbed best response dynamic is described by the following law of motion. ˜ p (Eσ) − σ p , σ˙ p = B

(BPB)

The current Bayesian strategy in society, σ, generates the social state Eσ. The payoff obtained under the current strategy by an agent of type tp is therefore F p (Eσ, tp ). The agent, when called 11 Any choice function that can be generated by some stochastic perturbation can also be generated by using some deterministic perturbation. But the converse is not true. See Sandholm (2011) for a counterexample.

11

upon to revise his strategy, respond to the state Eσ by playing the perturbed best response ˜ p (Eσ)(tp ) = M ˜ p (F p (Eσ, tp )). Hence, for a type tp agent, the change in the Bayesian strategy is B given by ˜ p (Eσ)(tp ) − σ p (tp ) = M ˜ p (F p (Eσ, tp )) − σ p (tp ). σ˙p (tp ) = B

(6)

Clearly, the set of rest points of the dynamic (BPB) coincide with the set of Bayesian perturbed equilibria of the corresponding Bayesian population game F . We denote this set of Bayesian perturbed equilibria as ΣP E . To complete the definition of the dynamic on Σ, we extend the L1 norm defined in Section 2 to ˆ This enables us to establish the following technical result about the Lipschitz continuity of all of Σ. (BPB). We use this result in Theorem 5.3 below to establish the fundamental properties of (BPB). Lemma 5.2 The vector field generated by the Bayesian perturbed best response dynamic (BPB) is Lipschitz continuous with respect to the L1 norm. Proof. Consider the dynamic (BPB). Let x and x0 be the social states generated by σ and σ 0 respectively. To prove the lemma, it is sufficient to show the following. ˜ ˜ 0 )k ≤ Kkσ − σ 0 k, kB(x) − B(x for some constant K. Under the L1 norm, ˜ ˜ 0 )k = kB(x) − B(x

XX p∈P i∈S p

˜p ˜ p (x0 ) E p B (x) − B i i

(7)

By the Lipschitz continuity of the perturbed best response function for each type, we have X p X p p 0 ˜ (x)(tp ) − B ˜ p (x0 )(tp ) ≤ K(tp ) B x − (x ) i

i

i

i∈S p

i

i∈S p

for some number K(tp ) that depends on the type. This implies that X i∈S p

¯p = where K

R

Tp

X p ˜p p 0 ˜ p (x0 ) ≤ K ¯p E p B (x) − B x − (x ) i i i i , i∈S p

¯ = maxp∈P K ¯ p . Hence, K p (tp ) µp (tp ). Let K XX p∈P i∈S p

X X p

˜p p 0 0 ˜ p (x0 ) ≤ K ¯ ¯ E p B (x) − B x − (x ) i i i i =K x−x

(8)

p∈P i∈S p

However, XX XX X X p

p p p 0 p 0 E p σip − (σip )0 = σ − σ 0 . xi − (xi ) = E (σi ) − E p (σi ) ≤

p∈P i∈S p

p∈P i∈S p

p∈P i∈S p

Combining the inequalities (7), (8), and (9), we obtain the desired result. 12

(9)

We now apply Lemma 5.2 and Theorem 3.1 in Ely and Sandholm (2005) to establish the fundamental properties of the existence and uniqueness of solution of the dynamic (BPB). Theorem 3.1 in Ely and Sandholm (2005) shows that under the following conditions, the dynamic f defined in (D) has a unique L1 solution trajectory {σt }t≥0 in Σ.12 • (LC) The vector field generated by f is L1 Lipschitz continuous on Σ. X p • (FI1) fi (σ) (tp ) = 0, for all σ ∈ Σ, tp ∈ T p , p ∈ P. i∈S p

• (FI2) For all σ ∈ Σ, tp ∈ T p , p ∈ P, fip (σ) (tp ) ≥ 0 whenever σip (tp ) = 0. • (UB) For all σ ∈ Σ, tp ∈ T p , p ∈ P, |f p (σ)(tp )| ≤ M . Condition (LC) is the standard Lipschitz continuity condition. Conditions (FI1) and (FI2) are forward invariance conditions that ensure that solution trajectories that begin in Σ do not leave Σ. The last condition imposes a uniform bound on f p (σ)(tp ) that is required because f (σ) is infinite dimensional. We now establish the following facts about the Bayesian perturbed best response dynamic. Theorem 5.3 (Basic properties of (BPB)) 1. There exists an L1 solution to (BPB) starting from each σ0 ∈ Σ. This solution is unique in the L1 sense: if σt and ρt are L1 solutions to (BPB)such that σ0 = ρ0 , µ − a.s., then σt = ρt , µ − a.s.. 2. If σt and ρt are L1 solutions to (BPB), then kσt − ρt k ≤ kσ0 − ρ0 keKt , where K is the Lipschitz constant of (BPB). 3. Solutions to (BPB) remain in Σ at all times t ∈ [0, ∞). Proof. By Theorem 3.1 in Ely and Sandholm (2005), any dynamic that satisfies conditions (LC), (FI1), (FI2), and (UB) exhibits the properties listed in the statement of the this theorem. 12

For ease of reference, we reproduce the definitions of the L1 limit and a L1 solution to a Bayesian dynamic from ˆ Ely and Sandholm (2005). Consider the dynamic (D) and let {σt }t≥0 be a L1 solution trajectory of f through Σ. 1 1 ˆ Then, σ ¯ ∈ Σ is the L limit of σs as s approaches t, denoted σ ¯ = L lims→t σs , if lim kσs − σ ¯ k = lim E|σs − σ ˆ |.

s→t

s→t

ˆ is defined as The trajectory σt is L1 continuous if σt = σ ¯ = L1 lims→t σs for all t. The L1 derivative σ˙ t ∈ Σ σ t+ε − σt σ˙ t = L1 lim . ε→0 ε ˆ is an L1 solution to the dynamic (D) if σ˙ t = f (σt ) for all t, where σ˙ t is interpreted as an A trajectory σ : R+ → Σ 1 L derivative.

13

Hence, to complete our proof, we simply need to show that (BPB) satisfies the properties (LC), (FI1), (FI2), and (UB). Condition (LC) follows from Lemma 5.2. Conditions (FI1) and (FI2) are obviously satisfied. If f is the (BPB) dynamic, then condition (UB) is satisfied with M = 2.

6

Aggregate Perturbed Best Response Dynamic

Our objective is to analyze the stability properties of the set of equilibria in Bayesian supermodular population games under the dynamic (BPB). It is, however, difficult to do this by working with the Bayesian dynamic directly since this dynamic operates in an abstract L1 space. Instead, we follow the strategy of Ely and Sandholm (2005) and develop the finite dimensional aggregate perturbed best response dynamic.13 Using results from Ely and Sandholm (2005), we establish the equivalence of stability concepts under the two dynamics. This allows us to focus on the more tractable finite dimensional dynamics to establish our desired results on stability. We define the aggregate perturbed best response dynamic on the state space X. Given x ∈ X, this dynamic is given by ˜ p (x)) − xp . x˙ p = E p (B

(APB)

˜ function as We can write (APB) in terms of the M p

Z

x˙ =

˜ p (F p (x, tp ))µp (dtp ) − xp . M

(10)

Tp

Suppose the Bayesian strategy σ generates the social state x. Under (APB), the social state moves towards the state generated by the perturbed best response to x. This dynamic therefore captures the change in the aggregate state when the change in the Bayesian strategy σ is given by dynamic (BPB). The right hand side of (APB) is a map from X to Rn . Hence, this dynamic is a finite dimensional ˜ imply that is dynamic is ordinary differential equation. The Lipschitz continuity of E and B Lipschitz continuous. Moreover, the state space X is forward invariant under this dynamic. Hence, solution trajectories to this dynamic from points in X exist, are unique, and do not leave X. ˜ x)). We call such a state x A rest point x ˜ of this dynamic satisfies the property x ˜ = E(B(˜ ˜ a perturbed equilibrium distribution and use the notation XP E ⊆ X to denote the set of such points. We now establish certain facts about the relationship between the two dynamics (BPB) and (APB). All these facts lead to the equivalence of the stability concepts under the two dynamics. The proofs of these assertions follow directly from the corresponding results in Ely and Sandholm (2005) once we replace, wherever necessary, “best response” with “perturbed best response”. We therefore refrain from giving detailed proofs of the results that follow, merely drawing the reader’s attention to the relevant results in Ely and Sandholm (2005). 13 Ely and Sandholm (2005) introduce the aggregate best response dynamic as the finite dimensional counterpart to their Bayesian best response dynamic.

14

The first fact establishes the one-to-one relationship between the rest points of the two dynamics. We state this relationship in the following theorem. Recall that we have defined the set of Bayesian perturbed equilibria of F (or equivalently, the set of rest points of the Bayesian perturbed best response dynamic ((BPB))), as ΣP E . ˜ : XP E → ΣP E . Theorem 6.1 The map E : ΣP E → XP E is a homeomorphism whose inverse in B Proof. See the proof of Theorem 4.1 in Ely and Sandholm (2005). Theorem 6.1 implies that if σ is a perturbed equilibrium, then Eσ is a perturbed equilibrium ˜ distribution. Conversely, if x is a perturbed equilibrium distribution, B(x) is a perturbed equilibrium. This result paves the way to the following results on the equivalence of stability under the two dynamics.14 Their proofs can be applied to the aggregate and Bayesian perturbed dynamics to establish the following results. Theorem 6.2 (Lyapunov Stability) The state x ˜ ∈ XP E is Lyapunov stable under (APB) if and ˜ x) is asymptotically stable under (BPB). only if the Bayesian strategy σ ˜ = B(˜ Proof. See the proof of Theorem 6.2 in Ely and Sandholm (2005).

Theorem 6.3 (Asymptotic stability) The state x ˜ ∈ XP E is asymptotically stable under (APB) if ˜ and only if the Bayesian strategy σ ˜ = B(˜ x) is asymptotically stable under (BPB). Proof. See the proof of Theorem 6.4 in Ely and Sandholm (2005). The result on local asymptotic stability can be extended to global asymptotic stability. Theorem 6.4 (Global Asymptotic stability) The state x ˜ ∈ XP E is globally asymptotically stable ˜ x) is globally asymptotically stable under under (APB) if and only if the Bayesian strategy σ ˜ = B(˜ (BPB). What is the nature of convergence in Σ under (BPB) to σ ˜ ? Asymptotic stability of x ˜ means that for any σ ∈ Σ, the trajectory {Eσt } converges to x ˜. Hence, in the L1 sense, σt converges to a rest point σ ˜ such that E σ ˜=x ˜. But by Theorem 6.1, there is only one σ ˜ such that E σ ˜=x ˜, and ˜ that is σ ˜ = B(˜ x). Hence, σt converges to σ ˜. 14

The same standard definition of Lyapunov stability and asymptotic stability applies to both the Bayesian dynamic ˆ k · k) and let the function h : Z → Zˆ be a and the aggregate dynamic. Let Z be a subset of a Banach space (Z, dynamic on Z defined by the law of motion z˙ = h(z). Suppose that Z is forward invariant under h(z) and let z ∗ be a rest point of the dynamic. We say that z ∗ is Lyapunov stable under h(z) if for each set A ⊂ Z containing z ∗ that is open (relative to Z), there is an open set A0 ⊂ A such that if {zt } is a solution to z, ˙ with z0 ∈ A0 , then {zt } ⊂ A. The ∗ rest point z is asymptotically stable under h(z) if it is Lyapunov stable and if there is an open set A with z ∗ ∈ A such that if z0 ∈ A, then limt→∞ zt = z ∗ .

15

7

Global Asymptotic Stability of Bayesian Perturbed Equilibria in Bayesian Supermodular Games

In order to determine stability of equilibria in Bayesian supermodular games, we first show that the set of aggregate perturbed equilibrium is globally asymptotically stable under the aggregate perturbed best response dynamic. We then use Theorem 6.4 to conclude that the set of Bayesian perturbed equilibrium is globally asymptotically stable under the Bayesian perturbed best response dynamic. Hofbauer and Sandholm (2007) establish global convergence to equilibria in common preference supermodular games under perturbed best response dynamics. They use the methodology of cooperative dynamics (Smith, 1995) to establish this result. Cooperative dynamics are defined by the property that increases in the value of any component of the state variable increase the speed of change of the value of all other components. The special characteristic of supermodular games — the monotonicity of best replies — establishes a relationship between these games and cooperative dynamics. In establishing global convergence under the dynamic (APB), we follow the method of Hofbauer and Sandholm (2007), as discussed in Sandholm (2011), and extend their result to a game with multiple types. In order to apply the theory of cooperative dynamics, we first apply a transformation to the state space X of the (APB) dynamic. Recall the definitions of the transformation matrices Γ and ˜ p introduced in Section 3. We now introduce another matrix from Sandholm (2011), Ωp ∈ Rnp ×np , Γ where

1 1 ···

···

1

0 0 · · · . p Ω = 0 0 . . .. .. . .

··· .. . .. .

0 .. . , .. .

0 0 ···

···

0

˜ p Γp + Ωp = I. and note that Γ We now apply the transformation Γp on X p to obtain the following set of transformed population states, p X p = Γp X p = χp ∈ Rn −1 : 1 ≥ χp1 ≥ χp2 ≥ · · · ≥ χpnp −1 ≥ 0 , and X = X 1 × X 2 · · · × X p . The inverse of the map Γp : X p → X p is as follows. ˜ p X p + xp , X p = Γp xp ⇔ xp = Γ

(11)

p

where xp = (1, 0, 0, · · · , 0) ∈ Rn is the lowest state in X p . We now apply the transformation Γ to (APB) to obtain the transformed perturbed best response

16

dynamic on X . p

p

Z

χ˙ = Γ

˜ + x, tp )µ (dtp ) − χp ˜ p (F p Γχ M

Tp

Z =

˜ + x, tp )µ (dtp ) − χp ˜ p (F p Γχ Γp M

Tp

˜ p (χ) − χp = E p Γp B

(TPB)

The following proposition verifies the obvious relationship between solutions to (TPB) and solutions to (APB). ˜ t + x} solves (APB) if and Proposition 7.1 (APB) and (TPB) are affinely conjugate: {xt } = {Γχ only if {χt } = {Γxt } solves (TPB). Our next task is to show that if F is a Bayesian supermodular population game, then (TPB) is a cooperative differential equation. Let us write the right hand side of (TPB) as V(χ). Then, (TPB) is a cooperative dynamic if

∂Vip (χ) ≥ 0, ∂χqj

(12)

for all (i, p) 6= (j, q). If inequality (12) holds with a strict inequality for all (i, p) and (j, q), then the dynamic is strongly cooperative. Results from Smith (1995), summarized in Sandholm (2011), imply that strongly cooperative differential equations converge to rest points from almost all initial conditions. Hence, if we can show that (TPB) is strongly cooperative, then we can conclude, by Proposition 7.1, that almost all solutions of (APB) converge to the set of perturbed equilibrium distributions. In order to establish that (TPB) is strongly cooperative, we impose an additional mild assumption on the payoffs of the game. In Theorem 3.4, we had shown that the Bayesian population game F is supermodular if and only if

˜p Γ

0

˜ q ≥ 0, for every tp ∈ T p , p, q ∈ P. Dxq F p (x, tp ) Γ

We now make the additional assumption that each column of

˜p Γ

0

˜ q contains at Dxq F p (x, tp ) Γ

least one strictly positive element. We say that F is irreducible if it satisfies this condition. We can now state the following theorem characterizing (TPB) as a strongly cooperative dynamic. The proof of this theorem is an extension of the proof in Theorem 3.3 in Hofbauer and Sandholm (2007) to a game with multiple types. Theorem 7.2 Let F be a C 1 irreducible Bayesian supermodular game, and let (APB) be the aggregate perturbed best response dynamic for F . Then the transformed dynamic (TPB) is strongly cooperative.

17

Proof. Let the right hand side of (TPB) be V(χ). The derivative of V(χ) is, ˜ (χ) (χ) − I. DV(χ) = DE ΓB

(13)

We want to show that all the off-diagonal elements of the Jacobian DV(χ) are strictly positive. Since all theof-diagonal elements of I are zero, it is sufficient to show that all the elements of ˜ (χ) are strictly positive. DE ΓB ˜ p (χ) to denote the derivative matrix of E p Γp B ˜ p (χ) For this, we use the notation Dχq E p Γp B ˜ (χ) are strictly positive, it is sufficient with respect to χq . To show that all elements of DE ΓB ˜ p (χ) are strictly positive, for all p and q. to show that all elements of Dχq E p Γp B D χq E

p

Z p p ˜ p F p Γχ ˜ + x, tp µp (dtp ) ˜ Γ B (χ) = Dχq Γ M p

Tp

Z =

˜ + x, tp µp (dtp ) ˜ p F p Γχ D χq Γ p M

Tp

Z =

˜ p (π p (tp )) Dxq F p (x, tp ) Γ ˜ q µp (dtp ) Γp DM

Tp

Z =

0 p p p p 0 ˜p p 0 ˜ ˜ q µp (dtp ) Γ DM (π (t )) (Γ ) Γ + (Ω ) Dxq F p (x, tp ) Γ p

Tp

Z =

0 ˜ p (π p (tp )) (Γp )0 Γ ˜ p Dxq F p (x, tp ) Γ ˜ q µp (dtp ) Γp DM

(14)

Tp

˜ p (π p (tp )) matrix listed in Lemma 5.1 imply all the components of The properties of the DM ˜ p (π p (tp )) (Γp )0 are strictly positive. Since F is supermodular and irreducible, Γp DM

˜p Γ

0

˜q Dxq F p (x, tp ) Γ

is non-negative with each column containing a positive element. Hence, the product of the two matrices has all positive elements. So, the right hand side of (14) is strictly positive. Therefore, the dynamic (TPB) is strictly cooperative. The properties of strongly cooperative dynamics imply the convergence of (TPB) to the set of transformed aggregate perturbed equilibria {Γ˜ x}x˜∈XP E . We therefore immediately obtain the following “almost global convergence” result under (APB). See Sandholm (2011) for the counterpart of this result for common preference games. Let ω(x) denote the ω−limit of x under the dynamic (APB).15 Recall that x ¯ and x are the highest and lowest social states in X. We can now state the following theorem. 15

Let {xt } = {xt }t≥0 be a solution trajectory to (APB) from the intitial point x. The ω−limit of x is the set of all

18

Theorem 7.3 Let F be a C 1 irreducible Bayesian supermodular population game, and let (APB) be the aggregate dynamic generated by a Bayesian stochastically perturbed best response dynamic for F . Then, ˜ 1. States x ¯ = ω(¯ x) and x ˜ = ω(x) exist and are the maximal and minimal elements of XP E . ˜] contains all ω−limit points of (APB), and is globally asymptotically stable. Moreover, [˜ x, x ¯ 2. Solutions to (APB) converge to states in XP E from an open, dense, full measure set of initial conditions in X. Proof. Follows from Proposition 7.1, Theorem 7.2 and the relevant results from Smith (1995) (summarized in in Sandholm, 2011). ˜¯ and x States x ˜ correspond to the minimal and maximal Nash equilibria of the Bayesian supermodular population game. From Theorem 7.3, we conclude that from almost all initial social states, solution trajectories of (APB) converge to the set of perturbed equilibrium distributions. We can therefore apply Theorem 6.4 to conclude that the set of Bayesian perturbed equilibria is globally asymptotically stable in a Bayesian supermodular population game under Bayesian perturbed best response dynamics. We state this formally in the following theorem, the main result of our paper. Theorem 7.4 Let F be a C 1 Bayesian supermodular game and let (BPB) be a Bayesian stochastically perturbed best response dynamic for F . Let (APB) be the aggregate dynamic generated by (BPB). Then, 1. From an open, dense, full measure set of initial conditions in Σ, solutions to (BPB) converge to the set of Bayesian perturbed equilibria. 2. An equilibrium σ ˜ is asymptotically stable under (BPB) if and only if x ˜ = Eσ ˜ is asymptotically stable under (APB). Proof. Follows from Theorem 7.3 and Theorem 6.4. The convergence result on (APB) hold for almost all initial conditions. Suppose the (APB) do not converge from some initial condition x0 . Then from all initial conditions σ0 ∈ Σ such that Eσ0 = x0 , solutions to (BPB) will not converge to the rest of Bayesian perturbed equilibria. The complement of the set of such points constitute an open, dense, full measure set of points in Σ. points that the trajectory {xt } approaches arbitrarily closely infinitely often: n o ω(x) = y ∈ X : there exists {tk }∞ k=1 with lim tk = ∞ such that lim xtk = y . t=∞

19

k=∞

8

Conclusion

Our motivation behind this paper was to give robust behavioural foundation to equilibria in Bayesian supermodular population games. We believe we have succeeded in that objective. We have shown in Theorem 7.4 that the set of equilibria in these games are globally asymptotically stable under Bayesian perturbed best response dynamics. In arriving at this conclusion, we have drawn upon results from Ely and Sandholm (2005) that establish equivalence between convergence under the Bayesian dynamic and a finite dimensional aggregate perturbed best response dynamic. We have then applied the theory of cooperative dynamics (Smith, 1995) to establish convergence under the aggregate dynamic. This enabled us to arrive at our desired result. We hope that our main result will give a fillip to research in the application of evolutionary game theory in economics. Supermodular games have already found wide application in economics. We hope that these models can be enriched by incorporating diversity of preferences, particularly in the context of the interaction of large populations of agents. Since small perturbations, a Bayesian perturbed equilibrium typically lies arbitrarily close to a Bayesian Nash equilibrium, our main result implies that in all such models, we should expect society to exhibit behaviour close to some Bayesian Nash equilibrium.

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