Iterated Generalized Half-dominance and Global Game Selection

Transcription

1 Iterated Generalized Half-dominance and Global Game Selection Ryota Iijima May 3, 15 Abstract We offer an equilibrium characterization of a general class of global games with strategic complementarities. The analysis highlights a form of acyclicity in the interim belief structure of global games, which allows us to formalize a selection criterion, iterated generalized half-dominance. This criterion is shown to be a unique global game selection when noise distributions satisfy a regularity condition. A similar logic also applies to the perfect foresight dynamics of Matsui and Matsuyama (1995); an iterated generalized half-dominant equilibrium is a unique globally stable state when agents are patient enough. The criterion is especially useful for games with more than two asymmetric players, and can be easily applied to local interaction games with an arbitrary network structure. 1 Introduction Many games admit multiple Nash equilibria, and there is now a vast body of literature that aims to select a unique equilibrium from them. The theory of global games is one of the leading approaches and has been used in various applications. 1 The global game approach extends a complete information game by allowing payoffs to depend on an unobservable state, where each player privately observes a noisy signal about the state. Carlsson and van Damme (1993) study games and show that as the noise level parameter goes to zero, the set of rationalizable actions in a global game shrinks to the risk-dominant action in the complete information game at each state. The key mechanism behind equilibrium selection is driven by the structure of interim beliefs held by threshold types, who are indifferent between two actions upon receiving a signal. Under symmetric payoffs and binary actions, a threshold type has a Laplacian belief, whereby opponents actions are uniformly distributed (Morris and Shin, 3). This property leads to a tractable characterization of equilibrium selection that is easily applicable to various applications. In this paper we allow for many asymmetric players and actions. While this generalization is clearly important for applications, its characterization is not fully developed in the literature, because one has to deal with more complicated belief structures that I am grateful to Mira Frick, Drew Fudenberg, Jonathan Libgober, Satoru Takahashi, the audience at Harvard, the editor, three anonymous referees, and to Stephen Morris and Daisuke Oyama in particular for their helpful advice and comments. Department of Economics, Harvard University, riijima@fas.harvard.edu. 1 See Morris and Shin (3) for a survey. Some recent papers incorporate endogenous information, which can prevent unique selection (e.g., Angeletos and Pavan, 13). 1

2 involve multiple threshold types. Our analysis uncovers a general logic underlying global game selection by highlighting a form of acyclicity in pairwise comparison of players interim beliefs. This inspires us to propose a equilibrium selection criterion of generalized half-dominance (GH-dominance). It requires each player s action to be strictly optimal if she believes that each marginal probability of her opponent choosing the equilibrium action is more than 1. As opposed to the p-dominance criterion that we will discuss shortly, this criterion is only concerned with beliefs about each opponent s action separately and does not involve beliefs about opponents collective action profiles. To illustrate the intuition behind why this reasoning about each opponent s separate action is relevant in global game selection, we consider the following example. It shows that, when a GHdominant equilibrium exists, players cannot coordinate on multiple equilibrium actions using payoff-irreverent private signals similar to a global-game type information structure. Example 1. Fix a complete information game, and suppose that there is a GH-dominant equilibrium, where a i denotes the equilibrium action of player i. We endow this game with a payoff-irrelevant state variable θ that follows the (improper) uniform prior over R; Each player i privately observes signal t i = θ + η i, where each η i is i.i.d noise with a positive density. Consider threshold strategies in which each player i chooses a i if t i < t i, and another action if t i > t i. We argue that such a strategy profile cannot be an equilibrium. Note that in such an equilibrium, i should be indifferent between choosing a i and another action at signal t i = t i. Take an agent i with t i = min i t i. If i observes signal t i = t i, she believes that for any i i, t i < t i holds with probability greater than 1. That is, she believes that each opponent i s marginal probability of choosing a i is more than 1. Since a i is a GH-dominant action, she has a strict incentive to choose it at signal t i, which is a contradiction. Figure 1: Threshold types The example illustrates how the contagion argument forces GH-dominant actions to be played at all interim types of players if these actions are known to be chosen at low enough signals. A key feature in this example is the existence of an extremal threshold type t i who believes each opponent s signal t i to be less than its threshold t i with marginal probability more than 1. As we will see, the conclusion carries over beyond i.i.d noises, as long as we impose a natural restriction on noise distributions (weak stochastic transitivity), which is satisfied under most applied global game settings in the literature. 3 In Section 3, we study general global games with strategic complementarities (Frankel et al., 3), and show that a GH-dominant equilibrium is selected under the This generalizes the notion of half-dominance (Morris et al., 1995), which is defined for symmetric two-player games. 3 This condition rules out a form of cyclical interim beliefs and ensures the existence of an extremal threshold type as t i in the above example.

3 stochastic transitivity condition. Moreover, the selection result holds more generally with an iterated version of GH-dominant equilibrium, which is especially relevant when there are many actions. In Section 3.1, we also obtain a noise-independent selection result by appropriately strengthening the GH-dominance condition. The GH-dominance criterion is particularly useful for studying games with many asymmetric players, and we illustrate this point using games on networks (Section.1). We show that, if there is an (iterated) half-dominant action in a component game, then it is an (iterated) GH-dominant equilibrium of a network game wherein every player chooses it. Because this claim is true for any network with any population size, an analyst does not need to know the network structure in order to obtain this prediction. The analysis of global games with asymmetric players has been mostly limited to particular classes of games. 4 A general class of global games with strategic complementarities is studied by Frankel et al. (3). For a nonnegative vector p, an action profile (a i ) i N is a p-dominant equilibrium (Kajii and Morris, 1997) if each player i has a strict incentive to follow the equilibrium action whenever she believes that a joint probability of the opponents following the equilibrium action profile is more than p i. Frankel et al. (3) show that a p-dominant equilibrium with i p i < 1 is selected in global games under any noise distributions. p-dominance is theoretically motivated by the critical path lemma of Kajii and Morris (1997), which is applicable to general incomplete information games. 5 On the other hand, the GH-dominance selection result exploits the belief structure that is specific to global games; as a consequence, GH-dominance can be easily applied to games with many players while the p-dominance condition with i p i < 1 cannot. The key logic of GH-dominance selection in global games carries over to the perfect foresight dynamics, which is introduced by Matsui and Matsuyama (1995). 6 In this model, there is a continuum population of forward-looking agents and each one is committed to an action for a stochastic length of time. Each agent s flow payoff depends on an aggregate action distribution in the population. We show that iterated GH-dominance is also selected by this model, that is, the population state in which every agent adopts an iterated GH-dominant action is a unique state that is globally stable (in a sense that will be formalized later). Following Takahashi (8), we can describe the perfect foresight dynamics (with patient agents) by a variant of global games that satisfy weak stochastic transitivity. Thus the coincidence of the GH-dominance selection results highlight the shared feature of the strategic beliefs of the two models (see Section 4 for details). Iterated GH-dominance Consider a normal form game G =< N, (A i, u i ) i N >. N = {1,,..., n} is a set of players. Each A i is a finite set of actions of player i N, and each u i : A N R is a payoff 4 Corsetti et al. (4) introduced a large trader into a population of small agents. Guimaraes and Morris (7) allow for heterogeneous payoffs in a model of currency attack. Sakovics and Steiner (1) generalize the Laplacian belief property in a class of binary-action games of regime change with heterogeneous payoffs. 5 That is, if i p i < 1 and an event E occurs with a high ex-ante probability (under a common prior), then there exists a high ex-ante probability event E E on which E is common p-believed. Kajii and Morris (1997) use this property to show that a p-dominant equilibrium with i p i < 1 is robust against general incomplete information perturbations. As shown by Oury and Tercieux (7), the robustness implies a global game selection in games with strategic complementarities. 6 Their result is generalized by subsequent papers including Hofbauer and Sorger (1999, ), Oyama (1), Terceiux (6), Kojima (6), and Oyama et al. (8). In particular, Oyama and Terceiux (9) show that iterated p-dominance with i p i < 1 is selected by perfect foresight dynamics. 3

4 function of player i N, where A N := i N A i. u i (a i, a i ) is the payoff of i from choosing a i A i against a i A i := j i A j. Assuming expected utility preferences, we extend the domain of the payoff functions to correlated action distributions, that is, u i (a i, x i ) denotes the expected payoff of i by choosing a i A i against x i (A i ). For any correlated action distribution µ (A N ) and i N, let µ i (A i ) and µ i (A i ) denote its marginal distributions over A i and A i, respectively. Definition 1. a A N is a GH-dominant equilibrium if for any µ (A N ) such that µ j (a j) 1 for all j, each a i uniquely maximizes u i (a i, µ i ). This requires that whenever each player i believes that a marginal probability of any other player j following a j is more than 1, it is strictly optimal to choose a i. This is a generalization of half-dominance (Morris et al., 1995), which is defined for two-payer symmetric games. A GH-dominant equilibrium might not exist, but there cannot be multiple GH-dominant equilibria. To see this, suppose there were different GH-dominant equilibria a and ã. Then, any player i s unique best response against the uniform randomization over a and ã should be both a i and ã i, which is a contradiction. When players have many actions, it might be difficult to ensure that a GH-dominant equilibrium exists. However, we can use an iterative argument as in Tercieux (6) to consider a more general criterion as follows: Definition. a A N is an iterated GH-dominant equilibrium if, for each i, there is a sequence A i = A i A 1 i A m i = {a i }, where for each l =,..., m 1 and any µ (A l N ) such that µ j(a l+1 j ) 1 for all j, arg max u i (a i, µ i ) A l+1 i. a i A l i To illustrate the iteration procedure in the definition, begin with the original action space A N = A N. At each step l =,..., m 1, we consider a restricted game with action space A l N. Select actions Al+1 N Al N that have a set-valued version of the GHdominance condition at step l; player i s optimal actions belong to A l+1 i whenever she believes each opponent j will choose an action from A l+1 i with probability more than 1. Then, eliminate remaining actions A l i \ A l+1 i for each i to construct the smaller game with A l+1 N. We repeat this argument until we reach the unique action profile a. An iterated GH-dominant equilibrium is unique if it exists, and thus it does not depend on the order of eliminations. 7 A GH-dominant equilibrium is a special case with a single iteration step m = 1. The example below illustrates that iterated GH-dominance is strictly more general. Example (Coordination with heterogeneous preferences). Let n be an even number for convenience, and A i = {, 1} for each i N. Payoffs are given by { 1 n j N u i (a i, a i ) = a j if a i = 1, w i if a i =, for each i N and a i A i. Suppose that < w i 1 for each i = 1,..., n/, and < w i 3 for each i = n/ + 1,..., n. Both (1,..., 1) and (,..., ) are strict Nash 4 equilibria. (1,..., 1) is an iterated GH-dominant equilibrium under A 1 i = {1} for each i = 1,..., n/, and A 1 i = {, 1} A i = {1} for each i = n/ + 1,..., n. But (1,..., 1) is not a GH-dominant equilibrium if w i > 1+n for i = n/ + 1,..., n. n 7 Its uniqueness directly follows from the selection result in Section 4. 4

5 Finally, let us compare GH-dominance with the p-dominance of Kajii and Morris (1997). For a vector p [, 1] n, (a i ) i N A N is a p-dominant equilibrium if for any i N and µ (A N ) such that µ i (a i) p i, a i uniquely maximizes u i (a i, µ i ). Note that p-dominance is useful only under the constraint i p i < 1, which makes it less practical for a large population size. 8 GH-dominance and p-dominance with i p i < 1 do not nest each other, 9 and they focus on different aspects of players strategic beliefs; GH-dominance is about each opponent s marginal probability of following the equilibrium action, while p-dominance is about the joint probability that opponents follow the equilibrium action profile..1 Games on Networks To illustrate the applicability of (iterated) GH-dominance, consider games on networks. Each game on a network is parametrized by a tuple (g, π), where g R+ n n is a weighted directed network, π : A A R is a payoff function in a symmetric two-player game, and A is a common action set. Player i s payoff takes the form u i (a i, a i ) = j i g ij π(a i, a j ). Note that g ij represents the impact of player j on player i. The following concept is introduced by Terceiux (6) who studies two-player symmetric games. An action h A is iterated half-dominant in π if there is a sequence A = A A 1 A m = {h }, where for each l =,..., m 1 and any µ (A l ) such that µ(a l+1 ) 1, arg max π(h, µ) A l+1. h A l Example 3 (Technology adoption). Consider the following game proposed by Goyal and Jansen (1997), in which each player has an incentive to adopt the same technology as her opponent. There are two technologies h and h with two strict Nash equilibria (h, h) and (h, h ). There is an option to introduce both, hh, which allows one to always coordinate on an opponent s technology by paying additional cost c >. h hh h h 4, 4 4, 4 c, hh 4 c, 4 4 c, 4 c 3 c, 3 h, 3, 3 c 3, 3 Table 1: Technology adoption By direct calculations, we can show that h is iterated half-dominant when cost c is small, under A 1 = {h, hh } and A = {h}. On the other hand, h is (iterated) halfdominant when c is sufficiently large. If we know that h is iterated half-dominant in π, then the symmetric action profile in which every player chooses h is an iterated GH-dominant equilibrium in any networked game. 8 For instance, (1,..., 1) in Example is not a p-dominant equilibrium with i p i < 1 if n is not small. In fact, it is also not an iterated p-dominant equilibrium with i p i < 1 (Oyama and Terciuex, 9). 9 Note, however, that a GH-dominant equilibrium is a (1/,...,1/)-dominant equilibrium. 5

6 Proposition 1. Let h A be iterated half-dominant in π. Then, for any network g, (h,.., h ) (A) N is an iterated GH-dominant equilibrium of the networked game. To apply this result, an analyst need not know either the network structure g or the population size N to make this conclusion. While network structures might be difficult to directly observe in practice, the result could provide a convenient robust prediction in such a situation. 3 Global Game Selection In this section we consider general global games with strategic complementarities, as in Frankel et al. (3) and Basteck et al. (13), and show that an iterated GH-dominant equilibrium is selected. The action set of each player is ordered by A i = {, 1,..., m i }. The global game G(v) is defined as follows. A state θ R is drawn from the real line according to a continuous density g with a full support. Each player i only observes a private signal t i = θ+vη i ; v > is a scale factor, and each noise term η i is distributed according to an atomless density f i with a support contained in the interval [ 1, 1 ]. The noise terms are independent. By normalization it is without loss to assume that each noise is unbiased E[η i ] =. Player i s pure strategy in the global game is a measurable function s i : R A i. U i : A N R R is the payoff function of player i. We make the following standard assumptions on payoffs. Assumption 1 (Strategic complementarities). If a i a i and a i a i, then for all θ, U i (a i, a i, θ) U i (a i, a i, θ) U i (a i, a i, θ) U i (a i, a i, θ). Assumption (Dominance regions). There exist θ < θ such that for all i and a i, U i (, a i, θ) > U i (a i, a i, θ) if a i and θ θ, and U i (m i, a i, θ) > U i (a i, a i, θ) if a i m i and θ θ. Assumption 3 (State monotonicity). There is a K > such that for all a i a i, a i, and θ, θ [θ, θ], θ θ, (U i (a i, a i, θ) U i (a i, a i, θ)) (U i (a i, a i, θ ) U i (a i, a i, θ )) K(a i a i)(θ θ ). We suppose that there exists θ such that U i (a i, a i, θ ) = u i (a i, a i ) holds for all i N, a i A i, and a i A i. That is, global game G(v) embeds a complete information supermodular game G at a particular state θ. The following result connects a global game G(v) and an embedded game G. Proposition (Frankel et al., 3; Basteck et al., 13). Suppose Assumptions 1-3. Then, there is a profile of increasing pure strategies s = (s i ) i N such that 1. if (s v i ) i N is an equilibrium of G(v) for each v >, then lim v s v i (t) = s i (t) for all i and t except at finitely many discontinuities;. s (θ ) = (s (θ )) i N A N is a Nash equilibrium of G. Such a profile s is unique up to the finitely many discontinuities. 6

7 We call an action profile s (θ ), a global game selection if each s i is continuous at θ. As shown by Basteck et al. (13), a global game selection depends only on the embedded game G and noise distributions (f i ) i N and not on other aspects of the model. We introduce a restriction on the class of noise distributions. For this purpose, for each i, let W i denote the family of random variables {w i + η i w i R}. Define a binary relation over i W i by w i + η i w j + η j iff Pr[w i + η i w j + η j ] 1. Assumption 4 (Weak stochastic transitivity). is transitive. Note that this condition is always satisfied in two-player games. 1 An intuitive condition that implies Assumption 4 is a pairwise unbiased condition in the following form: Pr[η i η j ] = Pr[η j η i ], i, j. (1) To see this, (1) ensures that Pr[w i + η i w j + η j ] 1 iff w i w j, such that is reduced to comparison of the constant values w i, w j in R, which is clearly transitive. Condition (1) is satisfied under either of the following: (i) each distribution f i has a symmetric shape around, (ii) there is a common distribution f = f i for all i. Note that either of these two is satisfied in most of the papers that applied global games. On the other hand, some complicated shapes of distributions could violate Assumption The next result shows the selection of an iterated GH-dominant equilibrium when noise distributions satisfy weak stochastic transitivity. The proof is based on the contagion argument as in Example 1 and the decomposition result in Basteck et al. (13). Proposition 3. Suppose Assumptions 1-4 hold. Let (a i ) i N be an iterated GH-dominant equilibrium of G. Then, it is a global game selection. 3.1 Noise-Independent Selection When weak stochastic transitivity does not hold, beliefs of threshold types might involve a form of cyclicality. The following example shows that a GH-dominant equilibrium is not necessarily a global game selection in such a case. Example 4 (Cyclical coordination). Consider the following three-player game: a 1 \a β 1, 1 + β, 1 + β 3,, 1 + β 3 1, 1 + β, 1,, a 1 \a β 1,,, 1, 1,, 1 1, 1, 1 Table : Cyclical coordination (Left: a 3 =, Right: a 3 = 1.) where β i > and A i = {, 1} for each i = 1,, 3. This game is supermodular and has two strict Nash equilibria, (,, ) and (1, 1, 1). (,, ) is a GH-dominant equilibrium 1 The terminology of weak stochastic transitivity is taken from the stochastic choice literature. If we interpret N as a choice set and t i + η i as a random utility of i, then the condition is as in the stochastic choice literature. That is, if i is chosen more than j at the menu {i, j} and j is chosen more than k at the menu {j, k}, then i should be chosen more than k at the menu {i, k}. 11 Possible violations of this condition are known in the literature on intransitive dice. Take the following discrete example: η 1, η, and η 3 are distributed uniformly over {, 4, 9}, {1, 6, 8}, and {3, 5, 7}, respectively. This leads to Pr[η 1 η ] = Pr[η η 3 ] = Pr[η 3 η 1 ] = 5/9. Then, we can approximate this example by continuous random variables that violate weak stochastic transitivity. 7

8 while (1, 1, 1) is not. In this game, each player i s incentive is only affected by player i + 1 (mod. 3). Take noise distributions f = (f 1, f, f 3 ) that violate weak stochastic transitivity, that is, Pr[η i+1 > η i ] > 1 for each i = 1,, 3 (mod. 3). Then we can show that (,, ) is not a global game selection under f if Pr[η i+1 > η i ] > 1+β i +β i holds for each i (see Appendix A.3 for details). However, there is an upper-bound on the extent of stochastic intransitivity in players interim beliefs, based on which we can obtain a noise-independent selection result by appropriately strengthening the GH-dominance condition; for q [, 1], we say that a A N is a generalized q-dominant equilibrium if for any µ (A N ) such that µ j (a j) q for all j, each a i uniquely maximizes u i (a i, µ i ). By Theorem of Trybula (1965), for each n, there exists a number q n ( 1 4, 3 4 ) such that min{pr[t i1 + η i1 < t i + η i ], Pr[t i + η i < t i3 + η i3 ]..., Pr[t im + η im < t i1 + η i1 ]} q n () holds under any vector (t i ) i N, distributions (f i ) i N, and a sequence of agents (i 1, i,..., i m ) with any length m n. 1 This implies that for any vector ( t i ) i N, there is a player i such that Pr[ t i +η i t i +η i ] 1 q n holds for all i i. Thus, under threshold strategies as in Example 1, if i observes signal t i = t i then she believes that any opponent i i chooses a i with marginal probability more than 1 q n. Based on this observation, we can obtain the following noise-independent selection result. The proof is a straightforward modification of that of Proposition 3, and omitted. 13 Proposition 4. Suppose Assumptions 1-3 hold. Let a A N be a generalized (1 q n )- dominant equilibrium of G. Then, it is a global game selection. 4 Perfect Foresight Dynamics In this section, we study the perfect foresight dynamics, initially introduced by Matsui and Matsuyama (1995). In this model, there is a continuum population that consists of subpopulations indexed by i N. Each member in subpopoulation i (i-agent, henceforth) has the same payoff function u i and action set A i. Every agent is committed to a pure action and receives an opportunity to revise it as per an independent Poisson clock. We assume that every agent has the common Poisson intensity rate that is normalized to be 1. Let r be the time discount rate that is common to all the agents. 14 Let X := i N (A i) be the state space of population dynamics. Each state x = (x i ) i N X specifies an action distribution x i (A i ) for each subpopulation i N. For each action profile a A N, let δ a X denote the state that puts all the mass on a. An i-agent who is committed to a i A i receives a flow payoff u i (a i, x i ) at state (x i, x i ) X. For each state x X, a path φ : [, ) X is feasible from x if φ() = x and for each i N, there exists a measurable function α φ i : [, ) (A i ) such that for all t, φ i (t) = e t φ i () + t e s t α φ i (s)ds 1 The value of q n is increasing in n, and converges to 3 4 as n. 13 A more general selection result of an iterated version of generalized (1 q n )-dominance also holds. 14 Agents objectives are well-defined even under r =. Further, our proof does not depend on the commonality of a discount rate, and the result holds even if discount rates are different across different subpopulations. 8

9 where φ(t) = (φ i (t)) i N and α φ i (t) (A i) is the distribution of actions that are newly chosen by i-agents who have revision opportunities at t. This can be written in a differential form as φ i (t) = α φ i (t) φ i(t) if φ i (t) is differentiable at t. Let Φ x be the set of feasible paths from x. Given feasible path φ, the discounted continuation payoff of an i-agent chooses a i during her commitment times is computed as V i (a i, t, φ) = (1 + r) e (1+r)s u i (a i, φ i (t + s))ds. (3) Definition 3. A feasible path φ from x is called a perfect foresight path from x if for all t [, ), i N, ) supp (α φi (t) arg max V i (a i, t, φ). a i A i That is, under a perfect foresight path, for any t, every agent who revises at t chooses an optimal action against the future path. As shown by Hofbauer and Sorger (), from any initial point x, there exists a perfect foresight path from x. For a state x X to be a credible prediction of the model, we require the following two conditions: Definition x X is globally accessible if for any x X, there exists a perfect foresight path φ from x such that lim t φ(t) = x holds.. x X is absorbing if there exists a neighborhood Y of x such that for any x Y and a perfect foresight path φ from x, lim t φ(t) = x holds. A globally accessible and absorbing state is unique if it exists. Thus, the joint requirement of these conditions is seen as the standard selection criteria in this literature. The following result shows that the point-mass state on an iterated GH-dominant equilibrium is absorbing and globally accessible if players are patient enough. Proposition 5. Let a A N be an iterated GH-dominant equilibrium. Then, 1. there is r > such that δ a is globally accessible if r r;. δ a is absorbing. We give a proof sketch. For global accessibility, fix an initial state x X and a large block-size T >. We use a fixed-point argument to find a perfect foresight path from x with the following property: any i-agent who has a revision opportunity at t choose from A k i if t T k. Under such a path, fractions of agents who employ actions other than a are eventually diminished over time and the state converges to δ a. We set T to be large enough to ensure that fractions of actions A \ A k get sufficiently small within each block [T k, T (k +1)). Further, r should be small enough to ensure that players put enough weight on future payoffs so that they have incentives to switch to new actions. We can then find bounds on T and r that are uniform in initial point x, under which players incentives along a path are satisfied. For absorbingness, take any initial point x close to δ a and consider any feasible path from x. Then, under the calculation of continuation payoff (3) at t =, the marginal weight put on each a j is more than 1. This implies that 9

10 any i-player does not have an incentive to choose an action from A i \ A 1 i at t =. Then, an inductive argument shows that any i-agent who has a revision opportunity at t = should choose from A l+1 i for any l =,..., m 1, and, thus, a i. The above result generalizes Hofbauer and Sorger (), who study a class of asymmetric games that have linear incentives. In their setting, GH-dominance corresponds to their equilibrium selection criterion. Kojima and Takahashi (8) show that p-dominant equilibrium with i p i < 1 is selected under the perfect foresight dynamics. Their proof is based on a version of the critical path lemma (Kajii and Morris 1997). Oyama and Tercieux (9) show that an iterated p-dominant equilibrium with i p i < 1 is selected. Their proof uses a form of generalized potential function with supermodularity, which extends the result by Oyama et al. (8). To understand the coincidence with the global game selection of GH-dominance (under weak stochastic transitivity) in Section 3, we note that Takahashi (8) finds that perfect foresight dynamics (when r = ) corresponds to a variant of global games in the following sense. Fix any x, and consider the following n-player incomplete information game. State θ is drawn by the (improper) uniform prior over R, and each player i = 1,,.., n only observes her type t i = θ+η i before choosing an action from A i, where η i is exponentially distributed with mean 1. Each player i has x i (A i ) as a dominant strategy if t i <, and otherwise has the complete information game payoff u i : A N R. Then, φ is a perfect foresight path from x that is induced by α if and only if a strategy profile defined by { α i (t) if t, σ i (t) = x i if t < for each i N is a Bayes Nash equilibrium in this incomplete information game. It is worthwhile to note that in this incomplete information game, noise distributions are i.i.d and thus satisfy weak stochastic transitivity. On the other hand, Assumption is violated, and thus this game can have multiple Bayes Nash equilibria. Therefore, the standard results in global games, which we used in Section 3, cannot be applied. 5 Concluding Remarks Our analysis uncovers the key logic of equilibrium selection in a general class of games that is shared by global games and perfect foresight dynamics. This is driven by the particular acyclic belief structure held by players, which leads to the notion of iterated GH-dominance. Morris (14) emphasizes the importance of using an incomplete information game framework to analyze dynamic models with action-revision frictions. Because agents types in his models are naturally ordered by time at which they move, as in perfect foresight dynamics, our approach is likely to be useful too. In this paper, we have focused on a uni-dimensional state space in global games, following Frankel et al. (3), Basteck et al. (13), and other application papers in the literature. Oury (13) shows that a noise-independent global game selection under a uni-dimensional state space implies its selection under a multi-dimensional state space. Thus, our selection result in Section 3.1 directly carries over to the multi-dimensional setting. However Proposition 3 requires weak stochastic transitivity, which itself depends on uni-dimensionality, and it is not clear how to generalize it. We leave this as an open issue. We finally add a few remarks on comparisons with stochastic evolutionary selection. 1

11 1. There are some games in which a GH-dominant equilibrium is not selected by the model of Kandori et al. (1993) and Young (1993). For example, consider a tree network with a central player i, g ij = g ji = 1, g jk =, j, k i, and a binary-action coordination game as a component game, π(h, h) =, π(h, h ) = 1 and π(h, h ) = π(h, h) =. Note that (h,..., h) is a GH-dominant equilibrium. However, as shown by Jackson and Watts (), both (h,..., h) and (h,..., h ) are stochastically stable in this game.. Peski (1) introduces the notion of (strict) cardinal GR-dominance, which implies stochastic stability under a broad class of dynamics, such as logit-dynamics. 15 (,, ) is a strict cardinal GR-dominant equilibrium in the cyclical coordination game (Example 4). On the other hand, its global game selection fails, depending on noise distributions. This finding answers the open question raised by Peski (1). A Appendix A.1 Proof of Proposition 1 Let a A be iterated half-dominant π, with an associated sequence of actions (A k ) k=,..,m. Fix any network g. Take any k =,..., m 1 and belief µ ((A k ) N ) such that µ j (A k+1 ) = a A a k+1 i µ i (a i )1 {aj =a} 1 for any j. Then player i s expected payoff is u i (a i, µ i ) = a i µ i (a i ) j g ij π(a i, a j ) = a ν(a)π(a i, a), where ν (A) is the aggregated action distribution defined by ν(a) = g ij µ i (a i )1 {aj =a} j a i for each a A. By assumption ν(a k+1 ) = ν(a) = g ij µ j (A k+1 ) 1, a A k+1 j i has a strict incentive to choose an action from A k+1. This shows that (a,..., a ) is an iterated GH-dominant equilibrium with associated sequence of actions (A k ) k=,..,m for each player. Q.E.D. A. Proof of Proposition 3 We only consider the smallest case of (a i ) i N = (,..., ). Then, the analogous argument proves the largest case (a i ) i N = (m 1,..., m n ) as well. In general, if (a i ) i N 15 a A N is called a strict cardinal GR-dominant equilibrium if u i (a i, a i ) + u i (a i, a i) > max u i (a i, a i ) + max u i (a i, a i) a i a i a i a i holds for any i N and any strategy profiles a, a A N such that a j = a j or a j = a j for each j N. 11

12 {(,..., ), (m 1,..., m n )}, consider two hypothetical games, in which each player i is restricted to choose from {,..., a i } and {a i,..., m i }, respectively. Because a is an iterated GH-dominant equilibrium in these restricted games as well, its global game selection obtains under these games (by the smallest and largest case result). Then, by Lemma of Basteck et al. (13), the global game selection of a in the general game follows. Step 1: Elaborations E and E Let a = (,..., ) be an iterated GH-dominant equilibrium with associated sequence of action sets (A l i) i N,l=1,..,m. As in Frankel et al. (3) and Basteck et al. (13), we can characterize the global game selection s (θ ) by studying the following simplified games. Consider an incomplete information game E, lower-elaboration, in which each player i has a payoff function Ũi(a i, a i, t i ) that depends directly on her signal t i, defined by Ũ i (a i, a i, t i ) = { a i if t i <, u i (a i, a i ) if t i. (4) That is, Ũi is equal to the original payoff u i in G for nonnegative signals, while a i = is a dominant action for negative signals. Players signals are given by t i = θ + η i, where θ is drawn uniformly from interval Θ := [ A N 1, A N + 1] and η i follows f i as in the original global game. By standard results of supermodular games, there is a highest pure strategy equilibrium s = (s i ) i N. It consists of monotonic step functions, where at each step at least one player switches to a higher action. s reaches a highest action profile for sufficiently high signals, and constant for signals above A N. Let (ā i ) i N := (s i ( A N )) i N denote the highest action profile. Similarly, we define E, upper-elaboration, which is analogous to E except that the payoff of each i is given by u i for signal t i and a i for t i >. Let (a i ) i N := (s i ( A N )) i N denote the lowest action profile attained in the lowest equilibrium s in E. Then Theorem 1 of Basteck et al. (13) implies that lim θ θ s i (θ) = ā i and lim θ θ s i (θ) = a i for each i N. Step : (ā i ) i N = (a i ) i N Let s denote the highest pure strategy equilibrium in lower-elaboration E. Define threshold t l i := sup{t i Θ t i t i, s(t i) A l i} for each i N and l = 1,..., m. Note that for each i and l, t l i holds since = a i A l i is a dominant action at negative signals. We will show by induction that t l i > A N for every i N and l = 1,..., m. First, we show t 1 i > A N for every i N. Towards a contradiction, suppose there is a set of players J N who have thresholds no greater than than A N. Fix any such player j J. She is indifferent between choosing between an action in A 1 j and another action in A j \ A 1 j after observing t j = t 1 j. Because a is an iterated GH-dominant equilibrium, there exists a player j 1 j, such that j believes that j 1 chooses an action in A 1 j 1 with probability strictly less than 1 when j observes t 1 j. 16 That is, Pr[t j1 > t 1 j 1 t j = t 1 j] > 1. As θ follows the uniform prior belief, this can be rewritten as Pr[η j1 t 1 j 1 > η j t 1 j] > This is because otherwise j has a strict incentive to choose an action from A 1 j at t j = t 1 j, as a is an iterated GH-dominant equilibrium, which is a contradiction. 1

13 Because such j 1 has threshold t j1 A N as well, the same argument ensures that there exists another player j such that Pr[η j t j > η j1 t j1 ] > 1. Because J is finite, continuing this argument implies that there exists a sequence of players with the form j = j, j 1,..., j n 1, j n = j such that Pr[η jm+1 t 1 j m+1 > η jm t 1 j m ] > 1 holds for each m =,..., n. This contradicts the assumption of weak stochastic transitivity. The remaining steps can be shown by using the same argument. That is, t l i > A N for every i N and l = 1,..., m. This implies ā i = a i for each i. Step 3: (a i ) i N = (a i ) i N Let s denote the lowest equilibrium in upper-elaboration E, where a i = s i ( A N ) for each i N. We only show that a i A 1 i for each i N. Then, the same argument ensures that a i A l i for each l =,..., m so that a i = a i. Fix any player i N and assume that A 1 i A i, as otherwise there is nothing to prove. Let s denote the lowest strategy profile in E, in which every player i chooses = a i at every signal. By the standard result in supermodular games, the lowest equilibrium s of E can be achieved by iterating the smallest best-response beginning with s. That is, we take a sequence of strategy profiles s, s 1, s,..., where s = s and s k i is i s smallest best-response against s k 1 i for k = 1,,... and i N. Let t k i := sup{t i R t i t i, s k i (t i) A 1 i } for each k = 1,,... and i N. By weak stochastic transitivity, for each k = 1,,..,, there is i(k) N such that t k j + η j t k i(k) + η i(k) for all j N. Suppose that there is some k such that t k i(k ) < 1. If there is no such k, then t k i holds for all k and i N, which ensures the desired claim a i = s i ( A N ) = a i for each i. Take any i N, and we now show that t k +1 i + η i t k i(k ) + η i(k ). Otherwise, t k j + η j t k +1 i + η i for all j and t k +1 i hold. Then, i has a strict incentive to choose a i at signal t k +1 i against strategy profile s k i, which leads to a contradiction. Therefore, as is complete and transitive, t k i + η i t k i(k ) + η i(k ) holds for any k k and i. This ensures the desired claim a i = s i ( A N ) = a i for each i. Q.E.D. A.3 Detail of Example 4 Consider the lower-elaboration E as defined in the proof of Proposition 3. We claim that the highest acton profile (ā i ) i N = (s i ( A N )) i N under the highest equilibrium s in the lower-elaboration E is (1, 1, 1). s is obtained by iterating the greatest best-response beginning with s, where s is the strategy profile in which 1 is chosen at every signal. That is, we take a sequence of strategy profiles s, s 1, s,..., where s = s and s k i is i s greatest best-response against s k 1 i for k = 1,,... and i N. We show by induction that s k i (t i ) = 1 for each k =, 1,,.., i N, and t i. The case of k = is by true by definition. Suppose that the claim is true up to k. Take any δ. At the strategy profile s k, conditional on observing t i = δ, player i believes that player i + 1 (mod. 3) chooses with probability less than Pr[t i+1 < t i = δ] Pr[t i+1 < t i = ] = Pr[η i+1 < η i ] < 1 + β i so that she has a strict incentive to choose 1. This shows that s k +1 i (t i ) = 1 for each i N and t i. Therefore lim θ θ s (θ) = (1, 1, 1), and (,, ) is not a global game selection. A.4 Proof of Proposition 5 13

14 1. δ a is globally accessible if r is sufficiently small. Fix T >. For x X, define a set of paths ) Φ x T = {φ Φ x i N, l = 1,..., m, t T (l 1), supp (α φi (t) A l i} Note that any path φ Φ x T is such that lim t φ(t) = δ a. This set Φ x T is compact under the topology of uniform convergence at compact time-intervals. Define the dynamic best-response correspondence Br: Φ x Φ x by ) Br(φ) = {ψ Φ x i N, t, supp (α ψi (t) arg max V i (a i, t, φ)} a i This correspondence is upper-hemicontinuous and convex-valued. Note that a fixed point of Br is a perfect foresight path from x. We show that there is r > such that, if r r, then Br(Φ x T ) Φx T holds for all sufficiently large T. Then, applying Kakutani fixed point theorem to the correspondence Br restricted to the domain Φ x T, it has a fixed point φ Φx T. That is, there is a perfect foresight path φ from x that converges to δ a. To show the above Br(Φ x T ) Φx T, take any φ Φx T, i N, l = 1,..., m and t T (l 1). Note that ) V i (a i, t, φ) = u i (a i, (1 + r) e (1+r)s φ i (t + s)ds Because supp (α j (s)) A l j holds for each j and s t, under the distribution (1+r) e (1+r)s φ i (t+s)ds (A i ) that appears in the above RHS, the marginal probability on A l j is (1 + r) = (1 + r) = e (1+r)s ( e s φ j (t)(a l j) + 1 e s) ds ( (1 q)φj (t)(a l j) + q ) (1 q) r dq ( (1 q)φj (t)(a l j) + q ) dq ( φj (t)(a l j) + 1 ) 1 as r. Likewise the marginal probability on A l 1 j 1 ( φj (t)(a l 1 j ) + 1 ) 1 ( 1 e T + 1 ) converges to as r. Therefore, if T is sufficiently large and r is sufficiently small, then arg max ai V i (a i, t, φ) A l i holds for all t T (l 1). Bounds on such r and T can be taken uniformly in φ, l = 1,..., m, i N. This implies Br(Φ x T ) Φx T. 14

15 Finally, because the relevant inequalities in the previous paragraph do not depend on x, we can also take r and T uniformly in x. This shows that for any x there is a perfect foresight path from x such that lim t φ(t) = δ a, i.e., δ a is globally accessible.. δ a is absorbing. Q.E.D. Take a neighborhood Y of δ a such that for any x Y, x i (a i ) 1 ɛ holds for each i N. We require ɛ > to be sufficiently small such that, for each i, (i) 1+r +r a i u i (a i, µ i ) A l i holds for any l = 1,..., m if µ j (A l j) 1 and µ j (A l 1 j ) 1+r 1 (1 ɛ) + hold for each j. +r +r Then, we show that any perfect foresight path φ from any x Y is such that α φ i (t) = δ a for all t, and therefore, lim t φ(t) = δ a, which ensures that δ a is absorbing. We prove this claim by induction. First, we show that supp (α i (t)) A 1 i for all i and t under any perfect foresight path from x. To see this, suppose there exists a perfect foresight path φ from some x Y and time t at which a i A 1 i is chosen by a positive mass of i-players at t. Take the infimum of such t. Note that ) V (a i, t, φ) = u i (a i, (1 + r) e (1+r)s φ i (t + s)ds s ) = u i (a i, (1 + r) e (1+r)s (e s φ i (t) + e s s α φ i (s )ds )ds Under the distribution (1+r) e (1+r)s (e s φ i (t)+ s s es α φ i (s )ds )ds (A i ) in the second line, for each j, the marginal probability on A 1 j is more than (1 + r) e (+r)s φ j (t)(a 1 j)ds = 1 + r + r φ j(t)(a 1 j) 1, 1 + r + r x j(a 1 j) This implies arg max ai V (a i, t, φ) A 1 i, a contradiction. ) Next, suppose we have shown that supp (α φi (t) A l i for all i, t, and l = 1,..., k under any perfect foresight path from x. Towards a contradiction, suppose there exists a perfect foresight path φ from some x Y and time t at which a i A k+1 i is chosen by a positive mass of i-players at t. Take the infimum of such t. Analogous to the previous paragraph, we can calculate the discounted probability distribution over A i such during future commitment periods. Under this distribution, the marginal probability on each A k+1 j is more than 1+rx +r j(a k+1 j ) 1. Also, using the inductive hypothesis, the marginal probability on A k j is equal to 1+rx +r j(a k j ) r This implies arg max ai V i (t, a i, φ) A k+1 i, a contradiction. Therefore, by induction, we have shown the desired claim that supp (αi α (t)) m l=1 Al i = {a i } for all i and t under any perfect foresight path from any x Y. 15

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