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Iterated Generalized Half-Dominance and Global Game Selection∗

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Iterated Generalized Half-Dominance and Global Game Selection∗ Iterated Generalized Half-dominance and Global Game Selection∗ Ryota Iijimay May 3, 2015 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 cri- terion, iterated generalized half-dominance. This criterion is shown to be a unique global game selection when noise distributions satisfy a regularity condition. A sim- ilar logic also applies to the perfect foresight dynamics of Matsui and Matsuyama (1995); an iterated generalized half-dominant equilibrium is a unique globally sta- ble 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 2 × 2 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, 2003). 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 gener- alization 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. yDepartment of Economics, Harvard University, e-mail: [email protected]. 1See Morris and Shin (2003) for a survey. Some recent papers incorporate endogenous information, which can prevent unique selection (e.g., Angeletos and Pavan, 2013). 1 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' in- terim 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 1 2 action is more than 2 . As opposed to the p-dominance criterion that we will discuss shortly, this criterion is only concerned with beliefs about each opponent's action sepa- rately 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 GH- dominant 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 ai 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 ti = θ + ηi, where each ηi is i.i.d noise with a positive ∗ ¯ density. Consider threshold strategies in which each player i chooses ai if ti < ti, and another action if ti > 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 ai and another ∗ ∗ action at signal ti = t¯i. Take an agent i with t¯i∗ = mini t¯i. If i observes signal ti∗ = t¯i∗ , ∗ ¯ 1 she believes that for any i 6= i , ti < ti holds with probability greater than 2 . That is, she ∗ 1 believes that each opponent i's marginal probability of choosing ai is more than 2 . Since ∗ ¯ ai∗ is a GH-dominant action, she has a strict incentive to choose it at signal ti∗ , 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 ti to be less than its threshold ¯ 1 ti with marginal probability more than 2 . 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., 2003), and show that a GH-dominant equilibrium is selected under the 2This generalizes the notion of half-dominance (Morris et al., 1995), which is defined for symmetric two-player games. 3This 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. 2 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 asym- metric players, and we illustrate this point using games on networks (Section 2.1). 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 par- ticular classes of games.4 A general class of global games with strategic complementarities ∗ is studied by Frankel et al. (2003). For a nonnegative vector p, an action profile (ai )i2N is a p-dominant equilibrium (Kajii and Morris, 1997) if each player i has a strict incen- tive to follow the equilibrium action whenever she believes that a joint probability of the opponents following the equilibrium action profile is more than pi. Frankel et al. (2003) P show that a p-dominant equilibrium with i pi < 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 P with many players while the p-dominance condition with i pi < 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 (2008), 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). 2 Iterated GH-dominance Consider a normal form game G =< N; (Ai; ui)i2N >. N = f1; 2; :::; ng is a set of players. Each Ai is a finite set of actions of player i 2 N, and each ui : AN ! R is a payoff 4Corsetti et al. (2004) introduced a large trader into a population of small agents. Guimaraes and Morris (2007) allow for heterogeneous payoffs in a model of currency attack. Sakovics and Steiner (2012) generalize the Laplacian belief property in a class of binary-action games of regime change with heterogeneous payoffs. 5 P That is, if i pi < 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 E0 ⊆ E on which E is common p-believed. Kajii and P Morris (1997) use this property to show that a p-dominant equilibrium with i pi < 1 is robust against general incomplete information perturbations.
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