Coalitional Stochastic Stability in Games, Networks and Markets

Coalitional stochastic stability in games, networks and markets Ryoji Sawa∗ Department of Economics, University of Wisconsin-Madison November 24, 2011 Abstract This paper examines a dynamic process of unilateral and joint deviations of agents and the resulting stochastic evolution of social conventions in a class of interactions that includes normal form games, network formation games, and simple exchange economies. Over time agents unilaterally and jointly revise their strategies based on the improvements that the new strategy profile offers them. In addition to the optimization process, there are persistent random shocks on agents utility that potentially lead to switching to suboptimal strategies. Under a logit specification of choice probabilities, we characterize the set of states that will be observed in the long-run as noise vanishes. We apply these results to examples including certain potential games and network formation games, as well as to the formation of free trade agreements across countries. Keywords: Logit-response dynamics; Coalitions; Stochastic stability; Network formation. JEL Classification Numbers: C72, C73. ∗Address: 1180 Observatory Drive, Madison, WI 53706-1393, United States., telephone: 1-608-262-0200, e-mail: [email protected]. The author is grateful to William Sandholm, Marzena Rostek and Marek Weretka for their advice and suggestions. The author also thanks seminar participants at University of Wisconsin-Madison for their comments and suggestions. 1 1 Introduction Our economic and social life is often conducted within a group of agents, such as people, firms or countries. For example, firms may form an R & D alliances and found a joint research venture rather than independently conducting R & D. When forming an alliance, two or more firms make a joint decision, or simultaneously make decisions while taking into account other firms’ decisions. In these settings, it is sometimes more appropriate to use a stronger solution concept than Nash equilibrium, one that accounts for joint deviations by groups of agents. In this paper, we introduce an evolutionary equilibrium selection approach for solution concepts stronger than Nash equilibrium, solution concepts that account for coalitional deviations. The importance of stronger solution concepts has been emphasized in recent years due to a grow- ing interest in social and economic networks. This is because Nash equilibrium has relatively weak predictive power in games on networks, particularly network formation games. Examples of such games are co-author networks (e.g. Jackson and Wolinsky (1996)), R & D networks among firms (e.g. Goyal and Moraga-Gonzalez´ (2001)), and trade agreement networks among countries (e.g. Furu- sawa and Konishi (2007)). When link formation between two agents requires both agents’ consent, Nash equilibrium fails to capture the fact that it may be beneficial for two agents to form a link. For instance, it is always a Nash equilibrium for no player to consent to any link, resulting in the empty network. It is thus natural in this context to use a stronger solution concept that is robust against deviations by pairs of agents as in the notion of pairwise stable equilibrium. Once joint deviations are admissible, it is reasonable to consider some settings which are not described by games because some unilateral deviations may be infeasible. For instance, consider a Gale-Shapley marriage problem. Suppose that an individual’s strategy is to choose whom to marry, and that a man and a woman will be married if and only if both agree to do so. This implies that the set of feasible strategy profiles is restricted: letting si denote individual i’s strategy, man m can choose sm = w if and only if woman w chooses sw = m. We call these generalizations of normal-form games interactions. Even when we apply stronger solution concepts to interactions, we may still face the problem of multiple equilibria. For example, in the Gale-Shapley marriage problem, any stable matching is a pairwise stable equilibrium. The average number of stable matchings grows more than proportion- ally as the population increases, making it difficult to predict which will emerge. To address this issue and single out one equilibrium, we apply the concept of stochastic stability. In the stochastic stability approach, we embed a static interaction in a dynamic process in which players revise their strategies based on the improvement that the new strategies offer them relative to the current strategies. We perform a stability test by adding stochastic noise to agents’ strategy choices in the above dynamic and studying the long-run outcomes as noise vanishes. This approach has been well developed to select an equilibrium among the set of Nash equilibria.1 However, despite the importance of decentralized but possibly joint decisions in many social and economic settings, the development of frameworks for analyzing stochastic stability in this setting is in its infancy. A pioneering paper by Jackson and Watts (2002) studied pairwise stochastically stable networks that are immune against random shocks in two players’ decisions. They restricted their 1See Chapter 12 of Sandholm (2010) for an overview. 2 attention to network formation games. The present paper follows a similar line to Jackson and Watts (2002), but extends the range of applications in three directions; (i) applying the stochastic stability approach to a broader class of environments, (ii) extending pairwise stochastic stability to stochastic stability with respect to an arbitrary set of feasible coalitions, and (iii) employing the logit-response dynamic, under which more serious mistakes are less likely to occur. Point (i) allows us to study stochastically stability in a broader class of environments. We define an interaction denoted by I = N, (Si, ui)i2N , S with player set N, strategy sets Si, payoff functions ui and the set of feasible strategy profiles S ⊆ S = ∏i2N Si. The restriction on strategy profiles allows us to consider settings where some unilateral deviations are prohibited or infeasible. For example, in an exchange economy, a consumer cannot unilaterally change her allocation, since doing so requires another consumer who is willing to exchange with her. Point (ii) allows us to apply our model in environments where more than two agents may form a coalition. Let R be a family of sets of players, representing the collection of feasible coalitions. We define an R-stable equilibrium of interaction I to be a feasible strategy profile which is immune against deviations by any coalition in R. Suppose for instance that S = S, so that I is a normal form game. Then if R is the family of singletons, R-stable equilibrium corresponds to Nash equilibrium; if R is the family of singletons and pairs, it corresponds to pairwise stable equilibrium; and if R is the family of all nonempty sets of players, it corresponds to strong equilibrium. If not all strategy profiles are feasible, then neither are all unilateral deviations, so some choice of R is necessary to define a meaningful notion of equilibrium. Point (iii) allows us to apply stochastic stability in settings where agents, while sometimes making mistakes, assign smaller probabilities to actions which deliver smaller payoffs. As an example, consider the probability that people send an email to unintended recipients. People will be more careful, and so make such a mistake with lower probability, when their email includes confidential information, e.g. a strategic move of their corporation. Our interest is to characterize strategy profiles that are stochastically stable against not only individual deviations but also coalitional deviations. We introduce a dynamic process in which players randomly form coalitions and revise their strategies in the following way. First, a new strategy profile for them is randomly proposed, and assessed by each agent. In the unperturbed updating process, an agent agrees to the new strategy profile if it yields her higher payoff than the original strategy profile. The new strategy profiles is accepted if all agents in the coalition agree. To this unperturbed dynamic process, we add stochastic noise that leads agents to agree to a new strategy profile accord- ing to the logit choice rule. Thus, coalitional deviations sometimes occur even if not all members of the coalition benefit, and this occurs with a probability that declines in the total cost to coalition members. As a consequence, the stochastic process visits every strategy profile repeatedly, and pre- dictions can be made concerning the relative amounts of time that the process spends at each. We consider the behavior of this system as the level of noise in agents’ choice becomes small, defining the stochastically stable strategy profiles to be those which are observed with positive frequency in the long run as noise vanishes. Our paper characterizes the stochastically stable strategy profiles of the logit-response dynamic when coalitional deviations are possible. The intuition for which strategy profiles are stochasti- 3 cally stable comes from the changes in payoffs by switching strategies. Under the logit choice rule, if agents face larger decreases in payoffs by switching to new strategies, it will be less likely that they agree to such revisions. Thus, the decrease in payoffs by a revision represents its unlikeliness. Roughly, the stochastically stable strategy profiles are those that minimize the sum of the payoff decreases by being switched from other strategy profiles. Those strategy profiles are harder to get away from and easier to get back to in the stochastic dynamic. We apply our main result (Theorem 1) to simple exchange economy and show that the stochastically stable allocations are those maximizing the sum of players’ utilities. Note that these allocations are Pareto efficient, but not uniquely so. We generalize our finding by characterizing a class of interactions that exhibit coalitional potential functions, showing that the stochastically stable outcomes are those maximizing the coalitional potential.

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