Equilibrium Concepts for Social Interaction Models

Arxiv preprint arXiv: …, 2007

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP).

Brazilian Review of Econometrics, 2007

We analyze the random field model applied to a non-cooperative social game of binary choices, as proposed by Durlauf (1997) and Brock and Durlauf (2001), but we allow for a more generic distribution of the agents' heterogeneity. We aim to show that their results still hold, i.e., there can be multiple equilibria when agents have some interest in adjusting their strategic behavior to that prevailing as the aggregate choice and we discuss some features such as positive feedbacks, the social multiplier, and the possibility of increasing returns on the outcomes of economic policy due to social interactions.

We consider the dynamics, existence and stability of the equilibrium states for large populations of individuals who can play various types of non--cooperative games. The players imitate the most attractive strategies, and the choice is motivated not only by the material payoffs of the strategies, but also by their popularity in the population. The parameter which determines the weights of both factors in the equilibrium states has the same analytical form for all types of considered games, and is identified with the sensitivity to reinforcements parameter in the Hernstein's Matching Law. We prove theorems of existence and uniqueness, and discuss examples of multiple locally stable polymorphic equilibria for the considered types of games.

This paper illustrates the effects of global or local social influences upon binary choice. Analytical results are summarized and an ACE (Agent based Computational Economics) approach is used to investigate the corresponding mechanisms of interdependence in the case of a coordination problem and finite size effects.

Econometrica, 1991

NOTES AND COMMENTS SOCIAL STABILITY AND EQUILIBRIUM BY ITZHAK GILBOA AND AKIHIKO MATSUI 1 1. INTRODUCTION IN THE FIELD OF NONCOOPERATIVE GAME THEORY, Nash equilibrium (Nash (1951)) has played a central role as a solution concept. In bold strokes, one may discern two major interpretations of Nash equilibrium in the context of rational players. The first, which is close to the "eductive" interpretation of Binmore (1987, 1988) and the "complete information" interpretation of Kaneko (1987), assumes that the game is played exactly once (if it is a repeated game, the repetition occurs once), and the players have sufficient knowledge and ability to analyze the game in a rational manner. Sometimes it is assumed that all players have consistent hierarchies of beliefs, where the game and their priors are common knowledge. Bayesian interpretation such as proposed by Aumann (1987) advanced this idea to the level that the players have a common prior. From this point of view, however, Nash equilibrium seems far from being satisfactory as it does not satisfy some requirements of "strategic stability."2 Thus, many studies have been made to refine the concept; among them are Selten (1975), Myerson (1978), Kalai and Samet (1984), and Kohlberg and Mertens (1986). Some studies (see, e.g., Brandenburger and Dekel (1987)) loosen the requirement of common knowledge, but still require some a priori knowledge. The second interpretation is sometimes referred to as the "evolutive" (Binmore) or "naive" interpretation (Kaneko). It does not require that participants in the game know its structure or other facts at the outset. According to this interpretation, a similar situation is repeated many times, and people use trials and errors in choosing better strategies on the basis of information they gradually acquire. A Nash equilibrium is considered as a stationary point in this repeated situation. At this point, it is worth noting that the price theory of an earlier age such as Walrasian economics shares the basic view of the world with the naive interpretation. It assumes rational participants in the economy but does not assume any common knowledge among participants. They do not know and do not have to know the entire structure of the economy; rather, they observe aggregated signals such as prices on the basis of which they determine their behavior. This "naive" price theory has solved many economic problems under some appropriate assumptions on the market structure. For example, in a perfect competition model, the assumption of price-takers results in that the participants have (usually unique) dominant strategies as a function of the price signal. The purpose of this paper is to apply similar analysis to general n-person normal-form games. In our model, we assume a large population out of which individual players are randomly matched to play a one-shot normal-form game; hence each one of them may consider oneself a "price-taker" and ignore one's effect on others' behavior. In price theory and game theory alike, there is interest in the stability of an equilibrium, and more generally, in the dynamics of processes which may or may not lead to an equilibrium. However, in our interpretation of a game, this question seems even 1 The authors wish to thank the participants of a seminar at Northwestern University as well as Professors Ehud Kalai, Andreu Mas-Colell, Kiminori Matsuyama, Robert Weber, two anonymous referees, and especially Dov Monderer for many useful comments and references. The first author gratefully acknowledges partial support from NSF Grant No. IRI-8814672. 2 See the discussion in Kohlberg and Mertens (1986).

Many real-life settings of consumer-choice involve social interactions, causing targeted policies to have spillover-effects. This paper develops novel empirical tools for analyzing demand and welfare-effects of policy-interventions in binary choice settings with social interactions. Examples include subsidies for healthproduct adoption and vouchers for attending a high-achieving school. We establish the connection between econometrics of large games and Brock-Durlauf-type interaction models, under both I.I.D. and spatially correlated unobservables. We develop new convergence results for associated beliefs and estimates of preference-parameters under increasing-domain spatial asymptotics. Next, we show that even with fully parametric specifications and unique equilibrium, choice data, that are sufficient for counterfactual demandprediction under interactions, are insufficient for welfare-calculations. This is because distinct underlying mechanisms producing the same interaction coefficient can imply different welfare-effects and deadweightloss from a policy-intervention. Standard index-restrictions imply distribution-free bounds on welfare. We illustrate our results using experimental data on mosquito-net adoption in rural Kenya.

2006

We investigate a class of binary choice models with social interactions. We propose a unifying perspective that integrates economic models using a utility function and psychological models using an impact function. A general approach for analyzing the equilibrium structure of these models within mean-field approximation is developed. It is shown that within a mean-field approach both the utility function and the impact function models are equivalent to threshold models. The interplay between heterogeneity and randomness in model formulation is discussed. A general framework is applied in a number of examples leading to some well-known models but also showing the possibility of more complex dynamics related to multiple equilibria. Our synthesis can provide a basis for many practical applications extending the scope of binary choice models.

Game Theory and Mathematical Economics, 2006

We discuss stochastic dynamics of finite populations of individuals playing games. We review recent results concerning the dependence of the long-run behavior of such systems on the number of players and the noise level. In the case of two-player games with two symmetric Nash equilibria, when the number of players increases, the population undergoes multiple transitions between its equilibria.

Lecture Notes in Economics and Mathematical Systems, 2006

This paper illustrates the effects of global or local social influences upon binary choice. Analytical results are summarized and an ACE (Agent based Computational Economics) approach is used to investigate the corresponding mechanisms of interdependence in the case of a coordination problem and finite size effects. 1 This approach is qualified as "sociophysics". For a discussion of the relationship with mechanical physics, see (Durlauf 1999), and (Phan and Nadal and Gordon 2004).