Large strategic dynamic interactions

Abstract

The paper presents a model of large strategic dynamic interactions in an environment with uncertain fundamentals. The interaction is among a large finite group of interdependent players, diversified in their preferences and information. We study an imagined-continuum equilibrium, a behavioral hybrid of games with a continuum of players and finite games. This equilibrium enables simple Bayesian reasoning and admits natural Markov-perfect equilibria. In addition, we establish bounds on the probabilistic discrepancies between players' beliefs that are derived from the continuum model and the actual finite reality.

Introduction

Large dynamic interactions play major roles in economic, political, online and other social systems. Typically, such an interaction involves a large finite number of participants and uncertain fundamentals that are an essential part of the functioning of the system. These uncertain fundamentals may have different effects on the preferences and information of the individual participants.

Papers that deal with large interactions cover a variety of theoretical issues and applications. Examples include markets (Novshek and Sonnenschein, 1978), bargaining (Mailath and Postlewaite, 1990), auctions (Olszewski and Siegel, 2016, Rustichini et al., 1994), voting (Feddersen and Pesendorfer, 1997, Myerson, 2000), electronic commerce (Gradwohl and Reingold, 2013), industrial organization and market design (Backus and Lewis, 2016, Weintraub et al., 2008).

However, these earlier papers have often been restricted to one-time interactions as opposed to ongoing interactions, to fully known fundamentals, and to perfect information and monitoring. To relax such restrictions this paper studies an uncertain large dynamic game, described as follows.

A finite large group of players plays a repeated stage game in which (1) player types, describing preferences and information, are correlated through an unknown state of fundamentals; (2) every period of play results in a random public outcome that depends on the aggregate distribution of players' actions and on the state of fundamentals, and; (3) players' period payoffs and information depend on the known public outcome combined with the players' private information.

Examples of such large dynamic games include repeated uncertain market games, in which the fundamentals may describe production and demand parameters and period outcomes may describe prices; repeated rush-hour-commute games, in which the fundamentals may describe road capacities and outcomes may describe driving times; and network-communication games, in which the fundamentals may describe technical parameters of communication devices and outcomes may describe data about their use.

We study a behavioral equilibrium concept that is tractable, yet sufficiently refined for the analysis of events in various applications. In this imagined-continuum equilibrium each of the n players behaves as if she is one of a continuum of players. However, the actual play-path of the game is generated in a finite n-player model.

In general terms, this paper extends the mission of the large-games literature to provide a microfoundation for large (macro) games, in Bayesian dynamic environments with bounded but highly rational players. Still, the model needs further extensions before it can be used in economic applications, as in finance and macro-economics.¹ For example, as it is now, the model cannot address issues of economic growth and shocks.

As an example, we use the model to describe a Markov-perfect equilibrium in a repeated Cournot game, when the traders have diverse preferences and interdependent information about uncertain fundamentals.

In a companion paper (Kalai and Shmaya, 2017), we use the imagined-continuum equilibrium to study the issues of learning, predicting, and stability in large games. We ask such questions as “To what extent prices in a trade period become predictable to economic agents who make plans prior to the start of the trade period?” and “to what extent rush-hour-driving-times in commute games become predictable to drivers who make their commute plans before the start of a rush hour?”.

In our large dynamic games, period outcomes (and thus payoffs and information) depend on population proportions, and not on the number of players and their names.² As a consequence, other than knowing that the number of players n is large, equilibrium play does not require precise knowledge of the value of n.

The imagined-continuum equilibrium we use throughout the paper is a behavioral hybrid of two standard models of large games: continuum and asymptotic games.

In accordance with the literature of continuum games, the imagined-continuum viewpoint assumes that every player in a large n-player game ignores the impact of her own actions on the outcome of the game and replaces population random variables by their expected values.

Consider, for example, a two-route commute game with a finite large number of drivers, each having to choose one of two routes, E or W. A continuum-imagining driver who believes that, independently of the others, every driver selects E over W with probabilities 2/3 to 1/3, assumes that with certainty, and regardless of her own choice, 2/3 of the drivers will drive on E and 1/3 on W.

But while the continuum viewpoint works well in many large one-shot games, one has to be more careful when dealing with repeated games. Consider, for example, the two-route commute game above with n players who play the following naive-best-response strategy: in the beginning and subsequent “balanced periods,” in which the proportion of W choosers was between 49 percent and 51 percent, E and W are chosen with equal probability; but following unbalanced periods, the less-traveled route is chosen in the next period.

Now there is a time-dependent discrepancy between the continuum view and the large-but-finite reality: for each of the n continuum-imagining players, the imagined play-path will be perfectly balanced forever. But since the actual number of players is finite, the actual play-path will eventually enter an alternating pattern: all-on-E, all-on-W, all-on-E... . In some applications, such as ones involving policy making and system design, researchers should consider the actual play-path, even if the players themselves subscribe to the continuum view.

Indeed, in the imagined-continuum equilibrium studied in this paper, players compute their best-response actions as if they are a part of a continuum. However, the outcome of the imagined-continuum equilibrium is defined to be the actual play-path: the one determined by the strategies of these n continuum-imagining players. We study this outcome asymptotically, as n becomes arbitrarily large.

This paper addresses the following issues:

(1) Equilibrium computations: How difficult are the Bayesian computations for the players and the analysts? We present an explicit description of the computations, showing that they are tractable, and that imagined-continuum equilibria include natural Markov-perfect equilibria despite the complex informational structure.

(2) Finite-infinite continuity: How large are the discrepancies between imagined probabilities of events computed along the imagined play-path, in comparison with their actual probabilities computed along the actual play-path? We present explicit upper bounds on these discrepancies that depend on the parameters of uncertainty, on the number of players, and on the period of play.

Combining (1) and (2) can simplify difficult computations. For example, we show that the play of an imagined-continuum equilibrium approximates the play of a regular Nash equilibrium when the number of players is large.

In an imagined-continuum equilibrium, each player chooses her strategy as if she were playing a continuum version of the game. Standard studies of continuum games include Schmeidler (1973), Judd (1985), and Sun (2006); dynamic versions are studied in the mean field games literature; see Guéant at al. (2011) and Adlakha et al. (2015).

However, unlike the literature cited above, in the current paper the continuum game is one of unknown fundamentals. Since the uncertainty over a single state of fundamentals cannot be aggregated away by the laws of large numbers, our continuum game must be turned into a Bayesian continuum game. In this sense the continuum component of our large dynamic games may be viewed as a Bayesian mean-field game, and our result about the existence of a Markov-perfect equilibrium is therefore contributions to such Bayesian mean-field game theory.

To compute the probability of events in an imagined equilibrium, we use an asymptotic analysis of a finite game in which the number of players n becomes arbitrarily large. There is a large literature that uses asymptotic methods for the analysis of large games. Such papers include Rashid (1983), McLean and Postlewaite (2002), Khan and Sun (2002), Kalai (2004), Carmona and Podczeck (2012), Khan et al. (2015), Carmona and Podczeck (2009), Yang (2017), Azrieli and Shmaya (2013).

In addition to the introduction of fundamental uncertainty, the asymptotic analysis in the imagined-continuum equilibrium differs from the papers cited above in two respects. First, since we deal with a repeated game, the time of play has to be incorporated into the analysis. But there is a second important conceptual and computational difference: the literature cited above studies the behavior of n standard players, i.e., each best responding to the $n-1$ opponents in the n-player game, while we study the behavior of n continuum imaginers, i.e., each best-responding to an imagined-continuum of opponents.

Our study of finite-infinite continuity is different from earlier papers for the same two reasons. First, discrepancies between the limits of probabilities in the finite games and the corresponding probability in the continuum games depends on the period of play, as we show explicitly in this paper. Second, the events we study in the finite games are determined by the behavior of n continuum imaginers and not by the behavior of n standard players.

Green's paper was pioneering in combining large games with repeated games. His paper studies large repeated strategic interactions restricted to complete information and pure strategies. Green and the later paper of Sabourian (1990) derive conditions under which the Nash correspondence is continuous in the sense that the equilibrium of the continuum game is a limit of the equilibria of the standard n-player games with an increasing n. In addition to Green's paper, an earlier version of the myopic property discussed in this paper was studied in Al-Najjar and Smorodinsky (2000).

The idea of compressing computations to expected values, which is used in imagined-continuum equilibria, appears in a variety of models in economics. See, for example, McAfee (1993), Angeletos et al. (2007), and Jehiel and Koessler (2008), all of which study a dynamic global game with fundamental uncertainty.