Positive feedback in coordination games: Stochastic evolutionary dynamics and the logit choice rule

Abstract

We study the problem of stochastic stability for evolutionary dynamics under the logit choice rule. We consider general classes of coordination games, symmetric or asymmetric, with an arbitrary number of strategies, which satisfies the marginal bandwagon property (i.e., there is positive feedback to coordinate). Our main result is that the most likely evolutionary escape paths from a status quo convention consist of a series of identical mistakes. As an application of our result, we show that the Nash bargaining solution arises as the long run convention for the evolutionary Nash demand game under the usual logit choice rule. We also obtain a new bargaining solution if the logit choice rule is combined with intentional idiosyncratic plays. The new bargaining solution is more egalitarian than the Nash bargaining solution, demonstrating that intentionality implies equality under the logit choice model.

Introduction

Conventions and customs are sometimes determining factors for formal contracts. For example, Young and Burke (2001) show that local custom is a driving force in setting up the crop sharing contract terms in the state of Illinois. Customary patterns of behaviors such as asymmetric norms between racial groups and genders can also produce a mechanism by which inequalities persist for a long period of time (Naidu et al., 2017). Changes in informal convention sometimes induce formal institutional changes which may contribute to long-run economic growth (Hwang et al., 2016; Acemoglu et al., 2005). Thus, understanding both the disruption and emergence of conventions can shed light on problems such as economic incentives, inequality, and long-run growth. Conventions or social norms that typically form over time and last for a long period of time are frequently modeled as long run equilibria in stochastic evolutionary dynamics, in which agents play myopic best responses subject to mistakes, errors, or idiosyncratic plays (Young, 1993a; Kandori et al., 1993; Bowles, 2004). Therefore identifying the most likely evolutionary paths escaping from an existing convention or transitioning between conventions is a key step in studying the disruption and emergence of conventions.

Recently, one of the behavioral rules of myopic agents, the logit choice model, has become popular among researchers because of its analytic convenience (see Section 2 for existing studies using the logit choice rule). Under the logit choice rule, the probability of an agent's mistake decreases log-linearly in the payoff losses incurred by such a mistake. Recent experimental literature also supports the hypothesis that mistake probabilities decrease in payoff losses (see Section 2). In the widely used uniform mistake model, in which all possible mistakes are equally likely, the more likely path can be easily determined by comparing the number of mistakes involved and, hence, the lengths of paths (e.g., Kandori et al. (1993)). However, under the logit choice rule, determining the most likely path is far from obvious, because the probability of a path depends on the kinds of mistakes involved as well as on the length of the path. For example, Hwang and Newton (2016) provide an example in which two different kinds of mistake plays are involved in the most likely escape path from a convention under a finite population logit model (see Example 1 in Hwang and Newton (2016)).

Because of the complexity of the logit choice rule, there has been, so far, no general way to analyze the optimal evolutionary paths from one convention to another. As a concrete example, it is unknown which kind of contract conventions will emerge and persist when agents play the evolutionary version of the familiar Nash bargaining game (Nash, 1953) under the logit rules(see Section 5). The goal of this paper is to fill in this gap in the literature for general classes of games. Two recent studies address similar questions. Hwang and Newton (2016) study two-population coordination evolutionary models with zero off-diagonal payoffs and an arbitrary number of strategies, for both finite and infinite populations. Sandholm and Staudigl (2016) study a one-population coordination evolutionary model with three strategies in the infinite population limit. These two papers are discussed in more detail in Section 2 but we note here that their results are limited to specific classes of games (either games with zero off-diagonal payoffs or games with three strategies).

One of the main novelties of this paper is a new method—we call it “comparison principles”—which can be applied to one- or two-population games with an arbitrary number of strategies and which allows greatly reducing the complexity of finding the most likely evolutionary paths under the logit choice rule. Specifically, we find that under the logit choice rule, the positive feedback of agents (to coordinate) plays a key role. Kandori and Rob (1998) introduce the “marginal bandwagon property” to capture the positive feedback aspect of network externality, requiring that the advantage of strategy i over j is greater when the other player is playing strategy i. In stochastic evolutionary game theory, the (un-)likeliness of a path is measured by a quantity called the “cost”: the less costly a path is, the more likely is the transition induced by that path. We show that for finite population models with the logit choice rule, (i) positive feedback (defined by the marginal bandwagon property) implies that along the minimum cost escape paths from a status quo convention, agents always deviate first from the status quo convention strategy before deviating from other strategies (Lemma 4.1 (i)), and (ii) the relative strength of the positive feedback effects implies that the transitions from the status quo convention to another convention must occur consecutively in the cost optimal escape paths (Lemma 4.1 (ii)).

We then apply these results to the exit problem—the problem of finding a cost minimum path escaping from a convention—under finite population models and characterize the candidates for the cost minimizing paths, as follows. The candidates consist of (possibly) different kinds of repeated identical mistakes deviating from the status quo convention strategy (Proposition 4.1). Finally, to pin down the exact minimum cost escaping path, we consider the infinite population limit as in Sandholm and Staudigl (2016) (Proposition 4.2) and show that the most likely escape paths from the status quo convention involve only one kind of repeated identical mistakes of agents (Proposition 4.3, Proposition 4.4, and Theorem 4.1). This result holds for any coordination games satisfying the marginal bandwagon properties and some regularity conditions with an arbitrary number of strategies, regardless of symmetric or asymmetric games (hence, one- or two-population models; Theorem 4.1, Theorem 5.1). To the best of our knowledge, this is a novel result.

As an application of our main results, we study the evolutionary bargaining convention for the Nash demand games under the logit choice rule and show that the Nash bargaining solution arises as the stochastically stable convention under the usual logit choice rule (called the unintentional logit dynamic). We also obtain a new bargaining convention when the logit choice rule is combined with intentional idiosyncratic (non-best response) plays (called the intentional logit dynamic). By intentional idiosyncratic plays, we mean that agents always experiment with strategies under which they would do better, should that strategy induce a convention (Naidu et al., 2010; Hwang et al., 2018). We show that the new solution under intentional logit dynamics is more egalitarian than the Nash bargaining solution, hence intentionality implies equality (Proposition 5.1). The reason for equality is as follows: under the unintentional logit rule, some transitions from the egalitarian convention (the equal division convention) to the Nash bargaining convention (the unequal division convention) are driven by a population who stands to lose by such transitions. Under the intentional logit rule, every transition is driven by the population who stands to benefit. Thus, some unfavorable transitions leading to the Nash bargaining convention are replaced by favorable transitions to the deviant population, leading to a more equal convention than the Nash bargaining convention. It can be easily seen that our comparison principle as well as our remaining arguments for the logit choice rule hold for the uniform mistake models under the assumption of the marginal bandwagon property. Thus, our comparison principles provide a unified framework for analyzing evolutionary dynamics, including the uniform mistake and logit models.

This paper is organized as follows. Section 2 discusses the related literature. Section 3 introduces the basic setup and discusses, in some detail, an example illustrating our methods. We present our main results for the exit problem for one population models in Section 4. In Section 5, we present results for two population models and analyze the Nash demand game. In the appendix, we show that our result for the exit problem can be used to study the stochastic stability problem. The appendix also provides the technical details and proofs of the paper's results.