On perturbed best response learning and equilibrium selection in homogeneous and Bayesian evolutionary games

Abstract

Evolutionary game theory is a sub-field of game theory concerned with the behavior of large populations of strategically interacting players who recurrently revise their strategies based on the availability of better payoff opportunities. In such games, rather than directly focusing on the equilibrium analysis (as in the case of static/one-shot games), the strategic behavior of the population is modeled via an evolutionary game dynamic (a differential equation on a suitable state space of the population). Consequently, the theory aims to study the asymptotic properties (such as stability, convergence) of these evolutionary dynamics. An important class of evolutionary dynamic studied in the literature is the class of ”perturbed best response dynamic.” In such a dynamic, the expected payoff of the population is perturbed using a “perturbation function” (in most cases, the Shannon entropy) and the population “best responds” to these perturbed expected payoffs. This thesis is primarily dedicated towards extending the literature of perturbed best response dynamics and deriving asymptotic stability and equilibrium selection results of the dynamic for various classes of games with a continuum of strategies and under incomplete information. To be more precise, the theory of evolutionary games can broadly be classified into four categories: • homogeneous population (players are of the same type) finite strategy games, • homogeneous population continuum strategy games, • heterogeneous population (players are of different types) finite strategy games, and • heterogeneous population continuum strategy games. As far as perturbed best response dynamic is concerned, the existing literature of evolutionary games has mostly been confined to the following cases where: the underlying population is homogeneous; the games are defined on finitely many strategies; and the perturbation function used is the Shannon entropy. Some works in this direction include Hofbauer (1995); Hofbauer and Sandholm (2002); Ely and Sandholm (2005); Hofbauer and Hopkins (2005); Hofbauer and Sorin (2006); Hofbauer and Sandholm (2007); Zusai (2023) to name a few. This leads us to the following set of questions: 1. Is it possible to extend the theory of perturbed best response dynamic to homogeneous population games with a continuum of strategies; and perhaps also for arbitrary perturbation functions? 2. Suppose that the underlying game has multiple equilibria. Is there a way to characterize which equilibrium among the many is most likely to be selected by the population in the long run? 3. Can we develop a theory of perturbed best response dynamic for Bayesian population games with finitely many strategies as well as for games with a continuum of strategies? This thesis aims to solve the aforementioned problems in a series of five chapters.