A comparison of the Moran Process and replicator equations for evolving social dilemma game strategies

Abstract

Social dilemma games are studied to gain insight into why humans cooperate with other unrelated people. The canonical game has cooperation and defection as the two strategies. Cooperation benefits the group, but a self-interested player can always do better by defecting. But if everybody defects, then the entire group loses. This tradeoff between cooperation and defection gives rise to the social dilemma. Social dilemma games need some method to evolve strategy changes between rounds. The two most widely accepted methods are a Moran process or replicator equations. Although both methods can predict how strategies evolve in a player population, no comparison of their performance has yet been made. In this paper we compare them in a public goods game which is an N-player version of prisoner's dilemma ($N>2$). Our results indicate only one of these methods should be used in future research efforts.

Introduction

Image a group of N people who are all engaged in an economic activity. During each round an individual interacts with one or more other individuals and receives a payoff. The amount of this payoff depends on the actions of the individual and the actions of the individuals they interact with. Individuals face a dilemma. They can choose to cooperate, which benefits everyone in the group, or they can choose to defect which only benefits the individual. Such economic situations are called social dilemmas.

All social dilemmas have two conflicting properties (Dawes, 1980): (1) defectors always do better than cooperators regardless of what actions others might take, and (2) mutual cooperation is the best group outcome. In particular, mutual cooperation has higher payoffs than mutual defection. However, the problem is that defection is quite tempting which makes mutual defection the inevitable outcome in all social dilemmas, at least as long as the maximization of personal utility is the only goal of interaction. Everybody loses if everybody defects.

Game theory provides an ideal framework for studying social dilemmas. The economic situation is formulated as a mathematical game which gives researchers something to play with. Different strategies can be evaluated to try and gain some insight into why humans cooperate and what circumstances promote cooperation in human populations. The most widely known social dilemma game is the iterated prisoner's dilemma and the most widely studied is the public goods game (PGG), which is its N-player version ($N>2$). Typically researchers posit some strategy that should promote cooperation and then run the game to see how various strategies change over time. Good performing strategies should increase in frequency within the population because other players switch to it to get the higher payoff. Thus the game needs some method to model players switching strategies. The two most common modeling methods used are the Moran process (Nowak et al., 2004; Taylor et al., 2004; Ohtsuki et al., 2007; Liu et al., 2017) and replicator equations (Schuster and Sigmund, 1983; Hofbauer and Sigmund, 2003; Hauert et al., 2006; Deng et al., 2016).

Both methods predict how strategy frequencies change over time but the underlying mathematics is completely different. The Moran process is purely stochastic whereas the replicator equations are purely deterministic. In this paper we empirically compare both methods in a PGG to see if they make the same predictions, both over time and in the final prediction, and explain why there are any differences. We believe only one of these methods should be used and provide a justification for that belief.