Games on graphs: A minor modification of payoff scheme makes a big difference

Many techniques developed in simulations of physical models have been adopted in studies of game theory by researchers including physicists and mathematicians. In this work, we show that a seemly non-essential mechanism – what we refer to as a “payoff scheme” have a large impact on strategic outcomes of some games. Payoff scheme refers to here that how each player’s payoff is calculated in each round after the states of all of the players are determined. Conventionally either the accumulated or the average payoff of a player is used, where its payoff is calculated from pairing up the player with all of its neighboring players. Here we consider to calculate the payoff from pairing up with only one random player from the neighboring players. The average payoff scheme that involves averaging over all of the neighbors should, in a sense, be equivalent to repeatedly randomly pairing up with one neighbor a time, which we refer to as the stochastic payoff scheme. However, our simulation of games on graphs shows that, in many cases, the two payoff schemes lead to qualitatively different levels of cooperation: Seemly non-essential modifications might have large impact on behavioral outcomes. We have also observed that results from the stochastic scheme are more robust than the average scheme: different updating rules and initial states of the players do not have a large impact on the final level of cooperation in the former case when compared with those in the latter case. Introduction. – The emergence of cooperation has been one of the central topics in game theory and its application in social studies, human behavior and biology [1, 2]. Understanding the relatively high level of cooperation among inherently selfish players remains a challenge, especially in situations in which there is a social dilemma, where the theories of game predict defection but not cooperation as the solution to the games (i.e., the theoretically expected game outcome). However, it is well recognized that, in many social dilemmas, cooperation is observed much more frequently than what the theories predict. Many natural and social scientists were inspired to investigate possible mechanisms of the emergence of cooperation [3,4]. Thus far, it turns out that evolutionary game theory [5–7] in well-mixed or heterogeneously localized (on lattices or networks) populations provides the most general theoretical framework for this line of investigation. In evolutionary game theory, symmetric 2 × 2 games, such as the Prisoner’s Dilemma (PD) and the Snowdrift Game (SG), have been used comprehensively as the underlying social dilemma for studies of the evolution of cooperation. A symmetric 2×2 game can be represented by a payoff bi-matrix as G = [ R,R S, T T, S P, P ]