Replicator Dynamics of Axelrod’s Norms Games

The optimal control of Axelrod's social norm game

“…In system , we note that x NP and x NL represent population fractions who follow the norm, whereas x DP and x DL represent fractions who defect the norm. According to known results in the work of Andrews et al, there are several equilibrium points of system . Moreover, the evolutionary stable states that arise in such a game involve defect and punish (DP) or defect and lenient (DL) strategies.…”

“…8 Andrews et al (2015) looked at Axelrod's norm and metanorm games using classic game theoretic frameworks, where the population of individuals is divided in fractions given by 4 pure strategies. 9 Using replicator dynamics, they showed that all Axelrod's conclusions hold, without the need of the agent-based framework. In this paper, we explore the applications of the control theory to the social norm game and its associated replicator dynamics, expanding on the work of Andrews et al 9 The optimal control problems started in the 1950s when researchers studied variational problems, 10 starting with the works of Pontryagin.…”

“…9 Using replicator dynamics, they showed that all Axelrod's conclusions hold, without the need of the agent-based framework. In this paper, we explore the applications of the control theory to the social norm game and its associated replicator dynamics, expanding on the work of Andrews et al 9 The optimal control problems started in the 1950s when researchers studied variational problems, 10 starting with the works of Pontryagin. 11 Since then, many researchers have contributed to the concept of optimal control with necessary conditions (see, for example, the works of Jacobson et al, 12 Girsanov,13 and Kirsch et al 14 ).…”

“…Andrews et al (2015) looked at Axelrod's norm and metanorm games using classic game theoretic frameworks, where the population of individuals is divided in fractions given by 4 pure strategies . Using replicator dynamics, they showed that all Axelrod's conclusions hold, without the need of the agent‐based framework.…”

“…Using replicator dynamics, they showed that all Axelrod's conclusions hold, without the need of the agent‐based framework. In this paper, we explore the applications of the control theory to the social norm game and its associated replicator dynamics, expanding on the work of Andrews et al…”