Evolutionary game dynamics of combining the Moran and imitation processes

doi:10.1088/1674-1056/28/2/020203

Abstract One of the assumptions of previous research in evolutionary game dynamics is that individuals use only one rule to update their strategy. In reality, an individual's strategy update rules may change with the environment, and it is possible for an individual to use two or more rules to update their strategy. We consider the case where an individual updates strategies based on the Moran and imitation processes, and establish mixed stochastic evolutionary game dynamics by combining both processes. Our aim is to study how individuals change strategies based on two update rules and how this affects evolutionary game dynamics. We obtain an analytic expression and properties of the fixation probability and fixation times (the unconditional fixation time or conditional average fixation time) associated with our proposed process. We find unexpected results. The fixation probability within the proposed model is independent of the probabilities that the individual adopts the imitation rule update strategy. This implies that the fixation probability within the proposed model is equal to that from the Moran and imitation processes. The one-third rule holds in the proposed mixed model. However, under weak selection, the fixation times are different from those of the Moran and imitation processes because it is connected with the probability that individuals adopt an imitation update rule. Numerical examples are presented to illustrate the relationships between fixation times and the probability that an individual adopts the imitation update rule, as well as between fixation times and selection intensity. From the simulated analysis, we find that the fixation time for a mixed process is greater than that of the Moran process, but is less than that of the imitation process. Moreover, the fixation times for a cooperator in the proposed process increase as the probability of adopting an imitation update increases; however, the relationship becomes more complex than a linear relationship.

Keywords: evolutionary game dynamics Moran process imitation process fixation probability fixation time

Received: 12 September 2018
Revised: 15 November 2018
Accepted manuscript online:

PACS:	02.50.Le	(Decision theory and game theory)
	02.50.Ga	(Markov processes)
	05.45.Pq	(Numerical simulations of chaotic systems)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 71871171, 71871173, and 71832010).

Corresponding Authors: Cui-Ling Gu
E-mail: 2017102060001@whu.edu.cn

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Xian-Jia Wang(王先甲), Cui-Ling Gu(顾翠伶), Shao-Jie Lv(吕少杰), Ji Quan(全吉) Evolutionary game dynamics of combining the Moran and imitation processes 2019 Chin. Phys. B 28 020203

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[1]	Evolutionary games in a generalized Moran process with arbitrary selection strength and mutation Quan Ji(全吉) and Wang Xian-Jia(王先甲) . Chin. Phys. B, 2011, 20(3): 030203.

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