Electronic Journal of Probability

Evolutionary games on the lattice: death-birth updating process

Stephen Evilsizor and Nicolas Lanchier

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This paper is concerned with the death-birth updating process. This model is an example of a spatial game in which players located on the $d$-dimensional integer lattice are characterized by one of two possible strategies and update their strategy at rate one by mimicking one of their neighbors chosen at random with a probability proportional to the neighbor’s payoff. To understand the role of space in the form of local interactions, the process is compared with its nonspatial deterministic counterpart for well-mixing populations, which is described by the replicator equation. To begin with, we prove that, provided the range of the interactions is sufficiently large, both strategies coexist on the lattice for a parameter region where the replicator equation also exhibits coexistence. Then, we identify parameter regions in which there is a dominant strategy that always wins on the lattice whereas the replicator equation displays either coexistence or bistability. Finally, we show that, for the one-dimensional nearest neighbor system and in the parameter region corresponding to the prisoner’s dilemma game, cooperators can win on the lattice whereas defectors always win in well-mixing populations, thus showing that space favors cooperation. In particular, several parameter regions where the spatial and nonspatial models disagree are identified.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 17, 29 pp.

Received: 23 June 2015
Accepted: 22 February 2016
First available in Project Euclid: 25 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 91A22: Evolutionary games

interacting particle systems evolutionary game theory evolutionary stable strategy death-birth updating process replicator equation prisoner’s dilemma cooperation

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Evilsizor, Stephen; Lanchier, Nicolas. Evolutionary games on the lattice: death-birth updating process. Electron. J. Probab. 21 (2016), paper no. 17, 29 pp. doi:10.1214/16-EJP4380. https://projecteuclid.org/euclid.ejp/1456412957

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