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2016 Evolutionary games on the lattice: death-birth updating process
Stephen Evilsizor, Nicolas Lanchier
Electron. J. Probab. 21: 1-29 (2016). DOI: 10.1214/16-EJP4380


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.


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Stephen Evilsizor. Nicolas Lanchier. "Evolutionary games on the lattice: death-birth updating process." Electron. J. Probab. 21 1 - 29, 2016.


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

zbMATH: 1336.91019
MathSciNet: MR3485359
Digital Object Identifier: 10.1214/16-EJP4380

Primary: 60K35 , 91A22

Keywords: cooperation , death-birth updating process , Evolutionary game theory , evolutionary stable strategy , interacting particle systems , prisoner’s dilemma , Replicator equation


Vol.21 • 2016
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