TY - JOUR
T1 - Evolutionary games on the lattice
T2 - Death-birth updating process
AU - Evilsizor, Stephen
AU - Lanchier, Nicolas
N1 - Funding Information:
Both authors were partially supported by NSA Grant MPS-14-040958.
Publisher Copyright:
© 2016, University of Washington. All Rights Reserved.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
KW - Cooperation
KW - Death-birth updating process
KW - Evolutionary game theory
KW - Evolutionary stable strategy
KW - Interacting particle systems
KW - Prisoner’s dilemma
KW - Replicator equation
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U2 - 10.1214/16-EJP4380
DO - 10.1214/16-EJP4380
M3 - Article
AN - SCOPUS:84963831210
SN - 1083-6489
VL - 21
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
ER -