Abstract
The best-response dynamics is an example of an evolutionary game where players update their strategy in order to maximize their payoff. The main objective of this paper is to study a stochastic spatial version of this game based on the framework of interacting particle systems in which players are located on an infinite square lattice. In the presence of two strategies, and calling a strategy selfish or altruistic depending on a certain ordering of the coefficients of the underlying payoff matrix, a simple analysis of the nonspatial mean-field approximation of the spatial model shows that a strategy is evolutionary stable if and only if it is selfish, making the system bistable when both strategies are selfish. The spatial and nonspatial models agree when at least one strategy is altruistic. In contrast, we prove that in the presence of two selfish strategies and in any spatial dimension, only the most selfish strategy remains evolutionary stable. The main ingredients of the proof are monotonicity results and a coupling between the best-response dynamics properly rescaled in space with bootstrap percolation to compare the infinite time limits of both systems.
Highlights
The framework of evolutionary game theory, which describes the dynamics of populations of individuals identified to players, has been initiated by theoretical biologist Maynard Smith and first appeared in his work with Price [7]
Each point of the d-dimensional square lattice is occupied by exactly one player who is characterized by her strategy
The spatial structure is included in the form of local interactions assuming that each player’s payoff only depends on the strategy of her 2d neighbors
Summary
The framework of evolutionary game theory, which describes the dynamics of populations of individuals identified to players, has been initiated by theoretical biologist Maynard Smith and first appeared in his work with Price [7]. Returning to general selfish-selfish interactions, the numerical simulations of the two-dimensional process displayed in Figure 1 suggest that, when a1 is slightly larger than a2 and the initial density p > 0 is small, the system fixates to a configuration in which the set of type 1 players consists of a union of disjoint rectangles, indicating that strategy 1 is unable to invade strategy 2 These simulations, are misleading due to the finiteness of the graph, and it can be proved that, in any dimension, the most selfish strategy always wins even when starting at a low density. The second ingredient is to show that, for the process starting from this reduced configuration, the set of type 1 players is a pure growth process, just like the Richardson model This strong monotonicity result is applied repeatedly to show that the best-response dynamics properly rescaled in space dominates stochastically bootstrap percolation with parameter d. From this domination and a result due to Schonmann [9, Theorem 3.1], we deduce that, unlike what Figure 1 suggests, the most selfish strategy invades the entire lattice
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
