Extension of a spatial evolutionary coordination game with neutral options

Abstract

The multiagent evolutionary games on a lattice are equivalent to a kinetic Ising model if the uniform pair interactions are defined by a two-strategy coordination game and the logit rule controls the strategy updates. Now we extend this model by allowing the players to use additional neutral strategies that provide zero payoffs for both players if one of them selects one of the neutral strategies. In the resulting n-strategy evolutionary games the analytical methods and numerical simulations indicate continuous order-disorder phase transitions when increasing the noise level if n does not exceed a threshold value. For larger n the system exhibits a first order phase transition at a critical noise level decreasing asymptotically as 2/ln(n).

Evolutionary games combining two or three pair coordinations on a square lattice

“…These numerical data suggest that the robustness of the Ising universality class is not disturbed by an additional coordinated strategy pair in the corresponding four-strategy evolutionary game. Here we emphasize that similar phenomena were observed for models where the coordinated strategy pair is extended by neutral strategies [30,31]. Fitting the power-law assumption to the 13 data points closest to the critical temperature yields the following approximations of the critical exponents:…”

“…In other words, the m-Ising-pair game is mean-field equivalent to the 2m-strategy elementary coordination game, whose mean-field behavior was previously studied in Ref. [30].…”

“…A general n-strategy coordination-type game's payoff matrix is a linear combination of payoff matrices of elementary coordination games that describe Ising-type coordination between strategies p and q (p < q), while the remaining n − 2 strategies are neutral, that is, choosing one of them always yields zero payoff for both players [16,30]. The elements of such an elementary payoff matrix are…”

“…Even among these very simple coordination-type games coordination can emerge in different ways. If strategy updates are governed by the logit rule [25][26][27][28][29] and the players are located at the sites of a square lattice, then the number of neutral strategies plays a decisive role [30]: When the number of strategies is below a threshold value (n < n th ), the system undergoes a continuous (Ising-type) order-disorder phase transition as the noise level is increased; otherwise, a first-order phase transition can be observed. Adding a self-dependent component to the elementary coordination game does not necessarily destroy coordination [31].…”

“…For example, Monte Carlo simulations have indicated that extending the Ising model with a sufficiently small number of neutral strategies and a properly chosen additional selfdependent game component does not change the system's critical properties [30,31]. Combining more elementary coordination games, however, results in models like clock models, Potts models, the Ashkin-Teller model, etc., which (with the notable exception of the two-Ising-pair game or four-state clock model [33,34]) belong to different universality classes.…”