Evolutionary potential games on lattices

Abstract

Game theory provides a general mathematical background to study the effect of pair interactions and evolutionary rules on the macroscopic behavior of multi-player games where players with a finite number of strategies may represent a wide scale of biological objects, human individuals, or even their associations. In these systems the interactions are characterized by matrices that can be decomposed into elementary matrices (games) and classified into four types. The concept of decomposition helps the identification of potential games and also the evaluation of the potential that plays a crucial role in the determination of the preferred Nash equilibrium, and defines the Boltzmann distribution towards which these systems evolve for suitable types of dynamical rules. This survey draws parallel between the potential games and the kinetic Ising type models which are investigated for a wide scale of connectivity structures. We discuss briefly the applicability of the tools and concepts of statistical physics and thermodynamics. Additionally the general features of ordering phenomena, phase transitions and slow relaxations are outlined and applied to evolutionary games. The discussion extends to games with three or more strategies. Finally we discuss what happens when the system is weakly driven out of the "equilibrium state" by adding non-potential components representing games of cyclic dominance.

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

“…We study evolutionary games on a square lattice of N = L × L sites with periodic boundary conditions [11,12,14]. Each of the sites hosts one of the N equivalent players who repeatedly play the same one-shot game with their four nearest neighbors.…”

“…where s x denotes the strategy chosen by player x, the summation runs over their nearest neighbors at sites x + δ, and the payoffs of the two-player game are defined by the n × n matrix A [11,12,19,20]. Between each round of playing one-shot games, players are allowed to change their strategy in random sequential order following the so-called logit rule: The probability of player x switching from strategy s x to any available pure strategy s x is…”

“…Methods for the evaluation of V (if it exists) are given in Ref. [12]. Potential games governed by the logit rule evolve into the Boltzmann distribution [35,36]: The probability p(s) of finding the system in strategy profile s becomes…”

“…Multiagent evolutionary games [10][11][12][13][14] are a family of model systems used to study the spontaneous emergence of cooperation. In these models, individual players, who are located at the sites of a lattice or a graph, repeatedly play two-player games with their neighbors and change their strategies according to a dynamical rule based on their payoffs.…”

“…The concept of matrix decomposition [12,[15][16][17][18] has revealed that payoff matrices, which define player incomes in two-player games [11,12,19,20], are linear combinations of basis matrices that form four orthogonal subsets and represent different types of interactions: self-dependence, cross-dependence, coordination, and cyclic dominance. These game classes are clearly distinguishable via a few key matrix properties.…”