Spatial Patterns in Winning Mediated Thermodynamic Strategy Evolution

The emergence of single strategy communities is often observed in spatial games played repeatedly. However, many social and biological systems exhibit a memory effect of successful strategy choices, represented by a caching of winnings which mediates future survival. We study a dynamical system defined by a repeated game on a lattice in one and two dimensions, where each agent stores their winnings as a measure of past success. Strategy updates are governed by a Boltzmann distribution, with the local energy given by the negative of the total local cache value for each strategy. Sites with higher cache values are effectively colder, and thus less likely to change strategy than sites with lower values. For a parameterized rock-paper-scissors game, we find a condition under which stationary local communities form, for which the domain sizes scale with the size of the system. We analyze a special case where community formation occurs but without fixed boundaries, leading to pattern migration. Using this analysis, we show which spatial structures are unstable in 1D. Comparison is made with numerical results for similar patterns in 2D.