Convergence Analysis and Strategy Control of Evolutionary Games With Imitation Rule on Toroidal Grid

This paper investigates discrete-time evolutionary games with a general stochastic imitation rule on the toroidal grid, which is a grid network with periodic boundary conditions. The imitation rule has been considered as a fundamental rule in the field of evolutionary game theory, while the grid is treated as the most basic network and has been widely used in the research of spatial evolutionary games. However, current investigations of evolutionary games on grids mainly use simulations or approximation methods, whereas few strict analyses are carried out on one-dimensional grids. To the best of the authors' knowledge, this paper is the first to prove the convergence of the evolutionary prisoner's dilemma, the evolutionary snowdrift game, and the evolutionary stag hunt game with the imitation rule on the two-dimensional grid. Furthermore, this paper studies the Minimum Agent Consensus Control (MACC) problem. It is demonstrated that for some evolutionary games on the toroidal grid, at least four fixed cooperation nodes are required to lead all nodes to reach cooperation almost surely.