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This paper investigates discrete-time evolutionary games with a generalstochastic imitation rule on the toroidal grid, which is a grid network withperiodic boundary conditions. The imitation rule has been considered as afundamental rule to the field of evolutionary game theory, while the grid istreated as the most basic network and has been widely used in the research ofspatial (or networked) evolutionary games. However, currently the investigationof evolutionary games on grids mainly uses simulations or approximationmethods, while few strict analysis is carried out on one-dimensional grids.This paper proves the convergence of evolutionary prisoner's dilemma,evolutionary snowdrift game, and evolutionary stag hunt game with the imitationrule on the two-dimensional grid, for the first time to our best knowledge.Simulations show that our results may almost reach the critical convergencecondition for the evolutionary snowdrift (or hawk-dove, chicken) game. Also,this paper provides some theoretical results for the strategy control ofevolutionary games, and solves the Minimum Agent Consensus Control (MACC)problem under some parameter conditions. We show that for some evolutionarygames (like the evolutionary prisoner's dilemma) on the toroidal grid, onefixed defection node can drive all nodes almost surely converging to defection,while at least four fixed cooperation nodes are required to lead all nodesalmost surely converging to cooperation.

Convergence Analysis and Strategy Control of Evolutionary Games with Imitation Rule on Toroidal Grid: A Full Version