  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 to 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 (or networked) evolutionary games. However, currently the investigation
of evolutionary games on grids mainly uses simulations or approximation
methods, 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 imitation
rule on the two-dimensional grid, for the first time to our best knowledge.
Simulations show that our results may almost reach the critical convergence
condition for the evolutionary snowdrift (or hawk-dove, chicken) game. Also,
this paper provides some theoretical results for the strategy control of
evolutionary games, and solves the Minimum Agent Consensus Control (MACC)
problem under some parameter conditions. We show that for some evolutionary
games (like the evolutionary prisoner's dilemma) on the toroidal grid, one
fixed defection node can drive all nodes almost surely converging to defection,
while at least four fixed cooperation nodes are required to lead all nodes
almost surely converging to cooperation.
