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

Title: 环形网格上具有模仿规则的进化博弈的收敛性分析与策略控制：完整版本

Abstract: 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. Show Chinese abstract

Abstract: 本文研究了在环面网格上具有通用随机模仿规则的离散时间进化博弈，该网格网络具有周期性边界条件。 模仿规则被认为是进化博弈论领域的一个基本规则，而网格被视为最基本的网络，并被广泛用于空间（或网络化）进化博弈的研究。 然而，目前对网格上的进化博弈的研究主要使用模拟或近似方法，而在一维网格上很少进行严格的分析。 据我们所知，本文首次证明了在二维网格上使用模仿规则的进化囚徒困境、进化雪堆博弈和进化猎鹿博弈的收敛性。 模拟结果显示，我们的结果可能几乎达到进化雪堆（或鹰鸽、懦夫）博弈的关键收敛条件。 此外，本文为进化博弈的战略控制提供了一些理论结果，并在某些参数条件下解决了最小代理共识控制（MACC）问题。 我们表明，在环面网格上的某些进化博弈（如进化囚徒困境）中，一个固定的背叛节点可以几乎肯定地使所有节点收敛到背叛，而至少需要四个固定的合作节点才能使所有节点几乎肯定地收敛到合作。