环面网格上带模仿规则的演化博弈的收敛性分析与策略控制：完整版
Convergence Analysis and Strategy Control of Evolutionary Games with Imitation Rule on Toroidal Grid: A Full Version
AI总结：
本文首次严格证明了环面网格上带随机模仿规则的演化博弈收敛性，并解决了最小智能体一致性控制问题，揭示了固定背叛与合作节点数量对系统收敛状态的影响。
摘要：
本文研究在环面网格（一种具有周期性边界条件的网格网络）上采用一般随机模仿规则的离散时间演化博弈。模仿规则被视为演化博弈论领域的基础规则，而网格被视为最基本的网络，并已广泛应用于空间（或网络化）演化博弈的研究中。然而，目前对网格上演化博弈的研究主要采用仿真或近似方法，而对一维网格进行严格分析的研究很少。据我们所知，本文首次证明了在二维网格上具有模仿规则的演化囚徒困境、演化雪堆博弈和演化猎鹿博弈的收敛性。仿真表明，我们的结果几乎达到了演化雪堆（或鹰鸽博弈、懦夫博弈）的临界收敛条件。此外，本文为演化博弈的策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智能体一致性控制（MACC）问题。我们证明，对于环面网格上的某些演化博弈（如演化囚徒困境），一个固定的背叛节点可以驱动所有节点几乎必然收敛于背叛，而至少需要四个固定的合作节点才能引导所有节点几乎必然收敛于合作。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
