标题： 非线性非局部扩散方程在连续协调与反协调博弈分析中的应用 显示英文标题

标题： Nonlinear Nonlocal Diffusion Equations for the Analysis of Continuous Coordination and Anti-Coordination Type Games

摘要： 经济学家、生态学家、社会学家以及研究集体行为中涌现的全局属性的其他人对具有明确空间或关系结构的协调博弈感兴趣。 当个体组成的集合通过近视最佳响应或其他复制者动力学试图相互协调行动时，由此产生的动力系统可以表现出许多丰富的行为。 然而，这些行为的研究仅限于玩家数量可数且关系结构离散描述的情况下。 通过将一类协调博弈（包括真正的协调博弈本身）扩展到连续环境中，我们可以开始使用来自偏微分方程和非局部方程的新工具来研究协调和合作行为。 在这项研究中，我们提出了一个有严格支持的结构化协调型博弈向连续空间和连续策略环境的扩展，并证明了在某些假设下，这些博弈的动力学可以用一个非线性、非局部扩散方程来描述。 我们进一步证明了初始值问题在未规定边界数据情况下的存在性和唯一性。 对于真正的协调博弈，我们进一步证明了一个最大值原理、弱正则性结果，以及一些了解协调方程解行为的数值结果。 我们给出了几个建模结果，通过严格的分析和数值实验刻画了稳态解，并以一个关于非齐次问题的结果作为结论。 显示英文摘要

摘要： Coordination games with explicit spatial or relational structure are of interest to economists, ecologists, sociologists, and others studying emergent global properties in collective behavior. When assemblies of individuals seek to coordinate action with one another through myopic best response or other replicator dynamics, the resulting dynamical system can exhibit many rich behaviors. However, these behaviors have been studied only in the case where the number of players is countable and the relational structure is described discretely. By giving an extension of a general class of coordination-like games, including true coordination games themselves, into a continuous setting, we can begin to study coordination and cooperative behavior with a new host of tools from PDEs and nonlocal equations. In this study, we propose a rigorously supported extension of structured coordination-type games into a setting with continuous space and continuous strategies and show that, under certain hypotheses, the dynamics of these games are described through a nonlinear, nonlocal diffusion equation. We go on to prove existence and uniqueness for the initial value problem in the case where no boundary data are prescribed. For true coordination games, we go further and prove a maximum principle, weak regularity results, as well as some numerical results toward understanding how solutions to the coordination equation behave. We present several modeling results, characterizing stationary solutions both rigorously and through numerical experiments and conclude with a result towards the inhomogeneous problem.