空间非齐次演化博弈
Spatially Inhomogeneous Evolutionary Games
摘要：
我们引入并研究了一个空间分布的玩家系统的平均场模型，该系统通过由复制动态驱动的进化博弈进行交互。策略的演变是由一个受位置和不同参与者之间的互动影响的复制动力学，并返回一个关于引导他们运动的速度场的反馈。我们的方法的主要新颖之一涉及整个系统的描述，它可以表示为一个不断发展的概率测度$Σ$在一个无限维的状态空间（对$(x,σ)$的位置和分布的战略）。我们提供了一个拉格朗日和欧拉描述的演变，我们证明了它们的等价性，连同存在性，唯一性和稳定性的解决方案。作为一个副产品的稳定性结果，我们也得到收敛的有限代理模型，我们的平均场公式，当玩家的数量$N$走向无穷大，和初始的离散分布的位置和策略收敛。为此，我们开发了一些基本的泛函分析工具来处理巴拿赫空间中的相互作用动力学和连续性方程，这可能是独立的兴趣。
英文摘要：
We introduce and study a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represented by an evolving probability measure $Σ$ on an infinite dimensional state space (pairs $(x,σ)$ of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean-field formulation, when the number $N$ of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces, that could be of independent interest.
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