arXiv 2403.169622024-03math.OCmath.PR

$N$人动态博弈的$α$-势博弈框架

An $α$-potential game framework for $N$-player dynamic games

Xin Guo, Xinyu Li, Yufei Zhang

备注：: 36 pages

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AI总结：

本文提出$α$-势博弈框架，将动态博弈寻找$α$-Nash均衡转化为最小化$α$-势函数，并给出解析刻画及在随机微分博弈中的应用。

摘要：

本文提出并研究了一种一般形式的动态$N$人非合作博弈，称为$α$-势博弈，其中参与者单方面偏离其策略时其价值函数的变化等于$α$-势函数的变化（误差为$α$）。类似于静态势博弈（对应于$α=0$），$α$-势博弈框架被证明能将寻找动态博弈$α$-Nash均衡的挑战性任务转化为最小化$α$-势函数。此外，建立了$α$-势函数的解析刻画，其中$α$用价值函数二阶导数不对称性的幅度表示。对于状态动态为受控扩散的随机微分博弈，$α$根据参与者数量、容许策略的选择、交互强度以及参与者间的异质性水平来刻画。分析了两类随机微分博弈，即分布式博弈和具有平均场交互的博弈，以突出$α$对超越平均场范式（关注同质参与者的$N$的极限）的一般博弈特征的依赖。为分析$α$-NE，将相关的优化问题嵌入到条件McKean-Vlasov控制问题中。建立了一个验证定理，基于无穷维Hamilton-Jacobi-Bellman方程的解来构造$α$-NE，对于线性二次博弈，该方程被简化为常微分方程组。

英文摘要：

This paper proposes and studies a general form of dynamic $N$-player non-cooperative games called $α$-potential games, where the change of a player's value function upon her unilateral deviation from her strategy is equal to the change of an $α$-potential function up to an error $α$. Analogous to the static potential game (which corresponds to $α=0$), the $α$-potential game framework is shown to reduce the challenging task of finding $α$-Nash equilibria for a dynamic game to minimizing the $α$-potential function. Moreover, an analytical characterization of $α$-potential functions is established, with $α$ represented in terms of the magnitude of the asymmetry of value functions' second-order derivatives. For stochastic differential games in which the state dynamic is a controlled diffusion, $α$ is characterized in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. Two classes of stochastic differential games, namely distributed games and games with mean field interactions, are analyzed to highlight the dependence of $α$ on general game characteristics that are beyond the mean-field paradigm, which focuses on the limit of $N$ with homogeneous players. To analyze the $α$-NE, the associated optimization problem is embedded into a conditional McKean-Vlasov control problem. A verification theorem is established to construct $α$-NE based on solutions to an infinite-dimensional Hamilton-Jacobi-Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.

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