控制的平均场对策：关于纳什均衡的收敛性
Mean Field Games of Controls: on the convergence of Nash equilibria
摘要：
在本文中，我们研究了一类平均场博弈，其中平均场的相互作用是通过控制状态和控制过程的联合（条件）分布来实现的。策略是$open\;loop$型的，波动系数$σ$是可控的.利用（受控）Fokker-Planck方程，引入了控制的平均场对策的测度值解的概念，并通过收敛性结果证明了这些解与$ε_N$ -Nash平衡之间的关系.结果表明，当$N$趋于无穷大时，$N$ -局中人博弈的$ε_N$-Nash均衡具有极限，并且每个极限都是控制的平均场博弈的测度值解.相反，任何测度值解都可以作为$N$-局中人博弈中$ε_N$-纳什均衡序列的极限而得到。换句话说，测度值解是$ε_N$ -Nash均衡的累积点。同样地，通过考虑控制的平均场对策的$ε$ --强解，证明了当$ε$趋于零时，其测度值解是这类解的聚集点，其中$ε$--强解是最优解的经典强解，其中最优解是通过允许小误差$ε,$获得的.最后，在无共同噪声的情况下，证明了控制的平均场对策测度值解的存在性。
英文摘要：
In this paper, we investigate a class of mean field games where the mean field interactions are achieved through the joint (conditional) distribution of the controlled state and the control process. The strategies are of $open\;loop$ type, and the volatility coefficient $σ$ can be controlled. Using (controlled) Fokker-Planck equations, we introduce a notion of measure-valued solution of mean-field games of controls, and through convergence results, prove a relation between these solutions on the one hand, and the $ε_N$--Nash equilibria on the other hand. It is shown that $ε_N$--Nash equilibria in the $N$--player games have limits as $N$ tends to infinity, and each limit is a measure-valued solution of the mean-field games of controls. Conversely, any measure-valued solution can be obtained as the limit of a sequence of $ε_N$--Nash equilibria in the $N$--player games. In other words, the measure-valued solutions are the accumulating points of $ε_N$--Nash equilibria. Similarly, by considering an $ε$--strong solution of mean field games of controls which is the classical strong solution where the optimality is obtained by admitting a small error $ε,$ we prove that the measure-valued solutions are the accumulating points of this type of solutions when $ε$ goes to zero. Finally, existence of measure-valued solution of mean-field games of controls are proved in the case without common noise.
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