论文标题
控制的平均野外游戏：关于纳什均衡的融合
Mean Field Games of Controls: on the convergence of Nash equilibria
论文作者
论文摘要
在本文中，我们研究了一类平均野外游戏，通过受控状态和控制过程的关节（条件）分布实现平均场相互作用。这些策略是$ open \; loop $ type，并且可以控制波动系数$σ$。使用（受控的）fokker-Planck方程，我们引入了一个均值控制控制的均值解决方案的概念，一方面证明了这些解决方案之间的关系，另一方面是$ε_n$ - NASH Equilibria。结果表明，$ε_n$ - $ n $中的nash equilibria在$ n $中具有限制，因为$ n $倾向于无穷大，每个限制都是对控件的平均场景游戏的测量值解决方案。相反，可以作为$ n $ - 玩家游戏中的$ε_n$ - nash equilibria的序列的限制获得任何测量值的解决方案。换句话说，测量值的解决方案是$ε_n$ - NASH EEFILIBRIA的累积点。同样，通过考虑$ε$ - 强大的控件平均野外游戏解决方案，这是经典强的解决方案，通过承认小误差$ε获得了最佳解决方案，我们证明，当$ε$转至零时，量度值值的解决方案是这种类型的解决方案的累积点。最后，在没有共同噪音的情况下证明了控制均值的均值游戏的测量值解决方案的存在。
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.
