潜在有限状态平均场对策的共同噪声消失选择
Selection by vanishing common noise for potential finite state mean field games
摘要：
本文的目标是提供一个选择原则，潜在的平均场游戏在有限的状态空间，并在这方面，以显示不最小化相应的平均场控制问题的平衡，应排除。我们的策略是偏微分方程的消失粘性方法的一个定制版本。在这里，粘性必须被理解为插入平均场博弈中的公共噪声的平方强度，或者等价地，作为主方程的相关抛物线版本中的扩散率参数。正如最近建立的，随机强制平均场游戏确实成为唯一可解的赖特-费舍尔共同噪声的相关选择，在主方程中的对应物是一个木村运营商的单纯形。我们在这里详细阐述了这个想法，使共同噪声的平均场游戏都是唯一可解的和潜在的，这意味着它的唯一的解决方案实际上是一个合适的随机平均场控制问题的唯一极小。取公共噪声强度为零的极限，得到了上述选择原理的严格证明。作为一个副产品，我们得到的经典解粘性主方程的平均场游戏与共同噪声收敛到的平均场控制问题的值函数的梯度没有共同噪声，因此，我们选择一个特定的弱解的主方程的原始平均场游戏。最后，我们建立了一个内在的唯一性准则，在一个合适的类内的解决方案。
英文摘要：
The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailored-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established recently, the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on this idea to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise; we hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class.
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