论文标题
具有非线性激活的有限状态平均现场游戏的主方程
Master equations for finite state mean field games with nonlinear activations
论文作者
论文摘要
我们在有限的状态空间上制定了一类平均现场游戏，其各种原则类似于连续状态的平均野外游戏。我们构建了一个受控的连续性方程，该方程在有限的可逆连续时间马尔可夫链引起的图上具有非线性激活函数。在这些图中，每个边缘都通过原始过程的过渡概率和不变性度量来加权。使用图形上的这些受控的动力学和值函数的动态编程原理，我们得出了几个关键组件：平均现场游戏系统，功能性汉密尔顿 - 雅各比方程以及潜在平均场游戏的有限概率空间上的主方程。通过密度 - 频率对的凸优化重新重新制定，可以确保对潜在平均现场游戏系统的解决方案的存在和唯一性。我们还得出了在连续的状态空间上的非电力游戏和混合游戏的主方程的变异原理。最后，我们在两点空间上提供了几个离散平均场游戏动力学的具体示例，并配有封闭式解决方案。这些示例包括离散的Wasserstein距离，平均现场计划和潜在的平均现场比赛。
We formulate a class of mean field games on a finite state space with variational principles resembling those in continuous-state mean field games. We construct a controlled continuity equation featuring a nonlinear activation function on graphs induced by finite-state reversible continuous time Markov chains. In these graphs, each edge is weighted by the transition probability and invariant measure of the original process. Using these controlled dynamics on the graph and the dynamic programming principle for the value function, we derive several key components: the mean field game systems, the functional Hamilton-Jacobi equations, and the master equations on a finite probability space for potential mean field games. The existence and uniqueness of solutions to the potential mean field game system are ensured through a convex optimization reformulation in terms of the density-flux pair. We also derive variational principles for the master equations of both non-potential games and mixed games on a continuous state space. Finally, we offer several concrete examples of discrete mean field game dynamics on a two-point space, complete with closed-formula solutions. These examples include discrete Wasserstein distances, mean field planning, and potential mean field games.
