| Abstract: |
| We formulate a class of mean field games on a finite state space with variational principles resembling continuous state mean field games. We construct a controlled continuity equation with a nonlinear activation function on graphs induced by finite reversible Markov chains. With this controlled dynamics on graph and the dynamic programming principle for value function, we derive the mean field game systems, the functional Hamilton-Jacobi equations and the master equations on the finite probability space for potential mean field games. We also give a variational derivation for the master equations of non-potential games and mixed games on a finite state space. Finally, several concrete examples of discrete mean field game dynamics on a two-point space are presented with closed formula solutions, including discrete Wasserstein distances, mean field planning, and potential mean field games. |
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