On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness
On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness
复制标题
平均场博弈中的收敛问题：无唯一性的二态模型
DOI：
10.1137/18m1222454
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Guglielmo Pelino
中科院分区：
文献类型：
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作者：
Alekos Cecchin;P. Pra;Markus Fischer;Guglielmo Pelino
我们考虑有限范围内连续时间内的 N 玩家游戏和平均场游戏，其中每个智能体的位置属于 {-1,1}。如果平均场博弈解存在唯一性，例如在单调性假设下，主方程具有光滑解，可用于证明 N 人博弈的价值函数和反馈纳什均衡的收敛性，以及相关最优轨迹的混沌特性的传播。我们在这里研究一个反单调成本的例子，并证明平均场博弈恰好有三个解。我们证明了价值函数收敛于主方程的熵解，在这种情况下可以写成一维标量守恒定律，并且最优轨迹有一个极限：它们选择一个平均场博弈解决方案，因此存在混沌传播。此外，将平均场博弈系统视为确定性控制问题最优的必要条件，我们证明 N 人博弈选择该问题的最优值。
We consider N-player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to {-1,1}. If there is uniqueness of mean field game solutions, e.g. under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the N-player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with anti-monotonous costs, and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game soution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N-player game selects the optimum of this problem.
影响因子：
1.4
作者：
F. Delarue;D. Lacker;K. Ramanan
通讯作者：
F. Delarue;D. Lacker;K. Ramanan
DOI：
10.1214/19-aop1359
发表时间：
2018-04
期刊：
The Annals of Probability
影响因子：
--
作者：
F. Delarue;D. Lacker;K. Ramanan
通讯作者：
F. Delarue;D. Lacker;K. Ramanan