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arXiv 0903.1620math.OCmath.DS
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离散平均场对策

Discrete mean field games

Diogo A. Gomes, Joana Mohr, Rafael R. Souza

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摘要：

本文研究了离散时间有限状态动态对策的平均场模型。这些模型出现的情况下，涉及到一个非常大量的代理从一个国家到另一个国家根据一定的最优性标准。最优控制和微分对策（连续状态和时间）的平均场方法由Lasry和Lions介绍。在这里，我们考虑一个离散版本的问题。我们的设置如下：我们假设有大量的相同的代理人，可以在有限数量的状态。由于代理的数量非常大，我们假设平均场假设，也就是说，唯一相关的信息，为全球的发展是在每个国家的分数$π^n_i$的球员$i$在时间$n$。代理寻求最小化运行成本，这取决于$π$加上终端成本$V^N$。与最优控制相比，通常只需要终端成本$V^N$来解决问题，在平均场博弈中，需要代理人的初始分布$π^0$和终端成本$V^N$来确定解，即$0\leq n\leq N$的参与者分布$π^n$和价值函数$V^n$。由于需要指定初始和终止数据，我们称这个问题为初始-终止值问题。解的存在性是不平凡的。然而，根据Lasry和Lions的思想，我们能够建立平稳和初终值问题的存在性和唯一性。在本文的最后一部分，我们证明了本文的主要结果，即指数收敛到一个平稳解的$(π^0, V^0)$，作为$N\to \infty$，初终值问题与（固定）数据$π^{-N}$和$V^N$。

英文摘要：

In this paper we study a mean field model for discrete time, finite number of states, dynamic games. These models arise in situations that involve a very large number of agents moving from state to state according to certain optimality criteria. The mean field approach for optimal control and differential games (continuous state and time) was introduced by Lasry and Lions. Here we consider a discrete version of the problem. Our setting is the following: we assume that there is a very large number of identical agents which can be in a finite number of states. Because the number of agents is very large, we assume the mean field hypothesis, that is, that the only relevant information for the global evolution is the fraction $π^n_i$ of players in each state $i$ at time $n$. The agents look for minimizing a running cost, which depends on $π$, plus a terminal cost $V^N$. In contrast with optimal control, where usually only the terminal cost $V^N$ is necessary to solve the problem, in mean-field games both the initial distribution of agents $π^0$ and the terminal cost $V^N$ are necessary to determine the solutions, that is, the distribution of players $π^n$ and value function $V^n$, for $0\leq n\leq N$. Because both initial and terminal data needs to be specified, we call this problem the initial-terminal value problem. Existence of solutions is non-trivial. Nevertheless, following the ideas of Lasry and Lions, we were able to establish existence and uniqueness, both for the stationary and for the initial-terminal value problems. In the last part of the paper we prove the main result of the paper, namely the exponential convergence to a stationary solution of $(π^0, V^0)$, as $N\to \infty$, for the initial-terminal value problem with (fixed) data $π^{-N}$ and $V^N$.

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