From the master equation to mean field game limit theory: Large deviations and concentration of measure
From the master equation to mean field game limit theory: Large deviations and concentration of measure
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DOI：
10.1214/19-aop1359
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发表时间：
2018-04
期刊：
影响因子：
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通讯作者：
F. Delarue;D. Lacker;K. Ramanan
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文献类型：
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作者：
F. Delarue;D. Lacker;K. Ramanan
研究了一类由特殊和常见噪声源驱动的对称$n$-玩家随机微分对策序列，其中玩家通过经验分布相互作用。已知的是，当$n$趋于无穷大时，$n$-博弈者博弈的唯一纳什均衡经验度量收敛到相关平均场博弈的唯一均衡。在适当的正则性条件下，在没有公共噪声的情况下，我们用n元玩家纳什均衡经验测度和平均场均衡之间的Wasserstein距离的非渐近集中界来补充这个大数定律结果。我们还证明了纳什均衡经验度量序列满足弱大偏差原理，只有在不存在共同噪声的情况下才能将其加强为完全大偏差原理。对于这两组结果，我们首先利用主方程，一个表征平均场博弈的值函数的无限维偏微分方程组，通过改进我们在伴文中获得的估计，构造了一个关联的McKean-Vlasov相互作用的$n$-粒子系统，它指数地接近于大$n$的$n$-参与者博弈的Nash均衡动力学。然后，我们建立了存在共同噪声的McKean-Vlasov系统的弱大偏差原理。在没有共同噪声的情况下，我们将其升级为完全大偏差原理，并得到了McKean-Vlasov系统的新的浓度估计。最后，在两个不满足我们主要定理的假设的具体例子中，我们展示了如何调整我们的方法来建立大偏差和集中结果。
We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with non-asymptotic concentration bounds for the Wasserstein distance between the $n$-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean-Vlasov interacting $n$-particle system that is exponentially close to the Nash equilibrium dynamics of the $n$-player game for large $n$, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean-Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean-Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.