Mean field game theory has been developed largely following two routes. One of them, called the direct approach, starts by solving a large-scale game and next derives a set of limiting equations as the population size tends to infinity. The second route is to apply mean field approximations and formalize a fixed point problem by analyzing the best response of a representative player. This paper addresses the connection and difference of the two approaches in a linear quadratic (LQ) setting. We first introduce an asymptotic solvability notion for the direct approach, which means for all sufficiently large population sizes, the corresponding game has a set of feedback Nash strategies in addition to a mild regularity requirement. We provide a necessary and sufficient condition for asymptotic solvability and show that in this case the solution converges to a mean field limit. This is accomplished by developing a re-scaling method to derive a low-dimensional ordinary differential equation (ODE) system, where a non-symmetric Riccati ODE has a central role. We next compare with the fixed point approach which determines a two-point boundary value (TPBV) problem, and show that asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true. We further address non-uniqueness in the fixed point approach and examine the long time behavior of the non-symmetric Riccati ODE in the asymptotic solvability problem.

中文翻译：

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线性二次平均场博弈：渐近可解性及其与不动点方法的关系


平均场博弈论的发展主要遵循两条路线。其中一种称为直接法，首先解决大规模博弈，然后随着人口规模趋于无穷大而推导出一组极限方程。第二条路线是应用平均场近似并通过分析代表性玩家的最佳响应来形式化固定点问题。本文讨论了线性二次 (LQ) 设置中两种方法的联系和区别。我们首先为直接方法引入渐近可解性概念，这意味着对于所有足够大的群体规模，除了温和的正则性要求之外，相应的博弈还具有一组反馈纳什策略。我们提供了渐进可解性的充分必要条件，并表明在这种情况下解收敛到平均场极限。这是通过开发重新缩放方法来导出低维常微分方程 (ODE) 系统来实现的，其中非对称 Riccati ODE 发挥着核心作用。接下来，我们与确定两点边界值（TPBV）问题的不动点方法进行比较，并表明渐近可解性意味着不动点方法的可行性，但反之则不然。我们进一步解决了定点方法中的非唯一性，并检查了渐进可解性问题中非对称 Riccati ODE 的长期行为。