ON THE CONVERGENCE OF CLOSED-LOOP NASH EQUILIBRIA TO THE MEAN FIELD GAME LIMIT

• 成果类型: Article
• 署名作者: Lacker, Daniel
• 署名单位: Columbia University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1541
• 发表日期: 2020
• 页码: 1693-1761
• 关键词: stochastic differential-games; vlasov systems; propagation; EXISTENCE; mimicking; EQUATIONS; chaos; state

摘要

This paper continues the study of the mean field game (MFG) conver- gence problem: In what sense do the Nash equilibria of n-player stochastic differential games converge to the mean field game as n -> infinity? Previous work on this problem took two forms. First, when the n-player equilibria are openloop, compactness arguments permit a characterization of all limit points of n-player equilibria as weak MFG equilibria, which contain additional randomness compared to the standard (strong) equilibrium concept. On the other hand, when the n-player equilibria are closed-loop, the convergence to the MFG equilibrium is known only when the MFG equilibrium is unique and the associated master equation is solvable and sufficiently smooth. This paper adapts the compactness arguments to the closed-loop case, proving a convergence theorem that holds even when the MFG equilibrium is nonunique. Every limit point of n-player equilibria is shown to be the same kind of weak MFG equilibrium as in the open-loop case. Some partial results and examples are discussed for the converse question, regarding which of the weak MFG equilibria can arise as the limit of n-player (approximate) equilibria.