On the connection between symmetric $N$-player games and mean field games

By Markus Fischer. 2014

145 citations recorded by the source corpus.

Mean field games are limit models for symmetric $N$-player games with interaction of mean field type as $N\to\infty$. The limit relation is often understood in the sense that a solution of a mean field game allows to construct approximate Nash equilibria for the corresponding $N$-player games. The opposite direction is of interest, too: When do sequences of Nash equilibria converge to solutions of an associated mean field game? In this direction, rigorous results are mostly available for stationary problems with ergodic costs. Here, we identify limit points of sequences of certain approximate Nash equilibria as solutions to mean field games for problems with It{o}-type dynamics and costs over a finite time horizon. Limits are studied through weak convergence of associated normalized occupation measures and identified using a probabilistic notion of solution for mean field games.

Subfields: Economics and Econometrics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c8bd77795059d58a92b838ba433133fe212e1015

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Citations in the corpus

The Master Equation and the Convergence Problem in Mean Field Games (2015, background): “The problem addressed here—concerning closed loop Nash equilibria—differs in a substantial way from [25, 45]: Indeed, we underline the surprising fact that the Nash system (1), which concerns equilibria in which the players observe each other, converges to an equation in which the players only need ”
From the master equation to mean field game limit theory: a central limit theorem (2018, background): “A law of large numbers has only recently come into focus [6, 31, 30, 19], clarifying how a MFG arises as the limit of a suitable sequence of n-player games as n→∞, in the presence of both idiosyncratic and common sources of noise.”
On the convergence of closed-loop Nash equilibria to the mean field game limit (2018, background): “The first comprehensive results came in [43, 26], but notably working with the distinct (and typically simpler to analyze) concept of open-loop controls.”
Mean field games via controlled martingale problems: Existence of Markovian equilibria (2014, background): “For example, concurrently with the finalization of this manuscript for publication, recent papers exploited relaxed control theory to study mean field games with common noise [14] as well as the convergence of equilbria of finite-player games [20, 34].”
MEAN FIELD GAMES: A TOY MODEL ON AN ERDOS-RENYI GRAPH (2017, background): “When the finite player equilibria are taken over open loop controls, compactness arguments, without any need for asymptotic uniqueness, may be used, see for instance [10, 13]; however, this strategy fails when equilibria are computed over controls in closed loop form.”
$N$-player games and mean-field games with absorption (2016, background): “Original citation: Campi, Luciano and Fischer, Markus (2018) N-player games and mean-field games with absorption.”
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “For the probabilistic approach to MFGs, in addition to the above mentioned works [13, 1, 14, 38, 43], we refer to Carmona and Delarue [12], Lacker [37], and Fischer [19].”