Mean field games via controlled martingale problems: Existence of Markovian equilibria

By Daniel Lacker. 2014

162 citations recorded by the source corpus.

Mean field games are studied in the framework of controlled martingale problems, and general existence theorems are proven in which the equilibrium control is Markovian. The framework is flexible enough to include degenerate volatility, which may depend on both the control and the mean field. The objectives need not be strictly convex, and the mean field interactions considered are nonlocal and Wasserstein-continuous. When the volatility is nondegenerate, continuity assumptions may be weakened considerably. The proofs first use relaxed controls to establish existence. Then, using a convexity assumption and measurable selection arguments, strict (non-relaxed) Markovian equilibria are constructed from relaxed equilibria.

Subfields: Demography, Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/691fe5e1fbd5931692bbbfd7dac4b0a4535a0ec8

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Citations in the corpus

Mean Field Games of Timing and Models for Bank Runs (2016, unknown): “Probabilistic methods in mean field game theory originated in [10], although our techniques are more closely related to the weak convergence and compactness arguments of [12, 28, 26].”
On the connection between symmetric $N$-player games and mean field games (2014, methodology): “…has been used by Carmona and Lacker (2015) to establish existence and uniqueness results for nondegenerate systems and, more recently, by Lacker (2015), where existence of solutions is established for mean field games of the type studied here; the assumptions on the coefficients are…”
Some remarks on mean field games (2018, background): “We refer to [12, 13] for examples of such a point of view and to [14, 15] for a complete presentation of this probabilistic approach.”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, methodology): “Similarly, this method works well for proving the existence of Markovian equilibria for mean field games [52, 9].”
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 [2, 13, 14, 38, 43], we refer to Carmona and Delarue [12], Lacker [37], and Fischer [19].”
On the convergence of closed-loop Nash equilibria to the mean field game limit (2018, methodology): “Relaxed controls were employed in an MFG context [42, 17, 43], and we will use them in the same way.”
A general characterization of the mean field limit for stochastic differential games (2014, methodology): “All of the results we need, well known or not, are summarized in the Appendices A and B of [29].”