Bruno Cordeiro

Bruno Cordeiro

Data { Scientist & Engineer }

Notes on Mean Field Game Theory

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Mean field game (MFG) theory was introduced independently by Lasry & Lions (2006) and Huang, Malhamé & Caines (2006) as a framework for studying Nash equilibria in stochastic differential games with a large number of interacting agents. The key insight is that as the number of agents grows, each individual’s influence on the aggregate becomes negligible. The interaction is mediated entirely through a mean field, the empirical distribution of the population’s states.

The Basic Setup

Consider $N$ agents, each with state $X_i^t$ evolving as:

\[dX_i^t = b(X_i^t, \alpha_i^t, \mu^t)\,dt + \sigma\,dW_i^t\]

where $\mu^t = \frac{1}{N}\sum_j \delta_{X_j^t}$ is the empirical distribution. As $N \to \infty$, the game decouples: each agent solves a single-agent stochastic control problem against a fixed mean field $\mu$, and the fixed point requires that $\mu$ is actually generated by that optimal behavior. This gives a coupled system of a Hamilton-Jacobi-Bellman equation (for the value function) and a Fokker-Planck equation (for the distribution).

Why This Is Interesting for Markets

Standard game theory struggles with markets because the number of participants is large but not infinite, and strategic interaction is complex. MFG offers a middle ground: tractable analysis that incorporates aggregate feedback effects without modeling every agent individually.

A few applications I’ve been thinking about:

  • Systemic risk: In a stylized bank run model, the probability of a bank’s default depends on the distribution of other banks’ health, which MFG can formalize naturally (Carmona et al., 2015).
  • Optimal execution: Large traders executing orders face a market impact that depends on the collective behavior of other traders. MFG gives a framework for the equilibrium impact.
  • Crowd motion in limit order books: The distribution of limit orders evolves as a mean field, and individual agents’ placement strategies depend on where they expect the crowd to be.

Open Questions I’m Sitting With

  1. How sensitive are MFG equilibria to the choice of interaction kernel? In practice, agents don’t interact through the full empirical distribution but through specific aggregates (price, index level). Does that simplification break the theory?

  2. Numerical tractability: The coupled PDE system is hard to solve for non-trivial dynamics. Machine learning approaches (deep Galerkin method, physics-informed networks) look promising but I haven’t worked through them carefully.

  3. Heterogeneous agents: The classical MFG assumes agents are ex-ante identical (same objective, same dynamics). Real markets have heterogeneous preferences and constraints. The extension to multi-population MFG is non-trivial.

References

  • Lasry, J.-M., & Lions, P.-L. (2006). Jeux à champ moyen. I – Le cas stationnaire. CRAS.
  • Huang, M., Malhamé, R. P., & Caines, P. E. (2006). Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communications in Information and Systems, 6(3), 221–251.
  • Carmona, R., Fouque, J.-P., & Sun, L.-H. (2015). Mean field games and systemic risk. Communications in Mathematical Sciences, 13(4), 911–933.
  • Carmona, R., & Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications. Springer.
  • Cardaliaguet, P. (2013). Notes on Mean Field Games (from P.-L. Lions’ lectures at Collège de France).