Large ranking games with diffusion control

Stefan Ankirchner (1) , Nabil Kazi-Tani (2) , Julian Wendt (1) , Chao Zhou (3)

1 Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany]
2 IECL - Institut Élie Cartan de Lorraine
3 NUS - National University of Singapore

We consider a symmetric stochastic differential game where each player can control the diffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best α ∈ (0, 1) of all states receive a fixed prize. Within the mean field limit version of the game we compute an explicit equilibrium, a threshold strategy that consists in choosing the maximal fluctuation intensity when the state is below a given threshold, and the minimal intensity else. We show that for large n the symmetric n-tuple of the threshold strategy provides an approximate Nash equilibrium of the n-player game. We also derive the rate at which the approximate equilibrium reward and the best response reward converge to each other, as the number of players n tends to infinity. Finally, we compare the approximate equilibrium for large games with the equilibrium of the two player case.

Mots clés: diffusion control; game; rank-based reward; mean field limit; oscillating Brownian motion

Domaines: Probabilités [math.PR]

Stefan Ankirchner, Nabil Kazi-Tani, Julian Wendt, Chao Zhou. Large ranking games with diffusion control. Mathematics of Operations Research, In press, ⟨10.1287/moor.2023.1373⟩. ⟨hal-03434678v3⟩