Quantum mean-field games

The Annals of Applied Probability. 2022. Vol. 32. No. 3. P. 2254–2288.

Vassili N. Kolokoltsov

n this paper we are merging the two new branches of game theory: quantum games and mean-field games (MFG). Building a quantum analog of MFGs requires the full reconstruction of its foundations and methodology, because in N-particle quantum evolution particles are not separated in individual dynamics and the key concept of the classical MFG theory, the empirical measure defined as the sum of Dirac masses of the positions of the players, is not applicable in quantum setting.

As a preliminary result we derive the new nonlinear stochastic Schrödinger equation, as the limit of the quantum filtering equation describing continuously observed and controlled system of a large number of interacting particles, the result that may have an independent value. We then show that to a control quantum system of interacting particles there corresponds a special system of classical interacting particles with the identical limiting MFG system, defined on an appropriate Riemanian manifold. Solutions of this system are shown to specify approximate Nash equilibria for N-agent quantum games.

Keywords: Mean field games, quantum control, Hamilton–Jacobi–Bellman equation, Belavkin equation, mild solutions, nonlinear stochastic Schrödinger equation, quantum interacting particles, Controlled diffusion on Riemannian manifolds, Quantum dynamic games

Publication based on the results of: Mathematical problems in the theory of evolution of biological systems and demography (2022)