Quantum Mean-Field Games with the Observations of Counting Type

Games. 2021. Vol. 12. Article 7.

Kolokoltsov V.

Quantum games and mean-field games (MFG) represent two important new branches of
game theory. In a recent paper the author developed quantum MFGs merging these two branches.
These quantum MFGs were based on the theory of continuous quantum observations and filtering
of diffusive type. In the present paper we develop the analogous quantum MFG theory based on
continuous quantum observations and filtering of counting type. However, proving existence and
uniqueness of the solutions for resulting limiting forward-backward system based on jump-type
processes on manifolds seems to be more complicated than for diffusions. In this paper we only
prove that if a solution exists, then it gives an e-Nash equilibrium for the corresponding N-player
quantum game. The existence of solutions is suggested as an interesting open problem.

Keywords: Mean field games, quantum games, quantum filtering, quantum mean field games, Belavkin equation, nonlinear stochastic Schroedinger equation

Publication based on the results of: Phase transitions and bifurcations in mathematical models of biophysics, quantum mechanics, disordered systems, magnetohydrodynamics, population dynamics, limit theorems of probability theory (2021)