Abstract
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 ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.
Highlights
In [1], two recently developed branches of game theory, quantum games and mean field games (MFGs), were merged, creating quantum mean-field games (MFG)
As a part of the construction we show that the limiting behavior of mean field interacting controlled quantum particles can be described by certain classical MFG forward-backward system of jump-type equations on manifolds, the forward part being given by a new kind of nonlinear jump-type stochastic Schrödinger equations
The main result states that any solution to this forward-backward system represents an approximate N −1/4 -Nash equilibrium for the initial N-player dynamic quantum game
Summary
In [1], two recently developed branches of game theory, quantum games and mean field games (MFGs), were merged, creating quantum MFGs. As a part of the construction we show that the limiting behavior of mean field interacting controlled quantum particles (or N-player quantum game) can be described by certain classical MFG forward-backward system of jump-type equations on manifolds, the forward part being given by a new kind of nonlinear jump-type stochastic Schrödinger equations. One of the objectives of the paper is to draw the attention of game theorists to this type of games and this type of forward-backward systems, which were not studied before, and no results even on the existence of solutions are available These objects are fully classical, but represent the limit of quantum games. The main result states that any solution to this forward-backward system represents an approximate N −1/4 -Nash equilibrium for the initial N-player dynamic quantum game. In the final section we state the problem of existence of the solutions, even in the simplest case of the control problem on a qubit
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