Infinitely repeated quantum games and strategic efficiency

Repeated quantum game theory addresses long-term relations among players who choose quantum strategies. In the conventional quantum game theory, single-round quantum games or at most finitely repeated games have been widely studied; however, less is known for infinitely repeated quantum games. Investigating infinitely repeated games is crucial since finitely repeated games do not much differ from single-round games. In this work, we establish the concept of general repeated quantum games and show the Quantum Folk Theorem, which claims that by iterating a game one can find an equilibrium strategy of the game and receive reward that is not obtained by a Nash equilibrium of the corresponding single-round quantum game. A significant difference between repeated quantum prisoner's dilemma and repeated classical prisoner's dilemma is that the classical Pareto optimal solution is not always an equilibrium of the repeated quantum game when entanglement is sufficiently strong. When entanglement is sufficiently strong and reward is small, mutual cooperation cannot be an equilibrium of the repeated quantum game. In addition, we present several concrete equilibrium strategies of the repeated quantum prisoner's dilemma.