We show that the maximum winning probability in the repeated CHSH game for classical strategies is at most $\left(\frac{1+\sqrt{5}}{4}\right)^n$ . Together with a matching lower bound by Barak et al. (FOCS'2008), this determines the asymptotic value of the repeated CHSH game exactly. We also show that, if an XOR game has a gap between classical and quantum values, there is also a gap between the asymptotic values for the repeated game.We show that the maximum winning probability in the repeated CHSH game for classical strategies is at most $\left(\frac{1+\sqrt{5}}{4}\right)^n$. Together with a matching lower bound by Barak et al. (FOCS'2008), this determines the asymptotic value of the repeated CHSH game exactly. We also show that, if an XOR game has a gap between classical and quantum values, there is also a gap between the asymptotic values for the repeated game.

quant-pharXiv:2608.16439Aug 18, 20260 citations

Repeated Quantum Games and Strategic Efficiency.

Shoto Aoki, Kazuki Ikeda

Repeated quantum game theory addresses long term relations among players who choose quantum strategies. In the conventional quantum game theory, only single round quantum games or at most finitely repeated games have been studied, therefore nothing is known for infinitely repeated quantum game. Investigating infinitely repeated games is crucial since finitely repeated games does 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. In addition we present several concrete equilibrium strategies of the repeated quantum prisoner's dilemma.Repeated quantum game theory addresses long term relations among players who choose quantum strategies. In the conventional quantum game theory, only single round quantum games or at most finitely repeated games have been studied, therefore nothing is known for infinitely repeated quantum game. Investigating infinitely repeated games is crucial since finitely repeated games does 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. In addition we present several concrete equilibrium strategies of the repeated quantum prisoner's dilemma.

May 12, 20202 citations

We show that the maximum winning probability in the repeated CHSH game for classical strategies is at most

\left(\frac{1+\sqrt{5}}{4}\right)^n

. Together with a matching lower bound by Barak et al. (FOCS'2008), this determines the asymptotic value of the repeated CHSH game exactly. We also show that, if an XOR game has a gap between classical and quantum values, there is also a gap between the asymptotic values for the repeated game.