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Proceedings of the American Mathematical Society

Nash equilibria in quantum games

Steven E. Landsburg

Published Apr 19, 2011• 40 citations
Publisher

Abstract

For any two-by-two game G \mathbf {G} , we define a new two-player game G Q \mathbf {G}^Q . The definition is motivated by a vision of players in game G \mathbf {G} communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins. In the game G Q \mathbf {G}^Q , each player’s (mixed) strategy set consists of the set of all probability distributions on the 3-sphere S 3 \mathbf {S}^3 . Nash equilibria in the game can be difficult to compute. Our main theorems classify all possible mixed-strategy equilibria. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of allowable geometric configurations.

DOI
10.1090/s0002-9939-2011-10838-4

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For any two-by-two game G \mathbf {G} , we define a new two-player game G Q \mathbf {G}^Q . The definition is motivated by a vision of players in game G \mathbf {G} communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins. In the game G Q \mathbf {G}^Q , each player’s (mixed) strategy set consists of the set of all probability distributions on the 3-sphere S 3 \mathbf {S}^3 . Nash equilibria in the game can be difficult to compute. Our main theorems classify all possible mixed-strategy equilibria. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of allowable geometric configurations.
For any two-by-two game $\G$, we define a new two-player game $\G^Q$. The definition is motivated by a vision of players in game $\G$ communicating via quantum technology according to a certain standard protocol originally introduced by Eisert and Wilkins [EW]. In the game $\G^Q$, each players' strategy set consists of the set of all probability distributions on the 3-sphere S3S^3S3. Nash equilibria in this game can be difficult to compute. Our main theorems classify all possible equilibria in $\G^Q$ for a Zariski-dense set of games $\G$ that we call {\it generic games}. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of geometric configurations. One easy consequence is that for zero-sum games, the payoff to either player in a mixed strategy quantum equilibrium must equal the average of that player's four possible payoffs.
We develop a rigorous mathematical framework for quantum game theory applied to static 2x2 games, extending classical concepts to the quantum setting where players may employ arbitrary unitary operations (pure strategies) or probability measures over the continuous group SU(2) (mixed strategies). The Eisert-Wilkens-Lewenstein protocol is introduced as the standard implementation of quantum 2x2 games. We prove the existence of Nash equilibria for continuous quantum mixed strategies via a fixed-point argument, generalising the classical Nash theorem to the quantum case.
The two-players N strategies games quantized according to the Eisert-Lewenstein-Wilkens scheme [1] are considered. It is shown that in the case of maximal entanglement no nontrivial pure Nash equilibrium exists. The proof relies on simple geometric properties of "chiral" group SU(N)×SU(N)SU (N)\times SU(N)SU(N)×SU(N) and is based on considering the stability subgroup of the initial state of the game. The explicit forms of neither the gate operator nor the payoff matrix are necessary.
Players’ choices in quantum game schemes are often correlated by a quantum state. This enables players to obtain payoffs that may not be achievable when classical pure or mixed strategies are used. On the other hand, players’ choices can be correlated due to a classical probability distribution, and if no player benefits by a unilateral deviation from the vector of recommended strategies, the probability distribution is a correlated equilibrium. The aim of this paper is to investigate relation between correlated equilibria and Nash equilibria in the MW-type schemes for quantum games.