Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0402117

Classification of Quantum Symmetric Non-zero Sum 2x2 Games in the Eisert Scheme

Álvaro Francisco Huertas-Rosero

Published Feb 17, 2004• 4 citations
arXivHTMLPDF

Abstract

A quantum game in the Eisert scheme is defined by the payoff matrix, plus some quantum entanglement parameters. In the symmetric nonzero-sum 2x2 games, the relevant features of the game are given by two parameters in the payoff matrix, and only one extra entanglement parameter is introduced by quantizing it in the Eisert scheme. The criteria adopted in this work to classify quantum games are the amount and characteristics of Nash equilibria and Pareto-optimal positions. A new classification based on them is developed for symmetric nonzero-sum classical 2x2 games, as well as classifications for quantum games with different restricted subsets of the total strategy set. Finally, a classification is presented taking the whole set of strategies into account, both unitary strategies and nonunitary strategies studied as convex mixures of unitary strategies. The classification reproduces features which have been previously found in other works, like appearance of multiple equilibria, changes in the character of equilibria, and entanglement regime transitions.

DOI
10.48550/arxiv.quant-ph/0402117

Related papers

Classification of Nash Equilibria inQuantum Games

Yoshitaka Sakagami

Abstract We consider general quantum two-player games and investigate the effect of the degree of entanglement on Nash equilibria. The strategy set is SU(2). Each player's payoffs of the payoff table are different from each other. The purpose of this paper is to classify Nash equilibria by their payoff patterns. We derive all different payoff patterns of Nash equlibria which may occur. Further, for each payoff pattern, we derive all Nash equilibria for which this pattern occurs.

•Aug 28, 2023•0 citations

Quantum game theory for 2 2 games: a mathematical framework

Gloria Ferraris, Veronica Umanità

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.

quant-pharXiv:2605.15747•May 18, 2026•0 citations

Nash Equilibria in Quantum Games

Steven E. Landsburg

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 $S^3$. 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.

arXiv:1110.1351•Oct 6, 2011•2 citations

Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space

Piotr Frąckiewicz, M. Szopa, Marcin Makowski +1 more

The aim of the paper is to examine pure Nash equilibria in a quantum game that extends the classical bimatrix game of dimension 2. The strategies of quantum players are specific types of two-parameter unitary operations such that the resulting quantum game is invariant under isomorphic transformations of the input classical game. We formulate general statements for the existence and form of Nash equilibria and discuss their Pareto efficiency. We prove that, depending on the payoffs of a classical game, the corresponding quantum game may or may not have Nash equilibria in the set of unitary strategies under study. Some of the equilibria cease to be equilibria if the players’ strategy set is the three-parameter special unitary group.

Applied Sciences•Nov 13, 2022•8 citations

Nash Equilibrium Improvement in Quantum Extensions of Symmetric Games

Anna Gorczyca–Goraj, Krzysztof Grzegorz Grzanka, Piotr Frąckiewicz +2 more

Abstract Quantum strategies can alter the equilibrium structure of classical games, but apparent improvements may depend on arbitrary strategy labels or on the particular quantum operations admitted. Here, we characterize payoff improvement across all permissible three-strategy extensions of renaming-symmetric \(2 \times 2\) games within the Eisert-Wilkens-Lewenstein framework. We preserve the classical game as a subgame and add one fixed unitary strategy, restricting attention to the three extension classes that remain invariant under isomorphic transformations of the classical game. Across six canonical payoff orderings, including coordination, anti-coordination, and dominant-strategy dilemmas, we derive the conditions under which the extended games admit pure Nash equilibria whose payoffs are no lower than those of the corresponding symmetric classical equilibrium. The classical-type \(A_0\) extension produces no strict improvement, while \(B_0\) does so only in restricted regions and generally asymmetrically. By contrast, the genuinely quantum \(C_0\) extension supports non-inferior equilibria across all six classes and yields strict, mutually beneficial improvements in identifiable payoff regions. These results move beyond isolated game examples by showing precisely when representation-invariant quantum extensions can improve equilibrium payoffs and when they cannot. These conditions provide a design map for quantum-enabled multi-agent protocols by identifying the payoff structures under which an added quantum strategy can improve equilibrium outcomes without altering the players' underlying preferences.

•Aug 6, 2026•0 citations

Nash equilibria in quantum games

Steven E. Landsburg

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.

Proceedings of the American Mathematical Society•Apr 19, 2011•40 citations
A quantum game in the Eisert scheme is defined by the payoff matrix, plus some quantum entanglement parameters. In the symmetric nonzero-sum 2x2 games, the relevant features of the game are given by two parameters in the payoff matrix, and only one extra entanglement parameter is introduced by quantizing it in the Eisert scheme. The criteria adopted in this work to classify quantum games are the amount and characteristics of Nash equilibria and Pareto-optimal positions. A new classification based on them is developed for symmetric nonzero-sum classical 2x2 games, as well as classifications for quantum games with different restricted subsets of the total strategy set. Finally, a classification is presented taking the whole set of strategies into account, both unitary strategies and nonunitary strategies studied as convex mixures of unitary strategies. The classification reproduces features which have been previously found in other works, like appearance of multiple equilibria, changes in the character of equilibria, and entanglement regime transitions.
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.
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.
The aim of the paper is to examine pure Nash equilibria in a quantum game that extends the classical bimatrix game of dimension 2. The strategies of quantum players are specific types of two-parameter unitary operations such that the resulting quantum game is invariant under isomorphic transformations of the input classical game. We formulate general statements for the existence and form of Nash equilibria and discuss their Pareto efficiency. We prove that, depending on the payoffs of a classical game, the corresponding quantum game may or may not have Nash equilibria in the set of unitary strategies under study. Some of the equilibria cease to be equilibria if the players’ strategy set is the three-parameter special unitary group.
Abstract Quantum strategies can alter the equilibrium structure of classical games, but apparent improvements may depend on arbitrary strategy labels or on the particular quantum operations admitted. Here, we characterize payoff improvement across all permissible three-strategy extensions of renaming-symmetric 2×22 \times 22×2 games within the Eisert-Wilkens-Lewenstein framework. We preserve the classical game as a subgame and add one fixed unitary strategy, restricting attention to the three extension classes that remain invariant under isomorphic transformations of the classical game. Across six canonical payoff orderings, including coordination, anti-coordination, and dominant-strategy dilemmas, we derive the conditions under which the extended games admit pure Nash equilibria whose payoffs are no lower than those of the corresponding symmetric classical equilibrium. The classical-type A0A_0A0​ extension produces no strict improvement, while B0B_0B0​ does so only in restricted regions and generally asymmetrically. By contrast, the genuinely quantum C0C_0C0​ extension supports non-inferior equilibria across all six classes and yields strict, mutually beneficial improvements in identifiable payoff regions. These results move beyond isolated game examples by showing precisely when representation-invariant quantum extensions can improve equilibrium payoffs and when they cannot. These conditions provide a design map for quantum-enabled multi-agent protocols by identifying the payoff structures under which an added quantum strategy can improve equilibrium outcomes without altering the players' underlying preferences.
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.