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Journal of Quantum Information Science

A Note on Special Nash Equilibria of Quantum Games

Yoshitaka Sakagami

Published Jan 1, 2025• 0 citations
Publisher

Abstract

This note revisits Sakagami in [1] and studies whether quantum 2 × 2 games with full SU (2) strategy space admit Nash equilibria whose payoff curves do not intersect the classical (no entanglement) payoffs. By allowing duplicate payoffs in each player’s table, we construct an example that yields such “special” equilibria for any entanglement and describe a further equilibrium when there is no entanglement. We also relate the example to strong isomorphism results by Frackiewicz in [2].

DOI
10.4236/jqis.2025.153007

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This note revisits Sakagami in [1] and studies whether quantum 2 × 2 games with full SU (2) strategy space admit Nash equilibria whose payoff curves do not intersect the classical (no entanglement) payoffs. By allowing duplicate payoffs in each player’s table, we construct an example that yields such “special” equilibria for any entanglement and describe a further equilibrium when there is no entanglement. We also relate the example to strong isomorphism results by Frackiewicz in [2].
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.
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 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.
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.
We study non-cooperative games in which each player chooses a unitary operator on a separable Hilbert space, possibly infinite-dimensional. We first show that no non-trivial subset of the unitary group is convex in the linear sense, so equilibrium existence cannot be obtained by applying a fixed-point theorem to convex sets of unitaries. We then prove existence of a Nash equilibrium in mixed strategies, i.e. Borel probability measures on strong-operator-topology (SOT) compact sets of unitaries, using the Fan-Glicksberg theorem. For the two-player Eisert-Wilkens-Lewenstein (EWL) quantum Prisoner's Dilemma we derive the payoff function in closed form and prove that the quantum profile (Q, Q) is a Nash equilibrium if and only if the entanglement parameter satisfies γ ≥ γ * = arcsin (t − r)/(t − s). Finally, we interpret the game as a reduced-form interbank liquidity game and show that, under an explicit payoff condition, the payoff of the Pareto-best pure equilibrium drops discontinuously from r to p as γ falls below γ *. The financial interpretation is stylized; its limits are stated explicitly.