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arXiv:2607.21103 · Quantum Information Processing

Entanglement, Evolutionary Stability, and Strategy-Space Dependence in an EWL Quantum Game

Azhar Iqbal, Derek Abbott

quant-phSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations

v1 · Jul 23, 2026 v2 · Oct 14, 2026

Abstract

We examine how Eisert--Wilkens--Lewenstein (EWL) quantization affects evolutionary stability in a symmetric $2\times2$ game whose interior mixed Nash equilibrium is not evolutionarily stable. In the restricted two-parameter EWL strategy space, the quantum equilibrium $s^*=(π/2,π/4)$ becomes a strict symmetric Nash equilibrium, and hence an evolutionarily stable strategy, for every $γ>0$. Extending the admissible pure strategies to the full three-parameter $SU(2)$ family preserves the Nash equilibrium but introduces a continuum of payoff-neutral mutants, causing strictness and the second ESS condition to fail. Thus, entanglement stabilizes the equilibrium only within the restricted EWL strategy space; the effect does not survive enlargement to full $SU(2)$.

Journal reference: Quantum Information Processing, Volume 25, article number 348 (2026)
DOI: 10.1007/s11128-026-05373-0
Comments: 11 pages, 2 figures

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We examine how Eisert--Wilkens--Lewenstein (EWL) quantization affects evolutionary stability in a symmetric

2\times2

game whose interior mixed Nash equilibrium is not evolutionarily stable. In the restricted two-parameter EWL strategy space, the quantum equilibrium

s^*=(π/2,π/4)

becomes a strict symmetric Nash equilibrium, and hence an evolutionarily stable strategy, for every

γ>0

. Extending the admissible pure strategies to the full three-parameter

SU(2)

family preserves the Nash equilibrium but introduces a continuum of payoff-neutral mutants, causing strictness and the second ESS condition to fail. Thus, entanglement stabilizes the equilibrium only within the restricted EWL strategy space; the effect does not survive enlargement to full

SU(2)

.

It is well-known that the evolutionary dynamics is widely studied in classical game settings. In this work, for the first time, the notion of evolutionary dynamics is studied in the context of quantum Cournot duopoly game using a modified EWL scheme. Unlike the well-known Eisert-Wilkens-Lewenstein scheme and the Marinatto-Weber scheme, the modified EWL scheme admits a wide range of two-qubit entangling operators which are essential to understand game theory from the perspective of quantum operators. By applying this scheme and by using replicator dynamics, it is observed that these exists only one evolutionary stable strategy (ESS). Further, the identified stable strategy plays a crucial role in replacing the strategy-profit maximization (played by majority of the population) with the mutant strategy-revenue maximization. Also, these results highlight the insignificance of the entangling operators in deciding the ESS.

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.

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)\times 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.

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.

We quantize the Hawk–Dove game by using the most general form of a pure initial state to investigate the existence of pure and mixed evolutionarily stable strategies (ESS). An example is considered to draw a comparison between the classical and quantum version of the game. Our choice of the most general initial quantum state enables us to make the game symmetric or asymmetric. We show that for a particular set of game parameters where there exists only mixed ESS in the classical version of the game, quantization allows even a pure strategy to be an ESS for the symmetric game in addition to mixed ESS. On the other hand only pure strategy ESS can exist for the asymmetric quantum version of the Hawk–Dove game.