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Quantum Nash Equilibria for Games with Unitary Strategies: Existence in Mixed Strategies, an Entanglement Threshold, and a Stylized Interbank Liquidity Game

Kim Khanh Nguyen

Published Sep 29, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23027808

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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.
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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.
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