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arXiv:1402.3932

Quantum two-players games, Entanglement and Nash equilibria

Katarzyna Bolonek-Lasoń

Published Feb 17, 2014• 3 citations
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Abstract

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.

DOI
10.48550/arxiv.1402.3932

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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.
Two-player |NNN|-strategy games quantized according to the Eisert–Lewenstein–Wilkens scheme [Phys. Rev. Lett. 83, 3077 (1999)] are considered. Group-theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three-strategy case.
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 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.
A well-known model in game theory, the Volunteer's Dilemma describes a group of nnn players who decide whether to volunteer for a collective benefit at a personal cost, or to abstain and risk forfeiting the benefit altogether. A quantum version of this dilemma, developed within the Eisert-Wilkens-Lewenstein framework, allows each player to manipulate one qubit of a shared entangled state, leading to symmetric Nash equilibria with higher expected payoffs than in the classical game. Existing analyses, however, assume maximal entanglement, which is experimentally demanding for hardware-constrained quantum systems. Within the same framework, we introduce a generalized Quantum Volunteer's Dilemma with a tunable entanglement parameter γγγ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating γγγ, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for 2≤n≤92\leq n\leq 92≤n≤9, and one for even nnn. We find that maximal entanglement is not required to sustain symmetric equilibria. Instead, equilibrium behavior persists above a threshold value, which we compute analytically in both cases. We also demonstrate that the threshold value directly depends on system size.