Entanglement in the Quantum Volunteer's Dilemma

arXiv:2606.08227v1 Announce Type: new Abstract: A well-known model in game theory, the Volunteer's Dilemma describes a group of $n$ 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. Within the same framework, we introduce a generalized Quantum Volunteer's Dilemma with a tunable entanglement parameter $\gamma$ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating $\gamma$, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for $2\leq n\leq 9$, and one for even $n$. 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. This characterization is directly relevant for implementations on resource-constrained quantum devices, where entanglement is inherently limited.

Abstract:A well-known model in game theory, the Volunteer's Dilemma describes a group of $n$ 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. Within the same framework, we introduce a generalized Quantum Volunteer's Dilemma with a tunable entanglement parameter $\gamma$ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating $\gamma$, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for $2\leq n\leq 9$, and one for even $n$. 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. This characterization is directly relevant for implementations on resource-constrained quantum devices, where entanglement is inherently limited.

Quantum Physics (quant-ph); Computer Science and Game Theory (cs.GT); Theoretical Economics (econ.TH); Mathematical Physics (math-ph)

arXiv:2606.08227 [quant-ph]

(or arXiv:2606.08227v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.08227 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Entanglement in the Quantum Volunteer's Dilemma

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