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arXiv:2606.08227 · Physical Review A

Entanglement in the Quantum Volunteer's Dilemma

Noah Dane Hebdon, Dax Enshan Koh

quant-phcs.GTecon.THmath-phSubmitted Jun 6, 2026• Announced Jun 9, 2026• 0 citations• Revised Sep 14, 2026 (v2)
v1 · Jun 6, 2026v2 · Sep 14, 2026
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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, 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 $γ$ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating $γ$, 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.

DOI
10.1103/2lb4-m4bg
Comments
19 pages, 2 figures

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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.
The volunteer's dilemma is a well-known game in game theory that models the conflict players face when deciding whether to volunteer for a collective benefit, knowing that volunteering incurs a personal cost. In this work we introduce a quantum variant of the classical volunteer's dilemma, generalizing it by allowing players to utilize quantum strategies. Employing the Eisert-Wilkens-Lewenstein quantization framework, we analyze a multiplayer quantum volunteer's dilemma scenario with an arbitrary number of players, where the cost of volunteering is shared equally among the volunteers. We derive analytical expressions for the players' expected payoffs and demonstrate the quantum game's advantage over the classical game. In particular, we prove that the quantum volunteer's dilemma possesses symmetric Nash equilibria with larger expected payoffs compared to the unique symmetric Nash equilibrium of the classical game, wherein players use mixed strategies. Furthermore, we show that the quantum Nash equilibria we identify are Pareto optimal. Our findings reveal distinct dynamics in volunteer's dilemma scenarios when players adhere to quantum rules, underscoring a strategic advantage of decision making in quantum settings.
A quantum approach to twice-repeated 2×22\times 22×2 game
Abstract This work proposes an extension of the well-known Eisert–Wilkens–Lewenstein scheme for playing a twice repeated 2×22\times 22×2 2 × 2 game using a single quantum system with ten maximally entangled qubits. The proposed scheme is then applied to the Prisoner’s Dilemma game. Rational strategy profiles are examined in the presence of limited awareness of the players. In particular, the paper considers two cases of a classical player against a quantum player game: the first case when the classical player does not know that his opponent is a quantum one and the second case, when the classical player is aware of it. To this end, the notion of unawareness was used, and the extended Nash equilibria were determined.
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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.
Abstract The maximally entangled (ME) Eisert–Wilkens–Lewenstein (EWL) quantization scheme efficiently solves the classical prisoner’s dilemma (PD). Despite the intensive research on the EWL scheme, little is known about the role of the arbiter in controlling the scheme properties. We redefine the EWL scheme by introducing an arbitrary phase shift into the ME initial state. Four basic initial states were analyzed in this study. It is found that the arbiter completely controls the characteristics of the arbitrary ME quantum PD. The arbiter can adjust the ME initial state to obtain an efficient symmetric Nash equilibrium. In contrast, the arbiter can nullify the quantization advantages. Moreover, it can preserve/violate the symmetry of the game or even turn it into a potential game.