Entanglement in the Quantum Volunteer's Dilemma

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≤ n≤ 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.