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 $\ensuremath{\gamma}$ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating $\ensuremath{\gamma}$, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for $2\ensuremath{\le}n\ensuremath{\le}9$ 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.