Quantum Volunteer's Dilemma

TL;DR

This work develops a quantum generalization of the $n$-player volunteer's dilemma using the Eisert–Wilkens–Lewenstein quantization framework. By encoding the game on an entangled $n$-qubit state and allowing each player to apply a two-parameter unitary before a joint measurement, the authors derive analytical expressions for quantum payoffs and show a clear quantum advantage over the classical mixed-strategy baseline. They establish symmetric Nash equilibria that yield the payoff $2-\frac{1}{n}$, with $Q^n$ being a NE for $n\le 9$ and an even-$n$-dependent NE $A^n$ for even $n$, both of which are Pareto optimal. This demonstrates that quantum strategies can enhance coordination in social-dilemma contexts and provides a foundation for exploring quantum decision-making in scalable, multi-agent networks.

Abstract

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.

Figures (5)

• Figure 1: Plots of the degree-$n$ polynomials $g_n(\alpha)$ versus $\alpha$ over the interval $\alpha \in [0,1]$, for $n=2$ to $n=10$. The unique root of $g_n(\alpha)$ within the interval $(0,1)$ is the strategy adopted by each player at the unique symmetric Nash equilibrium in the classical volunteer's dilemma with mixed strategies.
• Figure 2: Plots of the roots $\alpha_n$ versus $n$ for $n$ ranging from 2 to 50. The root $\alpha_n$, which is the unique root of the polynomial $g_n$ within the open interval $(0,1)$, represents the strategy adopted by each player at the unique symmetric Nash equilibrium in the classical volunteer's dilemma with mixed strategies. The approximation $\omega^*/n$ estimates $\alpha_n$, with an error term of $O(n^{-2})$.
• Figure 3: Payoffs at symmetric Nash equilibria for both the quantum volunteer's dilemma and the classical volunteer's dilemma with mixed strategies, as a function of the number of players $n$. The classical payoff asymptotically approaches $2(1-\mathrm{e}^{-w^*})\approx 1.43042$, while the quantum payoff, which is $2-\frac{1}{n}$, asymptotically approaches 2, demonstrating a larger value.
• Figure 4: Circuit diagram for the quantum volunteer's dilemma. First, the entangling gate $J$ is applied to the $n$-qubit computational basis state $|0\rangle^{\otimes n}$. Each player $i\in [n]$ then applies the single-qubit unitary $U(\theta_i,\phi_i)$ to qubit $i$. Next, the entangling gate $J^\dag$ is applied, followed by a computational basis measurement.
• Figure 5: Graph of the function $f(x) = \sin^2(\uppi x)-x$ for $x$ in the interval $\left[0,\tfrac{1}{2} \right]$. The graph $f(x)$ is negative for $x \in \left(0,\tfrac{1}{10}\right]$ and positive for $x \in \left[\tfrac{1}{9},\tfrac{1}{2}\right]$.

Theorems & Definitions (19)

• Definition 1: $n$-player game
• Definition 2: Nash Equilibrium
• Definition 3: Symmetric Nash equilibrium
• Definition 4: Pareto optimal
• Theorem 5: Weesie and Franzen weesie1998cost
• Theorem 6
• proof
• Lemma 7
• proof
• Lemma 8