This paper investigates the strategic implications of quantum mechanics on the N-Player Public Goods Game (PGG), a canonical model for the "Tragedy of the Commons." Classically, the PGG is dominated by a single, socially suboptimal Nash Equilibrium where all players 'defect' (free ride), leading to a collective loss. We investigate whether quantum strategies, specifically those that utilize multi-particle entanglement, can overcome this problem. We construct and evaluate three different strategies—a classical 'All Defect' benchmark, an 'Independent Quantum' strategy using superposition, and a 'Coordinated Quantum' strategy using an N-qubit entangled state—for N = 4 and N=5 player games and different pot multipliers (m = 1. 5, 2. 0). These techniques are run on a noise model calibrated to the IBM ibmq_manila quantum device, giving an honest estimate of performance on near-term hardware. Methodologically, we use repeated simulation for statistical robustness, reporting numbers with 95% confidence intervals. Our results exhibit a deep and statistically significant quantum benefit: the entangled approach systematically forms a cooperative Quantum Nash Equilibrium that achieves almost optimal payoffs, far surpassing the classical and independent quantum methods. Nevertheless, an extensive examination of the noise mechanisms at play shows that the scalability of this quantum solution is inherently limited in the Noisy Intermediate-Scale Quantum (NISQ) era. The interaction between error accumulation in the CNOT cascade needed for state preparation and the intrinsic decoherence of the entangled GHZ state produces an escalating failure mechanism that puts a bound on the realizable size of the game. This paper thereby offers both evidence of quantum advantage over a social dilemma and a realistic appraisal of the hardware-dictated constraints that will have to be overcome before such solutions are scalable.
This paper investigates the strategic implications of quantum mechanics on the N-Player Public Goods Game (PGG), a canonical model for the "Tragedy of the Commons." Classically, the PGG is dominated by a single, socially suboptimal Nash Equilibrium where all players 'defect' (free ride), leading to a collective loss. We investigate whether quantum strategies, specifically those that utilize multi-particle entanglement, can overcome this problem. We construct and evaluate three different strategies—a classical 'All Defect' benchmark, an 'Independent Quantum' strategy using superposition, and a 'Coordinated Quantum' strategy using an N-qubit entangled state—for N = 4 and N=5 player games and different pot multipliers (m = 1. 5, 2. 0). These techniques are run on a noise model calibrated to the IBM ibmq_manila quantum device, giving an honest estimate of performance on near-term hardware. Methodologically, we use repeated simulation for statistical robustness, reporting numbers with 95% confidence intervals. Our results exhibit a deep and statistically significant quantum benefit: the entangled approach systematically forms a cooperative Quantum Nash Equilibrium that achieves almost optimal payoffs, far surpassing the classical and independent quantum methods. Nevertheless, an extensive examination of the noise mechanisms at play shows that the scalability of this quantum solution is inherently limited in the Noisy Intermediate-Scale Quantum (NISQ) era. The interaction between error accumulation in the CNOT cascade needed for state preparation and the intrinsic decoherence of the entangled GHZ state produces an escalating failure mechanism that puts a bound on the realizable size of the game. This paper thereby offers both evidence of quantum advantage over a social dilemma and a realistic appraisal of the hardware-dictated constraints that will have to be overcome before such solutions are scalable.
We present an experimental implementation of the multiplayer Quantum Volunteer's Dilemma on noisy intermediate-scale quantum (NISQ) hardware, executed on the ibm_kingston backend via Qiskit Runtime. The game is evaluated for N = 2 to 9 players under four transpiler optimization levels, with 20 independent repetitions per configuration and 2048 shots per circuit, including post-processing readout error correction via mthree. Target-state fidelity decays with system size but remains above 70% (corrected) through N = 9. With readout correction, the global average payoff reproduces the quantum theoretical benchmark exactly for N <= 6 and exceeds the classical Nash equilibrium across the full tested range. Optimization level 2 is selected as the reference configuration after gate count analysis reveals that levels 2 and 3 produce identical transpiled circuits, with level 2 achieving superior fidelity stability. A Hamming distance analysis of raw measurement counts shows that single-qubit errors dominate at small N, with multi-qubit contributions growing beyond N = 6. A calibration-based digital twin captures global payoff trends but exhibits a linear fidelity decay profile that diverges from the hardware behavior at large N, exposing the limits of first-order independent per-qubit noise models. These results demonstrate that aggregate quantum advantage in multiplayer games is robust to NISQ noise conditions across the full tested range, while the practical observability of state-level advantage is constrained to N <= 8 under post-processed readout correction.
What happens when an infinite number of players play a quantum game? In this paper, we will answer this question by looking at the emergence of cooperation, in the presence of noise, in a one-shot quantum prisoner’s dilemma (QuPD). We will use the numerical agent-based model (ABM) and compare it with the analytical Nash equilibrium mapping (NEM) technique. To measure cooperation, we consider five indicators, i.e., game magnetization, entanglement susceptibility, correlation, player’s payoff average, and payoff capacity, respectively. In quantum social dilemmas, entanglement plays a non-trivial role in determining the behavior of the quantum players (or qubits) in the thermodynamic limit, and for QuPD, we consider the existence of bipartite entanglement between neighboring quantum players. For the five indicators in question, we observe first-order phase transitions at two entanglement values, and these phase transition points depend on the payoffs associated with the QuPD game. We numerically analyze and study the properties of both the quantum and the defect phases of the QuPD via the five indicators. The results of this paper demonstrate that both ABM and NEM, in conjunction with the chosen five indicators, provide insightful information on cooperative behavior in an infinite-player one-shot quantum prisoner’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.
Abstract We present an experimental implementation of the multiplayer Quantum Volunteer's Dilemma on noisy intermediate-scale quantum (NISQ) hardware, executed on the ibm_kingston backend via Qiskit Runtime. The game is evaluated for N = 2 to 9 players under four transpiler optimization levels, with 20 independent repetitions per configuration and 2048 shots per circuit, including post-processing readout error correction via mthree. Target-state fidelity decays with system size but remains above 70% (corrected) through N = 9. With readout correction, the global average payoff reproduces the quantum theoretical benchmark exactly for N ≤ 6 and exceeds the classical Nash equilibrium across the full tested range. Optimization level 2 is selected as the reference configuration after gate count analysis reveals that levels 2 and 3 produce identical transpiled circuits, with level 2 achieving superior fidelity stability. A Hamming distance analysis of raw measurement counts shows that single-qubit errors dominate at small N, with multi-qubit contributions growing beyond N = 6. A calibration-based digital twin captures global payoff trends but exhibits a linear fidelity decay profile that diverges from the hardware behavior at large N, exposing the limits of first-order independent per-qubit noise models. These results demonstrate that aggregate quantum advantage in multiplayer games is robust to NISQ noise conditions across the full tested range, while the practical observability of state-level advantage is constrained to N ≤ 8 under post-processed readout correction.
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.
The N-player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for N-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players' payoffs and mixed Nash equilibrium are determined. Players' N x N payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the N-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general N ≥ 2. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford's geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.