Quantum prisoners’ dilemma under enhanced interrogation

Abstract

In the quantum version of prisoners' dilemma, each prisoner is equipped with a single qubit that the interrogator can entangle. We enlarge the available Hilbert space by introducing a third qubit that the interrogator can entangle with the other two. We discuss an enhanced interrogation technique based on tripartite entanglement and analyze Nash equilibria. We show that for tripartite entanglement approaching a W-state, there exist Nash equilibria that coincide with the Pareto optimal choice where both prisoners cooperate. Upon continuous variation between a W-state and a pure bipartite entangled state, the game is shown to have a surprisingly rich structure. The role of bipartite and tripartite entanglement is explored to explain that structure.Quantum games as a field received a lot of attention from the early works of Meyer [1], and has grown steadily ever since. Connections exist between quantum games and various other fields, such as Bell non-locality [3] and quantum logic [2], to name a few. Various aspects of quantum games, including the role of entanglement and multiple player extensions, explored by different authors can be found in references [4][5][6]. A solution to the quantum prisoners' dilemma in which players have a Nash equilibrium that is Pareto optimal ignited interest in quantum games [7]. However, this initial formulation drew criticism because it dramatically restricted the strategy space of the players and did not persist under maximal entanglement if arbitrary quantum strategies were allowed [8].In this work, we enlarge the Hilbert space in a minimal way in order to arrive at a Nash equilibrium (NE) that is Pareto-optimal with maximal entanglement. In addition to the two player qubits, we consider a resource qubit that the referee, or interrogator, has access to. In the three-qubit Hilbert space, one can introduce tripartite entanglement which will lead to a much richer Nash equilibrium structure. Interestingly, for tripartite entanglement close to maximum (approaching a W-state), we obtain Nash equilibria that coincide with Pareto optimal points. The bipartite entanglement between the players' qubits partially explains the structure. In addition, it is shown that there is no NE for a GHZ state, which has a fundamentally different type of entanglement [9].We briefly review the standard formulation of the prisoners' dilemma.

Coevolution of actions, personal norms and beliefs about others in social dilemmas

“…Nevertheless several approaches successfully capturing certain aspects of human decision-making have been developed. These include classical (Fudenberg and Tirole, 1992 ), evolutionary (Sandholm, 2010 ), mean-field (Tembine, 2017 ) and quantum (Piotrowski and Sladkowski, 2003 ; Siopsis et al, 2018 ) game theories focusing on the effects of material payoffs, social influence models focusing on the dynamics of consensus formation (or fragmentation) in social networks as a result of social learning and imitation (DeGroot, 1974 ; Watts, 2002 ; Friedkin et al, 2016 ; Redner, 2019 ; Galesic and Stein, 2019 ; Zino et al, 2020 ; Kashima et al, 2021 ), models of strategic deliberation (Golman et al, 2020 ), models of normative behaviour (Azar, 2004 ; S. Gavrilets and Richerson, 2017 ; S. Gavrilets, 2020 ) and models of foresight (Perry et al, 2018 ; Perry and Gavrilets, 2020 ). Each of these approaches concentrates on specific forces shaping human behaviour and beliefs while neglecting many other important factors.…”

Quantum Conditional Strategies and Automata for Prisoners’ Dilemmata under the EWL Scheme

“…Siopsis et al discuss an improved interrogation technique based on tripartite entanglement in order to analyze the Nash equilibrium. They calculate the Nash equilibrium for tripartite entanglement and show that it coincides with the Pareto-optimum choice of cooperation between the players [44].…”

Zero-correlation entanglement vs. Schmidt rank

“…In the quantum world, the term quantum correlation has multiple meanings [4,5] and associated metrics thereof, including discord and metrics based on relative entropy. It is then important to make the distinction between those intrinsic quantum correlations and the emergent (classical) correlations resulting from quantum phenomena, which lead to the resource theories of non-locality and coherence [4] and applications such as random access coding [6,7], changing the equilibria of communication complexity problems such as prisoner's dilemma [8] and other quantum games [9], perhaps most surprisingly raising the odds in the game of bridge [10] even under traditional rules.…”