Relativistic and noise effects on multiplayer Prisoners’ dilemma with entangling initial states

Abstract

Three-players Prisoners' dilemma (Alice, Bob and Colin) is studied in the presence of a single collective environment effect as a noise. The environmental effect is coupled with final states by a particular form of Kraus operators \(K_0\) and \(K_1\) through amplitude damping channel. We introduce the decoherence parameter \(0\le p\le 1\) to the corresponding noise matrices, in order to controling the rate of environment influence on payoff of each players. Also, we consider the Unruh effect on the payoff of player, who is located at a noninertial frame. We suppose that two players (Bob and Colin) are in Rindler region I from Minkowski space-time, and move with same uniform acceleration ( \(r_b=r_c\) ) and frequency mode. The game is begun with the classical strategies cooperation (C) and defection (D) accessible to each player. Furthermore, the players are allowed to access the quantum strategic space (Q and M). The quantum entanglement is coupled with initial classical states by the parameter \(\gamma \in [0,\pi /2]\) . Using entangled initial states by exerting an unitary operator \(\hat{J}\) as entangling gate, the quantum game (competition between Prisoners, as a three-qubit system) is started by choosing the strategies from classical or quantum strategic space. Arbitrarily chosen strategy by each player can lead to achieving profiles, which can be considered as Nash equilibrium or Pareto optimal. It is shown that in the presence of noise effect, choosing quantum strategy Q results in a winning payoff against the classical strategy D and, for example, the strategy profile (Q, D, C) is Pareto optimal. We find that the unfair miracle move of Eisert from quantum strategic space is an effective strategy for accelerated players in decoherence mode ( \(p=1\) ) of the game.

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