Quantum Traveler's dilemma and the role of non-maximal entanglement

In this work, we introduce a quantum analogue of two-player Traveler's dilemma game. Traveler's dilemma is an entirely new type of game in comparison to Prisoner's dilemma because of the role of backward induction chain. The game presents a unique challenge as we strive to reduce this classical game to a quantum game without com-promising its integrity and complexity. In classical TD with a strict game-theoretical approach, backward induction causes descent to a Nash equilibrium with the worst payoff. Interestingly, experiments have shown that individuals and groups choose strategies that demonstrably provide much higher payoffs than what the pure classical analysis predicts. We have shown in this paper that with the quantum model, the pay-offs are Pareto-optimal at the Nash equilibrium with maximally entangled particles. As the entanglement is made non-maximal, it is observed that the quantum strate-gies do not always produce the Pareto-optimal Nash equilibrium at all values of the entanglement parameter ?. We observe phase transition like behavior for the Nash equilibria. The behavior of the TD game with a generalized payoff matrix and the effect of the entanglement parameter are analyzed in detail in this work. The relation between the Eisert-Wilkens-Lewenstein entangler parameter ? and the von Neumann entropy S of the resultant state is analyzed. As can be anticipated in an actual experi-ment, entanglement may not always be maximal and therefore it becomes important to know the value of the parameter ? before the players finalize a strategy. We show that the Traveler's dilemma is completely resolved in the quantum model.