Three-player Prisoner's Dilemma Research Articles

Zero-determinant strategies in infinitely repeated three-player prisoner's dilemma game

The “iterated prisoner's dilemma” (IPD) game is considered to be a well-known model for studying the evolution of cooperative manner among rational individuals over the long term. Analytical studies in IPD mostly focus on the two-player IPD game since the game's relations and dynamics become more complicated when increasing players' numbers. Here, we extend investigating the “Zero-Determinant” (ZD) strategies to the symmetric infinitely repeated IPD game involving three players with two actions for each player. We propose two simultaneous models of the IPD game with three players to investigate ZD strategies. Further, we analytically extract the boundaries of the equalizer and extortion ZD strategies in both proposed models and provide numerical examples demonstrating such boundaries.

Disbandment rule sways the evolution of tolerance

How cooperation could have evolved is a puzzle for evolutionary ecologists. If cooperators try to keep the partnership with those who cooperated and try to break the partnership with those who defected, the evolution of cooperation can be facilitated by natural selection. In dyadic relationships, the number of the opponent player is only one and the situation is simple; therefore, it is obvious how players should behave. When more than two individuals make interactions, the situation under which both cooperators and defectors are contained in the opponent players can take place. Thus, the situation becomes complicated, and under such circumstances it is not intuitive how players should behave. Previous theoretical work studied the case where players can choose either to try to keep the partnership or try to break the partnership based on what the opponent players did, in social interactions involving more than two individuals, and investigated the evolution of tolerance. However, all of the previous studies only dealt with a disbandment rule in which the interaction stops if at least one player hopes to break the partnership with the current partners (i.e., one vote is enough to dissolve a group). In this article, we study the evolution of tolerance for a variety of disbandment rules, in which the interaction can continue even if somebody hopes to break the partnership with the current partners. By analyzing three-player prisoner's dilemma games, we reveal that the evolution of tolerance can be sensitive to different disbandment rules.

Three-player repeated games with an opt-out option

The existence of cooperation is problematic for those who view the world through an evolutionary lens. Although natural selection disfavors unconditional cooperation, cooperators who can distinguish cooperators from defectors and discontinue their interactions with the latter can be favored. Economic experiments have indicated that human beings indeed adopt such strategies when they detect defectors. Experiments have also shown that human beings also use those strategies in interactions involving more than two individuals as both cooperators and defectors may be present among opponent players. In such cases, it is not obvious whether it is better to continue or stop the interaction. In this study, I analyzed three-player prisoner's dilemma games. My findings algebraically reveal that both tolerant cooperators, who want to continue interacting with their current partners comprising one cooperator and one defector, and intolerant cooperators, who choose to discontinue such interactions, can survive when the cost of cooperation is small. In addition, my numerical analysis indicates that the coexistence of tolerant and intolerant cooperators cannot be an evolutionary outcome.

Communication aspects of a three-player Prisoner's Dilemma quantum game

We present a quantization scheme for a three-player Prisoner's Dilemma game. It is shown that entanglement plays a dominant role in the three-player quantum game. Four different types of payoffs are identified on the basis of different combinations of initial state and measurement basis entanglement parameters. A relation among these different payoffs is also established. We also study the communication aspects of the three-player game. By exploiting different combinations of initial state and measurement basis entanglement parameters, we establish a relationship for the information shared among the parties. It is seen that the strategies of the players act as carriers of information in quantum games.

Noise effects in a three-player prisoner's dilemma quantum game

We study the three-player prisoner's dilemma game under the effect of decoherence and correlated noise. It is seen that the quantum player is always better off than the classical players. It is also seen that the game's Nash equilibrium does not change in the presence of correlated noise in contradiction to the effect of decoherence in the multiplayer case. Furthermore, it is shown that for maximum correlation the game does not behave as a noiseless game and the quantum player is still better off for all values of the decoherence parameter p which is not possible in the two-player case. In addition, the payoffs reduction due to decoherence is controlled by the correlated noise throughout the course of the game.

Stability Criteria for Social Norms with Applications to the Prisoner's Dilemma

This article introduces four criteria for characterizing social norms in both cooperative and noncooperative games. The criteria are hybrids of von Neumann and Morgenstern's notion of stability and Greenberg's theory of social situations. When applied to the three-player prisoner's dilemma, these criteria illustrate that Nash and strong Nash equilibrium behavior do not rule out the possibility of unilateral defection as a social norm. We conclude with a new type of equilibrium behavior that induces coalition building and leads to social norms that rule out unilateral defection and allow for cooperation.