Observations on Cooperation

With the help of both an experiment and analytical techniques, Axelrod and Hamilton [1] showed that cooperation can evolve in a Prisoner’s Dilemma game when pairs of individuals interact repeatedly. They also demonstrated that, when pairing of individual is not completely random, cooperating behaviour can evolve in a world initially dominated by defectors. This result leads us to address the following question: Since non-random pairing is a powerful mechanism for the promotion of cooperation in a repeated Prisoner’s Dilemma game, can this mechanism also promote the evolution of cooperation in a nonrepeated version of the game? Computer simulations are used to study the relation between non-random pairing and the maintenance of cooperative behaviour under evolutionary dynamics. We conclude that non-random pairing can secure cooperation also when the possibility of repeated interaction among the same pairs of individuals is ruled out.

The Journal of Mathematical Sociology

Explaining cooperation in social dilemmas is a central issue in behavioral science, and the prisoner's dilemma (PD) is the most frequently employed model. Theories assuming rationality and selfishness predict no cooperation in PDs of finite duration, but cooperation is frequently observed. We therefore build a model of how individuals in a finitely repeated PD with incomplete information about their partner's preference for mutual cooperation decide about cooperation. We study cooperation in simultaneous and sequential PDs. Our model explains three behavioral regularities found in the literature: (i) the frequent cooperation in one-shot and finitely repeated N-shot games, (ii) cooperation rates declining over the course of the game, and (iii) cooperation being more frequent in the sequential PD than in the simultaneous PD.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 2013

The fixation properties of a simple prisoner's dilemma game in the presence of "cooperation facilitators" have recently been investigated in finite and well-mixed populations for various dynamics [M. Mobilia, Phys. Rev. E, 86, 011134 (2012)]. In a Comment, Miekisz claims that for cooperation to be favored by selection in the standard prisoner's dilemma games with facilitators it suffices that f C > f D (where f C/D are the respective fitnesses of cooperators/defectors). In this Reply, we show that in generic prisoner's dilemma games with cooperation facilitators, it is generally not sufficient that a single cooperator has a higher fitness than defectors to ensure that selection favors cooperation. In fact, it is also necessary that selection promotes the replacement of defection by cooperation in a population of size N , which requires the fixation probability of a single cooperator exceeds (N −) −1. This replacement condition is independent of f C > f D and, when the payoff for mutual defection is negative, it is shown to be more stringent than the invasion condition. Our results, illustrated by a series of examples, considerably generalize those reported in [Phys. Rev. E, 86, 011134 (2012)] and in the aforementioned Comment whose claims are demonstrated to be relevant only for a special subclass of prisoner's dilemma games.

We experimentally study the effect of asymmetry on cooperation in a 40 period prisoner's dilemma game in fixed partner design. We distinguish between a high and low payoff symmetric prisoner's dilemma and an asymmetric game combined out of both symmetric ones. Asymmetry significantly decreases cooperation, as low-type players are more likely to defect after mutual cooperation while high-type players initiate cooperation more often than the former. Asymmetry also has a significant negative effect on the stability of cooperation rendering long sequences of mutual cooperation extremely rare.

2020

Abstract: In many repeated interactions, repetition is not guaranteed but instead must be agreed upon. We formulate a model of voluntary repetition by introducing outside options to a repeated Prisoner's Dilemma and investigate how the structure of outside options affects the sustainability of mutual cooperation. When the outside option is deterministic and greater than the value of mutual defection, the lower bound of the discount factors that sustain repeated cooperation is greater than the one for ordinary repeated Prisoner's Dilemma, making cooperation more difficult. However, stochastic outside options with the same mean may reduce the lower bound of discount factors as compared to the deterministic case. This is possible when the stochasticity of the options increases the value of the cooperation phase more than the value of the punishment phase. Necessary and sufficient conditions for this positive effect are given under various option structures.

Direct reciprocity and conditional cooperation are important mechanisms to prevent free riding in social dilemmas. However, in large groups, these mechanisms may become ineffective because they require single individuals to have a substantial influence on their peers. However, the recent discovery of zero-determinant strategies in the iterated prisoner's dilemma suggests that we may have underestimated the degree of control that a single player can exert. Here, we develop a theory for zero-determinant strategies for iterated multiplayer social dilemmas, with any number of involved players. We distinguish several particularly interesting subclasses of strategies: fair strategies ensure that the own payoff matches the average payoff of the group; extortionate strategies allow a player to perform above average; and generous strategies let a player perform below average. We use this theory to describe strategies that sustain cooperation, including generalized variants of Tit-for-Tat and Win-Stay Lose-Shift. Moreover, we explore two models that show how individuals can further enhance their strategic options by coordinating their play with others. Our results highlight the importance of individual control and coordination to succeed in large groups. evolutionary game theory | alliances | public goods game | volunteer's dilemma | cooperation C ooperation among self-interested individuals is generally difficult to achieve (1-3), but typically the free rider problem is aggravated even further when groups become large (4-9). In small communities, cooperation can often be stabilized by forms of direct and indirect reciprocity (10-17). For large groups, however, it has been suggested that these mechanisms may turn out to be ineffective, as it becomes more difficult to keep track of the reputation of others and because the individual influence on others diminishes (4-8). To prevent the tragedy of the commons and to compensate for the lack of individual control, many successful communities have thus established central institutions that enforce mutual cooperation (18-22).

In many repeated interactions, repetition is not guaranteed but instead must be agreed upon. We formulate a model of voluntary repetition by introducing outside options to a repeated Prisoner’s Dilemma and investigate how the structure of outside options affects the sustainability of mutual cooperation. When the outside option is deterministic and greater than the value of mutual defection, the lower bound of the discount factors that sustain repeated cooperation is greater than the one for ordinary repeated Prisoner’s Dilemma, making cooperation more difficult. However, stochastic outside options with the same mean may reduce the lower bound of discount factors as compared to the deterministic case. This is possible when the stochasticity of the options increases the value of the cooperation phase more than the value of the punishment phase. Necessary ansufficient conditions for this positive effect are given under various option structures.

Journal of Economic Theory, 1982

A common observation in experiments involving finite repetition of the prisoners' dilemma is that players do not always play the single-period dominant strategies ("finking"), but instead achieve some measure of cooperation. Yet finking at each stage is the only Nash equilibrium in the finitely repeated game. We show here how incomplete information about one or both players' options, motivation or behavior can explain the observed cooperation. Specifically, we provide a bound on the number of rounds at which Fink may be played, when one player may possibly be committed to a "Tit-for-Tat" strategy.

Decision, 2017

The Prisoner's Dilemma (PD) is a classic decision problem where two players simultaneously must decide whether to cooperate or to act in their own narrow self-interest. The PD game has been used to model many naturally occurring interactive situations, at the personal, organizational and social levels, in which there exists a tension between individual material gain and the common good. At least two factors may influence the emergence of cooperative behavior in this well known collective action problem: the incentive structure of the game itself, and the intrinsic social preferences of each of the players. We present a framework that integrates these two factors in an effort to account for patterns of high or low cooperation from repeated choice interactions. In an experiment using a collection of different PD games, and a measure of individual social preferences, we identify regions of PD games in which: (1) cooperation is independent of social preferences; (2) nice people can be exploited; and (3) being nice is consistently rewarded.

Scientific reports, 2012

Reciprocity or conditional cooperation is one of the most prominent mechanisms proposed to explain the emergence of cooperation in social dilemmas. Recent experimental findings on networked games suggest that conditional cooperation may also depend on the previous action of the player. We here report on experiments on iterated, multi-player Prisoner's dilemma, on groups of 2 to 5 people. We confirm the dependence on the previous step and that memory effects for earlier periods are not significant. We show that the behavior of subjects in pairwise dilemmas is qualitatively different from the cases with more players; After an initial decay, cooperation increases significantly reaching values above 80%. The strategy of the players is rather universal as far as their willingness to reciprocate cooperation is concerned, whereas there is much diversity in their initial propensity to cooperate. Our results indicate that, for cooperation to emerge and thrive, three is a crowd.