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Concept: Prisoner's dilemma


Recent work has revealed a new class of “zero-determinant” (ZD) strategies for iterated, two-player games. ZD strategies allow a player to unilaterally enforce a linear relationship between her score and her opponent’s score, and thus to achieve an unusual degree of control over both players' long-term payoffs. Although originally conceived in the context of classical two-player game theory, ZD strategies also have consequences in evolving populations of players. Here, we explore the evolutionary prospects for ZD strategies in the Iterated Prisoner’s Dilemma (IPD). Several recent studies have focused on the evolution of “extortion strategies,” a subset of ZD strategies, and have found them to be unsuccessful in populations. Nevertheless, we identify a different subset of ZD strategies, called “generous ZD strategies,” that forgive defecting opponents but nonetheless dominate in evolving populations. For all but the smallest population sizes, generous ZD strategies are not only robust to being replaced by other strategies but can selectively replace any noncooperative ZD strategy. Generous strategies can be generalized beyond the space of ZD strategies, and they remain robust to invasion. When evolution occurs on the full set of all IPD strategies, selection disproportionately favors these generous strategies. In some regimes, generous strategies outperform even the most successful of the well-known IPD strategies, including win-stay-lose-shift.

Concepts: Kin selection, Biology, Nash equilibrium, Natural selection, Evolution, The Evolution of Cooperation, Prisoner's dilemma, Game theory


The strong reciprocity model of the evolution of human cooperation has gained some acceptance, partly on the basis of support from experimental findings. The observation that unfair offers in the ultimatum game are frequently rejected constitutes an important piece of the experimental evidence for strong reciprocity. In the present study, we have challenged the idea that the rejection response in the ultimatum game provides evidence of the assumption held by strong reciprocity theorists that negative reciprocity observed in the ultimatum game is inseparably related to positive reciprocity as the two sides of a preference for fairness. The prediction of an inseparable relationship between positive and negative reciprocity was rejected on the basis of the results of a series of experiments that we conducted using the ultimatum game, the dictator game, the trust game, and the prisoner’s dilemma game. We did not find any correlation between the participants' tendencies to reject unfair offers in the ultimatum game and their tendencies to exhibit various prosocial behaviors in the other games, including their inclinations to positively reciprocate in the trust game. The participants' responses to postexperimental questions add support to the view that the rejection of unfair offers in the ultimatum game is a tacit strategy for avoiding the imposition of an inferior status.

Concepts: The Evolution of Cooperation, Reciprocity, Hypothesis, Experimental economics, Ultimatum game, Dictator game, Prisoner's dilemma, Game theory


We study evolutionary game dynamics on structured populations in which individuals take part in several layers of networks of interactions simultaneously. This multiplex of interdependent networks accounts for the different kind of social ties each individual has. By coupling the evolutionary dynamics of a Prisoner’s Dilemma game in each of the networks, we show that the resilience of cooperative behaviors for extremely large values of the temptation to defect is enhanced by the multiplex structure. Furthermore, this resilience is intrinsically related to a non-trivial organization of cooperation across the network layers, thus providing a new way out for cooperation to survive in structured populations.

Concepts: Structure, Nash equilibrium, Evolutionary game theory, Evolutionarily stable strategy, Game theory, The Evolution of Cooperation, Sociology, Prisoner's dilemma


Humans and animals face decision tasks in an uncertain multi-agent environment where an agent’s strategy may change in time due to the co-adaptation of others strategies. The neuronal substrate and the computational algorithms underlying such adaptive decision making, however, is largely unknown. We propose a population coding model of spiking neurons with a policy gradient procedure that successfully acquires optimal strategies for classical game-theoretical tasks. The suggested population reinforcement learning reproduces data from human behavioral experiments for the blackjack and the inspector game. It performs optimally according to a pure (deterministic) and mixed (stochastic) Nash equilibrium, respectively. In contrast, temporal-difference(TD)-learning, covariance-learning, and basic reinforcement learning fail to perform optimally for the stochastic strategy. Spike-based population reinforcement learning, shown to follow the stochastic reward gradient, is therefore a viable candidate to explain automated decision learning of a Nash equilibrium in two-player games.

Concepts: Prisoner's dilemma, Risk, Trembling hand perfect equilibrium, Matching pennies, Equilibrium, Game theory, Decision theory, Nash equilibrium


Zero-determinant strategies are a new class of probabilistic and conditional strategies that are able to unilaterally set the expected payoff of an opponent in iterated plays of the Prisoner’s Dilemma irrespective of the opponent’s strategy (coercive strategies), or else to set the ratio between the player’s and their opponent’s expected payoff (extortionate strategies). Here we show that zero-determinant strategies are at most weakly dominant, are not evolutionarily stable, and will instead evolve into less coercive strategies. We show that zero-determinant strategies with an informational advantage over other players that allows them to recognize each other can be evolutionarily stable (and able to exploit other players). However, such an advantage is bound to be short-lived as opposing strategies evolve to counteract the recognition.

Concepts: Players, Strategy, Evolution, Prisoner's dilemma, Probability theory, Nash equilibrium, Player, Game theory


The two-player Iterated Prisoner’s Dilemma game is a model for both sentient and evolutionary behaviors, especially including the emergence of cooperation. It is generally assumed that there exists no simple ultimatum strategy whereby one player can enforce a unilateral claim to an unfair share of rewards. Here, we show that such strategies unexpectedly do exist. In particular, a player X who is witting of these strategies can (i) deterministically set her opponent Y’s score, independently of his strategy or response, or (ii) enforce an extortionate linear relation between her and his scores. Against such a player, an evolutionary player’s best response is to accede to the extortion. Only a player with a theory of mind about his opponent can do better, in which case Iterated Prisoner’s Dilemma is an Ultimatum Game.

Concepts: The Evolution of Cooperation, Mind, Stag hunt, Evolutionarily stable strategy, Metaphysics, Nash equilibrium, Prisoner's dilemma, Game theory


Common sense suggests that networks are not random mazes of purposeless connections, but that these connections are organized so that networks can perform their functions well. One function common to many networks is targeted transport or navigation. Here, using game theory, we show that minimalistic networks designed to maximize the navigation efficiency at minimal cost share basic structural properties with real networks. These idealistic networks are Nash equilibria of a network construction game whose purpose is to find an optimal trade-off between the network cost and navigability. We show that these skeletons are present in the Internet, metabolic, English word, US airport, Hungarian road networks, and in a structural network of the human brain. The knowledge of these skeletons allows one to identify the minimal number of edges, by altering which one can efficiently improve or paralyse navigation in the network.

Concepts: Solution concept, Brain, Evolutionarily stable strategy, Human brain, Nervous system, Prisoner's dilemma, Game theory, Nash equilibrium


Axelrod’s celebrated Prisoner’s Dilemma computer tournaments, published in the early 1980s, were designed to find effective ways of acting in everyday interactions with the strategic properties of the iterated Prisoner’s Dilemma game. The winner of both tournaments was tit-for-tat, a program that cooperates on the first round and then, on every subsequent round, copies the co-player’s choice from the previous round. This has been interpreted as evidence that tit-for-tat is an effective general-purpose strategy. By re-analyzing data from the first tournament and some more recent data, we provide new results suggesting that the efficacy of tit-for-tat is contingent on the design of the tournament, the criterion used to determine success, and the particular values chosen for the Prisoner’s Dilemma payoff matrix. We argue that this places in doubt the generality of the results and the policy implications drawn from them.

Concepts: Machine code, Robert Axelrod, Tit for tat, Nash equilibrium, The Evolution of Cooperation, Game theory, Prisoner's dilemma


Preserving global public goods, such as the planet’s ecosystem, depends on large-scale cooperation, which is difficult to achieve because the standard reciprocity mechanisms weaken in large groups. Here we demonstrate a method by which reciprocity can maintain cooperation in a large-scale public goods game (PGG). In a first experiment, participants in groups of on average 39 people play one round of a Prisoner’s Dilemma (PD) with their two nearest neighbours on a cyclic network after each PGG round. We observe that people engage in “local-to-global” reciprocity, leveraging local interactions to enforce global cooperation: Participants reduce PD cooperation with neighbours who contribute little in the PGG. In response, low PGG contributors increase their contributions if both neighbours defect in the PD. In a control condition, participants do not know their neighbours' PGG contribution and thus cannot link play in the PD to the PGG. In the control we observe a sharp decline of cooperation in the PGG, while in the treatment condition global cooperation is maintained. In a second experiment, we demonstrate the scalability of this effect: in a 1,000-person PGG, participants in the treatment condition successfully sustain public contributions. Our findings suggest that this simple “local-to-global” intervention facilitates large-scale cooperation.

Concepts: Contribution margin, Contributions, Game theory, Public good, Prisoner's dilemma, Public goods game, The Evolution of Cooperation


The interplay of social structure and cooperative behavior is under much scrutiny lately as behavior in social contexts becomes increasingly relevant for everyday life. Earlier experimental work showed that the existence of a social hierarchy, earned through competition, was detrimental for the evolution of cooperative behaviors. Here, we study the case in which individuals are ranked in a hierarchical structure based on their performance in a collective effort by having them play a Public Goods Game. In the first treatment, participants are ranked according to group earnings while, in the second treatment, their rankings are based on individual earnings. Subsequently, participants play asymmetric Prisoner’s Dilemma games where higher-ranked players gain more than lower ones. Our experiments show that there are no detrimental effects of the hierarchy formed based on group performance, yet when ranking is assigned individually we observe a decrease in cooperation. Our results show that different levels of cooperation arise from the fact that subjects are interpreting rankings as a reputation which carries information about which subjects were cooperators in the previous phase. Our results demonstrate that noting the manner in which a hierarchy is established is essential for understanding its effects on cooperation.

Concepts: Prisoner's dilemma, Ranking, Public goods game, The Evolution of Cooperation, Sociology, Structure, Social stratification, Hierarchy