Iterated games are a fundamental component of economic and evolutionary game theory. They describe situations where two players interact repeatedly and have the ability to use conditional strategies that depend on the outcome of previous interactions, thus allowing for reciprocation. Recently, a new class of strategies has been proposed, so-called "zero-determinant" strategies. These strategies enforce a fixed linear relationship between one's own payoff and that of the other player. A subset of those strategies allows "extortioners" to ensure that any increase in one player's own payoff exceeds that of the other player by a fixed percentage. Here, we analyze the evolutionary performance of this new class of strategies. We show that in reasonably large populations, they can act as catalysts for the evolution of cooperation, similar to tit-fortat, but that they are not the stable outcome of natural selection. In very small populations, however, extortioners hold their ground. Extortion strategies do particularly well in coevolutionary arms races between two distinct populations. Significantly, they benefit the population that evolves at the slower rate, an example of the so-called "Red King" effect. This may affect the evolution of interactions between host species and their endosymbionts.replicator dynamics | adaptive dynamics T he Iterated Prisoner's Dilemma (IPD) has a long history as a model for the cultural and biological evolution of cooperation (1-9). A new class of so-called "zero-determinant" (ZD) strategies has recently attracted considerable attention (10-12). Such strategies allow players to enforce a linear relation unilaterally between one player's own payoff and the coplayer's payoff. A subset consists of the so-called "equalizer" strategies, which assign to the coplayer's score a predetermined value, independent of the coplayer's strategy (13). Another subset consists of the extortion strategies, which guarantee that one player's own surplus exceeds the coplayer's surplus by a fixed percentage. Press and Dyson (10) have explored the power of ZD strategies to manipulate any "evolutionary" opponent (i.e., any coplayer able to learn and to adapt).In Stewart and Plotkin's (11) commentary to the article by Press and Dyson (10), they ask: "What does the existence of ZD strategies mean for evolutionary game theory: Can such strategies naturally arise by mutation, invade, and remain dominant in evolving populations?" In evolutionary game theory, it is the population that adapts: More and more players switch to the more successful strategies. From the outset, it may seem that the opportunities for extortion strategies are limited. If a strategy is successful, it will spread, and therefore be more likely to be matched against its like, but any two extortioners hold each other down to surplus zero. In a homogeneous population of extortioners, it is thus better to deviate by cooperating. Extortion is therefore evolutionarily unstable (12). However, we shall see that if the two players engaged in an IPD game b...
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)(19)(20)(21)(22).However, a recent discovery suggests that we may have underestimated the amount of control that single players can exert in repeated games. For the repeated prisoner's dilemma, Press and Dyson (23) have shown the existence of zero-determinant strategies (or ZD strategies), which allow a player to unilaterally enforce a linear relationship between the own payoff and the coplayer's payoff, irrespective of the coplayer's actual strategy. The class of zero-determinant strategies is surprisingly rich: for example, a player who wants to ensure that the own payoff will always match the coplayer's payoff can do so by applying a fair ZD strategy, like Titfor-Tat. On the other hand, a player who wants to outperform the respective opponent can do so by slightly tweaking the Tit-for-Tat strategy to the own advantage, thereby giving rise to extortionate ZD strategies. The discovery of such strategies has prompted several theoretical studies, exploring how dif...
Cooperation in public good games is greatly promoted by positive and negative incentives. In this paper, we use evolutionary game dynamics to study the evolution of opportunism (the readiness to be swayed by incentives) and the evolution of trust (the propensity to cooperate in the absence of information on the co-players). If both positive and negative incentives are available, evolution leads to a population where defectors are punished and players cooperate, except when they can get away with defection. Rewarding behaviour does not become fixed, but can play an essential role in catalysing the emergence of cooperation, especially if the information level is low.
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