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, 110 (17), 6913-8

Evolution of Extortion in Iterated Prisoner's Dilemma Games

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Evolution of Extortion in Iterated Prisoner's Dilemma Games

Christian Hilbe et al. Proc Natl Acad Sci U S A.

Abstract

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-for-tat, 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.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Payoffs formula image and formula image of players I and II if both players use memory-one strategies in an IPD game (with formula image, formula image, formula image, and formula image). In each graph, the strategy of player I is fixed to some p, whereas the strategy q of the coplayer II can vary, sampling the 4D cube of memory-one strategies (the blue dots correspond to formula image different realizations of q). (A) In general, the payoff-pairs of the two players cover a 2D area, as here, when player I applies the strategy of WSLS (i.e., formula image and formula image). (B) However, if player I adopts a ZD strategy, the possible payoff-pairs are restricted to a line. Two special classes of ZD strategies were highlighted by Press and Dyson (10): equalizers [strategies that set the coplayer’s score to a fixed value (the line of payoffs has slope zero)] (C), and extortioners [strategies that guarantee the surplus of player I is the χ-fold of the surplus of player II (i.e., formula image, with formula image (the line of payoffs has a positive slope and intersects the diagonal at P)] (D).
Fig. 2.
Fig. 2.
Reactive strategies (formula image, formula image) for the donation game. All reactive strategies (the square formula image) are ZD strategies. The equalizer strategies are those on the segment between “generous TFT” (GTFT) (formula image, formula image) (16) and “Miser” (formula image, formula image) (30), the extortion strategies are those between Miser and TFT (formula image, formula image), and the “compliant” strategies (Discussion and ref. 11) are those between GTFT and TFT.
Fig. 3.
Fig. 3.
Evolutionary competition between some important strategies in the IPD. For various population sizes M, the graphs show the frequency of each strategy in the mutation-selection equilibrium. We consider two mutation regimes: the limit of rare mutations formula image (Upper), for which the equilibrium can be calculated analytically using the method of Fudenberg and Imhof (23), and a regime with mutation rate formula image (Lower), which is explored by individual-based simulations. For the copying process, we assume that individuals A and B are chosen randomly. A switches to B’s strategy with a probability given by formula image, where formula image and formula image are the corresponding payoff values and formula image corresponds to “selection strength” (cf. ref. 21). (A) If All D competes with WSLS, the population is mostly in the defector’s state, independent of population size and the mutation rate. (B and C) However, once formula image or TFT is added, WSLS succeeds if populations are sufficiently large. TFT works slightly better than formula image. (D and E) Adding All C only leads to minor changes in the stationary distribution, which now slightly favors formula image. The parameters are formula image, formula image, formula image, and formula image.
Fig. 4.
Fig. 4.
Statistics of the evolutionary dynamics for memory-one strategies for a range of different population sizes. We have calculated the relative abundance of extortioners, equalizers, and ZD strategies (i.e., the time spent in a δ-neighborhood) divided by the volume of the intersection of that neighborhood with the set of memory-one strategies (A), the average strategy of the population (B), and the average payoff (C). Extortioners, equalizers, and ZD strategies are only favored for small population sizes. As the population size increases, individuals tend to apply WSLS-like strategies and to cooperate only after mutual cooperation or mutual defection. As a result, the average payoff increases with population size. For the simulations, formula image mutant strategies were randomly drawn from the space of memory-one strategies. As in the study by Imhof and Nowak (22), the switch from a monomorphic population using strategy X to a monomorphic population using strategy Y occurred with the probability of fixation of a single Y mutant in a population of X residents. The parameters are formula image, formula image, formula image, and formula image.
Fig. 5.
Fig. 5.
Evolution of extortion in host–symbiont interactions. The graphs show two typical simulation runs for a population of 40 hosts, with each having a subpopulation of 20 symbionts. For each simulation run, one graph (Upper) shows the average payoff for each population, whereas the other graph (Lower) shows the Euclidean distance of each population to the set of extortioners (which can be 1.5275 at most). In the initial population, all individuals cooperate unconditionally. Further evolution depends on the RER. (A) If RER = 1, both species converge toward All D and no population is able to extort the other. (B) For RER = 200, symbionts evolve much more quickly. In the short term, they can thus increase their average payoff by switching to a noncooperative strategy. However, in the long term, hosts apply extortion strategies to force their symbionts to cooperate. Eventually, the hosts’ payoff exceeds formula image, whereas the symbionts’ payoff is close to zero. To model the evolutionary process, we followed the method of Damore and Gore (26). Whenever a symbiont reproduces, its offspring remains associated with the same host. Whenever the host reproduces, the new host offspring acquires its symbionts from other hosts (horizontal transmission). Mutations occur with probability formula image, by adding Gaussian noise to an entry of the memory-one strategy of the parent (formula image). The process is run for 2,000 host generations (corresponding to more than formula image reproduction events for RER = 1 and more than formula image reproduction events for RER = 200). The other parameters are formula image, formula image, and formula image.

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