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, 110 (38), 15348-53

From Extortion to Generosity, Evolution in the Iterated Prisoner's Dilemma

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From Extortion to Generosity, Evolution in the Iterated Prisoner's Dilemma

Alexander J Stewart et al. Proc Natl Acad Sci U S A.

Abstract

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.

Keywords: altruism; evolution of cooperation; evolutionary stability; nash.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Evolution from extortion to generosity within the space of ZD strategies. Populations were simulated in the regime of weak mutation. The figure shows the ensemble mean value of κ in the population, plotted over time. The expression formula image corresponds to the extortion strategies of Press and Dyson (1), whereas formula image corresponds to the generous ZD strategies. Each population was initialized at an extortion strategy formula image, with χ drawn uniformly from the range formula image. Given a resident strategy in the population, mutations to κ were proposed as normal deviates of the resident strategy, truncated to constrain formula image, whereas mutations to formula image were drawn uniformly from formula image with ϕ drawn uniformly within the feasible range, given κ and χ. A proposed mutant strategy replaces the resident strategy with a fixation probability dependent on their respective payoffs, as in the work of Hilbe et al. (12) and Traulsen et al. (20). The mean κ among 103 replicate populations is plotted as a function of time. Parameters are formula image, formula image, formula image, and selection strength formula image.
Fig. 2.
Fig. 2.
Relationship between ZD and good strategies in the IPD. The intersection between ZD and good is precisely the set of generous ZD strategies. Not all good strategies are generous. As a result, only a strict subset of good strategies is evolutionary robust, just as a strict subset of ZD strategies is evolutionary robust. Extortion strategies are neither generous nor evolutionary robust. Also shown are the locations of the classic IPD strategies (19) win-stay-lose-shift, tit-for-tat, and generous tit-for-tat.
Fig. 3.
Fig. 3.
Space of all cooperative IPD strategies, projected onto the parameters χ and λ. The boundary of the simplex delineates the set of feasible strategies with formula image. Strategies colored light blue or dark blue are good, whereas strategies colored dark blue are both good and evolutionary robust, under weak selection. Setting formula image recovers the space of cooperative ZD strategies (red line). Note that all robust strategies are generous (i.e., formula image, formula image). Each point in the figure, formula image, has an associated range of ϕ values, and thus corresponds to multiple IPD strategies. However, the evolutionary robust good strategies resist replacement by any other strategy, regardless of the choice of ϕ. The figure illustrates the robust region for a large population size, whereas Eq. 3 gives the exact N-dependent conditions for robustness. Also shown are the locations of several classic IPD strategies. Tit-for-tat formula image and generous tit-for-tat formula image are limiting cases of generous ZD strategies, but they are not robust. Likewise, win-stay-lose-shift is good but not robust.
Fig. 4.
Fig. 4.
Generous strategies are favored by selection in evolving populations. We simulated a population under weak mutation, proposing mutant strategies drawn uniformly from the full set of memory-1 IPD strategies. We calculated the time spent in the δ-neighborhood (12) of ZD and extortion strategies, as well as robust ZD strategies, good strategies, and robust good strategies, relative to their random (neutral) expectation. For small population sizes, extortioners are abundant and generous strategies are nearly absent. As population size increases, the frequency of generous strategies and good strategies is strongly amplified by selection, whereas extortion strategies, and ZD strategies in general, are disfavored, as previously reported (12). Simulations were run until the population fixed 107 mutations. Parameters are formula image, formula image, formula image, and selection strength formula image.

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