Direct reciprocity in structured populations

Abstract

Reciprocity and repeated games have been at the center of attention when studying the evolution of human cooperation. Direct reciprocity is considered to be a powerful mechanism for the evolution of cooperation, and it is generally assumed that it can lead to high levels of cooperation. Here we explore an openended, infinite strategy space, where every strategy that can be encoded by a finite state automaton is a possible mutant. Surprisingly, we find that direct reciprocity alone does not lead to high levels of cooperation. Instead we observe perpetual oscillations between cooperation and defection, with defection being substantially more frequent than cooperation. The reason for this is that "indirect invasions" remove equilibrium strategies: every strategy has neutral mutants, which in turn can be invaded by other strategies. However, reciprocity is not the only way to promote cooperation. Another mechanism for the evolution of cooperation, which has received as much attention, is assortment because of population structure. Here we develop a theory that allows us to study the synergistic interaction between direct reciprocity and assortment. This framework is particularly well suited for understanding human interactions, which are typically repeated and occur in relatively fluid but not unstructured populations. We show that if repeated games are combined with only a small amount of assortment, then natural selection favors the behavior typically observed among humans: high levels of cooperation implemented using conditional strategies. repeated prisoner's dilemma | game theory T he problem of cooperation in its simplest and most challenging form is captured by the Prisoners' Dilemma. Two people can choose between cooperation and defection. If both cooperate, they get more than if both defect, but if one defects and the other cooperates, the defector gets the highest payoff and the cooperator gets the lowest. In the one-shot Prisoners' Dilemma, it is in each person's interest to defect, even though both would be better off had they cooperated. This game illustrates the tension between private and common interest.However, people often cooperate in social dilemmas. Explaining this apparent paradox has been a major focus of research across fields for decades. Two important explanations for the evolution of cooperation that have emerged are reciprocity (1-19) and population structure (20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32). If individuals find themselves in a repeated Prisoner's Dilemma-rather than a one-shot version-then there are Nash equilibria where both players cooperate under the threat of retaliation in future rounds (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). The existence of such equilibria is a cornerstone result in economics (1-3), and the evolution of cooperation in repeated games is of shared interest for biology (4-10), economics (11)(12)(13)(14), psychology (15), and sociology (16), with applications that range from antitrust laws (17) to sticklebacks (18), alt...

The mechanics of stochastic slowdown in evolutionary games

“…The study of cooperation in an evolutionary context mainly focuses on mechanism that allow the emergence and maintenance of cooperation (see e.g., Axelrod and Hamilton, 1981;Maynard Smith, 1982b;Nowak, 2006b;Santos et al, 2006;Sigmund et al, 2010;Hilbe and Traulsen, 2012;van Veelen et al, 2012). In well-mixed populations, cooperation cannot be maintained when defection emerges unless mechanisms promoting cooperation are present, such as repetition, punishment, or rewarding.…”

Adaptive Dynamics with Interaction Structure

“…While early formulations of evolutionary game theory focused on notions of evolutionary stability (Maynard Smith and Price, 1973) and on frequency dynamics in infinite populations (Taylor and Jonker, 1978), much recent attention has been devoted to individual-based evolutionary game modelswhich we abbreviate as "IBEG models"-in which births, deaths, game interactions, and other events are represented explicitly (Nowak and May, 1992;Durrett and Levin, 1994;Killingback and Doebeli, 1996;Nakamaru et al, 1997;van Baalen and Rand, 1998;Mitteldorf and Wilson, 2000;Nowak et al, 2004;Santos and Pacheco, 2005;Ohtsuki et al, 2006;Traulsen and Nowak, 2006;Szabó and Fáth, 2007;Taylor et al, 2007a;Antal et al, 2009;Tarnita et al, 2009a;Nowak et al, 2010a;Perc and Szolnoki, 2010;van Veelen et al, 2012). In contrast to traditional evolutionary game theory, IBEG models allow investigation of how localized interaction, population size, and other factors affect evolutionary game competition.…”

“…Well-studied examples include evolution in spatially structured populations (Wright, 1943;Kimura and Weiss, 1964;Barton and Slatkin, 1986;Durrett and Levin, 1994;Tilman and Kareiva, 1997;Dieckmann et al, 2000;Rousset, 2004;Lieberman et al, 2005;Lion and van Baalen, 2008) in group-or demestructured populations (Wright, 1931;Wilson, 1977;Taylor, 1992;Hanski and Gilpin, 1997;Traulsen and Nowak, 2006;Ohtsuki, 2010), and in populations with active assortment by kin (Hamilton, 1971;Eshel and Cavalli-Sforza, 1982) or phenotype (Antal et al, 2009). These forms of localized interaction have significant consequences for the evolution of cooperation (Nowak and May, 1992;Killingback and Doebeli, 1996;Nakamaru et al, 1997;Mitteldorf and Wilson, 2000;Santos and Pacheco, 2005;Ohtsuki et al, 2006;Traulsen and Nowak, 2006;Taylor et al, 2007a;Fletcher and Doebeli, 2009;Helbing and Yu, 2009;van Veelen et al, 2012), host-parasite interactions (Hassell et al, 1991;Herre, 1993;Boots and Sasaki, 1999;Haraguchi and Sasaki, 2000;van Baalen, 2002;Read and Keeling, 2003;Boots et al, 2004), signaling …”

Evolutionary prisoner׳s dilemma game on graphs and social networks with external constraint

“…Pure cooperation, as well as the coexistence of ALLD and TFT, is possible (Zhang et al, 2009). This result, confined by the three strategies, not only is a special case but also belongs to direct reciprocity in structured populations, as van Veelen et al (2012) summarized that natural selection favors cooperation, provided that the repeated game is combined with an intermediate population structure. Specifically, the effect of the presence of constraint makes the dynamics of the system more diverse, depending on the parameter values.…”

“…This is of course an extremely complex topic as diverse factors including social, physiological, and emotional are involved. However, the essence of conditional strategies under social networks can be summarized in one sentence recipe of human cooperation: "a strong dose of repetition and a pinch of population structure" (van Veelen et al, 2012). Recently, the evolution of extortion in the iterated prisoner's dilemma game has been investigated by Hilbe and co-workers where they have shown that extortioners can act as catalysts for the evolution of cooperation in reasonably large populations (Hilbe et al, 2013).…”