Robert Hoffmann (2000)
Twenty Years on: The Evolution of Cooperation Revisited
Journal of Artificial Societies and Social Simulation
vol. 3, no. 2,
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Received: 26-Mar-00 Published: 31-Mar-00
Figure 1. The Prisoner's Dilemma. The payoffs obey T>R>P>S. Players have a choice between cooperation (c) and defection (d). The dominant strategy equilibrium (P,P) for the one-shot game is Pareto-dominated by (R, R) |
^{2}The discussion of all literature in this area is beyond the scope of a single review. A overview of this literature up to 1988 is contained in Axelrod and Dion (1988). Axelrod's website at http://pscs.physics.lsa.umich.edu/Software/ComplexCoop.html provides a comprehensive annotated bibliography.
^{3}The strategy that cooperates in round one of any game and subsequently repeats the opponent's previous move.
^{4}A lot has been written about this dynamic and its properties in the theoretical literature. It was originally developed in theoretical biology. Its first application to game theory is due to Taylor and Jonker (1978). See also Zeeman (1981), Schuster and Sigmund (1983), Hofbauer and Sigmund (1988), Sigmund (1993).
^{5}The effort of decision making has been considered in psychology for some time (e.g. Johnson and Payne 1985). A trade-off between decision efficacy and cognitive effort required is said to exist. In game theory, a strategy's memory requirement provides a natural vehicle for examining this issue (see, for example, Rubinstein 1986, Binmore and Samuelson 1992).
^{6}The former two are the strategies that cooperate (ALL-C) and defect (ALL-D) in all rounds of a given game.
^{7}Axelrod introduced a bias into the tournaments by pairing all strategies not just with every other strategy, but also with their own respective twins. This favours cooperative strategies that do well against each other, while strategies that defect fare badly against their own kind.
^{8}A number of authors have made this point. See Axelrod (1984, p.48), Sigmund (1993, p.196), see also Hirshleifer and Coll (1988, p.370), Dawkins (1989, p.215), Nachbar (1992, p.308), Linster (1992, p.881).
^{9}A three-dimensional shape resembling a doughnut.
^{10}The strategy that cooperates until a single act of defection by its opponents causes it to defect for the remainder of the game. It is sometimes known as the GRIM-strategy.
^{11}Axelrod (1984) follows Taylor (1976) to claim that strategies that are best replies to themselves can resist invasion. However, alternative best replies can emerge and spread neutrally until a third form of behaviour can emerge and prosper. As a result, only strategies that are strictly their own unique best replies (or better replies to alternative ones) can be evolutionary stable (Maynard Smith 1982). In the standard RPD, no pure strategy has this quality (Boyd and Lorberbaum 1987). However, noise and complexity cost generate payoff differentials between alternative best replies and may therefore re-establish evolutionary stability (Sugden 1986, May 1987, Boyd 1989).
^{12}The backward induction argument suggests that defection in every round is the only rational outcome of the finite RPD with complete information. The reason is that the identification of the final round leads to defecting endgame behaviour which ultimately unravels the game.
^{13}See, for example, Axelrod and Dion (1988, p.1387), Sigmund (1993 p.192), Bendor (1993), Bendor, Kramer and Stout (1991), Hirshleifer and Coll (1988), Molander (1985), Sugden (1986 p.109), Mueller (1988), Lomborg (1996).
^{14}In the paper first mentioning the prisoner's dilemma, Flood (1958, p.16) reports a private communication with John Nash in which the latter explains that two TRIGGER-strategies are in equilibrium in the infinite version of the game. The general result pertaining to infinite games is known as the Folk Theorem.
^{15}The parameters considered by Axelrod (1984) include the initial population (p.48), population structure (chapter 8), payoff variation (p.133), repetition (p.59, p.126) and noise (p.182-183). The issues of population dynamics and agent representation are examined in his later article (Axelrod 1987).
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