Indirect reciprocation for evolutionary games

In evolutionary games, it is no longer possible to postulate that players settle on an equilibrium which is sustained by their anticipation of the payoff obtained when they deviate unilaterally. Players are not assumed to be rational, or able to think ahead, deliberate, or coordinate. Strategies are simple behavioral programs; they are supposed to spread within the population if they are successful in the sense of yielding a high payoff (see e.g. Hofbauer and Sigmund, 1998). Typically, one assumes that such strategies arise randomly within a small minority of the population, by mutation or some other process. The question then becomes whether simple trial-and-error mechanisms resembling natural selection are able to lead, in the long run, to the emergence of cooperative behaviour.

The first papers in this field, by Nowak and Sigmund (1998a,b), led to a number of theoretical and experimental investigations. Roughly speaking, by now the fact that cooperative behaviour based on indirect reciprocity can emerge through evolutionary mechanisms is no longer in doubt, but there is debate on which strategy it is most likely to be based.

In the evolutionary version of the indirect reciprocity game, one considers populations of players which are endowed with some simple strategies. Whenever two players meet in one round of that game, one of them is randomly assigned the role of the donor and the other the role of the recipient. The donor can give help to the recipient: in this case, the recipient's payoff increases by a benefit b whereas the donor's payoff decreases by -c, the cost of giving (with c < b). The donor can, alternatively, refuse to help, in which case the payoffs of both players are not affected. A player's strategy specifies under which conditions the player should give help, when in the role of the donor.

From time to time, players leave the population and are replaced by new players. The probability that a new player inherits a given strategy occuring within the population is proportional to its frequency, and to the average payoff achieved by players using this strategy. This mimicks selection, but it can just as well be interpreted as a learning process: in that case, players switch their strategies without actually having to die. Some models of evolutionary games also incorporate mutations, which introduce small numbers of players using strategies which were not present in the resident population.

The first model by Nowak and Sigmund (1998a) was based on the concept of a score, a numerical value for reputation. A player's score, at any given time, is defined as difference between the number of decisions to give help, and the number of decisions to refuse help, up to that time. The score of a player entering the game is zero: it then increases or decreases by one point in each round in which the player is in the position of a donor. The range of the score is the set of all integers. This is called the 'full score' model. In a second, 'binary' model, discussed in Nowak and Sigmund (1998b), the range is reduced to two numbers only, 0 (bad) or 1 (good). This reflects only the players' behaviour in their previous round as a donor. One can, of course, conceive many other ways for keeping score: for instance, by considering neither all the previous actions of the players, nor their last action only, but their last five or ten actions, etc. The decision whether to give help or not should then be based on the scores of the players involved. In particular, a recipient with a high score should be more likely to receive help.

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