Abstract
Game Theory has been developed as a theory of rational behavior in interpersonal conflict situations, with economics and the other social sciences being the intended fields of application. Since the theory is based on an idealized picture of human rationality, it is by no means obvious that it can be applied to situations in which the players cannot be attributed any intellectual capabilities. However, in their seminal paper ‘The logic of animal conflict’, Maynard Smith and Price showed that animal contests can be modeled as games and that game theory can be applied successfully in biology. The objective of this chapter is to review some of the developments in this biological branch of Game Theory and to point out the distinctions and similarities with the classical branch (also see Parker and Hammerstein [1985] ). The main emphasis will be on the mathematics involved, lack of space prevents an extensive discussion of the underlying biological assumptions as well as an analysis of specific examples. For these, the reader is referred to the very stimulating book ‘Evolution and the Theory of Games’ by John Maynard Smith.
Recall our convention that we identify a symmetric bimatrix game with its associated fitness matrix of player 1. Hence, in Fig. 9.2.1 only player l’s payoffs are displayed.
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Notes
A recent survey of the material discussed in this chapter is Hammerstein and Selten [1991]. A survey that covers some topics discussed in the first part of this chapter to some greater dept is Hines [1987].
For reformulation of the ESS conditions for the case of finite population, see Riley [1979a] and Schaffer [1988]. Schaffer shows that in finite populations the ESS leads to ‘over-competitive’ behavior: An ESS-strategist is not trying to maximize his payoff, he is trying to ‘beat the average’ (also cf. Shubik [1954]).
For a discussion on the role of sex, see Eshel [1989] and the references cited therein.
A more general model is the so-called ‘playing the field model’ introduced in Maynard Smith [1982, pp. 23-27]. In this model each individual’s payoff is determined by the individual’s own strategy and the frequencies of the other individuals’ strategies, and fitness need not be linear in the latter frequencies. By means of this model one can incorporate ‘n-player’ — or ‘n-population’ — interactions. See Crawford [1990a] for the relationship between Nash equilibria and ESS in large — and finite population ‘playing the field models’. Crawford shows that if the fitness of a strategy depends discontinuously on the population’s strategy, then there may be strict Nash equilibria that are not ESS. Crawford [1990b] discusses an interesting game (due to Van Huyck, Battalio and Beil) where such a discontinuity exists.
Theorem 9.3.4 was not contained in Bomze [1986].
For an excellent survey on the mathematical theory of evolution and the dynamical systems involved, see Hofbauer and Sigmund [1988]. For a complete overview of the discrepancies between evolutionary stability and dynamic stability in a class of 3 × 3 games, see Weissing [1989].
Recent papers that discuss material that is related to the topic of this section are Friedman and Rosenthal [1986], Friedman [1987], Samuelson [1988], Nachbar [1990] and Boylan [1990]. Friedman/Rosenthal consider a dynamic equation, different from (9.4.3) of which the steady states need not be Nash equilibria. Friedman does not insist on the rate of increase of a strategy being proportional to how much this strategy is better than the average, hence, he studies a class of dynamical processes that include (9.4.3) as a special case. He finds that the Corollaries 9.4.2 and 9.4.5 remain true for a large class of dynamics. Samuelson also studies a class of dynamics that is more general than (9.4.3) and he establishes a generalization of Theorem 9.4.6. Nachbar works in the same spirit as Friedman and Samuelson. He also gives an interesting example of a dominance solvable game of which the solution (i. e. the strategy pair that remains after all weakly dominated strategies have been iteratively eliminated) is an unstable fixed point of the replicator equation. Finally, Boylan defines the concept of ‘evolutionary equilibria’ (as roughly those stationary points of processes more general than (9.4.3) that are stable against small mutation rates). He shows that every regular equilibrium is an evolutionary equilibrium and that every evolutionary equilibrium is perfect.
This question is also addressed in Bomze and Van Damme [1990] and Thomas [1985].
Schlag [1990] defines a generalized concept of eESS for games in which equivalent strategies exist (The ‘e’ stands for ‘equivalent’), Schlag’s basic idea is to look at the ESS of the ‘reduced game’ in which a player can pick an equivalence class of strategies. He shows that with this generalized definition the problems due to spurious duplication disappear.
In Samuelson [1989b] it is shown that, in a truly asymmetric contest, a strategy b* is a limit ESS if and only if (i) (b*, b*) is a pure strategy perfect equilibrium and (ii) there is no strategy b+b * that is payoff equivalent to b * (cf. Theorem 9.6.2). See Schlag [1990] for a discussion that focuses on equivalent strategies.
For examples of ESS analysis in extensive form games, see Gardner and Morris [1989 a, b].
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© 1991 Springer-Verlag Berlin Heidelberg
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van Damme, E. (1991). 9 Evolutionary Game Theory. In: Stability and Perfection of Nash Equilibria. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58242-4_9
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