Evolutionary game theory attempts to predict individual behavior (whether of humans or other species) when interactions between individuals are modeled as a noncooperative game. Most dynamic analyses of evolutionary games are based on their normal forms, despite the fact that many interesting games are specified more naturally through their extensive forms.
The remainder of the chapter assumes that tness is additive. Every N-locus system with additive tness can be represented in extensive form as in gure 5.2.1 above. This consists of an initial move by nature (player 0) that takes each of N possible actions with probability 1/N. The kth action by nature leads to a two-player subgame with the symmetric payoff matrix2 Mkk which models natural selection at a single locus as considered in chapter 2.8. Overall, we then have the extensive form of a truly symmetric two-player game considered in chapter 4.6 with |Sk| = nk for all 1 k N. We will adopt the notation used there to write the state space of gametic frequencies as (cid:6)(S1 SN).
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Remark 5.2.1 The preceding discussion shows that the asymmetric replicator dynamic for a truly symmetric game with N information situations where each subgame has a symmetric payoff matrix is the model of N-locus natural selection where tness is additive among loci and there is no recombination. 2. Strictly speaking, the entries in each Mkk should be scaled by a factor N in order that the notation of chapter 4.6 match (5.2.1) above. We have ignored this nuisance factor since it plays no role in our analysis. 157 (5.2.1) cress-79032 book January 27, 2003 11:11 158
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Chapter 5 Natural Selection with Multiple Loci The case of no recombination is usually dismissed as uninteresting in the population genetics literature, through the simple observation that the theory developed in chapter 2.8 applies, since this case is equivN k=1 nk alleles. However, alent to single-locus natural selection with considered in this traditional way as a single-locus model, the system is highly nongeneric (in the set of symmetric normal form games with symmetric payoff matrix M) compared to the typical system in chapter 2.8.2 where most initial population polymorphisms evolve to an ESS of M. (cid:2) Example 5.2.2 Consider a three-locus, two-allele system with additive tness and no recombination. Suppose that each of the three loci separately have an interior ESS. Then, by theorem 4.6.2, the 8 8 payoff matrix has a single four-dimensional ESSet and no ESS. Specically, if ), Mkk = whereas the only ESSet of M is (cid:7) (cid:8) 1 2 2 1
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2.1 for k {1, 2, 3}, each Mkk has ESS ( 1 2 , 1 2 E = (cid:9) p (cid:6)8 (cid:10) (cid:10) (cid:10) (cid:10) 1 2 = (cid:4) p1 = (cid:4) p1 = (cid:4)
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