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Evolutionary Dynamics for Bimatrix Games: A Hamiltonian system?



J. Math. Biol. (1996) 34: 675—688 Evolutionary dynamics for bimatrix games: A Hamiltonian system? Josef Hofbauer Institut fu¨r Mathematik, Universita¨t Wien, Strudlhofgasse 4, A-1090 Vienna, Austria Received 5 January 1994; received in revised form 1 September 1994 Abstract. We review some properties of the evolutionary dynamics for asymmetric conflicts, give a simplified approach to them, and present some new results on the stability and bifurcations occurring in these conservative systems. In particular, we compare their dynamics to those of Hamiltonian systems. Key words : Evolutionary game theory — Replicator dynamics — Bimatrix games — Hamiltonian systems — Volume-preserving flows — Bipartite systems — Stability 1 Introduction In the standard situation of evolutionary game theory, as initiated by Maynard Smith and Price, there is one population of players. However, there are situations, called asymmetric conflicts in [MS], where interactions or conflicts take place only between two separate populations. The resulting evolutionary games correspond to the bimatrix games of classical game theory: Suppose the first population has a repertoire of n#1 pure strategies E 0 , . . . , E n , occurring with relative frequencies x 0 , . . . , x n , and the second population plays strategies F 0 , . . . , F m with frequencies y 0 , . . . , y m , respec- tively. After a contest E i versus F j , the payoff for the first player is a ij , and for the second player b ji . For such games the following evolutionary dynamics was introduced by [SS] and [SSHW], see also ([HS, Chs. 17, 27]): xR i "x i ((Ay) i !x ·Ay) i"0 , . . . , n yR j "y j ((Bx) j !y ·Bx) j"0 , . . . , m . (1.1) It is the analog of the replicator equation for bimatrix games. It is a differential equation on the product S n ]S m of two probability simplices, where S n "Mx3Rn`1: x i 70, +x i "1N. The essential assumptions for this dynamics are: (1) A strategy not played at time 0 is n

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