Gradients for the evolution of bimatrix games
The evolutionary dynamics of bimatrix games is studied for rescaled partnership games and zero sum games. The former case leads to gradient systems. The selection equations for sexual and asexual reproduction of genotypes corresponding to mixed strategies are analysed. As examples, the origins of anisogamy and cyclic chases for predator-prey coevolution are studied. Specifically, a bimatrix game (A,B) is given by two payoff matrices \(A=(a_{ij})\), \(B=(b_{ji})\), \((i=1,...,n\); \(j=1,...,m)\). If \(x=(x_ i)\), \(y=(y_ j)\) denote mixed strategies of the two players then the payoffs are given by xAy resp. yBx. The game (A,B) is called a partnership game if \(B=A'\) (the transpose of A), and (A,B) is called a rescaled partnership game or a c-partnership game (with c positive) if there exist \(c_{ij}\), \(c_ j\), \(d_ i\) s.t. \(a_{ij}=c_{ij}+c_ j\), \(b_{ji}=cc_{ij}+d_ i\) for all i,j (i.e. (A,B) is equivalent to the partnership game (C,C')). We interpret the players as two populations divided into types of pure strategists, and \(x_ i\), \(y_ j\) as relative frequencies of the corresponding pure strategies. Assuming that the frequency of a pure strategy increases if its success is above average, we obtain the game dynamical equations: \[ (1)\quad \dot x_ i=x_ i((Ay)_ i-xAy),\quad \dot y_ j=y_ j((Bx)_ j-yBx). \] It is shown that if (A,B) is a c- partnership game, then (1) is a gradient system.
scientific article; zbMATH DE number 4047487