Gradients for the evolution of bimatrix games
type: Item
instance of: scholarly article
title: Gradients for the evolution of bimatrix games
zbMATH Open document ID: 0642.92010
DOI: 10.1007/BF00275498
author: Maria Koth
author: Karl Sigmund
published in: Journal of Mathematical Biology
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 92D15
Mathematics Subject Classification ID: 91A40
Mathematics Subject Classification ID: 91A80
Mathematics Subject Classification ID: 92D50
zbMATH DE Number: 4047487
zbMATH Keywords: evolutionary dynamics of bimatrix games
zbMATH Keywords: rescaled partnership games
zbMATH Keywords: zero sum games
zbMATH Keywords: gradient systems
zbMATH Keywords: reproduction of genotypes
zbMATH Keywords: mixed strategies
zbMATH Keywords: anisogamy
zbMATH Keywords: predator-prey coevolution are studied
Wikidata QID: Q68673502
reviewed by: Manfred Nermuth
MaRDI profile type: Publication
cites work: Q3879120
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full work available at URL: https://doi.org/10.1007/bf00275498
OpenAlex ID: W2044369620
description: scientific article; zbMATH DE number 4047487