A theory for the evolutionary game
The authors present an evolutionary game theory with a continuum of strategies (this includes matrix games with mixed strategies), and require an ESS to be stable against invasion by more than one mutant. Let U be the set of strategies (e.g. an interval), and denote by \(u^ 1,u^ 2,...,u^ r\in U\) the distinct strategies used in the population. The vector \(\tilde u=(u^ 1,u^ 2,...,u^ r)\) is partitioned into \(\tilde u=(u^ 0,u^ m)\), where \(u^ 0=(u^ 1,...,u^{\sigma})\) and \(u^ m=(u^{\sigma +1},...,u^ r)\) \((u^ m\) are the mutant strategies). \(N_ i(t)\) [resp. \(p_ i(t)]\) denotes the number [resp. relative frequency] of individuals using strategy \(u^ i\) at time t, with total population size \(N(t)=\sum^{r}_{i=1}N_ i(t)\), and \(p(t)=(p_ 1(t),...,(p_ r(t))\). By def., \(p_ i=N_ i/N\), and \(p(t)\in P\), where P is the unit simplex in \({\mathbb{R}}^ r\). Denote by Q the subset of P which gives positive probability precisely to the non- mutant strategies \(u^ 1,...,u^{\sigma}.\) H\({}_ i[\tilde u,p_ r(t),N(t)]\) are the individual fitness functions and \(\bar H(t)= \sum^{r}_{i=1}p_ i(t)H_ i[\tilde u,p(t),N(t)]\) is the average fitness of the population (note that this allows for both frequency and density dependent selection). The dynamics of the system is then given by \[ N_ i(t+1)=N_ i(t)\cdot H_ i[\tilde u,p(t),N(t)]\quad or,\quad equivalently, \] \[ (1)\quad p_ i(t+1)=p_ i(t)H_ i[\tilde u,p(t),N(t)]/\bar H(t),\quad N(t+1)=N(t)\bar H(t). \] A vector of strategies \(u^ 0\) is called a coalition vector if the dynamic system (1) converges to a unique nontrivial stationary state \(p^*\in Q\), \(N^*>0\), provided there are no mutants present (i.e. for \(p(0)\in Q\), and arbitrary \(u^ m)\). Now consider the same dynamics with mutants, i.e. \(p(0)\in P\), and call \(p_ 0(t)=\sum^{\sigma}_{i=1}p_ i(t)\) the coalition frequency. A coalition vector \(u^ 0\) is called an ESS if there exists a time \(\bar t\) such that for arbitrary \(u^ m\) and \(p(0)\in P\), the coalition frequency \(p_ 0(t)\) is monotone increasing for \(t\geq \bar t\), and \(p(t)\to p^*\), \(N(t)\to N^*.\) ESS's are then characterized by means of ''generating functions'' (Th. 2.1), and the new definition is applied to matrix games. A strategy which is an ESS according to Maynard Smith's original definition need not be stable against invasion by two mutant strategies. In fact there is never an interior ESS for a matrix game with more than one mutant.
frequency and density dependent selection
scientific article; zbMATH DE number 4003969
scientific article