Strictly copositive matrices and ESS's
Let A be some real \(n\times n\)-matrix and let \(x^ TAy\) denote the corresponding bilinear form \((x,y\in {\mathbb{R}}^ n)\). A is called strictly copositive if \(x^ TAx>0\) for all \(x\in [0,\infty)^ n\setminus \{(0,...,0)\}\). Put \(S_ n:=\{(x_ 1,...,x_ n)\in {\mathbb{R}}^ n| \quad x_ 1,...,x_ n\geq 0;\) \(x_ 1+...+x_ n=1\}\). \(p\in S_ n\) is called an evolutionarily stable strategy (ESS) (with respect to A) if \((p-x)^ TAp\geq 0\) for all \(x\in S_ n\) and if in addition \((p-x)^ TAx>0\) for all \(x\in S_ n\setminus \{p\}\) for which \((p-x)^ TAp=0\). A simple proof of an already known result concerning the characterization of strictly copositive matrices of dimension 3 [cf. \textit{K. P. Hadeler}, Linear Algebra Appl. 49, 79-89 (1983; Zbl 0506.15016)] is given and it is shown how knowing if a certain matrix is strictly copositive helps in deciding if some element of \(S_ n\) of the form (0,...,0),1,0,...,0) is an ESS.
scientific article; zbMATH DE number 4091666