On the extreme points of the set of bistochastic operators

Abstract

In [1], a class of quadratic stochastic operators mapping the finite-dimensional simplex into itself was singled out, and these operators were called bistochastic quadratic operators. They are closely related to the notion of majorization and are applied not only to problems in population genetics [1], [2], but also to the problems in economics [3]. In mathematical economics, the bistochastic quadratic operator is called thewelfare operator. Bistochastic quadratic operators were first studied in [1], where a necessary and sufficient condition for the bistochasticity of operators was obtained. This theorem will be presented below and used throughout the paper. In the present paper, it is shown that the set of bistochastic quadratic operators is a convex polyhedron. Therefore, an analog of Birkhoff’s theorem on the extreme points of the set of bistochastic matrices [4] is of interest. In this paper, the problem is partially solved and, more precisely, we obtain a sufficient condition for points to be extreme and, for the two-dimensional simplex, a necessary and sufficient condition as well. Moreover, we find the number of extreme points of the set of bistochastic quadratic operators in the two-dimensional simplex. Let us now pass to some necessary notions.