Equivalence of Probabilistic Tournament and Polynomial Ranking Selection

TL;DR

This paper mathematically demonstrates the equivalence between probabilistic tournament selection and polynomial rank schemes in evolutionary algorithms, providing explicit translation methods and highlighting the practical relevance of these methods.

Contribution

It establishes a formal equivalence between probabilistic tournament and polynomial rank selection, with explicit translation operators and practical implications.

Findings

Probabilistic tournaments are equivalent to unique polynomial rank schemes.

Most linear and quadratic rank schemes are probabilistic tournaments.

Explicit operators enable translation between the two selection methods.

Abstract

Crucial to an Evolutionary Algorithm's performance is its selection scheme. We mathematically investigate the relation between polynomial rank and probabilistic tournament methods which are (respectively) generalisations of the popular linear ranking and tournament selection schemes. We show that every probabilistic tournament is equivalent to a unique polynomial rank scheme. In fact, we derived explicit operators for translating between these two types of selection. Of particular importance is that most linear and most practical quadratic rank schemes are probabilistic tournaments.