Characterizing the Value Functions of Polynomial Games

TL;DR

This paper characterizes the set of value functions for polynomial games, showing they are exactly the continuous piecewise rational functions, thus providing a complete mathematical description of possible game outcomes.

Contribution

It offers a precise characterization of value functions in polynomial games, linking them to continuous piecewise rational functions, which was previously unknown.

Findings

Value functions are exactly continuous piecewise rational functions.

Provides a necessary and sufficient condition for a function to be a polynomial game value.

Enhances understanding of the structure of polynomial game solutions.

Abstract

We provide a characterization of the set of real-valued functions that can be the value function of some polynomial game. Specifically, we prove that a function $u : \dR \to \dR$ is the value function of some polynomial game if and only if $u$ is a continuous piecewise rational function.