The level sets of typical games

TL;DR

This paper explores the structure of level sets in non-cooperative games, revealing their connection to real algebraic geometry and providing insights into game dynamics and paradoxes like the plankton paradox.

Contribution

It introduces the fundamental connection between non-cooperative game theory and real algebraic geometry, proving a general structural result about level sets.

Findings

Level sets in non-cooperative games have a specific mathematical structure.

The results have implications for understanding game dynamics and paradoxes.

The work encourages interdisciplinary research between game theory and algebraic geometry.

Abstract

In a non-cooperative game, players do not communicate with each other. Their only feedback is the payoff they receive resulting from the strategies they execute. It is important to note that within each level set of the total payoff function the payoff to each player is unchanging, and therefore understanding the structure of these level sets plays a key role in understanding non-cooperative games. This note, intended for both experts and non-experts, not only introduces non-cooperative game theory but also shows its fundamental connection to real algebraic geometry. We prove here a general result about the structure of the level sets, which although likely to be known by experts, has interesting implications, including our recent application to provide a new mathematical explanation for the "paradox of the plankton." We hope to encourage communication between these interrelated areas and stimulate further work in similar directions.