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The level sets of typical games

Julie Rowlett

TL;DR

This work investigates the geometric structure of level sets of the total payoff in non-cooperative games, linking game-theoretic payoff maps to real algebraic geometry. Under typical game assumptions, the payoff image is a finite-dimensional semialgebraic set, and for almost every point in this image, the corresponding level sets are high-dimensional submanifolds with positive or infinite Hausdorff measure, revealing that many strategy profiles yield the same payoff. The paper also shows a simpler linear-payoff case where level sets are affine subspaces and provides a novel application to biology, offering a mathematical explanation for the paradox of the plankton via large, equivalent-payoff level sets that support biodiversity. By bridging game theory, semialgebraic geometry, and biological modeling, the work highlights the significance of level-set structures beyond classical equilibria. The results motivate further cross-disciplinary exploration of typical-game geometry and its implications for complex adaptive systems.

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.

The level sets of typical games

TL;DR

This work investigates the geometric structure of level sets of the total payoff in non-cooperative games, linking game-theoretic payoff maps to real algebraic geometry. Under typical game assumptions, the payoff image is a finite-dimensional semialgebraic set, and for almost every point in this image, the corresponding level sets are high-dimensional submanifolds with positive or infinite Hausdorff measure, revealing that many strategy profiles yield the same payoff. The paper also shows a simpler linear-payoff case where level sets are affine subspaces and provides a novel application to biology, offering a mathematical explanation for the paradox of the plankton via large, equivalent-payoff level sets that support biodiversity. By bridging game theory, semialgebraic geometry, and biological modeling, the work highlights the significance of level-set structures beyond classical equilibria. The results motivate further cross-disciplinary exploration of typical-game geometry and its implications for complex adaptive systems.

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.

Paper Structure

This paper contains 10 sections, 3 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A beautiful mind
  4. Linear payoff functions
  5. Main result
  6. Bibliographical Note
  7. Applications
  8. Biodiversity of Micro-Organisms
  9. Compatibility
  10. Concluding Remarks

Key Result

Theorem 1

There exists at least one equilibrium strategy in $\mathfrak{S}$.

Figures (1)

  • Figure 1: For a two-player, symmetric, zero-sum game with one dominant strategy, the level sets of the total payoff function are line segments. This depicts the general idea of Theorem \ref{['lots']}; the level sets of most games almost always have positive $j$-dimensional Hausdorff measure for some $j \geq 1$.

Theorems & Definitions (6)

  • Theorem 1: Nash
  • Proposition 1
  • proof
  • Theorem 2: Nash
  • proof
  • Remark 1