Dependency equilibria: Boundary cases and their real algebraic geometry

TL;DR

This paper explores the algebraic geometry of dependency equilibria, providing new definitions, conditions for pure equilibria, and analyzing their structure in small games, connecting game theory with real algebraic geometry.

Contribution

It introduces two alternative definitions of dependency equilibria, establishes conditions for pure equilibria, and analyzes their geometric structure within real algebraic geometry.

Findings

Every Nash equilibrium lies on the Spohn variety.

For generic games, the real points of the Spohn variety are Zariski dense.

In (2x2)-games, the geometric structure of dependency equilibria is thoroughly analyzed.

Abstract

This paper is a significant step forward in understanding dependency equilibria within the framework of real algebraic geometry encompassing both pure and mixed equilibria. In alignment with Spohn's original definition of dependency equilibria, we propose two alternative definitions, allowing for an algebro-geometric comprehensive study of all dependency equilibria. We give a sufficient condition for the existence of a pure dependency equilibrium and show that every Nash equilibrium lies on the Spohn variety, the algebraic model for dependency equilibria. For generic games, the set of real points of the Spohn variety is Zariski dense. Furthermore, every Nash equilibrium in this case is a dependency equilibrium. Finally, we present a detailed analysis of the geometric structure of dependency equilibria for $(2\times2)$-games