An algebro-geometric glimpse on a concept from cooperative game theory

The classical game theory notion of Nash equilibrium imposes on the players the tacit assumption of acting independently from each other. However, in real-life situations this might not at all be a natural restriction. In 2003, the philosopher Wolfgang Spohn introduced the concept of dependency equilibrium (DE) which allows cooperation of the players. His definition leads to a system of equations in many real variables involving rational expressions and limits.

We try to handle these equations employing tools from algebraic geometry. Among other things, we show that the games whose set of DE equals the non-negative real part of the Spohn variety, an algebraic variety recently introduced by Portakal and Sturmfels, form a Zariski open set in the affine space of all games of a fixed size. We explicitly determine this set for games with two players who have two pure strategies each, and we prove that, in general, the Spohn variety contains its real points as a Zariski dense set.

This is joint work with Irem Portakal.