Elliptic curves in game theory

TL;DR

This paper studies Spohn curves, elliptic curves arising from game theory models, analyzing their geometric properties, reducibility, and invariants to classify game equilibria.

Contribution

It provides a full classification of Spohn curves' reducibility, analyzes their real points, and introduces a $j$-invariant based equivalence for $2\times 2$ games.

Findings

Real points are dense on Spohn curves in all cases.

Classification of conditions for reducibility of Spohn curves.

Methodology for computing $j$-invariants of intersection elliptic curves.

Abstract

We investigate Spohn curves, the algebro-geometric models of totally mixed dependency equilibria for $2 \times 2$ normal-form games. These curves arise as the intersection of two quadrics in $\mathbb{P}^3$ and are generically elliptic curves. We examine the reduction of Spohn curves to plane curves, providing a full classification of conditions under which they are reducible. Notably, we prove that the real points are dense on the Spohn curve in all cases, which is relevant for applications. These computations are further supported by Macaulay2 and stored in Mathrepo. We review methods to compute the $j$-invariants of elliptic curves arising as the intersection of quadrics in $\mathbb P^3$ which we apply to the case of Spohn curves aimed at game theorists. We propose a definition of equivalence of generic $2\times 2$ games based on the $j$-invariant of the Spohn curve.