Elliptic Curves in Game Theory

We explain the concept of dependency equilibria in game theory, which generalize the well-known concept of Nash equilibria and allow for better results via assumptions of dependencies. They can be described nicely by an algebraic variety, the so-called Spohn variety. The new "GameTheory" package for Macaulay2 (currently still in development) can be used to compute this Spohn variety for a given game and also to intersect it with a conditional independence model.

For 2x2 games, the Spohn variety generically takes the form of an elliptic curve that is the intersection of two quadrics in P^3. We examine the reduction of Spohn curves to plane curves, analyzing conditions under which they are reducible via computations in Macaulay2. We then prove that the real points are dense on the Spohn curve in all cases, which is relevant since we are of course interested in real probabilities.