Nash conditional independence curve

Abstract

We study the Spohn conditional independence (CI) variety CX of an n-player game X for undirected graphical models on n binary random variables consisting of one edge. For a generic game, we show that CX is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety (P1)n−2×P3 and we give an explicit formula for its degree and genus. We prove two universality theorems for CX: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in Rm defined by at most m−1 polynomials is isomorphic to an affine open subset of CX for some game X.