Nash Conditional Independence Curve

In the setting of normal form games, the Nash equilibrium analyses when no player can increase their expected payoffs by changing their strategy while assuming the other players have fixed strategies. In this case, each player acts independently, without any communication to the other players. In contrast, the concept of dependency equilibrium, introduced by philosopher Wolfgang Spohn in 2003, studies the case where the players simultaneously maximize their conditional expected payoffs. The Spohn variety is the algebraic interpretation of dependency equilibria. We view these two concepts of equilibria in terms of Bayesian networks. The Nash conditional independence curve (CI) is defined as the intersection of the Spohn variety with the statistical model of one-edge Bayesian networks. In other words, the Nash (CI) curve arises when only the choices of two players are dependent on each other. In this talk, we will explore certain algebro-geometric features of this curve as the genus, the degree, smoothness, etc. This is a joint-work with Irem Portakal.