On the Geometry of Nash and Correlated Equilibria with Cumulative Prospect Theoretic Preferences

It is known that the set of all correlated equilibria of an $n$-player non-cooperative game is a convex polytope and includes all of the Nash equilibria. Furthermore, the Nash equilibria all lie on the boundary of this polytope. We study the geometry of both these equilibrium notions when the players have cumulative prospect theoretic (CPT) preferences. The set of CPT correlated equilibria includes all of the CPT Nash equilibria, but it need not be a convex polytope. We show that it can, in fact, be disconnected. However, all of the CPT Nash equilibria continue to lie on its boundary. We also characterize the sets of CPT correlated equilibria and CPT Nash equilibria for all $2\times 2$ games, with the sets of correlated and Nash equilibria in the classical sense being a special case.