Mixed-stategy Nash equilibrium under prospect theory

Question

I know that for Expected Utility Theory we have at least one mixed-Strategy Nash equilibrium for finite games, but is there a proof to generalize the existence of NE under prospect theory? Specifically, I use framing effect in PT to calculate utility and now, I wonder if we have a Mixed-Strategy Nash equilibrium for players. I also appreciate any reference that discusses games under PT thoroughly.

Answer 1

It is my understanding that under prospect theory only utilities will change versus the standard case. So long as your utility functions, extended to the space of randomizations, are still quasi-concave for all players you will have a Nash equilibrium.

Theorem: Let $G=\big(N, (\Delta(S_i), u_i)_{i\in N}\big)$ be a finite (i.e. $|N|< \infty$ and $|S_i|<\infty$ for all $i$) strategic form game where, $u_i : \times_{i \in N} \Delta(S_i) \to \mathbb{R}$ is quasi-concave for all agents. Then there exists a Nash equilibrium.

Proof: By Berge's Theorem best-replies are upper-hemicontinuous and compact-valued for all agents. By quasiconcavity, these best-replies are convex valued correspondences (to see this, recall that for a quasiconcave function all upper contour sets are convex, hence so too is the argmax as it is the upper contour set of the maximal value). Hence the joint best-reply map for all agents inherits all of these properties, and thus by Kakutani's fixed point theorem an equilibrium exists. QED