  We consider repeated games where the players behave according to cumulative
prospect theory (CPT). We show that, when the players have calibrated
strategies and behave according to CPT, the natural analog of the notion of
correlated equilibrium in the CPT case, as defined by Keskin, is not enough to
capture all subsequential limits of the empirical distribution of action play.
We define the notion of a mediated CPT correlated equilibrium via an extension
of the stage game to a so-called mediated game. We then show, along the lines
of the result of Foster and Vohra about convergence to the set of correlated
equilibria when the players behave according to expected utility theory that,
in the CPT case, under calibrated learning the empirical distribution of action
play converges to the set of all mediated CPT correlated equilibria. We also
show that, in general, the set of CPT correlated equilibria is not approachable
in the Blackwell approachability sense. We observe that a mediated game is a
specific type of a game with communication, as introduced by Myerson, and as a
consequence we get that the revelation principle does not hold under CPT.
