Learning in Games with Cumulative Prospect Theoretic Preferences

TL;DR

This paper explores how players with cumulative prospect theory preferences learn in repeated games, introducing mediated CPT correlated equilibria and showing convergence properties and limitations of traditional equilibrium concepts.

Contribution

It extends the concept of correlated equilibrium to CPT preferences via mediated games and analyzes convergence and approachability properties under calibrated learning.

Findings

Empirical distribution converges to mediated CPT correlated equilibria.

CPT correlated equilibria are not always approachable via Blackwell approachability.

Revelation principle does not hold under CPT preferences.

Abstract

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.