Learning in Games with Cumulative Prospect Theoretic Preferences

Abstract

We consider repeated games where players behave according to cumulative prospect theory (CPT). We show that a natural analog for 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 when players have calibrated strategies and behave according to CPT. We define the notion of a mediated CPT correlated equilibrium via an extension of the game to a so-called mediated game. We then show, along the lines of Foster and Vohra's result, that 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 mediated games are a specific type of games with communication introduced by Myerson, and as a consequence we get that the revelation principle does not hold under CPT.

Mechanism Design for Cumulative Prospect Theoretic Agents: A General Framework and the Revelation Principle

“…To see this, consider a type θi for player i such that V θ i i (L i ) = V θ i i (L i ) > V θ i i (0.5L i + 0.5L i ), for lotteries L i , L i ∈ ∆(Γi). See Phade and Anantharam [2018] for an example of CPT preferences and lotteries (over 4 outcomes) that satisfy the above condition. Let there be two allocations α and α such that ζi(α ) = L i and ζi(α ) = L i .…”

“…Under EUT, the set of all correlated equilibria of a game is characterized as the union over all possible message sets and mediator distributions, of the sets of joint distributions on the action profiles of all players, arising from all the Nash equilibria for the resulting game. See Phade and Anantharam [2018] for a discussion on the revelation principle for correlated equilibrium when players have CPT preferences. Myerson [1986] has considered a further generalization to games with incomplete information in which each player first reports her type.…”