Scalarization methods and expected multi-utility representations

We characterize the class of (possibly incomplete) preference relations over lotteries that can be represented by a compact set of (continuous) expected utility functions which preserve both indifferences and strict preferences. This finding contrasts with the representation theorem of Dubra, Maccheroni and Ok (2004) which either delivers a noncompact set of utility functions, or at least one function that does not respect strict preferences (unless the completeness axiom holds). For a preference relation of the sort we consider, our representation theorem reduces the problem of recovering the induced choice correspondence over convex sets of lotteries to a scalarvalued, parametric optimization exercise. Several applications of this observation are presented. Most notably, in an otherwise standard game with incomplete preferences, the collection of pure strategy equilibria that one can find using this scalarization method is found to correspond to a refinement of the notion of Nash equilibrium that requires the (deterministic) action of each player be undominated by any mixed strategy that she can follow, given others’ actions. Along similar lines, we also obtain a refinement of Walrasian equilibrium and provide an equilibrium existence theorem assuming a continuum of traders.