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Recently, Halpern and Leung suggested representing uncertainty by a weightedset of probability measures, and suggested a way of making decisions based onthis representation of uncertainty: maximizing weighted regret. Their paperdoes not answer an apparently simpler question: what it means, according tothis representation of uncertainty, for an event E to be more likely than anevent E'. In this paper, a notion of comparative likelihood when uncertainty isrepresented by a weighted set of probability measures is defined. Itgeneralizes the ordering defined by probability (and by lower probability) in anatural way; a generalization of upper probability can also be defined. Acomplete axiomatic characterization of this notion of regret-based likelihoodis given.

Weighted regret-based likelihood: a new approach to describing uncertainty