Universal pareto dominance and welfare for plausible utility functions

Perhaps one of the most fundamental notions in economics is that of Pareto efficiency. We study Pareto efficiency in a setting that involves two kinds of uncertainty: Uncertainty over the possible outcomes is modeled using probability distributions (lotteries) whereas uncertainty over the agents’ preferences over lotteries is modeled using sets of plausible preference relations. A lottery is universally undominated if there is no other lottery that Pareto dominates it for all plausible preference relations. A lottery is potentially efficient if it is Pareto efficient for some vector of plausible preference relations. It is easily seen that every potentially efficient lottery is universally undominated. We show that, under fairly general conditions, the converse holds as well, i.e., the set of universally undominated and the set of potentially efficient lotteries coincide. Special cases of our result include the ordinal efficiency welfare theorem by McLennan [2002] and a generalization of the latter by Carroll [2010]. In contrast to their work which is based on von Neumann-Morgenstern (vNM) utility theory, we assume that preferences over lotteries are given by sets of skew-symmetric bilinear (SSB) utility functions. SSB utility theory is a generalization of vNM utility theory that neither requires transitivity nor the controversial independence axiom [see, e.g., Fishburn 1988]. Sets of plausible utility functions are typically interpreted as incomplete information on behalf of the social planner. Indeed, it seems quite natural to assume that the social planner’s information about the agents’ utility functions is restricted to ordinal preferences, top choices, or subsets of pairwise comparisons. Further conditions might be implied by domain restrictions. Three particularly interesting classes of plausible utility functions arise when contemplating that only ordinal preferences over pure outcomes are known. For a given preference relation Ri, we consider

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