Expected Utility from a Constructive Viewpoint

Abstract

This paper introduces a space of variable lotteries and proves a constructive version of the expected utility theorem. The word ``constructive'' is used here in two senses. First, as in constructive mathematics, the logic underlying proofs is intuitionistic. In a second sense of the word, ``constructive'' is taken to mean ``built up from smaller components.'' Lotteries as well as preferences vary continuously over some topological space. The topology encodes observability or verifiability restrictions -- the open sets of the topology serve as the possible truth values of assertions about preference and reflect constraints on the ability to measure, deduce, or observe. Replacing an open set by a covering of smaller open sets serves as a notion of refinement of information. Within this framework, inability to compare arises as a phenomenon distinct from indifference and this gives rise to the constructive failure of the classical expected utility theorem. A constructive version of the theorem is then proved, and accomplishes several things. First, the representation theorem uses continuous real-valued functions as indicators of preference for variable lotteries and these functions reflect the inability to compare phenomenon. Second, conditions are provided whereby local representations of preference over open sets can be collated to provide a global representation. Third, the proofs are constructive and do not use the law of the excluded middle, which may not hold for variable lotteries. Fourth, a version of the classical theorem is obtained by imposing a condition on the collection of open sets of the topology which has the effect of making the logic classical.