  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.
