Decision Analysis Working Paper Abstract
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WP030022
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Title: Utility-Probability Duality
Authors: Ali
Abbas Stanford University and
James
E. Matheson Chairman, SmartOrg, Inc. and Stanford University
Date: October 2003
Status: Working Paper
This paper introduces duality between probability distributions and utility
functions. The primal problem is to maximize the expected utility over
a set of probability distributions. To develop the dual problem,
we scale the utility function between zero and one, so that it obeys the
same mathematical properties as a (cumulative) probability function. We
show that reversing the roles of the two functions in the expected utility
formulation provides a natural “dual” problem. Many of the known
results for the primal problem can be reinterpreted in the dual problem.
For example, we introduce a new quantity, the aspiration equivalent, as
the “dual” of the certain equivalent. The aspiration equivalent provides
a new method for choosing between lotteries and a win-win situation for
principal-agent delegation when used as a target. We also show several
new dual results such as utility dominance relationships as dual to stochastic
dominance relationships and introduce a new saddle-point method for allocating
lotteries to decision makers.
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