Towards Efficient Prediction of Decisions under Interval Uncertainty

In many practical situations, users select between n alternatives a1, . . . , an, and the only information that we have about the utilities vi of these alternatives are bounds vi ≤ vi ≤ vi. In such situations, it is reasonable to assume that the values vi are independent and uniformly distributed on the corresponding intervals [vi, vi]. Under this assumption, we would like to estimate, for each i, the probability pi that the alternative ai will be selected. In this paper, we provide efficient algorithms for computing these probabilities. 1 Formulation of the Problem Making a decision when we know the exact values of the maximized quantity. Let us assume that we want to select an alternative with the largest possible value of a certain quantity. If for n alternatives a1, . . . , an, we know the exact values v1, . . . , vn of the corresponding quantity, then the decision maker will select the alternative ai for which the corresponding value vi is the largest. How to predict this decision. When we know the values v1, . . . , vn, then predicting a decision means computing the index in of the largest value vi. This can be done in time O(n), by the following iterative process. At each iteration k (k = 1, . . . , n), ik will be index of the largest of the first k values v1, . . . , vk. In the first iteration k = 1, we naturally take i1 = 1. Once we got ik, on the next (k + 1)-st iteration, we compare the largest-so-far value vik with the new value