An information theory for preferences

Recent literature in the last Maximum Entropy workshop introduced an analogy between cumulative probability distributions and normalized utility functions [1]. Based on this analogy, a utility density function can de defined as the derivative of a normalized utility function. A utility density function is non-negative and integrates to unity. These two properties of a utility density function form the basis of a correspondence between utility and probability, which allows the application of many tools from one domain to the other. For example, La Place’s principle of insufficient reason translates to a principle of insufficient preference. The notion of uninformative priors translates to uninformative utility functions about a decision maker’s preferences. A natural application of this analogy is a maximum entropy principle to assign maximum entropy utility values. Maximum entropy utility interprets many of the common utility functions based on the preference information needed for their assignment, and helps assign utility values based on partial preference information. This paper reviews maximum entropy utility, provides axiomatic justification for its use, and introduces further results that stem from the duality between probability and utility, such as joint utility density functions, utility inference, and the notion of mutual preference. INTRODUCTION In many decision situations, we are faced with multiple and conflicting attributes. When the decision situation is deterministic, each decision alternative is described by a single prospect (consequence). The problem of choosing the best decision alternative is that of ordering the prospects present or, alternatively, assigning a value function over the attributes of each prospect. The optimal decision alternative is the one that has the largest value as determined by the value function or the highest order in the ranked list. A normative justification for the order requirement is that a decision maker who cannot order the prospects is vulnerable to being a “money pump” (we would choose prospect A over prospect B, but also prospect B over prospect A, and be willing to pay money to move from one prospect to the other) When uncertainty is present, a decision alternative is characterized by several prospects and a probability distribution representing the probability of their occurrence. In this case, the rank order of the prospects alone is insufficient to determine the optimal decision alternative, and the Von Neumann and Morgenstern utility values need to be elicited. [2]. To elicit these utility values, a decision maker is first asked to order the prospects of the decision situation from best to worst. Once the order is provided, the next step is to assign a utility value for each prospect. For any three ordered prospects, , the decision maker assigns a probability, 1 2 P P P3 π , for which she is indifferent to receiving for sure and a deal where she would receive 2 P 1 P with probability π and with probability 3 P (1 ). π − If the utility values of and are one and zero respectively, then the utility value of , also called the preference probability of , will be equal to 1 P 3 P 2 P . 2 P (1) (1 )(0) π π π + − = The higher the prospect is in the ranked list, the larger is the utility value assigned to it. The optimal decision alternative is now the one, which has the highest expected utility for its prospects.