A Direct Approach to Choice under Uncertainty

In a problem of decision making under uncertainty, the payoff is random, obeying some probability distribution. The distribution depends upon both the problem faced and the action taken by the decision maker. For example, in the selection of a portfolio, funds are allocated among securities, each of which has an associated distribution over the return it yields. The allocation results in a composite probability distribution over the total return from the entire portfolio. Although it is typical to select an action, which in turn implies a resulting distribution, we suggest in the present paper that it may sometimes be appropriate to select a probability distribution directly, or to view the problem abstractly as though the distribution were to be developed directly. We envision the decision maker selecting a most preferred distribution over payoffs via modification of an existing probability distribution at a direct monetary cost. Thus the family of distributions from which the decision maker can choose is implicitly defined by initial wealth, the original distribution, and the modification cost function. A formal statement of a class of problems for which our direct approach may be appropriate is developed, followed by examples. The illustrations cover problems of preventive maintenance for a machine subject to failure, pricing of a product under threat of rival entry, and planning a risky R and D project.