Bounds on the value of information in uncertain decision problems

The cost of obtaining good information regarding the various probability distributions needed for the solution of most stochastic decision problems is considerable. It is important to consider questions such as: (1) what minimal amounts of information are sufficient to determine optimal decision rules; (2) what is the value of obtaining knowledge of the actual realization of the random vectors; and (3) what is the value of obtaining some partial information regarding the actual realization of the random vectors. This paper is primarily concerned with questions two and three when the decision maker has an a priori knowledge of the joint distribution function of the random variables. Some remarks are made regarding results along the lines of question one. Mention is made of assumptions sufficient so that knowledge of means, or of means, variances, co-variances and n-moments are sufficient for the calculation of optimal decision rules. The analysis of the second question leads to the development of bounds on the value of perfect information. For multiperiod problems it is important to consider when the perfect information is available. Jensen's inequality is the key tool of the analysis. The calculation of the bounds requires the solution of nonlinear programs and the numerical evaluation of certain functions. Generally speaking, tighter bounds may be obtained only at the expense of additional information and computational complexity. Hence, one may wish to compute some simple bounds to decide upon the advisability of obtaining more information. For the analysis of the value of partial information it is convenient to introduce the notion of a signal. Each signal represents the receipt of certain information, and these signals are drawn from a given probability distribution. When a signal is received, it alters the decision maker's perception of the probability distributions inherent in his decision problem. The choice between different information structures must then take into account these probability distributions as well as the decision maker's preference function. A hierarchy of bounds may be determined for partial information evaluation utilizing the tools of the multiperiod perfect information case. However, the calculation of these bounds is generally considerably more dicult than the calculation of similar boulids in the perfect information case. Most of the analysis is directed towards problems in which the decision maker has a linear utility function over profits, costs or some other numerical variable. However, some of the bounds generalize to the case when the utility function is strictly increasing and concave.