In this paper we describe and compare two models of the familiar cost-loss ratio situation. This situation involves a decision maker who must decide whether or not to take protective action, with respect to some activity or operation, in the face of uncertainty as to whether or not weather adverse to the activity will occur. The original model, first described by J.C. Thompson, is based in part upon the (implicit) assumption that taking protective action completely eliminates the loss associated with the occurrence of adverse weather. In the model formulated in this paper, on the other hand, it is assumed that taking protective action may reduce or eliminate this loss. The original model, then, is a special am of this “generalized” model. We show that the decision rule in each model depends upon a cost-loss ratio and that in both models this ratio is simply the cost of protection divided by the protectable portion of the loss. Thus the two models are equivalent from a decision-making point of view. This result also implies that the original model is applicable to a wider class of decision-making situations than has generally been recognized heretofore.
We also formulate measures of the value of probability forecasts within the frameworks of these models. First, the expenses (i.e., costs and losses) are translated into utilities, which are assumed to express the decision maker's preferences for the consequences. Then,. probabilistic specifications of the utilities are briefly discussed and general expressions are presented for the appropriate measures of value in cost-loss ratio situations with such specifications, namely expected-utility measures. Finally, we formulate the expected-utility measure associated with each model when the relevant utilities are assumed to possess a uniform probability distribution. Both measures are then shown to be equivalent (i.e., linearly related) to the Brier, or probability, score, a (familiar measure of the accuracy of probability forecasts. These results provide additional support for the use of the probability score as an evaluation measure.