Constructing decision functions with augmented ordinal information

Our concern is with the problem of constructing decision functions to aid in making decision under uncertainty. We discuss the tradeoff that has to be made, when selecting a scale for representing our possible payoffs, between the power of the scale and the burden of the scale. We consider here the situation in which our basic scale is an ordinal scale, however we augment this scale by allowing an additional notion, a classification of payoffs as to whether they are acceptable or not. This allows us to have information such as A is preferred to B but both are acceptable. We indicate that this formally corresponds to an ordinal scale with a denoted element and call such a scale a denoted ordinal scale (DOS). It is shown that this augmentation of the ordinal scale increases the power of the scale and therefore allows us to build more sophisticated decision models.