Value functions when decision criteria are not totally substitutable

A necessary condition for the widely used additive value function is total preferential independence, or somewhat equivalently, total substitutability among the decision criteria. The paper considers cases where total substitutability is absent, and studies the value functions that are applicable to such cases. First it takes the case of total nonsubstitutability, and proves that the maximum value function is appropriate for it. This result easily extends to the closely related maximax value function. Next the paper considers the case where there is neither total substitutability nor total nonsubstitutability, and shows how a minsum value function can be applicable. A minsum function is one that uses only addition and minimum extraction operations. The paper explains how the structure of a minsum function can be inferred from substitutability information. In the process, certain subsets of criteria are encountered, which we call chains and cuts.