Utility Theory Violations by Multi-Criteria

This paper discusses two violations of utility theory that can arise with some common multi-criteria hierarchical weighting methods when employed for selection of the best of several alternatives considering multiple criteria. First, these hierarchical weighting methods are found to violate utility theory's principle of independence of irrelevant alternatives. Proof of this violation is made by counter-example. Second, a common violation of additive utility exists even when only two alternatives are considered, a case without irrelevant alternatives. The frequency of this incompatibility of hierarchical weighting with additive utility is explored through a numerical experiment and found to range between 4% and 13% for different problem sizes. These problems do not exist for application of weighting methods to single-objective problems. INTRODUCTION Over the last few decades, a number of multi-criteria weighting methods have become popular for evaluating alternative decisions. While these procedures are typically easy to implement, their compliance with fairly fundamental principles of rational decision-making, as represented by utility theory, has received little examination until recently. It seems appropriate to attempt to find the general forms of utility functions which would be implied by particular multi-criteria weighting methods. It is often presumed that multi-criteria weighting methods conform to the conditions of utility theory, particularly additive utility (e.g., Palmer and Lund, 1985). While this has been shown to be false in earlier literature (Hobbs, 1980; deNeufville, 1990; Dyer 1990), this finding has not greatly deterred the use of multi-criteria weighting methods. This short paper demonstrates the general incompatibility of hierarchical weighting methods with utility theory when used for the selection of a single best alternative. This incompatibility arises from violation of utility theory's principle of the independence of irrelevant alternatives. This is shown by counterexample. A further incompatibility of hierarchical weighting methods with additive utility theory is also shown when both methods are used for selection of a single best alternative. This effect is distinct from violation of the independence axiom of utility theory. The results of numerical experiments are provided which show the frequency with which multi-criteria weighting methods lead to selection of a different alternative than would direct use of additive multi-criteria utility functions. Conditions of certainty are assumed throughout the paper.