Estimation of Attribute Weights from Preference Comparisons

The multi-attribute utility model serves as a basis for many marketing decisions such as new product planning and advertising message selection. The estimation of individuals' attribute weights can be performed using several data types and estimation techniques. There is evidence to suggest that the estimates derived from ordinal preference data through linear programming show greater stability and predictive validity. In this paper we address two fundamental issues which have not been addressed in the context of this latter type estimation: the theoretical foundations for estimating cardinal utility functions from ordinal preference data and the properties of the linear programming estimators. First, we establish the theoretical foundations from economics, mathematical psychology, and decision analysis of obtaining a cardinal (interval scaled) multi-attribute function from ordinal data. This leads us to recommend that in addition to the collection of paired preference comparisons, also comparisons of pairs of pairs be collected. We then describe the type of errors which are likely to arise in the measurement stage, and their relationship to the phenomenon of intransitivities. We formulate a linear program, LINPAC, for the estimation of attribute weights from the above preference data. The previously proposed LINMAP procedure is a special case of this formulation when only the information on the paired preferences is utilized. Next, the statistical properties of the estimators, such as uniqueness, unbiasedness, consistency and efficiency, are examined. Then, through a simulation study we examine the rate of convergence of the estimated weights to the true weights as a function of the number of brands. In the simulation study we also examine the conditions under which the estimators outperform equal weights and compare the estimates derived from LINPAC with those derived from LINMAP. Finally, the estimation procedures are examined with actual data while the simulation results, an equal weights model, and a stated weights model serve as benchmarks.