A theoretical note on the number of free parameters in the Elimination-By-Aspects model

Tversky (1972b) stated that the number of free parameters in the Elimination-By-Aspects (EBA) model is ^2m - 3 (the number of proper nonempty subsets of the full choice set [T], minus an arbitrary unit of measurement), where m is the number of objects in the total set of objects under consideration. However, for m = 3, Batsell, Polking, Cramer, and Miller (2003) proved that the ^2m - 3 condition would imply 5 free parameters, when there are actually 4. Furthermore, they proposed a linear version of the EBA model and conjectured that the number of free EBA parameters is ^2m - m - 1 based on numerical experiments. The experiments were conducted with their least squares estimation method that was developed for the linear version of the EBA model. This paper corrects the error of Batsell etal.'s (2003) estimation method and demonstrates that the number of free parameters is min [ (m - 2) ^{2m -1} + 1, ^2m - 2] - 1. The formula presented in this paper indicates that Tversky's statement is correct for m > 3 whereas Batsell and his colleagues' proof is correct for m = 3.