# Séminaire Donnees et APprentissage Artificiel

# From Cardinal to Ordinal Preference Models - Validation of ordinal models by psychological experiments

09/17/2009

Speaker(s) : Michio Sugeno (Doshisha University, Kyoto, J ; European Center for Soft Computing, Mieres, ES

Generally speaking, there are two types of models for human subjective preferences where in cardinal models, we use addition/multiplication calculations on real numbers and in ordinal models, we use min/max calculations on orders. The conventional preference models are cardinal ones. As for the cardinal models, we can go back to Bernoulli’s idea of utility functions in 18 C. Since then we have had expected utility theory based on objective probabilities by von Neumann and Morgenstern for preferences under risk and Savage’s theory based on subjective probabilities for preferences under uncertainty. However in order to solve counter examples to these theories shown by Allais and Ellsberg , we need Schmeidler’s seminal idea on Choquet expected utilities based on Choquet integrals with respect to fuzzy measures. Kahneman received a Nobel Prize in economics in 2002 for his Cumulative Prospect Theory (CPT) using Choquet integral models on a real line with two fuzzy measures. On the other hand, in the ordinal models we can use S-integrals with respect to fuzzy measures. Using S-integrals, we can also discuss subjective preference in parallel with the cardinal models. However in order to cope with Kahneman’s Cumulative Prospect Theory, we have to extend S-integrals onto a bipolar scale. Grabisch has suggested CPT S-integrals. We have conducted psychological experiments to verify ordinal models. The results have shown that we can model subjective preferences by S-integrals and CPT S-integrals. There are some cases in preferences which cannot be explained even by CPT-type models. For these cases we can use Grabisch’s bi-capacities as extended fuzzy measures.

Thomas.Baerecke (at) nulllip6.fr