Choice Axioms and a Geometrical Analysis of Rankings

The discrete utility theory is treated in this chapter. In view of mainstream, the alternative one to the traditional choice theory must be the probabilistic choice theory of utility given by Luce (1959). Later, this method became well known by McFadden’s Nobel Prize in 2007 (McFadden, 1974) in the academic world but before the event rather famous in the practical field. Shortly after the release of Luce (1959), unfortunately, Debreu (1960), one of the most respectful general equilibrium theorist, declared that “Luce’s choice axiom” is inconsistent. Subsequently, therefore, the theoretical study of Luce’s choice axioms had only sporadic development in the academic world, and this type of Luce utility theory was rather limited to practical analysis. When Luce’s utility is algebraically considered, this is currently dealt with as the multinomial logit model, which can illustrate either successive choices or mode choices. Alternatively, it may be possible to derive the multinomial logit model type utility from the neoclassical assumptions. A restrictive assumption like Independence from Irrelevant Alternatives (IIA) is often not only required to be imposed but also to depend on the assumption that the set/subset of preference are universal. However, Saari pointed out that Debreu’s criticism was off the point, as Saari found (2005), that Debreu’s example on Luce’s theory was discussing a problem in which the number of constraint expressions was too excessive than the number of unknowns to deal with, and then that nothing could be determined. On the contrary, Saari escaped the narrow deterministic arguments to expand significantly the theoretical coverage of choice theory. In his framework, even Arrow’s Possibility Theorem (Namba, 1951) is simply a special case.