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A new axiomatization of Luce’s model of choice and ranking

Authors
Dr. Donald Bamber
University of California, Irvine ~ Cognitive Sciences
I. R. Goodman
Independent Scholar
Hung T. Nguyen
Chiang Mai University, Thailand
Abstract

Suppose that an agent is asked to rank the elements of a finite set U, starting with the most preferred and ending with the least. These rank orders may vary stochastically from occasion to occasion. Let PUrank denote the agent’s probability distribution of rankings of U. Duncan Luce’s well-known Choice Axiom, together with his Ranking Postulate, imply that the PUrank distribution will be a member of the Plackett-Luce family.We derive Luce’s Choice Axiom, rather than assuming it. Suppose that T and S are any disjoint sets whose union is U. Let Tspec and Sspec denote any specifications of of the preference order over the elements of T and S, respectively. Let Sfirst denote the event that every element of S is preferred to every element of T. Our Axiom of Independence from the Past (IFP) states that the events Tspec and Sspec-intersection-Sfirst are independent under PUrank. This axiom implies that PUrank is a Plackett-Luce distribution.Our Rational Choice Axiom states that, when the agent chooses an element from a subset T of U, the agent consults its preference ranking over U and selects the element of T that is highest ranked. Together, this axiom and the IFP Axiom imply Luce’s Choice Axiom.In addition, we formulate a ranking mechanism, based on Goodman and Nguyen’s Product Space Conditional Event Algebra, whose behavior conforms to the IFP Axiom.

Tags

Keywords

Luce's Choice Axiom
preference ranking
Plackett-Luce distribution
Product Space Conditional Event Algebra

Topics

Cognitive Modeling
Probabilistic Models
Theory development
Discussion
New
Cool! Last updated 2 months ago

Hey Don: Very felicitous and trenchant result! It's amazing to me how that simple axiom of Duncan's from so many years ago, continues to bear tasty fruit and into the bargain, to teach us about novel potential dependencies in choice and ranking! Thank you, Jim

James T. Townsend 0 comments