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