JP Doignon | PubMed Health

PubMed1998

The Choice Probabilities of the Latent-Scale Model Satisfy the Size-Independent Model When n Is Small.

Two probabilistic models for subset choices are compared. The first one, due to Marley (1993), was dubbed the latent-scale model by Regenwetter, Marley, and Joe (1996). The second one is Falmagne and Regenwetter's (1996) size-independent model of approval voting. We show that for up to five choice alternatives, the choice probabilities generated by the latent-scale model can be explained also by the size-independent model. The proof uses the König-Hall theorem of graph theory and the characterization of the size-independent model by the approval-voting polytope of Doignon and Regenwetter (1997). The problem remains open for the case with more than five choice alternatives. Copyright 1998 Academic Press.

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PubMed1997

An Approval-Voting Polytope for Linear Orders

A probabilistic model of approval voting on n alternatives generates a collection of probability distributions on the family of all subsets of the set of alternatives. Focusing on the size-independent model proposed by Falmagne and Regenwetter, we recast the problem of characterizing these distributions as the search for a minimal system of linear equations and inequalities for a specific convex polytope. This approval-voting polytope, with n! vertices in a space of dimension 2(n), is proved to be of dimension 2(n)-n-1. Several families of facet-defining linear inequalities are exhibited, each of which has a probabilistic interpretation. Some proofs rely on special sequences of rankings of the alternatives. Although the equations and facet-defining inequalities found so far yield a complete minimal description when n<=4 (as indicated by the PORTA software), the problem remains open for larger values of n.

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