An Approval-Voting Polytope for Linear Orders

✍ Scribed by Jean-Paul Doignon; Michel Regenwetter

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 542 KB
• Volume: 41
• Category: Article
• ISSN: 0022-2496
• DOI: 10.1006/jmps.1997.1155

✦ Synopsis

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.