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A new perspective on implementation by voting trees
Author(s) -
Fischer Felix,
Procaccia Ariel D.,
Samorodnitsky Alex
Publication year - 2011
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20336
Subject(s) - tournament , approval voting , voting , mathematics , vertex (graph theory) , combinatorics , degree (music) , tree (set theory) , discrete mathematics , computer science , condorcet method , graph , physics , politics , political science , acoustics , law
Voting trees describe an iterative procedure for selecting a single vertex from a tournament. They provide a very general abstract model of decision‐making among a group of individuals, and it has therefore been studied which voting rules have a tree that implements them, i.e., chooses according to the rule for every tournament. While partial results concerning implementable rules and necessary conditions for implementability have been obtained over the past 40 years, a complete characterization of voting rules implementable by trees has proven surprisingly hard to find. A prominent rule that cannot be implemented by trees is the Copeland rule, which singles out vertices with maximum degree. In this paper, we suggest a new angle of attack and re‐examine the implementability of the Copeland solution using paradigms and techniques that are at the core of theoretical computer science. We study the extent to which voting trees can approximate the maximum degree in a tournament, and give upper and lower bounds on the worst‐case ratio between the degree of the vertex chosen by a tree and the maximum degree, both for the deterministic model concerned with a single fixed tree, and for randomizations over arbitrary sets of trees. Our main positive result is a randomization over surjective trees of polynomial size that provides an approximation ratio of at least 1/2. The proof is based on a connection between a randomization over caterpillar trees and a rapidly mixing Markov chain. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 59–82, 2011