A new perspective on implementation by voting trees

Voting trees provide an abstract model of decision-making among a group of individuals in terms of an iterative procedure for selecting a single vertex from a tournament. A family of voting trees is said to implement a given voting rule if for every tournament it chooses according to the rule. While partial results concerning implementable rules and necessary conditions for implementability have been obtained, 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 at the core of theoretical computer science. We study the extent to which voting trees can approximate the maximum degree, 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.