A $O(\log m)$, deterministic, polynomial-time computable approximation of Lewis Carroll's scoring rule

TL;DR

This paper introduces deterministic polynomial-time algorithms that approximate Dodgson's and Young's voting rules within a logarithmic factor, providing efficient and natural scoring methods for complex voting rules.

Contribution

It presents the first deterministic, polynomial-time approximation algorithms for Dodgson's and Young's scoring rules with tight bounds, improving computational feasibility.

Findings

Approximate Dodgson's rule within a logarithmic factor

Approximate Young's rule with a logarithmic factor

Algorithms are simple, natural, and greedy in design

Abstract

We provide deterministic, polynomial-time computable voting rules that approximate Dodgson's and (the ``minimization version'' of) Young's scoring rules to within a logarithmic factor. Our approximation of Dodgson's rule is tight up to a constant factor, as Dodgson's rule is $\NP$-hard to approximate to within some logarithmic factor. The ``maximization version'' of Young's rule is known to be $\NP$-hard to approximate by any constant factor. Both approximations are simple, and natural as rules in their own right: Given a candidate we wish to score, we can regard either its Dodgson or Young score as the edit distance between a given set of voter preferences and one in which the candidate to be scored is the Condorcet winner. (The difference between the two scoring rules is the type of edits allowed.) We regard the marginal cost of a sequence of edits to be the number of edits divided by the number of reductions (in the candidate's deficit against any of its opponents in the pairwise race against that opponent) that the edits yield. Over a series of rounds, our scoring rules greedily choose a sequence of edits that modify exactly one voter's preferences and whose marginal cost is no greater than any other such single-vote-modifying sequence.