Cycles and Intractability in a Large Class of Aggregation Rules

Abstract

We introduce the (j, k)-Kemeny rule -a generalization of Kemeny's voting rule that aggregates j-chotomous weak orders into a k-chotomous weak order. Special cases of (j, k)-Kemeny include approval voting, the mean rule and Borda mean rule, as well as the Borda count and plurality voting. Why, then, is the winner problem computationally tractable for each of these other rules, but intractable for Kemeny? We show that intractability of winner determination for the (j, k)-Kemeny rule first appears at the j = 3, k = 3 level. The proof rests on a reduction of max cut to a related problem on weighted tournaments, and reveals that computational complexity arises from the cyclic part in the fundamental decomposition of a weighted tournament into cyclic and cocyclic components. Thus the existence of majority cycles -the engine driving both Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem -also serves as a source of computational complexity in social choice.

An axiomatic characterization of the Borda mean rule

“…In particular, it applies when votes are given by linear orders, or when they are given by arbitrary weak orders. It also applies to j -chotomous weak orders whenever ; thus, our result characterizes Zwicker’s ( 2018 ) ( j , 2)- Kemeny rule for each , which is the Borda mean rule as applied to profiles of j -chotomous weak orders. More generally, our proof works whenever the domain of allowed preference orders forms a McGarvey domain , that is, whenever every possible weighted majority tournament (with only even weights or only odd weights) can be induced by a profile using such orders.…”

“…This rule was defined by Duddy et al. ( 2014 ) and further discussed by Duddy et al ( 2016 , Section 5) and Zwicker ( 2018 ). Notice that the Borda mean rule satisfies reversal symmetry.…”

“…Zwicker ( 2018 ) introduced the idea of using Kemeny scores to define aggregation rules for other output types. For example, suppose we maximize the Kemeny score over the family of relations that have a unique most-preferred element and that are indifferent between all other alternatives.…”

“…This yields precisely the relations whose most-preferred element is a winner of Borda’s rule . In his paper, Zwicker ( 2018 ) proposed the k - Kemeny rule which returns the k -chotomous weak orders of highest Kemeny score; a weak order is called k - chotomous if its induced indifference relation partitions A into at most k indifference classes. 2 2-chotomous orders are usually called dichotomous ; these are the orders that partition the alternatives into a set of approved and a set of disapproved alternatives.…”

An Informational Basis for Voting Rules

“…Special cases of this generalization include the Borda rule and plurality voting, which are computationally tractable. Zwicker (2018) showed that computational complexity of the Kemeny rule arises from the cyclic part in the fundamental decomposition of a weighted tournament into cyclic and co-cyclic components. This cyclic part is associated with the Condorcet paradox.…”