Cycles and Intractability in a Large Class of Aggregation Rules
Abstract
We introduce the -Kemeny rule -- a generalization of Kemeny's voting rule that aggregates -chotomous weak orders into a -chotomous weak order. Special cases of -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 -Kemeny rule first appears at the , 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.
Cite
@article{arxiv.1608.03999,
title = {Cycles and Intractability in a Large Class of Aggregation Rules},
author = {William S. Zwicker},
journal= {arXiv preprint arXiv:1608.03999},
year = {2018}
}
Comments
25 pages, 8 figures. Uses jair.sty style file. Two earlier versions had a slightly different title, "Cycles and Intractability in Social Choice Theory." The first of these did not include Theorem 1.2; it was presented in COMSOC 2016 (Sixth International Workshop on Computational Social Choice). The second version was posted to arXiv, and this is the final (published) version