Winner determination in voting trees with incomplete preferences and weighted votes

Abstract

In multiagent settings where agents have different preferences, preference aggregation can be an important issue. Voting is a general method to aggregate preferences. We consider the use of voting tree rules to aggregate agents' preferences. In a voting tree, decisions are taken by performing a sequence of pairwise comparisons in a binary tree where each comparison is a majority vote among the agents. Incompleteness in the agents' preferences is common in many real-life settings due to privacy issues or an ongoing elicitation process. We study how to determine the winners when preferences may be incomplete, not only for voting tree rules (where the tree is assumed to be fixed), but also for the Schwartz rule (in which the winners are the candidates winning for at least one voting tree). In addition, we study how to determine the winners when only balanced trees are allowed. In each setting, we address the complexity of computing necessary (respectively, possible) winners, which are those candidates winning for all completions (respectively, at least one completion) of the incomplete profile. We show that many such winner determination problems are computationally intractable when the votes are weighted. However, in some cases, the exact complexity remains unknown. Since it is generally computationally difficult to find the exact set of winners for voting trees and the Schwartz rule, we propose several heuristics that find in polynomial time a superset of the possible winners and a subset of the necessary winners which are based on the completions of the (incomplete) majority graph built from the incomplete profiles.This article is a revised and extended version of the conference papers [23,28].

Possible Winners in Approval Voting

“…The main difference with our setting is that in all these works (up to one exception, discussed below), the voting rule used takes a classical profile, that is, a collection of rankings, as input, and the incomplete information consists of a collection of partial orders: a possible (resp. necessary) winner is then a candidate that wins in some completion (respectively, all completions) of this collection of partial orders [21,32,4,3,33,10,5,1,22,18]. An exception is [33], which, in Section 4, states a characterization of possible winners in approval voting, given an initial approval ballot over an initial set of candidates, and given a number of new candidates to be added; the nature of the incomplete information about approval ballots in their setting and ours (an approval profile over a subset of candidates vs. a ranking profile over all candidates) is totally different, and results cannot easily be compared.…”

Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty

“…Necessary) Winner problem with weighted voters when the given tree is balanced. Pini et al [32] and Lang et al [23] show that the Possible (resp. Necessary) Winner problem with weighted voters is intractable for a constant number of voters (see Table 1).…”

“…m" stands for "fixed-parameter tractable with respect to m" and means that if the number m of alternatives is a constant, then the relevant problem is polynomial-time solvable and the degreee of the polynomial does not depend on m. The result marked with ♥ also follows from the work of Miller [27]. Those marked with ♠ follow from the work of Pini et al [32] and Lang et al [23]. Entries containing statements of the form "NP-c (z)" (resp.…”

On the complexity of bribery and manipulation in tournaments with uncertain information

“…Perhaps the closest notion is that of a possible winnera notion intrinsic to reasoning under uncertainty introduced by Konczak and Lang [28], with further studies by Lang et al [29]. Recent studies of the complexity of computing possible winners include those by Lang et al [30], Bachrach et al [5], Conitzer et al [11] and Xia et al [57].…”