Winner determination in voting trees with incomplete preferences and weighted votes
Autonomous Agents and Multi-Agent Systems
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
... ssues 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]. 132 Auton Agent Multi-Agent Syst (2012) 25:130-157 Voting trees, the Schwartz rule, and fair winners We define some basic notions related to voting trees. We first focus on unweighted profiles and then extend all the notions to weighted votes. Preferences, profiles and majority graphs We assume that each agent's preferences are specified by a strict total order (TO), that is, by an asymmetric, transitive and complete order, on a set of m candidates. The candidates are taken from a set , and they represent the possible options over which agents vote. Definition 1 (profile) A profile P on is a collection of n strict total orders over , i.e., P = (P 1 , . . . , P n ), where P i is the preference relation (or vote) of agent (also called voter) i. For the sake of simplicity we assume throughout the article that the number n of voters is odd. Our results can be extended to the case where n is even, but at the price of some complications that arise with the handling of ties. Profiles are denoted using the following notation: ((A > B > C); (A > C > B); (C > A > B)) means that voter 1 prefers A to B and B to C, etc. Definition 2 (voting rule and correspondence) A (deterministic) voting rule is a mapping from the set of profiles to the set of candidates. A voting correspondence is a mapping from the set of profiles to the set of nonempty sets of candidates. A voting rule computes a single winner. On the other hand, a voting correspondence can compute multiple winners. For the sake of simplicity, we will use the word "rule" even for correspondence where it does not cause any ambiguity. Given a profile P, the induced majority graph M(P) is defined as follows. Definition 3 (majority graph) Let P be a profile. The majority graph induced by P, denoted by M(P), is the graph whose set of vertices is the set of candidates and where, for all A, B ∈ , there is a directed edge from A to B (denoted by A > P m B, or by the abbreviated form A > m B) if and only if a strict majority of voters prefers A to B. The majority graph M(P) induced from any profile P is asymmetric, but it is not necessarily transitive. Moreover, since the number of voters is odd, M(P) is complete: for each A and B = A, either A > m B or B > m A holds. Therefore, M(P) is a complete and asymmetric graph, also called a tournament on  .