Towards a dichotomy for the Possible Winner problem in elections based on scoring rules

Abstract

To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, whether a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2, 1, . . . , 1, 0), while it is solvable in polynomial time for plurality and veto.

Selections from ordered sets

“…Conversely, relays in track and field (especially the 4 × 400) approximately satisfy Axiom 5. 14 Young (1995) includes an intuitive introduction; among the many more recent examples, Davenport andKalagnanam (2004), or Betzler andDorn (2010); Davenport and Kalagnanam (2004).…”

Winner determination in voting trees with incomplete preferences and weighted votes

“…Xia and Conitzer showed that, for unweighted votes and an unbounded number of candidates, for most commonly used voting rules computing possible winners is NP-hard, whilst computing necessary winners is coNP-hard for some voting rules but polynomial for others [38]. Also, Betzler and Dorn [3,4], and Baumeister and Rothe [2], showed that, for an unbounded number of candidates and unweighted votes, computing possible winners is NP-hard for some classes of scoring rules and polynomial for others.…”

“…Let P be an incomplete profile, A is a possible winner for M(P) and r if there exists a completion M of M(P) such that 4 respectively the set of possible and necessary winners for M(P) and r . Note that these notions of possible and necessary winners only apply to rules that are based on the majority graph.…”

“…The notions of possible and necessary winners for voting rules, as well as the notions of possible and necessary Condorcet winners, were introduced by Konczak and Lang [21]. The computational complexity of computing possible and necessary winners for many common voting rules was thoroughly studied by Xia and Conitzer [38], by Betzler and Dorn [4], and by Baumeister and Rothe [2]. We complete the picture by adding results concerning voting trees and Schwartz.…”

“…The computational aspects of incomplete tournaments have been thoroughly investigated by Brandt et al [6,7]. Voting under incomplete knowledge has received even more attention [2,3,5,4,8,10,19,21,27,36,38]. We will discuss some of these papers in Sect.…”