We study the impact on strategic voting of tie-breaking by means of considering the order of tied candidates within a random vote. We compare this to another non deterministic tie-breaking rule where we simply choose candidate uniformly at random. In general, we demonstrate that there is no connection between the computational complexity of computing a manipulating vote with the two different types of tie-breaking. However, we prove that for some scoring rules, the computational complexity of computing a manipulation can increase from polynomial to NP-hard. We also discuss the relationship with the computational complexity of computing a manipulating vote when we ask for a candidate to be the unique winner, or to be among the set of co-winners.

In their groundbreaking paper, Bartholdi, Tovey and Trick [1989] argued that many well-known voting rules, such as Plurality, Borda, Copeland and Maximin are easy to manipulate. An important assumption made in that paper is that the manipulator’s goal is to ensure that his preferred candidate is among the candidates with the maximum score, or, equivalently, that ties are broken in favor of the manipulator’s preferred candidate. In this paper, we examine the role of this assumption in the easiness results of [Bartholdi et al., 1989]. We observe that the algorithm presented in [Bartholdi et al., 1989] extends to all rules that break ties according to a fixed ordering over the candidates. We then show that all scoring rules are easy to manipulate if the winner is selected from all tied candidates uniformly at random. This result extends to Maximin under an additional assumption on the manipulator’s utility function that is inspired by the original model of [Bartholdi et al., 1989]. In contrast, we show that manipulation becomes hard when arbitrary polynomial-time tie-breaking rules are allowed, both for the rules considered in [Bartholdi et al., 1989], and for a large class of scoring rules.

In elections, an alternative is said to be a Condorcet winner if it is preferred to any other alternative by a majority of voters. While this is a very attractive solution concept, many elections do not have a Condorcet winner. In this paper, we propose a setvalued relaxation of this concept, which we call a Condorcet winning set: such sets consist of alternatives that collectively dominate any other alternative. We also consider a more general version of this concept, where instead of domination by a majority of voters we require domination by a given fraction theta of voters; we refer to this concept as theta-winning set. We explore social choice-theoretic and algorithmic aspects of these solution concepts, both theoretically and empirically.

Sequential voting rules and correspondences provide a way for agents to make group decisions when the set of available options has a multi-issue structure. One important question about sequential voting rules (correspondences) is whether they satisfy two crucial criteria, namely neutrality and efficiency. Recently, Benoit and Kornhauser established an important result about seat-by-seat voting rules (which are a special case of sequential voting rules): they proved that if the multi-issue domain satisfies some properties, then the only seat-by-seat rules being either efficient or neutral are dictatorships. However, there are still some cases not covered by their results, including a very important and interesting case—voting correspondences. In this paper, we extend the impossibility theorems by Benoit and Kornhauser to voting correspondences, and obtain a dichotomy theoremon the existence of efficient or neutral sequential (seat-by-seat) voting rules and correspondences. Therefore, the question of whether sequential (seat-by-seat) voting rules (correspondences) can be efficient or neutral is now completely answered.