On the Gap between Outcomes of Voting Rules

Predicting winner and estimating margin of victory in elections using sampling

The concept of distance rationalizability has several applications within social choice. In the context of voting, it allows one to define (rationalize) voting rules via a consensus class (roughly, a set of elections in which it is obvious who should win) and a distance function: namely, a candidate is said to be an election winner if it is ranked first in one of the nearest (with respect to the given distance) consensus elections. It is known that many classic voting rules can be represented in this manner. In this paper, we provide new results on distance rationalizability of several well-known voting rules such as all scoring rules, Approval, Young's rule and Maximin. We also show that a previously published proof of distance rationalizability of Young's rule is incorrect: the consensus notion and the distance function used in that proof give rise to a voting rule that is similar to---but distinct from---the Young's rule. Finally, we demonstrate that some voting rules cannot be rationalized via certain notions of consensus. To the best of our knowledge, these are the first non-distance-rationalizability results for voting rules.

When choosing a voting rule to make subsequent decisions, the members of a committee may wish this rule to be self-selected when it is the object of a choice among a menu of different possible voting rules. Such concepts have recently been explored in Social Choice theory, and a menu of voting rule is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this article the menu constituted by the three well-known scoring rules {Borda, Plurality, and Antiplurality}. Under the Impartial Culture assumption, which proposes an a priori model to estimate the likelihood of the profiles, we will derive a probability for the stability of this triplet of voting rules.

Lu and Boutilier proposed a novel approach based on "minimax regret" to use classical score based voting rules in the setting where preferences can be any partial (instead of complete) orders over the set of alternatives. We show here that such an approach is vulnerable to a new kind of manipulation which was not present in the classical (where preferences are complete orders) world of voting. We call this attack "manipulative elicitation." More specifically, it may be possible to (partially) elicit the preferences of the agents in a way that makes some distinguished alternative win the election who may not be a winner if we elicit every preference completely. More alarmingly, we show that the related computational task is polynomial time solvable for a large class of voting rules which includes all scoring rules, maximin, Copeland α for every α ∈ [0,1], simplified Bucklin voting rules, etc. We then show that introducing a parameter per pair of alternatives which specifies the minimum number of partial preferences where this pair of alternatives must be comparable makes the related computational task of manipulative elicitation NP-complete for all common voting rules including a class of scoring rules which includes the plurality, k-approval, k-veto, veto, and Borda voting rules, maximin, Copeland α for every α ∈ [0,1], and simplified Bucklin voting rules. Hence, in this work, we discover a fundamental vulnerability in using minimax regret based approach in partial preferential setting and propose a novel way to tackle it.

Explaining the outcome of an election is a crucial task to address, especially in the case of complex voting rules. For those without a background in social choice, understanding the result of an election with a complex voting rule can be difficult. One possible way of explaining a voting rule is by using a decision tree structure, allowing the reader to follow the reasoning behind the outcome. This work proposes a methodology for explaining voting rules using decision-tree-based classifiers. Using simple features, the classifiers can be trained to a high accuracy while maintaining a human-readable size. We test this framework with well-established voting rules -- Copeland, Kemeny-Young, Ranked Pairs and Schulze -- to generate explanations for each election's outcome. We experiment with different decision tree algorithms on a synthetic dataset to generate explanations for the election outcome. We find that Copeland and Schulze under three candidates can be learned perfectly using an optimized decision tree algorithm, while cases of other rules have high accuracy experimentally.

We determine the communication complexity of the common voting rules. The rules (sorted by their communication complexity from low to high) are plurality, plurality with runoff, single transferable vote (STV), Condorcet, approval, Bucklin, cup, maximin, Borda, Copeland, and ranked pairs. For each rule, we first give a deterministic communication protocol and an upper bound on the number of bits communicated in it; then, we give a lower bound on (even the nondeterministic) communication requirements of the voting rule. The bounds match for all voting rules except STV and maximin.

Various conditions from social choice theory are discussed as they apply to actual voting procedures. It is found that none of these conditions makes a very interesting distinction among actual voting rules. Further, many of these conditions are argued to be not universally acceptable. It is then shown that, especially in cases in which there are more than two candidates, strategies are an essential part of voting, and particularly strategies of the type associated with ‘coordination problems’ as studied by Thomas C. Schelling. Some tentative acceptability conditions for voting rules are set forth which attempt to take coordination strategies into account. These are also found to be not universally acceptable.

Voting rules are powerful tools that allow multiple agents to aggregate their preferences in order to reach joint decisions. A common flaw of some voting rules, known as the no-show paradox, is that agents may obtain a more preferred outcome by abstaining an election. Whenever a rule does not suffer from this paradox, it is said to satisfy participation. In this paper, we initiate the study of participation in probabilistic social choice, i.e., for voting rules that yield probability distributions over alternatives. We consider three degrees of participation based on expected utility, the strongest of which even requires that an agent is strictly better off by participating at an election. While the latter condition is prohibitive in non-probabilistic social choice, we show that it can be met by reasonable probabilistic functions. More generally, we study to which extent participation and Pareto efficiency are compatible. To the best of our knowledge, this is the first work in this direction.

Colman (1984) has proposed a new justification for studying the weak Borda effect. He argues that it represents the violation by the plurality procedure of the weak dominance axiom. Weak dominance is a condition which some social choice theorists consider that a fair and reasonable collective choice procedure should satisfy. However, there are fundamental difficulties associated with this new position. The criterion defined by Colman as underlying the weak Borda effect differs considerably from the weak dominance axiom. It is also inconsistent with the definition of the weak Borda effect itself. Moreover, it can be argued that the criterion underlying the weak Borda effect is in fact an unreasonably severe standard. It confounds an undesirable requirement (that only tied winners should be permitted when there is a cyclical majority) with a desirable requirement (the Condorcet criterion). Thus, only social choice procedures with a tendency to produce indecisive outcomes can satisfy the criterion. Other problems with the approach are pointed out. It is concluded that the weak Borda effect is of little relevance to social choice theory.

Classical results in voting theory show that strategic manipulation by voters is inevitable if a voting rule simultaneously satisfies certain desirable properties. Motivated by this, we study the relevant question of how often a voting rule is manipulable. It is well known that elections with a large number of voters are rarely manipulable under impartial culture (IC) assumption. However, the manipulability of voting rules when the number of candidates is large has hardly been addressed in the literature and our article focuses on this problem. First, we propose two properties: (i) asymptotic strategy-proofness and (ii) asymptotic collusion-proofness, with respect to new voters, which makes the two notions more relevant from the perspective of computational problem of manipulation. In addition to IC, we explore a new culture of society where all score vectors of the candidates are equally likely. This new notion has its motivation in computational social choice and we call it impartial scores culture (ISC) assumption. We study asymptotic strategy-proofness and asymptotic collusion-proofness for plurality, veto and [Formula: see text]-approval, and Borda voting rules under IC as well as ISC assumptions. Specifically, we prove bounds for the fraction of manipulable profiles when the number of candidates is large. Our results show that the size of the coalition and the tie-breaking rule play a crucial role in determining whether or not a voting rule satisfies the above two properties.

Election databases are the main elements of a recently introduced framework that aims to create bridges between the computational social choice and the data management communities. An election database consists of incomplete information about the preferences of voters, in the form of partial orders, alongside with standard database relations that provide contextual information. Earlier work in computational social choice focused on the computation of possible winners and necessary winners that are determined by the available incomplete information and the voting rule at hand. The presence of the relational context, however, permits the formulation of sophisticated queries about voting rules, candidates, potential winners, issues, and positions on issues. Such queries can be given possible answer semantics and necessary answer semantics on an election database, where the former means that the query is true on some completion of the given partial orders and the latter means that the query is true on every such completion. %In this paper, łooseness = -1 We carry out a systematic investigation of query evaluation on election databases by analyzing how the interaction between the partial preferences, the voting rules and the relational context impacts on the complexity of query evaluation. To this effect, we focus on positional scoring rules and unions of conjunctive queries. We establish a number of results that delineate the complexity of the possible answers and of the necessary answers for different positional scoring rules and for various classes of unions of conjunctive queries. Furthermore, we show that query evaluation is fixed-parameter tractable, where the parameter is the number of candidates in the election.

In contrast to a social choice function, a social choice procedure is proposed which depends both on the way a set of alternatives is broken up into the subsets and the sequence in which each of these subsets is taken up for consideration. This article investigates the Arrow question in this generalized framework.

Chapter Fifteen - Competitive Market Mechanisms as Social Choice Procedures

Optimal defense against election control by deleting voter groups

We introduce a general problem about bribery in voting systems. In the $\mathcal{R}$-Multi-Bribery problem, the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the perturbed election under the voting rule R. Voters assign prices for withdrawing their vote, for swapping the positions of two consecutive candidates in their preference order, and for perturbing their approval count to favour candidates. As our main result, we show that $\mathcal{R}$-Multi-Bribery is fixed-parameter tractable parameterized by the number of candidates for many natural voting rules $\mathcal{R}$, including Kemeny rule, all scoring protocols, maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves the parameterized complexity of $\mathcal{R}$-Swap Bribery for all those voting rules, thereby solving a long-standing open problem and "Challenge #2" of the "Nine Research Challenges in Computational Social Choice" by Bredereck et al. Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlier double-exponential time algorithm by Dorn and Schlotter that is restricted to the uniform-cost case for all scoring protocols, the maximin rule, and Bucklin rule.