Optimal defense against election control by deleting voter groups

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.

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.

Standard voting rules usually assume that the preferences of voters are provided in the form of complete rankings over a fixed set of alternatives. This assumption does not hold in applications like recommendation systems where the set of alternatives is extremely large and only partial preferences can be elicited. In this paper, we study the problem of strategic manipulation of voting rules that aggregate voter preferences provided in the form of pairwise comparisons between alternatives. Our contributions are twofold: first, we show that any onto pairwise voting rule is manipulable in principle. Next, we analyze how the computational complexity of manipulation of such rules varies with the structure of the graph induced by the pairs of alternatives that the manipulator is allowed to vote over and the type of the preference relation. Building on natural connections between the pairwise manipulation and sports elimination problems (including a mixed-elimination variant that we introduce in this paper), we show that manipulating pairwise voting rules can be computationally hard even in the single-manipulator setting, a setting where most standard voting rules are known to be easy to manipulate.

This is an advance summary of a forthcoming article in the Oxford Research Encyclopedia of Politics. Please check back later for the full article.Narrowly understood, social choice theory is a specialized branch of applied logic and mathematics that analyzes abstract objects called preference aggregation functions, social welfare functions, and social choice functions. But more broadly, social choice theory identifies, analyzes, and evaluates rules that may be used to make collective decisions. So understood, social choice is a subfield of the social sciences that examines what may be called “voting rules” of various sorts. While social choice theory typically assumes a finite set of alternatives over which voter preferences are unrestricted, the spatial model of social choice assumes that policy alternatives can be represented by points in a space of one or more dimensions, and that voters have preferences that are plausibly shaped by this spatial structure.Social choice theory has considerable relevance for the study of legislative (as well as electoral) institutions. The concepts and tools of social choice theory make possible formal descriptions of legislative institutions such as bicameralism, parliamentary voting procedures, effects of decision rules (e.g., supramajority vs. simple majority rule and executive veto rules), sincere vs. strategic voting by legislators, agenda control, and other parliamentary maneuvers. Spatial models of social choice further enrich this analysis and raise additional questions regarding policy stability and change. Spatial models are used increasingly to guide empirical research on legislative institutions and processes.

A voting rule maps voter preferences into outcomes, and is called sophisticated if there exists a voting tree whose sophisticated outcomes coincide with the voting rule for every voter preference. As yet, no complete characterization of such rules is available. In this paper, we provide an important step toward this characterization by completely solving the problem when there are two possible sets of voter preferences.

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.

Various voting rules (or social choice procedures) have been proposed to select a winner from the preferences of an entire population: Plurality, veto, Borda, Minimax, Copeland, etc. Although in theory, these rules may yield drastically different outcomes, for real-world datasets, behavioral social choice analyses have found that the rules are often in perfect agreement with each other! This work attempts to give a mathematical explanation of this phenomenon. We quantify the gap between the outcomes of two voting rules by the pairwise margin between their winners. We show that for many common voting rules, the gap between them can be almost as large as 1 when the votes are unrestricted. As a counter, we study the behavior of voting rules when the vote distribution is a uniform mixture of a small number of multinomial logit distributions. This scenario corresponds to a population consisting of a small number of groups, each voting according to a latent preference ranking. We show that for any such voting profile on g groups, at least 1/2g fraction of the population prefers the winner of a Borda election to any other candidate.

The concept of distance rationalizability of voting rules has been explored in recent years by several authors. Roughly speaking, we first choose a consensus set of elections (defined via preferences of voters over candidates) for which the result is specified a priori (intuitively, these are elections on which all voters can easily agree on the result). We also choose a measure of distance between elections. The result of an election outside the consensus set is defined to be the result of the closest consensual election under the distance measure. Most previous work has dealt with a definition in terms of preference profiles. However, most voting rules in common use are anonymous and homogeneous. In this case there is a much more succinct representation (using the voting simplex) of the inputs to the rule. This representation has been widely used in the voting literature, but rarely in the context of distance rationalizability. We show exactly how to connect distance rationalizability on profiles for anonymous and homogeneous rules to geometry in the simplex. We develop the connection for the important special case of votewise distances, recently introduced and studied by Elkind, Faliszewski and Slinko in several papers. This yields a direct interpretation in terms of well-developed mathematical concepts not seen before in the voting literature, namely Kantorovich (also called Wasserstein) distances and the geometry of Minkowski spaces. As an application of this approach, we prove some positive and some negative results about the decisiveness of distance rationalizable anonymous and homogeneous rules. The positive results connect with the recent theory of hyperplane rules, while the negative ones deal with distances that are not metrics, controversial notions of consensus, and the fact that the $\ell^1$-norm is not strictly convex.

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.

In this paper, we study the metric distortion of deterministic social choice rules that choose a winning candidate from a set of candidates based on voter preferences. Voters and candidates are located in an underlying metric space. A voter has cost equal to her distance to the winning candidate. Ordinal social choice rules only have access to the ordinal preferences of the voters that are assumed to be consistent with the metric distances. Our goal is to design an ordinal social choice rule with minimum distortion, which is the worst-case ratio, over all consistent metrics, between the social cost of the rule and that of the optimal omniscient rule with knowledge of the underlying metric space. The distortion of the best deterministic social choice rule was known to be between $3$ and $5$. It had been conjectured that any rule that only looks at the weighted tournament graph on the candidates cannot have distortion better than $5$. In our paper, we disprove it by presenting a weighted tournament rule with distortion of $4.236$. We design this rule by generalizing the classic notion of uncovered sets, and further show that this class of rules cannot have distortion better than $4.236$. We then propose a new voting rule, via an alternative generalization of uncovered sets. We show that if a candidate satisfying the criterion of this voting rule exists, then choosing such a candidate yields a distortion bound of $3$, matching the lower bound. We present a combinatorial conjecture that implies distortion of $3$, and verify it for small numbers of candidates and voters by computer experiments. Using our framework, we also show that selecting any candidate guarantees distortion of at most $3$ when the weighted tournament graph is cyclically symmetric.

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.

In many real-life scenarios, a group of agents needs to agree on a common action, e.g., on a location for a public facility, while there is some consistency between their preferences, e.g., all preferences are derived from a common metric space. The facility location problem models such scenarios and it is a well-studied problem in social choice. We study mechanisms for facility location on unweighted undirected graphs, which are resistant to manipulations (strategyproof, abstention-proof, and false-name-proof ) by both individuals and coalitions and are efficient (Pareto optimal). We define a family of graphs, ZV -line graphs, and show a general facility location mechanism for these graphs which satisfies all these desired properties. Moreover, we show that this mechanism can be computed in polynomial time, the mechanism is anonymous, and it can equivalently be defined as the first Pareto optimal location according to some predefined order. Our main result, the ZV -line graphs family and the mechanism we present for it, unifies the few current works in the literature of false-name-proof facility location on discrete graphs, including the preliminary (unpublished) works we are aware of. Finally, we discuss some generalizations and limitations of our result for problems of facility location on other structures.

The problem ofjustifying democracy arises when a society sees the need for cooperative, collective action. Collective action may be needed to solve coordination problems, public goods problems, Prisoner's Dilemmas, and other structural problems of human interaction or perhaps to realize common ideals of justice in concrete political institutions. Once the need for collective action is established, one must ask how the goals of a cooperative endeavor are to be secured. Cooperation requires that individuals, at least over a specified range of activity, pursue a joint strategy. To assure voluntary compliance in a joint venture, cooperation must be to each person's advantage. However, each person may have an incentive to induce others to cooperate and to defect from the joint strategy in the hope of enjoying the fruits of cooperation without incurring the opportunity costs of compliance. This is the essence of the Prisoner's Dilemma, and considerations of this sort suggest that if collective action is to succeed, policies or strategies formed collectively must be coercively enforceable. Coercion, however, requires justification. Because solving the problem of rational noncompliance requires that collective decisions be coercively enforceable, the rules by which collective decisions are reached require justification. Douglas Rae neatly puts the problem of justification that emerges when political solutions are coercively enforceable as follows: "Once a political community has decided which of its members are to participate directly in the making of collective policy, an important question remains: 'How many of them must agree before a policy is imposed on the community?"" This is essentially the question to which the principle of democratic rule provides an answer: by what process are collective decisions to be made? Answering Rae's question requires a normative framework. We could say that a procedure for making collective decisions is justified if and

The theory of social choice is concerned with the problem of aggregating the preferences of several persons into a single preference order. The problem has the same structure as that of aggregating the preference orders of a single person with regard to several aspects of alternatives into a single preference order on the set of alternatives. For this reason, the theory of social choice comes logically not under the topic of collective action (as one might suppose) but rather under the topic of multi-objective decisions. This is so because the several participants in a situation defined as a problem of social choice do not really select courses of action among several available ones when they present their preference orders on a set of alternatives. Therefore they are not ‘actors’, as actors are defined in the theory of decision. To be sure, there are situations where voters do choose strategies, that is, choose among different preference orders to present, which may or may not represent their true preferences but which they believe will be more likely to lead to a more desirable aggregated order. These situations can be considered as n-person decision problems, and we will examine them in the context of non-cooperative n-person games (see Chapter 14. At this time, however, we will suppose that each voter acts ‘sincerely’, that is, presents his ‘actual’ preference order. The decision is now up to some agency, which must combine all these submitted preference orders into a ‘social’ preference order.KeywordsSocial ChoicePreference OrderPareto OptimalitySocial Welfare FunctionSimple MajorityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.