We study computational aspects of a well-known single-winner voting rule called the Schulze method [Schulze, 2003] which is used broadly in practice. In this method the voters give (weak) ordinal preference ballots which are used to define the weighted majority graph (WMG) of direct comparisons between pairs of candidates. The choice of the winner comes from indirect comparisons in the graph, and more specifically from considering directed paths instead of direct comparisons between candidates. When the input is the WMG, to our knowledge, the fastest algorithm for computing all possible winners in the Schulze method uses a folklore reduction to the All-Pairs Bottleneck Paths (APBP) problem and runs in $O(m^{2.69})$ time, where $m$ is the number of candidates. It is an interesting open question whether this can be improved. Our first result is a combinatorial algorithm with a nearly quadratic running time for computing all possible winners. If the input to the possible winners problem is not the WMG but the preference profile, then constructing the WMG is a bottleneck that increases the running time significantly; in the special case when there are $O(m)$ voters and candidates, the running time becomes $O(m^{2.69})$, or $O(m^{2.5})$ if there is a nearly-linear time algorithm for multiplying dense square matrices. To address this bottleneck, we prove a formal equivalence between the well-studied Dominance Product problem and the problem of computing the WMG. We prove a similar connection between the so called Dominating Pairs problem and the problem of verifying whether a given candidate is a possible winner. Our paper is the first to bring fine-grained complexity into the field of computational social choice. Using it we can identify voting protocols that are unlikely to be practical for large numbers of candidates and/or voters, as their complexity is likely, say at least cubic.

A voting center is in charge of collecting and aggregating voter preferences. In an iterative process, the center sends comparison queries to voters, requesting them to submit their preference between two items. Voters might discuss the candidates among themselves, figuring out during the elicitation process which candidates stand a chance of winning and which do not. Consequently, strategic voters might attempt to manipulate by deviating from their true preferences and instead submit a different response in order to attempt to maximize their profit. We provide a practical algorithm for strategic voters which computes the best manipulative vote and maximizes the voter's selfish outcome when such a vote exists. We also provide a careful voting center which is aware of the possible manipulations and avoids manipulative queries when possible. In an empirical study on four real-world domains, we show that in practice manipulation occurs in a low percentage of settings and has a low impact on the final outcome. The careful voting center reduces manipulation even further, thus allowing for a non-distorted group decision process to take place. We thus provide a core technology study of a voting process that can be adopted in opinion or information aggregation systems and in crowdsourcing applications, e.g., peer grading in Massive Open Online Courses (MOOCs).

We develop a novel framework that aims to create bridges between the computational social choice and the database management communities. This framework enriches the tasks currently supported in computational social choice with relational database context, thus making it possible to formulate sophisticated queries about voting rules, candidates, voters, issues, and positions. At the conceptual level, we give rigorous semantics to queries in this framework by introducing the notions of necessary answers and possible answers to queries. At the technical level, we embark on an investigation of the computational complexity of the necessary answers. We establish a number of results about the complexity of the necessary answers of conjunctive queries involving positional scoring rules that contrast sharply with earlier results about the complexity of the necessary winners.

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We study computational problems for two popular parliamentary voting procedures: the amendment procedure and the successive procedure. While finding successful manipulations or agenda controls is tractable for both procedures, our real-world experimental results indicate that most elections cannot be manipulated by a few voters and agenda control is typically impossible. If the voter preferences are incomplete, then finding possible winners is NP-hard for both procedures. Whereas finding necessary winners is coNP-hard for the amendment procedure, it is polynomial-time solvable for the successive one.

Manipulation can be performed when intermediate voting results are known; voters might attempt to vote strategically and try and manipulate the results during an iterative voting process. When only partial voting preferences are available, preference elicitation is necessary. In this paper, we combine two approaches of iterative processes: iterative preference elicitation and iterative voting and study the outcome and performance of a setting where manipulative voters submit partial preferences. We provide practical algorithms for manipulation under the Borda voting rule and evaluate those using different voting centers: the Careful voting center that tries to avoid manipulation and the Naive voting center. We show that in practice, manipulation happens in a low percentage of the settings and has a low impact on the final outcome. The Careful voting center reduces manipulation even further.

Understanding the nature of strategic voting is the holy grail of social choice theory, where game-theory, social science and recently computational approaches are all applied in order to model the incentives and behavior of voters. In a recent paper, Meir et al.[EC'14] made another step in this direction, by suggesting a behavioral game-theoretic model for voters under uncertainty. For a specific variation of best-response heuristics, they proved initial existence and convergence results in the Plurality voting system. This paper extends the model in multiple directions, considering voters with different uncertainty levels, simultaneous strategic decisions, and a more permissive notion of best-response. It is proved that a voting equilibrium exists even in the most general case. Further, any society voting in an iterative setting is guaranteed to converge to an equilibrium. An alternative behavior is analyzed, where voters try to minimize their worst-case regret. As it turns out, the two behaviors coincide in the simple setting of Meir et al.[EC'14], but not in the general case.

While voting schemes provide an effective means for aggregating preferences, methods for the effective elicitation of voter preferences have received little attention. We address this problem by first considering approximate winner determination when incomplete voter preferences are provided. Exploiting natural scoring metrics, we use max regret to measure the quality or robustness of proposed winners, and develop polynomial time algorithms for computing the alternative with minimax regret for several popular voting rules. We then show how minimax regret can be used to effectively drive incremental preference/vote elicitation and devise several heuristics for this process. Despite worst-case theoretical results showing that most voting protocols require nearly complete voter preferences to determine winners, we demonstrate the practical effectiveness of regret-based elicitation for determining both approximate and exact winners on several real-world data sets.

In some voting situations, some new candidates may show up in the course of the process. In this case, we may want to determine which of the initial candidates are possible winners, given that a fixed number k of new candidates will be added. Focusing on scoring rules, we give complexity results for the above possible winner problem.

In sports competitions, teams can manipulate the result by, for instance, throwing games. We show that we can decide how to manipulate round robin and cup competitions, two of the most popular types of sporting competitions in polynomial time. In addition, we show that finding the minimal number of games that need to be thrown to manipulate the result can also be determined in polynomial time. Finally, we show that there are several different variations of standard cup competitions where manipulation remains polynomial.