Empirical Evaluation of Real World Tournaments

Computational Social Choice (ComSoc) is a rapidly developing field at the intersection of computer science, economics, social choice, and political science. The study of tournaments is fundamental to ComSoc and many results have been published about tournament solution sets and reasoning in tournaments. Theoretical results in ComSoc tend to be worst case and tell us little about performance in practice. To this end we detail some experiments on tournaments using real wold data from soccer and tennis. We make three main contributions to the understanding of tournaments using real world data from English Premier League, the German Bundesliga, and the ATP World Tour: (1) we find that the NP-hard question of finding a seeding for which a given team can win a tournament is easily solvable in real world instances, (2) using detailed and principled methodology from statistical physics we show that our real world data obeys a log-normal distribution; and (3) leveraging our log-normal distribution result and using robust statistical methods, we show that the popular Condorcet Random (CR) tournament model does not generate realistic tournament data.Comment: 2 Figure

How Hard Is It to Control an Election by Breaking Ties?

We study the computational complexity of controlling the result of an election by breaking ties strategically. This problem is equivalent to the problem of deciding the winner of an election under parallel universes tie-breaking. When the chair of the election is only asked to break ties to choose between one of the co-winners, the problem is trivially easy. However, in multi-round elections, we prove that it can be NP-hard for the chair to compute how to break ties to ensure a given result. Additionally, we show that the form of the tie-breaking function can increase the opportunities for control. Indeed, we prove that it can be NP-hard to control an election by breaking ties even with a two-stage voting rule.Comment: Revised and expanded version including longer proofs and additional result

Who Can Win a Single-Elimination Tournament?

A single-elimination (SE) tournament is a popular way to select a winner in both sports competitions and in elections. A natural and well-studied question is the tournament fixing problem (TFP): given the set of all pairwise match outcomes, can a tournament organizer rig an SE tournament by adjusting the initial seeding so that their favorite player wins? We prove new sufficient conditions on the pairwise match outcome information and the favorite player, under which there is guaranteed to be a seeding where the player wins the tournament. Our results greatly generalize previous results. We also investigate the relationship between the set of players that can win an SE tournament under some seeding (so called SE winners) and other traditional tournament solutions. In addition, we generalize and strengthen prior work on probabilistic models for generating tournaments. For instance, we show that \emph{every} player in an $n$ player tournament generated by the Condorcet Random Model will be an SE winner even when the noise is as small as possible, $p=\Theta(\ln n/n)$; prior work only had such results for $p\geq \Omega(\sqrt{\ln n/n})$. We also establish new results for significantly more general generative models.Comment: A preliminary version appeared in Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI), 201

Election with Bribed Voter Uncertainty: Hardness and Approximation Algorithm

Bribery in election (or computational social choice in general) is an important problem that has received a considerable amount of attention. In the classic bribery problem, the briber (or attacker) bribes some voters in attempting to make the briber's designated candidate win an election. In this paper, we introduce a novel variant of the bribery problem, "Election with Bribed Voter Uncertainty" or BVU for short, accommodating the uncertainty that the vote of a bribed voter may or may not be counted. This uncertainty occurs either because a bribed voter may not cast its vote in fear of being caught, or because a bribed voter is indeed caught and therefore its vote is discarded. As a first step towards ultimately understanding and addressing this important problem, we show that it does not admit any multiplicative $O(1)$-approximation algorithm modulo standard complexity assumptions. We further show that there is an approximation algorithm that returns a solution with an additive-$\epsilon$ error in FPT time for any fixed $\epsilon$.Comment: Accepted at AAAI 201

How Hard is Bribery in Elections with Randomly Selected Voters

Many research works in computational social choice assume a fixed set of voters in an election and study the resistance of different voting rules against electoral manipulation. In recent years, however, a new technique known as random sample voting has been adopted in many multi-agent systems. One of the most prominent examples is blockchain. Many proof-of-stake based blockchain systems like Algorand will randomly select a subset of participants of the system to form a committee, and only the committee members will be involved in the decision of some important system parameters. This can be viewed as running an election where the voter committee (i.e., the voters whose votes will be counted) is randomly selected. It is generally expected that the introduction of such randomness should make the election more resistant to electoral manipulation, despite the lack of theoretical analysis. In this paper, we present a systematic study on the resistance of an election with a randomly selected voter committee against bribery. Since the committee is randomly generated, by bribing any fixed subset of voters, the designated candidate may or may not win. Consequently, we consider the problem of finding a feasible solution that maximizes the winning probability of the designated candidate. We show that for most voting rules, this problem becomes extremely difficult for the briber as even finding any non-trivial solution with non-zero objective value becomes NP-hard. However, for plurality and veto, there exists a polynomial time approximation scheme that computes a near-optimal solution efficiently. The algorithm builds upon a novel integer programming formulation together with techniques from n-fold integer programming, which may be of a separate interest

The number of parties and decision-making in legislatures

This paper proposes a model of a legislature, formed by several parties, which has to vote for or against a certain bill in the presence of a lobbyist interested in a certain vote outcome. We show that the ease with which the lobbyist can manipulate a legislature decision increases with the number of elected parties, and, consequently, decreases with an electoral threshold. On the other hand, a lower electoral threshold increases the representativeness of a legislature. We combine these two effects in a notion of fairness. We show the existence of an electoral threshold that optimizes the fairness of a political system, which is close to 1–5%. Namely, the optimal threshold (in our sense) is close to thresholds that exist in most parliamentary democracies. © 2021 by the authors. Licensee MDPI, Basel, Switzerland