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

Robust Draws in Balanced Knockout Tournaments

Balanced knockout tournaments are ubiquitous in sports competitions and are also used in decision-making and elections. The traditional computational question, that asks to compute a draw (optimal draw) that maximizes the winning probability for a distinguished player, has received a lot of attention. Previous works consider the problem where the pairwise winning probabilities are known precisely, while we study how robust is the winning probability with respect to small errors in the pairwise winning probabilities. First, we present several illuminating examples to establish: (a)~there exist deterministic tournaments (where the pairwise winning probabilities are~0 or~1) where one optimal draw is much more robust than the other; and (b)~in general, there exist tournaments with slightly suboptimal draws that are more robust than all the optimal draws. The above examples motivate the study of the computational problem of robust draws that guarantee a specified winning probability. Second, we present a polynomial-time algorithm for approximating the robustness of a draw for sufficiently small errors in pairwise winning probabilities, and obtain that the stated computational problem is NP-complete. We also show that two natural cases of deterministic tournaments where the optimal draw could be computed in polynomial time also admit polynomial-time algorithms to compute robust optimal draws

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

Single-Elimination Brackets Fail to Approximate Copeland Winner

Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature

Approximately Strategyproof Tournament Rules: On Large Manipulating Sets and Cover-Consistence

We consider the manipulability of tournament rules, in which n teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all binom{n}{2} matches. Prior work defines a tournament rule to be k-SNM-? if no set of ? k teams can fix the ? binom{k}{2} matches among them to increase their probability of winning by >? and asks: for each k, what is the minimum ?(k) such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) k-SNM-?(k) tournament rule exists? A simple example witnesses that ?(k) ? (k-1)/(2k-1) for all k, and [Jon Schneider et al., 2017] conjectures that this is tight (and prove it is tight for k=2). Our first result refutes this conjecture: there exists a sufficiently large k such that no Condorcet-consistent tournament rule is k-SNM-1/2. Our second result leverages similar machinery to design a new tournament rule which is k-SNM-2/3 for all k (and this is the first tournament rule which is k-SNM-(<1) for all k). Our final result extends prior work, which proves that single-elimination bracket with random seeding is 2-SNM-1/3 [Jon Schneider et al., 2017], in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is 2-SNM-1/3 and cover-consistent (the winner is an uncovered team with probability 1)

May a weak tennis player win?

The winner of a competition depends on the choice of actual matches played. We assume that each match is played between two players. Our goal is to examine which players can be made winners of a competition if we know any match result in advance. We only consider competitions in which the winner of a single match progresses to the next round and the loser leaves the competition. We focus on tennis competitions and use real data downloaded from atpworldtour.com. The final winner of a competition depends on the choice of matches in the first round — we call it a bracket. We would like to determine possible competition winners and for every winner π construct an appropriate bracket in which π is the winner. Apart from that we also study how tight are the sufficient conditions for a player to become a winner, as described in the paper Fixing a Tournament (Williams, AAAI 2010). For instance, one of our results is that a player whose relative rank is between 1 and 36 can with high probability be made a winner in a competition of 64 players

May a weak tennis player win?

The winner of a competition depends on the choice of actual matches played. We assume that each match is played between two players. Our goal is to examine which players can be made winners of a competition if we know any match result in advance. We only consider competitions in which the winner of a single match progresses to the next round and the loser leaves the competition. We focus on tennis competitions and use real data downloaded from atpworldtour.com. The final winner of a competition depends on the choice of matches in the first round — we call it a bracket. We would like to determine possible competition winners and for every winner π construct an appropriate bracket in which π is the winner. Apart from that we also study how tight are the sufficient conditions for a player to become a winner, as described in the paper Fixing a Tournament (Williams, AAAI 2010). For instance, one of our results is that a player whose relative rank is between 1 and 36 can with high probability be made a winner in a competition of 64 players

May a weak tennis player win?

The winner of a competition depends on the choice of actual matches played. We assume that each match is played between two players. Our goal is to examine which players can be made winners of a competition if we know any match result in advance. We only consider competitions in which the winner of a single match progresses to the next round and the loser leaves the competition. We focus on tennis competitions and use real data downloaded from atpworldtour.com. The final winner of a competition depends on the choice of matches in the first round — we call it a bracket. We would like to determine possible competition winners and for every winner π construct an appropriate bracket in which π is the winner. Apart from that we also study how tight are the sufficient conditions for a player to become a winner, as described in the paper Fixing a Tournament (Williams, AAAI 2010). For instance, one of our results is that a player whose relative rank is between 1 and 36 can with high probability be made a winner in a competition of 64 players