77 research outputs found
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
A theory of knockout tournament seedings
This paper provides nested sets and vector representations of knockout tournaments. The paper introduces classification of probability domain assumptions and a new set of axioms. Two new seeding methods are proposed: equal gap seeding and increasing competitive intensity seeding. Under different probability domain assumptions, several axiomatic justifications are obtained for equal gap seeding. A discrete optimization approach is developed. It is applied to justify equal gap seeding and increasing competitive intensity seeding. Some justification for standard seeding is obtained. Combinatorial properties of the seedings are studied
Weak transitivity and agenda control for extended stepladder tournaments
A tournament graph over n players is weakly transitive at player p if it contains a Hamiltonian path (p1,p2,…,pn) with p1=p such that for all odd integers i≤n−2 there is an arc from pi to pi+2. We show that weak transitivity at p suffices to make player p win any extended stepladder tournament of degree at most two
How to Design a Stable Serial Knockout Competition
We investigate a new tournament format that consists of a series of
individual knockout tournaments; we call this new format a Serial Knockout
Competition (SKC). This format has recently been adopted by the Professional
Darts Corporation. Depending on the seedings of the players used for each of
the knockout tournaments, players can meet in the various rounds (eg first
round, second round, ..., semi-final, final) of the knockout tournaments.
Following a fairness principle of treating all players equal, we identify an
attractive property of an SKC: each pair of players should potentially meet
equally often in each of the rounds of the SKC. If the seedings are such that
this property is indeed present, we call the resulting SKC stable. In this note
we formalize this notion, and we address the question: do there exist seedings
for each of the knockout tournaments such that the resulting SKC is stable? We
show, using a connection to the Fano plane, that the answer is yes for 8
players. We show how to generalize this to any number of players that is a
power of 2, and we provide stable schedules for competitions on 16 and 32
player
A paradox of tournament seeding
A mathematical model of seeding is analysed for sports tournaments where the
qualification is based on round-robin contests. The conditions of
strategyproofness are found to be quite restrictive: if each team takes its own
coefficient (a measure of its past performance), only one or all of them should
qualify from every round-robin contest. Thus the standard draw system creates
incentives for tanking in order to be assigned to a stronger pot as each team
prefers to qualify with teams having a lower coefficient. Major soccer
competitions are shown to suffer from this weakness. Strategyproofness can be
guaranteed by giving to each team the highest coefficient of all teams that are
ranked lower in its round-robin contest. The proposal is illustrated by the
2020/21 UEFA Champions League.Comment: 23 pages, 3 table
Selecting the Best? Spillover and Shadows in Elimination Tournaments
We consider how past, current, and future competition within an elimination tournament affect the probability that the stronger player wins. We present a two-stage model that yields the following main results: (1) a shadow effect—the stronger the expected future competitor, the lower the probability that the stronger player wins in the current stage and (2) an effort spillover effect—previous effort reduces the probability that the stronger player wins in the current stage. We test our theory predictions using data from high-stakes tournaments. Empirical results suggest that shadow and spillover effects influence match outcomes and have been already been priced into betting markets.
Balancing the Game: Comparative Analysis of Single Heuristics and Adaptive Heuristic Approaches for Sports Scheduling Problem
Sport timetabling problems are Combinatorial Optimization problems which involve the creation of schedules that determine when and where teams compete against each other. One specific type of sports scheduling, the double round-robin (2RR) tournament, mandates that each team faces every other team twice, once at their home venue and once at the opponent’s. Despite the relatively small number of teams involved, the sheer volume of potential scheduling combinations has spurred researchers to employ various techniques to find efficient solutions for sports scheduling problems. In this thesis, we present a comparative analysis of single and adaptive heuristics designed to efficiently solve sports scheduling problems. Specifically, our focus is on constructing time-constrained double round-robin tournaments involving 16 to 20 teams, while adhering to hard constraints and minimizing penalties for soft constraints violations. The computational results demonstrate that our adaptive heuristic approach not only successfully finds feasible solutions for the majority of instances but also outperforms the single heuristics examined in this study.Master's Thesis in InformaticsINF399MAMN-INFMAMN-PRO
Rank-Order Tournaments as Optimum Labor Contracts
This paper analyzes compensation schemes which pay according to an individual's ordinal rank in an organization rather than his output level. When workers are risk neutral, it is shown that wages based upon rank induce the same efficient allocation of resources as an incentive reward scheme based on individual output levels. Under some circumstances, risk-averse workers actually prefer to be paid on the basis of rank. In addition, if workers are heterogeneous inability, low-quality workers attempt to contaminate high-quality firms, resulting in adverse selection. However, if ability is known in advance, a competitive handicapping structure exists which allows all workers to compete efficiently in the same organization.
Making the Rules of Sports Fairer
The rules of many sports are not fair-they do not ensure that equally skilled competitors have the same probability of winning. As an example, the penalty shootout in soccer, wherein a coin toss determines which team kicks first on all five penalty kicks, gives a substantial advantage to the first-kicking team, both in theory and in practice. We show that a so-called Catch-Up Rule for determining the order of kicking would not only make the shootout fairer but is also essentially strategyproof. By contrast, the so-called Standard Rule now used for the tiebreaker in tennis is fair. We briefly consider several other sports, all of which involve scoring a sufficient number of points to win, and show how they could benefit from certain rule changes which would be straightforward to implement
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