Fair draws for group rounds in sport tournaments

We propose two draw systems for the group round of sport tournaments where there are some geographical and/or seeding restrictions. One of the systems, related to the equal-sum partition problem, is "perfect, " since it yields perfectly balanced groups. The other system, which uses the classical scheme of extracting teams from pots, is heuristic and gives results where the groups have very similar scores. We apply our results to Federation Internationale de Football Association (FIFA) Soccer World Cups and show that our proposals are much better than the FIFA system and also outperform other recently developed systems

The "No Justice in the Universe" phenomenon: why honesty of effort may not be rewarded in tournaments

In 2000 Allen Schwenk, using a well-known mathematical model of matchplay tournaments in which the probability of one player beating another in a single match is fixed for each pair of players, showed that the classical single-elimination, seeded format can be "unfair" in the sense that situations can arise where an indisputibly better (and thus higher seeded) player may have a smaller probability of winning the tournament than a worse one. This in turn implies that, if the players are able to influence their seeding in some preliminary competition, situations can arise where it is in a player's interest to behave "dishonestly", by deliberately trying to lose a match. This motivated us to ask whether it is possible for a tournament to be both honest, meaning that it is impossible for a situation to arise where a rational player throws a match, and "symmetric" - meaning basically that the rules treat everyone the same - yet unfair, in the sense that an objectively better player has a smaller probability of winning than a worse one. After rigorously defining our terms, our main result is that such tournaments exist and we construct explicit examples for any number n >= 3 of players. For n=3, we show (Theorem 3.6) that the collection of win-probability vectors for such tournaments form a 5-vertex convex polygon in R^3, minus some boundary points. We conjecture a similar result for any n >= 4 and prove some partial results towards it.Comment: 26 pages, 2 figure

Principles of Stakes Fairness in Sport

Fairness in sport is not just about assigning the top prizes to the worthiest competitors. It is also about the way the prize structure itself is organised. For many sporting competitions, although it may be acceptable for winners to receive more than losers, it can seem unfair for winners to take everything and for losers to get nothing. Yet this insight leaves unanswered some difficult questions about what stakes fairness requires and which principles of stakes fairness are appropriate for particular competitions. In this article I specify a range of different principles of stakes fairness (ten in total) that could regulate sporting competitions. I also put forward a theoretical method for pairing up appropriate principles of stakes fairness with given sporting competitions. Specifically, I argue that the underlying rationales for holding sporting competitions can provide useful guides for identifying appropriate principles of stakes fairness. I then seek to clarify and work through some of the implications of this method for a sample of real world controversies over sporting prize structures. I also attempt to refine the method in response to two possible objections from indeterminacy and relativism. Finally, I compare and contrast my conclusions with more general philosophical debates about justice

Comparing league formats with respect to match importance in Belgian football

Recently, most clubs in the highest Belgian football division have become convinced that the format of their league should be changed. Moreover, the TV station that broadcasts the league is pleading for a more attractive competition. However, the clubs have not been able to agree on a new league format, mainly because they have conflicting interests. In this paper, we compare the current league format, and three other formats that have been considered by the Royal Belgian Football Association. We simulate the course of each of these league formats, based on historical match results. We assume that the attractiveness of a format is determined by the importance of its games; our importance measure for a game is based on the number of teams for which this game can be decisive to reach a given goal. Furthermore, we provide an overview of how each league format aligns with the expectations and interests of each type of club

Mitigating the risk of tanking in multi-stage tournaments

Multi-stage tournaments consisting of a round-robin group stage followed by a knockout phase are ubiquitous in sports. However, this format is incentive incompatible if at least two teams from a group advance to the knockout stage where the brackets are predetermined. A model is developed to quantify the risk of tanking in these contests. Our approach is applied to the 2022 FIFA World Cup in order to uncover how its design could have been improved by changing the group labelling policy (a reform that has received no attention before) and the schedule of group matches. The proposed interventions can halve the level of unfairness without changing any fundamental aspect of the tournament.Comment: 16 pages, 2 figures, 4 table

Stop Simulating! Efficient Computation of Tournament Winning Probabilities

In the run-up to any major sports tournament, winning probabilities of participants are publicized for engagement and betting purposes. These are generally based on simulating the tournament tens of thousands of times by sampling from single-match outcome models. We show that, by virtue of the tournament schedule, exact computation of winning probabilties can be substantially faster than their approximation through simulation. This notably applies to the 2022 and 2023 FIFA World Cup Finals, and is independent of the model used for individual match outcomes.Comment: Working paper; first draft published prior to WWC202

The effects of draw restrictions on knockout tournaments

The paper analyses how draw constraints influence the outcome of a knockout tournament. The research question is inspired by European club football competitions, where the organiser generally imposes an association constraint in the first round of the knockout phase: teams from the same country cannot be drawn against each other. Its effects are explored in both theoretical and simulation models. An association constraint in the first round(s) is found to increase the likelihood of same nation matchups to approximately the same extent in each subsequent round. If the favourite teams are concentrated in some associations, they will have a higher probability to win the tournament under this policy but the increase is less than linear if it is used in more rounds. Our results can explain the recent introduction of the association constraint for both the knockout round play-offs with 16 teams and the Round of 16 in the UEFA Europa League and UEFA Europa Conference League.Comment: 18 pages, 5 figures, 4 table