arXiv subjects

László Csató

Publications and source records attributed to László Csató.

At least 37 records · Page 2Linked to original sources

arXiv2025

Ranking matters: Does the new format select the best teams for the knockout phase in the UEFA Champions League?

Starting in the 2024/25 season, the Union of European Football Associations (UEFA) has fundamentally changed the format of its club competitions: the group stage has been replaced by a league phase played by 36 teams in an incomplete round robin format. This makes ranking the teams based on their results challenging because teams play against different sets of opponents, whose strengths vary. In this research note, we apply several well-known ranking methods for incomplete round robin tournaments to the 2024/25 UEFA Champions League league phase in order to check the robustness of the official ranking, as well as to call the attention of organizers to the non-trivial issue of ranking in these competitions. Our results show that it is doubtful whether the currently used point-based system provides the best ranking of the teams.

physics.soc-ph↗

arXiv2025

Competitive balance in the UEFA Champions League group stage: Novel measures show no evidence of decline

Competitive balance, which refers to the level of control teams have over a sports competition, is a crucial indicator for tournament organisers. According to previous studies, competitive balance has significantly declined in the UEFA Champions League group stage over the recent decades. Our paper introduces alternative indices to investigate this issue. Two ex ante measures are based on Elo ratings, and four dynamic concentration indicators compare the final group ranking to reasonable benchmarks. Using these indices, we find no evidence of any long-run trend in the competitive balance of the UEFA Champions League group stage between the 2003/04 and 2023/24 seasons.

econ.GN↗

arXiv2025

Random matching in balanced bipartite graphs: The (un)fairness of draw mechanisms used in sports

The draw of some knockout tournaments requires finding a perfect matching in a balanced bipartite graph. The problem becomes challenging with draw constraints: the two draw procedures used in sports are known to be non-uniformly distributed (the feasible matchings are not equally likely), which may threaten fairness. We compare the biases of both mechanisms, each of them having two forms, for reasonable subsets of balanced bipartite graphs up to 16 nodes. An optimal mechanism exist in the draw of quarterfinals under reasonable restrictions. The UEFA Champions League Round of 16 draw is verified to apply the least distorted design among the four available options between the 2003/04 and 2023/24 seasons. However, a considerable scope remains to improve these randomisation procedures.

physics.soc-ph↗

arXiv2025

Misaligned incentives in sports: A mathematical analysis of the post-2024 UEFA Champions League qualification

UEFA declares that it is committed to respecting the fundamental values of sports. However, the qualification rules of the post-2024 UEFA Champions League are shown to be unfair: a game with misaligned incentives was narrowly avoided in the 2023/24 German Bundesliga. We develop a mathematical model to reveal how incentives for losing can be reduced or eliminated. Since UEFA repeatedly commits the same theoretical mistake in designing the qualification system of its competitions, governing bodies in sports are called to work more closely together with the scientific community.

physics.soc-ph↗

arXiv2025

Attractiveness and equal treatment in a group draw

National teams from different continents can play against each other only in afew sports competitions. Therefore, a reasonable aim is maximising the number of intercontinental games in world cups, as done in basketball and football, in contrast to handball and volleyball. However, this objective requires additional draw constraints that imply the violation of equal treatment. In addition, the standard draw mechanism is non-uniformly distributed on the set of valid assignments, which may lead to further distortions. Our paper analyses this novel trade-off between attractiveness and fairness through the example of the 2025 World Men's Handball Championship. We introduce a measure of inequality, which enables considering 32 sets of reasonable geographical restrictions to determine the Pareto frontier. The proposed methodology can be used by policy-makers to select the optimal set of draw constraints.

math.OC↗

arXiv2024

The logarithmic least squares priorities and ordinal violations in the best-worst method

The best-worst method is an increasingly popular approach to solving multi-criteria decision-making problems. However, the usual prioritisation techniques may result in an ordinal violation if the best (worst) alternative identified in the first step does not receive the highest (lowest) weight. The current paper gives two sufficient conditions for the logarithmic least squares method, applied to an incomplete best-worst method matrix, to guarantee the lack of ordinal violations. Our results provide another powerful argument for using the logarithmic least squares priorities in the best-worst method.

math.OC↗

arXiv2024

Tournament design: A review from an operational research perspective

Every sport needs rules. Tournament design refers to the rules that determine how a tournament, a series of games between a number of competitors, is organized. This study aims to provide an overview of the tournament design literature from the perspective of operational research. Three important design criteria are discussed: efficacy, fairness, and attractiveness. Our survey classifies the papers discussing these properties according to the main components of tournament design: format, seeding, draw, scheduling, and ranking. We also outline several open questions and promising directions for future research.

physics.soc-ph↗

arXiv2024

The fairness of the group draw for the FIFA World Cup

Several sports tournaments contain a round-robin group stage where the teams are assigned to groups subject to some constraints. Hence, the organisers usually use a computer-assisted random draw to avoid any dead end, a situation when the teams still to be drawn cannot be assigned to the remaining empty slots. This procedure is known to be unfair: the feasible allocations are not equally likely, that is, the draw does not have a uniform distribution. We quantify the implied unfairness of the 2018 FIFA World Cup draw and evaluate its effect on the probability of qualification for the knockout stage for each national team. The official draw order of Pot 1, Pot 2, Pot 3, Pot 4 turns out to be a significantly better option than the 23 other draw orders with respect to the unwanted distortions. Nonetheless, the non-uniform draw distorts the probability of qualification by more than one percentage point for two countries. Our results call attention to the non-negligible role of draw order and make it possible for policymakers to decide whether using fairer draw mechanisms is justified.

physics.soc-ph↗

arXiv2024

A clustering approach for pairwise comparison matrices

We consider clustering in group decision making where the opinions are given by pairwise comparison matrices. In particular, the k-medoids model is suggested to classify the matrices since it has a linear programming problem formulation that may contain any condition on the properties of the cluster centres. Its objective function depends on the measure of dissimilarity between the matrices but not on the weights derived from them. Our methodology provides a convenient tool for decision support, for instance, it can be used to quantify the reliability of the aggregation. The proposed theoretical framework is applied to a large-scale experimental dataset, on which it is able to automatically detect some mistakes made by the decision-makers, as well as to identify a common source of inconsistency.

math.OC↗

arXiv2024

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. The suggested approach is applied to the 2022 FIFA World Cup to uncover how its design could have been improved by changing group labelling (a reform that has received no attention before) and the schedule of group matches. Scheduling is found to be a surprisingly weak intervention compared to previous results on the risk of collusion in a group. The probability of tanking, which is disturbingly high around 25\%, cannot be reduced by more than 3 percentage points via these policies. Tournament organisers need to consider more fundamental changes against tanking.

physics.soc-ph↗

arXiv2024

On head-to-head results as tie-breaker and consequent opportunities for collusion

The outcome of some football matches has benefited both teams at the expense of a third team because head-to-head results were used for breaking ties. Inspired by these examples, our mathematical analysis identifies all possible collusion opportunities caused by this particular ranking rule in a single round-robin tournament with four teams. According to a simulation model based on the 2024 UEFA European Football Championship, merely the tie-breaking rule increases the probability of reaching a situation vulnerable to collusion by between 11.5 and 14.8 percentage points. This risk can scarcely be mitigated by a static match schedule. Therefore, tournament organisers are strongly encouraged to choose goal difference as the primary tie-breaking rule, similar to the official policy of FIFA.

physics.soc-ph↗

arXiv2024

The allocation of FIFA World Cup slots based on the ranking of confederations

Qualifications for several world championships in sports are organised such that distinct sets of teams play in their own tournament for a predetermined number of slots. Inspired by a recent work studying the problem with the tools from the literature on fair allocation, this paper provides an alternative approach based on historical matches between these sets of teams. We focus on the FIFA World Cup due to the existence of an official rating system and its recent expansion to 48 teams, as well as to allow for a comparison with the already suggested allocations. Our proposal extends the methodology of the FIFA World Ranking to compare the strengths of five confederations. Various allocations are presented depending on the length of the sample, the set of teams considered, as well as the frequency of rating updates. The results show that more European and South American teams should play in the FIFA World Cup. The ranking of continents by the number of deserved slots is different from the ranking implied by FIFA policy. We recommend allocating at least some slots transparently, based on historical performances, similar to the access list of the UEFA Champions League.

econ.GN↗

arXiv2023

How to measure consumer's inconsistency in sensory testing?

Standard methods, standard test conditions and the use of good sensory practices are key elements of sensory testing. However, while compliance assessment by trained and expert assessors is well developed, few information is available on testing consumer consistency. Therefore, we aim to answer the following research questions: What type of metrics can be used to characterise (in)consistent evaluation? How to rank assessors and attributes' evaluations based on (in)consistency? Who can be considered an (in)consistent assessor? What is the difference between consistent and inconsistent assessors' evaluations and use of scale? What is the impact of consistent and inconsistent assessors' evaluations on the overall liking estimate? The proposed detection of (in)consistency requires evaluations on two connected scales. We reveal how assessors and attributes can be ranked according to the level of inconsistency, as well as how inconsistent assessors can be identified. The suggested approach is illustrated by data from sensory tests of biscuits enriched with three pollens at different levels. Future consumer tests are recommended to account for possible inconsistent evaluations.

stat.AP↗

arXiv2023

How to choose a completion method for pairwise comparison matrices with missing entries: An axiomatic result

Since there exist several completion methods to estimate the missing entries of pairwise comparison matrices, practitioners face a difficult task in choosing the best technique. Our paper contributes to this issue: we consider a special set of incomplete pairwise comparison matrices that can be represented by a weakly connected directed acyclic graph, and study whether the derived weights are consistent with the partial order implied by the underlying graph. According to previous results from the literature, two popular procedures, the incomplete eigenvector and the incomplete logarithmic least squares methods fail to satisfy the required property. Here, the recently introduced lexicographically optimal completion combined with any of these weighting methods is shown to avoid ordinal violation in the above setting. Our finding provides a powerful argument for using the lexicographically optimal completion to determine the missing elements in an incomplete pairwise comparison matrix.

math.OC↗

arXiv2023

Club coefficients in the UEFA Champions League: Time for shift to an Elo-based formula

One of the most popular club football tournaments, the UEFA Champions League, will see a fundamental reform from the 2024/25 season: the traditional group stage will be replaced by one league where each of the 36 teams plays eight matches. To guarantee that the opponents of the clubs are of the same strength in the new design, it is crucial to forecast the performance of the teams before the tournament as well as possible. This paper investigates whether the currently used rating of the teams, the UEFA club coefficient, can be improved by taking the games played in the national leagues into account. According to our logistic regression models, a variant of the Elo method provides a higher accuracy in terms of explanatory power in the Champions League matches. The Union of European Football Associations (UEFA) is encouraged to follow the example of the FIFA World Ranking and reform the calculation of the club coefficients in order to avoid unbalanced schedules in the novel tournament format of the Champions League.

stat.AP↗

arXiv2023

Tournament schedules and incentives in a double round-robin tournament with four teams

In a round-robin tournament, a team may lack the incentive to win if its final rank does not depend on the outcome of the matches still to be played. This paper introduces a classification scheme to determine these weakly (where one team is indifferent) or strongly (where both teams are indifferent) stakeless matches in a double round-robin contest with four teams. The probability that such matches arise can serve as a novel fairness criterion to compare and evaluate match schedules. Our approach is illustrated by the UEFA Champions League group stage. A simulation model is built to compare the 12 valid schedules for the group matches. Some schedules are shown to be dominated by other schedules. It is found that the strongest team should play at home in the last round against one of the middle teams, depending on the preferences of the tournament organiser. Choosing an optimal sequence of matches with respect to the proposed metric can help to avoid uninteresting matches.

math.OC↗

arXiv2023

Right-left asymmetry of the eigenvector method: A simulation study

The eigenvalue method, suggested by the developer of the extensively used Analytic Hierarchy Process methodology, exhibits right-left asymmetry: the priorities derived from the right eigenvector do not necessarily coincide with the priorities derived from the reciprocal left eigenvector. This paper offers a comprehensive numerical experiment to compare the two eigenvector-based weighting procedures and their reasonable alternative of the row geometric mean with respect to four measures. The underlying pairwise comparison matrices are constructed randomly with different dimensions and levels of inconsistency. The disagreement between the two eigenvectors turns out to be not always a monotonic function of these important characteristics of the matrix. The ranking contradictions can affect alternatives with relatively distant priorities. The row geometric mean is found to be almost at the midpoint between the right and inverse left eigenvectors, making it a straightforward compromise between them.

math.OC↗

arXiv2023

Bibliometric indices as a measure of performance and competitive balance in the knockout stage of the UEFA Champions League

We argue for the application of bibliometric indices to quantify the long-term uncertainty of outcome in sports. The Euclidean index is proposed to reward quality over quantity, while the rectangle index can be an appropriate measure of core performance. Their differences are highlighted through an axiomatic analysis and several examples. Our approach also requires a weighting scheme to compare different achievements. The methodology is illustrated by studying the knockout stage of the UEFA Champions League in the 20 seasons played between 2003 and 2023: club and country performances as well as three types of competitive balance are considered. Measuring competition at the level of national associations is a novelty. All results are remarkably robust concerning the bibliometric index and the assigned weights. Since the performances of national associations are more stable than the results of individual clubs, it would be better to build the seeding in the UEFA Champions League group stage upon association coefficients adjusted for league finishing positions rather than club coefficients.

stat.AP↗