arXiv subjects

László Csató

Publications and source records attributed to László Csató.

At least 55 records · Page 3Linked to original sources

arXiv2023

On the coincidence of optimal completions for small pairwise comparison matrices with missing entries

Incomplete pairwise comparison matrices contain some missing judgements. A natural approach to estimate these values is provided by minimising a reasonable measure of inconsistency after unknown entries are replaced by variables. Two widely used inconsistency indices for this purpose are Saaty's inconsistency index and the geometric inconsistency index, which are closely related to the eigenvector and the logarithmic least squares priority deriving methods, respectively. The two measures are proven to imply the same optimal filling for incomplete pairwise comparison matrices up to order four but not necessarily for order at least five.

math.OC↗

arXiv2023

A lexicographically optimal completion for pairwise comparison matrices with missing entries

Estimating missing judgements is a key component in many multi-criteria decision making techniques, especially in the Analytic Hierarchy Process. Inspired by the Koczkodaj inconsistency index and a widely used solution concept of cooperative game theory called the nucleolus, the current study proposes a new algorithm for this purpose. In particular, the missing values are substituted by variables, and the inconsistency of the most inconsistent triad is reduced first, followed by the inconsistency of the second most inconsistent triad, and so on. The necessary and sufficient condition for the uniqueness of the suggested lexicographically optimal completion is proved to be a simple graph-theoretic notion: the undirected graph associated with the pairwise comparisons, where the edges represent the known elements, should be connected. Crucially, our method does not depend on an arbitrarily chosen measure of inconsistency as there exists essentially one reasonable triad inconsistency index.

math.OC↗

arXiv2023

A note on the UEFA Champions League Round of 16 draw

A paper published in a leading journal of management science has recently analysed the mechanism used for the UEFA Champions League Round of 16 draw. Since the authors have not been familiar with the related sports analytics literature, this note attempts to summarise what has already been done, and what can still be done on the issue of group draw in sports tournaments.

physics.soc-ph↗

arXiv2022

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 might 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.

physics.soc-ph↗

arXiv2022

How to avoid uncompetitive games? The importance of tie-breaking rules

If the final position of a team is already secured independently of the outcomes of the remaining games in a round-robin tournament, it might play with little enthusiasm. This is detrimental to attendance and can inspire collusion and match-fixing. We demonstrate that tie-breaking rules might affect the occurrence of such a situation. Its probability is quantified via simulations for the four groups of the 2022/23 UEFA Nations League A under two well-established tie-breaking rules, goal difference and head-to-head records. In these home-away round-robin contests with four teams and 12 matches, the competitiveness of the final four games can be promoted by giving priority to goal difference, which reduces the chance of a fixed position in the group ranking by at least two and usually five percentage points in the last round. Our findings, supported by sensitivity analysis in a theoretical model, provide important lessons on how to design ranking systems.

physics.soc-ph↗

arXiv2022

A paradox of tournament seeding

We analyse a mathematical model of seeding for sports contests with round-robin qualifying tournaments. The standard seeding system based on coefficients measuring the historical performance of the teams is shown to be unfair as it might potentially punish a team for its better results by having to face stronger opponents on average in the next stage. Major football competitions are revealed to suffer from this weakness. Incentive compatibility can be guaranteed by providing each qualified team with the highest coefficient of all teams that are ranked lower in its qualifying tournament for seeding purposes. Our proposal is illustrated by the 2020/21 UEFA Champions League.

physics.soc-ph↗

arXiv2022

The unfairness of the revenue distribution system used in the UEFA club competitions

The Union of European Football Associations (UEFA) distributes commercial revenue from its club competitions according to a complex mechanism. Coefficient-based amounts are paid such that the participating teams are ranked on the basis of performances over the last 10 years. Due to the non-anonymity of this allocation rule, all clubs are interested in qualification together with lower-ranked teams, which makes the procedure incentive incompatible. In particular, the English club Arsenal is found to lose 132 thousand Euros prize money in the 2022/23 UEFA Europa League because of exerting full effort in its domestic championship. Two strategy-proof alternatives are proposed to divide fairly this pillar of the revenue distribution system.

physics.soc-ph↗

arXiv2022

A comparative study of scoring systems by simulations

Scoring rules aggregate individual rankings by assigning some points to each position in each ranking such that the total sum of points provides the overall ranking of the alternatives. They are widely used in sports competitions consisting of multiple contests. We study the tradeoff between two risks in this setting: (1) the threat of early clinch when the title has been clinched before the last contest(s) of the competition take place; (2) the danger of winning the competition without finishing first in any contest. In particular, four historical points scoring systems of the Formula One World Championship are compared with the family of geometric scoring rules, recently proposed by an axiomatic approach. The schemes used in practice are found to be competitive with respect to these goals, and the current rule seems to be a reasonable compromise close to the Pareto frontier. Our results shed more light on the evolution of the Formula One points scoring systems and contribute to the issue of choosing the set of point values.

stat.OT↗

arXiv2022

Group draw with unknown qualified teams: A lesson from the 2022 FIFA World Cup draw

The draw for the 2022 FIFA World Cup has been organised before the identity of three winners of the play-offs is revealed. Seeding has been based on the FIFA World Ranking released on 31 March 2022 but these three teams have been drawn from the weakest Pot 4. We show that the official seeding policy does not balance the difficulty levels of the groups to the extent possible: a better alternative would have been to assign the placeholders according to the highest-ranked potential winner, similar to the rule used in the UEFA Champions League qualification. Our simulations reinforce that this is the best strategy in general to create balanced groups in the FIFA World Cup.

physics.soc-ph↗

arXiv2022

The efficacy of tournament designs

Tournaments are a widely used mechanism to rank alternatives in a noisy environment. This paper investigates a fundamental issue of economics in tournament design: what is the best usage of limited resources, that is, how should the alternatives be compared pairwise to best approximate their true but latent ranking. We consider various formats including knockout tournaments, multi-stage championships consisting of round-robin groups followed by single elimination, and the Swiss-system. They are evaluated via Monte-Carlo simulations under six different assumptions on winning probabilities. Comparing the same pair of alternatives multiple times turns out to be an inefficacious policy. While seeding can increase the efficacy of the knockout and group-based designs, its influence remains marginal unless one has an unrealistically good estimation on the true ranking of the players. The Swiss-system is found to be the most accurate among all these tournament formats, especially in its ability to rank all participants. A possible explanation is that it does not eliminate a player after a single loss, while it takes the history of the comparisons into account. The results can be especially interesting for emerging esports, where the tournament designs are not yet solidified.

stat.AP↗

arXiv2022

Quantifying incentive (in)compatibility: A case study from sports

For every sports tournament, it is an important requirement to provide contestants with the appropriate incentives to perform. However, incentive compatibility is usually considered an all or nothing (binary) concept in the academic literature, that is, the rules are proved to be either strategy-proof or vulnerable to manipulation. Our paper aims to present a method for quantifying the violation of this theoretical property through the example of the European Qualifiers for the 2022 FIFA World Cup. Even though that competition is known to be incentive incompatible since lower-ranked Nations League group winners are interested in the success of their higher-ranked peers, the extent of the problem has remained unexplored until now. Computer simulations reveal that the threat of tanking can be substantially mitigated by adding a carefully chosen set of draw restrictions, which offers a justifiable and transparent solution to improve fairness. Sports governing bodies are encouraged to take our findings into account.

physics.soc-ph↗

arXiv2022

Fairness in penalty shootouts: Is it worth using dynamic sequences?

The sequence of moves in a dynamic team tournament may distort the ex-ante winning probabilities and harm efficiency. This paper compares seven soccer penalty shootout rules that determine the kicking order, from a theoretical perspective. Their fairness is evaluated in a reasonable model of First Mover Advantage. We also discuss the probability of reaching the sudden death stage. In the case of stationary scoring probabilities, dynamic mechanisms are not better than static rules. However, it is worth compensating the second-mover by making it the first-mover in the sudden death stage. Our work has the potential to impact decision-makers who can guarantee fairer outcomes in dynamic tournaments by a carefully chosen sequence of actions.

cs.GT↗

arXiv2022

Quantifying the unfairness of the 2018 FIFA World Cup qualification

This paper investigates the fairness of the 2018 FIFA World Cup qualifying competition via Monte Carlo simulations. The qualifying probabilities are calculated for 102 nations, all teams except for African and European countries. A method is proposed to quantify the degree of unfairness. Although the qualifications within four FIFA confederations are constructed fairly, serious differences are found between the continents: for instance, a South American team could have tripled its chances by playing in Asia. Choosing a fixed matchup in the inter-continental play-offs instead of the current random draw can reduce the unfairness of the competition. The move of Australia from the Oceanian to the Asian zone is shown to increase its probability of participating in the 2018 FIFA World Cup by about 65%. Our results provide important insights for the administrators on how to reallocate the qualifying berths.

physics.soc-ph↗

arXiv2021

UEFA against the champions? An evaluation of the recent reform of the Champions League qualification

The UEFA Champions League is the major European club football competition, however, most national champions have to play qualification matches to receive a slot in the group stage of the tournament. The paper evaluates the impact of the only reform in the Champions Path of the qualifying system since 2009, effective from the 2018/19 season. While it is anticipated that the reduction in the number of berths decreases the probability of advancing to the group stage, the distribution of the effects among the national associations is revealed here via Monte-Carlo simulations. In contrast to previous studies, our methodology considers five seasons instead of only one to filter out any possible season-specific attributes. Almost all countries are found to gain less prize money on average. Several champions, including the Bulgarian, the Scottish, and, especially, the Swiss, might lose million Euros. As the consequences depend to a large extent on the arbitrary cutoffs in the access list, we propose to introduce some randomness into the determination of entry stages in the qualification.

stat.AP↗

arXiv2021

How to design a multi-stage tournament when some results are carried over?

The paper discusses the strategy-proofness of sports tournaments with multiple group stages, where the results of matches already played in the previous round against teams in the same group are carried over. These tournaments, widely used in handball and other sports, are shown to be incentive incompatible in the sense that a team can be strictly better off by not exerting full effort in a game. Historical examples are presented when a team was ex ante disinterested in winning by a high margin. We propose a family of incentive compatible designs. Their main characteristics are compared to the original format via simulations. Carrying over half of the points scored in the previous round turns out to be a promising policy.

physics.soc-ph↗

arXiv2021

Inconsistency thresholds for incomplete pairwise comparison matrices

Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices for similar incomplete data sets and no reasonable measure has an associated threshold. This paper generalises the famous rule of thumb for the acceptable level of inconsistency, proposed by Saaty, to incomplete pairwise comparison matrices. The extension is based on choosing the missing elements such that the maximal eigenvalue of the incomplete matrix is minimised. Consequently, the well-established values of the random index cannot be adopted: the inconsistency of random matrices is found to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly built into decision-making software and used by practitioners as a statistical criterion for accepting or rejecting an incomplete pairwise comparison matrix.

math.ST↗

arXiv2021

On the role of tournament design in sporting success: A study of the North, Central American and Caribbean qualification for the 2022 FIFA World Cup

Playing in the FIFA World Cup finals is an ambition shared by several nations. Since, besides luck and skill, the probability of qualification depends on the design of the qualifiers, the study of these competitions forms an integral part of sports analytics. The Confederation of North, Central America and Caribbean Association Football (CONCACAF) announced a novel qualifying format for the 2022 FIFA World Cup in July 2019. However, the COVID-19 pandemic forced the organisers to return to a more traditional structure. The present chapter analyses how this reform impacted the chances of the national teams to qualify. It is found that the probability of participating in the FIFA World Cup finals can change by more than 5 percentage points under the assumption of fixed strengths for the teams. The idea behind the original design, to divide the contestants into two distinct sets, is worth considering due to the increased competitiveness of the matches played by the strongest and the weakest teams. We recommend mitigating the sharp nonlinearity caused by the seeding policy via a probabilistic rule to the analogy of the NBA draft lottery system.

physics.soc-ph↗

arXiv2021

Coronavirus and sports leagues: obtaining a fair ranking when the season cannot resume

Many sports leagues are played in a tightly scheduled round-robin format, leaving a limited time window to postpone matches. If the season cannot resume due to an external shock such as the coronavirus pandemic in 2020, the ranking of the teams becomes non-trivial: it is necessary to account for schedule imbalances and possibly for the different number of matches played. First in the literature, we identify a set of desired axioms for ranking in these incomplete tournaments. It is verified that the generalised row sum, a parametric family of scoring rules, satisfies all of them. In particular, the well-established least squares method maximizes the influence of the strength of opponents on the ranking. Our approach is applied for six major premier European soccer competitions, where the rankings are found to be robust concerning the weight of adjustment for the strength of the opponents. Some methodologically simpler alternative policies are also discussed.

stat.AP↗