Tournament Designs Research Articles

Psychoacoustic ranking and selection using modified knockout tournaments.

This paper introduces a ranking and selection approach to psychoacoustic and psychophysical experimentation, with the aim of identifying top-ranking samples in listening experiments with minimal pairwise comparisons. We draw inspiration from sports tournament designs and propose to adopt modified knockout (KO) tournaments. Two variants of modified KO tournaments are described, which adapt the tree selection sorting algorithm and the replacement selection algorithm known from computer science. To validate the proposed method, a listening experiment is conducted, where binaural renderings of seven chamber music halls are compared regarding loudness and reverberance. The rankings obtained by the modified KO tournament method are compared to those obtained from a traditional round-robin (RR) design, where all possible pairs are compared. Moreover, the paper presents simulations to illustrate the method's robustness when choosing different parameters and assuming different underlying data distributions. The study's findings demonstrate that modified KO tournaments are more efficient than full RR designs in terms of the number of comparisons required for identifying the top ranking samples. Thus, they provide a promising alternative for this task. We offer an open-source implementation so that researchers can easily integrate KO designs into their studies.

Effort and risk-taking in tournaments with superstars – evidence for teams

ABSTRACT Competition is often observed between heterogenous teams. Heterogeneity in ability is critical for the effectiveness of different tournament designs. I investigate how the presence of a dominant competitor or superstar does affect the decisions of other competitors in team contests. In particular, I use novel data from top-level professional basketball to investigate the consequences of superstar presence on effort and risk-taking of teams in rank-order contests. To identify the effect, I exploit exogenous variation in superstar dominance, induced by an institutional reform. My results demonstrate that the effect of competing with a superstar on effort does depend on the level of superstar dominance and the number of dominant teams in the contest. For tournaments with a single and clearly dominant superstar team, I find a sizable and significant negative effect of superstar presence on effort of other competing teams. Reduced dominance by the superstar, however, is found to result in a positive peer effect. I also present results for a robust association between competing with a superstar and risk-taking, indicating that a dominant superstar decreases risk-taking. Presence of a weak superstar, however, is estimated to increase risk-taking among other competitors.

The existence of partitioned balanced tournament designs

AbstractE. R. Lamken proved that there exists a partitioned balanced tournament design of side , PBTD(), for a positive integer, , except possibly for . In this article, we establish the existence of PBTD() for . As a consequence, the existence of PBTD() has now been completely determined.

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.

When neither team wants to win: A flaw of recent UEFA qualification rules

Tanking, the act of deliberately dropping points or losing a game in order to gain some other advantage, is usually seen as being against the spirit of sports. It can be even more serious if playing a draw is a (weakly) dominant strategy for both teams in a match, since this may lead to collusion. We show that such a situation occurred in a particular football match. As our generalisation reveals, the root of the problem resides in the incentive incompatibility of certain UEFA qualification rules. The governing bodies of major sports should choose strategy-proof tournament designs because of several reasons. First, they may lead to the elimination of a third, innocent team. Second, incentive incompatible rules may discourage both teams from scoring goals, and the players could be interested in improving other match statistics than the number of goals.

A simulation comparison of tournament designs for the World Men's Handball Championships

AbstractThe study aims to compare different designs for the World Men's Handball Championships. This event, organized in every two years, has adopted four hybrid formats consisting of knockout and round‐robin stages in recent decades, including a change of design between the two recent championships in 2017 and 2019. They are evaluated under two extremal seeding policies with respect to various outcome measures through Monte Carlo simulations. We find that the ability to give the first four positions to the strongest teams, as well as the expected quality and outcome uncertainty of the final, is not necessarily a monotonic function of the number of matches played: the most frugal format is the second best with respect to these outcome measures, making it a good compromise in an unavoidable trade‐off. A possible error is identified in a particular design. The relative performance of the formats is independent of the seeding rules and the competitive balance of the teams. The recent reform is demonstrated to have increased the probability of winning for the top teams. Our results have useful implications for the organizers of hybrid tournaments.

The efficacy of league formats in ranking teams

The efficacy of different league formats in ranking teams according to their true latent strength is analysed. To this end, a new approach for estimating attacking and defensive strengths based on the Poisson regression for modelling match outcomes is proposed. Various performance metrics are estimated reflecting the agreement between latent teams’ strength parameters and their final rank in the league table. The tournament designs studied here are used in the majority of European top-tier association football competitions. Based on numerical experiments, it turns out that a two-stage league format comprising of the three round-robin tournament together with an extra single round-robin is the most efficacious setting. In particular, it is the most accurate in selecting the best team as the winner of the league. Its efficacy can be enhanced by setting the number of points allocated for a win to two (instead of three that is currently in effect in association football).

A DESIGNER'S CHOICE BETWEEN SINGLE‐PRIZE AND PARALLEL TOURNAMENTS

In this article, I theoretically and experimentally compare a designer's profits from two tournament designs. The first design is a standard winner‐take‐all tournament with a single prize. The second design features two winner‐take‐all (parallel) tournaments with different prizes where individuals choose which tournament to enter before competing. I develop a simple model that illustrates how the relative performances of these designs change as contestants' abilities differ. The theoretical model shows that the designer's profit is higher (lower) in the parallel tournament when contestants' abilities differ greatly (are similar). I complement these findings with experimental evidence. The experiments show that the parallel tournament is more profitable under high heterogeneity, whereas under low heterogeneity, the designer is better off with the single‐prize tournament. Furthermore, high‐ability agents under‐participate and low‐ability agents over‐participate in the high‐prize tournament relative to the theoretical prediction. (JEL C72, D82, J33, M51, M52)

Bayesian optimal design of fixed knockout tournament brackets

AbstractWe present a methodology for finding globally optimal knockout tournament designs when partial information is known about the strengths of the players. Our approach involves maximizing an expected utility through a Bayesian optimal design framework. Given the prohibitive computational barriers connected with direct computation, we compute a Monte Carlo estimate of the expected utility for a fixed tournament bracket, and optimize the expected utility through simulated annealing. We demonstrate our method by optimizing the probability that the best player wins the tournament. We compare our approach to other knockout tournament designs, including brackets following the standard seeding. We also demonstrate how our approach can be applied to a variety of other utility functions, including whether the best two players meet in the final, the consistency between the number of wins and the player strengths, and whether the players are matched up according to the standard seeding.

Asymptotic existence of frame-GBTDs

Generalized balanced tournament designs (GBTDs) are an equivalent characterization of a class of equitable symbol weight codes. Motivated by the construction of GBTDs, we establish in this paper an asymptotic existence theorem for frame-GBTDs of type g n and block size k via decompositions of edge-colored complete digraphs into prescribed edge-colored subgraphs.

Generalized Balanced Tournament Packings and Optimal Equitable Symbol Weight Codes for Power Line Communications

AbstractGeneralized balanced tournament packings (GBTPs) extend the concept of generalized balanced tournament designs introduced by Lamken and Vanstone (1989). In this paper, we establish the connection between GBTPs and a class of codes called equitable symbol weight codes (ESWCs). The latter were recently demonstrated to optimize the performance against narrowband noise in a general coded modulation scheme for power line communications. By constructing classes of GBTPs, we establish infinite families of optimal ESWCs with code lengths greater than alphabet size and whose narrowband noise error‐correcting capability to code length ratios do not diminish to zero as the length grows.

Generalized Balanced Tournament Designs with Block Size Four

In this paper, we remove the outstanding values $m$ for which the existence of a GBTD$(4,m)$ has not been decided previously. This leads to a complete solution to the existence problem regarding GBTD$(4,m)$s.

The Existence of Near Generalized Balanced Tournament Designs

In this paper, we complete the existence of near generalized balanced tournament designs (NGBTDs) with block size 3 and 5. As an application, we obtain new classes of optimal constant composition codes.

Construction of Hamilton Path Tournament Designs

A Hamilton path tournament design involving n teams and n/2 stadiums, is a round robin schedule on n − 1 days in which each team plays in each stadium at most twice, and the set of games played in each stadium induce a Hamilton path on n teams. Previously, Hamilton path tournament designs were shown to exist for all even n not divisible by 4, 6, or 10. Here, we give an inductive procedure for the construction of Hamilton path tournament designs for n = 2 p ≥ 8 teams.

Near generalized balanced tournament designs with block sizes 4 and 5

A near generalized balanced tournament design, or an NGBTD(k,m) in short, is a (km + 1, k, k − 1)-BIBD defined on a (km +1)-set V. Its blocks can be arranged into an m × (km + 1) array in such a way that (1) the blocks in every column of the array form a partial parallel class partitioning V[x] for some point x, and (2) every element of V is contained in precise k cells of each row. In this paper, we completely solve the existence of NGBTD(4,m) and almost completely solve the existence of NGBTD(5,m) with four exceptions.

Existence of (2, 8) GWhD(v) and (4, 8) GWhD(v) with $${v \equiv 0,1 (mod 8)}$$

(2, 8) Generalized Whist tournament Designs (GWhD) on v players exist only if $${v \equiv 0,1 (mod 8)}$$. We establish that these necessary conditions are sufficient for all but a relatively small number of (possibly) exceptional cases. For $${v \equiv 1 (mod 8)}$$ there are at most 12 possible exceptions: {177, 249, 305, 377, 385, 465, 473, 489, 497, 537, 553, 897}. For $${v \equiv 0 (mod 8)}$$ there are at most 98 possible exceptions the largest of which is v = 3696. The materials in this paper also enable us to obtain four previously unknown (4, 8)GWhD(8n+1), namely for n = 16,60,191,192 and to reduce the list of unknown (4, 8) GWhD(8n) to 124 values of v the largest of which is v = 3696.

Generalized balanced tournament designs and related codes

A generalized balanced tournament design, or a GBTD(k, m) in short, is a (km, k, k ? 1)-BIBD defined on a km-set V. Its blocks can be arranged into an m × (km ? 1) array in such a way that (1) every element of V is contained in exactly one cell of each column, and (2) every element of V is contained in at most k cells of each row. In this paper, we present a new construction for GBTDs and show that a GBTD(4, m) exists for any integer m ? 5 with at most eight possible exceptions. A link between a GBTD(k, m) and a near constant composition code is also mentioned. The derived code is optimal in the sense of its size.

Multiple signals in chameleon contests: designing and analysing animal contests as a tournament

Traditionally, studies of intrasexual selection have focused on single traits that are more exaggerated in males. Relatively little is known about systems in which traits are larger in females or the role of multiple traits in male contests. We used a tournament design in which each male encounters a series of different opponents, in conjunction with the structured Bradley–Terry model, to examine the role of multiple male traits in contests between male Cape dwarf chameleons, Bradypodion pumilum. Females are larger but males have relatively longer tails, larger and more ornamented heads and a larger central flank patch, all of which are emphasized during agonistic displays. We found no evidence that larger body size confers an advantage in male contests, despite high levels of aggression and escalated encounters. However, both the height of the casque (head ornament) and relative area of the flank patch were positively associated with fighting ability, and not correlated with each other, suggesting that they may represent independent sources of information about an opponent's ability. We discuss these results in relation to the role of male contest competition in the evolution of multiple male signals and sexual dimorphism in dwarf chameleons. In addition, we show that the use of tournament designs, in conjunction with the structured Bradley–Terry model, has important advantages over traditional designs and methods of analysing animal contests.

New Z-cyclic triplewhist frames and triplewhist tournament designs

Triplewhist tournaments are a specialization of whist tournament designs. The spectrum for triplewhist tournaments on v players is nearly complete. It is now known that triplewhist designs do not exist for v = 5 , 9 , 12 , 13 and do exist for all other v ≡ 0 , 1 ( mod 4 ) except, possibly, v = 17 . Much less is known concerning the existence of Z-cyclic triplewhist tournaments. Indeed, there are many open questions related to the existence of Z-cyclic whist designs. A (triple)whist design is said to be Z-cyclic if the players are elements in Z m ∪ A where m = v , A = ∅ when v ≡ 1 ( mod 4 ) and m = v - 1 , A = { ∞ } when v ≡ 0 ( mod 4 ) and it is further required that the rounds also be cyclic in the sense that the rounds can be labelled, say, R 1 , R 2 , … in such a way that R j + 1 is obtained by adding + 1 ( mod m ) to every element in R j . The production of Z-cyclic triplewhist designs is particularly challenging when m is divisible by any of 5 , 9 , 11 , 13 , 17 . Here we introduce several new triplewhist frames and use them to construct new infinite families of triplewhist designs, many for the case of m being divisible by at least one of 5 , 9 , 11 , 13 , 17 .

A Comparison of Potential Playoff Systems for NCAA I-A Football

This article discusses the properties of various knockout tournament designs and presents theoretical results. Potential playoff schemes for Division I-A football are examined via simulation studies. Several metrics are used to assess the relative merits of playoff scenarios, which differ in number, selection, and seeding of playoff teams. Most suggest that college football would benefit from a limited playoff system. Interestingly, for the class of playoff systems examined, the number of teams influences the performance far more than does the seeding procedure.