Set Of Tournaments Research Articles

Flashes and Rainbows in Tournaments

Colour the edges of the complete graph with vertex set {1,2,⋯,n}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\{1, 2, \\dotsc , n\\}}$$\\end{document} with an arbitrary number of colours. What is the smallest integer f(l, k) such that if n>f(l,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n > f(l,k)$$\\end{document} then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that f(l,k)=lk-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(l, k) = l^{k - 1}$$\\end{document} and proved this for l⩾(3k)2k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l \\geqslant (3 k)^{2 k}$$\\end{document}. We prove the conjecture for l⩾k3(logk)1+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l \\geqslant k^3 (\\log k)^{1 + o(1)}$$\\end{document} and establish the general upper bound f(l,k)⩽k(logk)1+o(1)·lk-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(l, k) \\leqslant k (\\log k)^{1 + o(1)} \\cdot l^{k - 1}$$\\end{document}. This reduces the gap between the best lower and upper bounds from exponential to polynomial in k. We also generalise some of these results to the tournament setting.

Safe sets and in-dominating sets in digraphs

A non-empty subset S of the vertices of a digraph D is a safe set if(i) for every strongly connected component M of D−S, there exists a strongly connected component N of D[S] such that there exists an arc from M to N; and(ii) for every strongly connected component M of D−S and every strongly connected component N of D[S], we have |M|≤|N| whenever there exists an arc from M to N.In the case of acyclic digraphs a set X of vertices is a safe set precisely when X is an in-dominating set, that is, every vertex not in X has at least one arc to X. We prove that, even for acyclic digraphs which are traceable (have a hamiltonian path) it is NP-hard to find a minimum cardinality safe (in-dominating) set. Then we show that the problem is also NP-hard for tournaments and give, for every positive constant c, a polynomial algorithm for finding a minimum cardinality safe set in a tournament on n vertices in which no strong component has size more than clog(n). Under the so called Exponential Time Hypothesis (ETH) this is close to best possible in the following sense: If ETH holds, then, for every ϵ>0 there is no polynomial time algorithm for finding a minimum cardinality safe set for the class of tournaments in which the largest strong component has size at most log1+ϵ(n). We also discuss bounds on the cardinality of safe sets in tournaments.

Locating-dominating sets in local tournaments

Locating-dominating sets in local tournaments

Balancing academics and athletics: School-level athletes’ results are positively associated with their academic performance

BackgroundUsing data from a nationwide high-school sample, the present study aimed to examine whether school-level athletes’ sporting results are positively or negatively associated with their academic performance. MethodsAthletic performance was assessed using the results of spring and summer regional qualifying interschool tennis tournaments in Japan among both male and female tournaments across all prefectures in Japan. Academic performance was assessed using the standardized rank scores for academic performance at the school level, with higher scores denoting superior academic levels; this is the most common measure of academic performance in Japanese high schools. Linear mixed models were conducted to compare the academic performance of high schools with winning and losing records, respectively, for all games together and for the spring and summer tournaments separately. The results of a total of 4,870 games were analyzed. ResultsHigh schools with winning records showed a significantly higher academic-performance score than those with losing records, and this association was stronger among boys than girls. The observed difference in academic-performance scores between the schools with winning and losing records, respectively, was replicated in both tournament settings (i.e., spring and summer). ConclusionsAthletic results are positively associated with academic performance at the school level. Given the cross-sectional nature of the present study, the present results do not necessarily imply that sports participation can improve academic performance; rather, the present study suggests that balancing academic and athletic commitments is feasible.

A NOVEL APPROACH FOR PREDICTING FOOTBALL MATCH RESULTS: AN EVALUATION OF CLASSIFICATION ALGORITHMS

The task of predicting the outcomes of football matches is rendered increasingly complex by the intricate nature of the game and the variety of variables that could affect how things turn out. In the recent past, machine learning algorithms have been applied to this challenge, with varying degrees of success. In this particular research paper, we have meticulously evaluated the performance of several classification algorithms with the objective of predicting the outcomes of football matches in a tournament setting. The algorithms that were thoroughly tested encompassed a diverse range of classification models, including logistic regression, support vector machines and random forests. The study employed a dataset of historical match data drawn from the FIFA World Cup, historical team ranking data and team strength data from FIFA games. In order to accurately assess the efficacy of the algorithms tested, the evaluation metrics used were accuracy, precision and recall. The results of the study highlight the fact that machine learning algorithms can indeed prove to be effective tools for predicting the outcomes of football matches.

Packing Feedback Arc Sets in Tournaments Exactly

Let [Formula: see text] be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set (FAS) if [Formula: see text] contains no cycles (directed). A collection [Formula: see text] of FASs (with repetition allowed) is called an FAS packing if each arc e is used at most w(e) times by the members of [Formula: see text]. The purpose of this paper is to give a characterization of all tournaments [Formula: see text] with the property that, for every nonnegative integral weight function w defined on A, the minimum total weight of a cycle is equal to the maximum size of an FAS packing. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11801266 and 11971228]; the Chinese Academy of Sciences [Grants XDA27000000 and ZDBS-LY-7008]; the Ministry of Science and Technology of China [Grant 2018AAA0101002]; the Research Grants Council of Hong Kong; the Fundamental Research Funds for the Central Universities [Grant 020314380035].

Cooperate or compete? The impact of vertical wage dispersion on employees’ behavior in tournaments

We investigate the effect of vertical wage dispersion, defined as the difference in wages between superiors and subordinates, on subordinates’ behaviors in competition. We propose that higher vertical wage dispersion increases subordinates’ desire to reduce the vertical pay gap through collusion against their superiors in a setting where collusion reduces subordinate effort while increasing subordinates’ pay. Our two experiments test our prediction in one-shot (Study 1) and repeated (Study 2) tournament settings. In Study 1, we find that rather than increasing collusion, high vertical wage dispersion increases competitiveness and effort contribution. In Study 2, we find support for our prediction that high vertical wage dispersion increases collusion and reduces effort contribution due to the trust building between subordinates that is facilitated by repeated tournaments. We contribute to the growing research on pay dispersion by studying how vertical wage dispersion affects lower-level employees’ interaction with their peers. We also extend tournament research by studying how a contextual variable outside the tournament, i.e., ex ante vertical wage dispersion, could affect employees’ willingness to compete or to collude in tournaments. An implication of our finding is that high vertical wage dispersion may make competitive incentives more or less effective, depending on the context.

MSE-optimal K-factor of the Elo rating system for round-robin tournament

Abstract The Elo rating system contains a coefficient called the K-factor which governs the amount of change to the updated ratings and is often determined by empirical or heuristic means. Theoretical studies on the K-factor have been sparse and not much is known about the pertinent factors that impact its appropriate values in applications. This paper has two main goals: to present a new formulation of the K-factor that is optimal with respect to the mean-squared-error (MSE) criterion in a round-robin tournament setting and to investigate the effects of the relevant variables, including the number of tournament participants n, on the optimal K-factor (based on the model-averaged MSE). It is found that n and the variability of the deviation between the true rating and the pre-tournament rating have a strong influence on the optimal K-factor. Comparisons between the MSE-optimal K-factor and the K-factors from Elo and from the US Chess Federation as a function of n are also provided. Although the results are applicable to other sports in similar settings, the study focuses on chess and makes use of the rating data and the K-factor values from the chess world.

Pitching Behaviors in Youth Baseball: Comparison With the Pitch Smart Guidelines.

Background:The Pitch Smart guidelines aim to limit youth baseball pitching behaviors associated with overuse injuries. Despite many youth baseball leagues being compliant with the guidelines, during tournaments, pitch count restrictions or guidelines are often not followed.Purpose:To perform a quantitative analysis of pitch counts in youth baseball players and evaluate compliance with regard to the Pitch Smart guidelines in the tournament setting.Study Design:Cross-sectional study; Level of evidence, 3.Methods:Included in the analysis were 100 youth baseball teams that competed in the 8-and-under to 14-and-under age divisions during the 2019 tournament season. Pitching data were compared with the Pitch Smart guidelines. Violations were identified as (1) exceeding maximum daily pitch count, (2) inadequate rest between pitching events, and (3) pitching more than 1 event on the same day. Pitcher and game factors were analyzed for possible relationships to guideline violations using mixed-effects negative binomial regression models, with comparisons of violations using rate ratios (RRs).Results:Analysis included 1046 pitchers and 2439 games. There were 1866 total Pitch Smart guideline violations, with 48.6% of pitchers having at least 1 violation. Inadequate rest was the most common reason for violation, with noncompliance occurring in 43.3% of pitchers. The highest rate of any violation (0.32 per appearance) occurred in the 8-and-under age division. High-volume pitchers had increased violation rates in each category compared with low-volume pitchers (P < .001). Violation rates were increased more than twice the rate when pitchers participated in ≥5 consecutive games without a rest day when compared with a single game (RR, 2.48; P < .001).Conclusion:Noncompliance with Pitch Smart guidelines in tournament settings occurred in more than 90% of teams and almost half of all pitchers. Factors associated with noncompliance included younger pitcher age, high-volume pitching, and pitching in multiple consecutive games. Education of tournament directors, coaches, parents, and athletes regarding pitching guidelines is warranted in order to limit the risk of injury.

Margin of victory for tournament solutions

Tournament solutions are frequently used to select winners from a set of alternatives based on pairwise comparisons between them. Prior work has shown that several common tournament solutions tend to select large winner sets and therefore have low discriminative power. In this paper, we propose a general framework for refining tournament solutions. In order to distinguish between winning alternatives, and also between non-winning ones, we introduce the notion of margin of victory (MoV) for tournament solutions. MoV is a robustness measure for individual alternatives: For winners, the MoV captures the distance from dropping out of the winner set, and for non-winners, the distance from entering the set. In each case, distance is measured in terms of which pairwise comparisons would have to be reversed in order to achieve the desired outcome. For common tournament solutions, including the top cycle, the uncovered set, and the Banks set, we determine the complexity of computing the MoV and provide bounds on the MoV for both winners and non-winners. We then reveal a number of structural insights on the MoV by investigating fundamental properties such as monotonicity and consistency with respect to the covering relation. Furthermore, we provide experimental evidence on the extent to which the MoV notion refines winner sets in tournaments generated according to various stochastic models. Our results can also be viewed from the perspective of bribery and manipulation.

Who won the social media March Madness bracket? Demand shifters for Twitter followers

Social media has become an essential set of platforms for sport teams and organizations to engage, interact, and connect with their fan bases. From an analytic approach to fan demand for sport tournaments, the authors examined the demand shifters motivating fans to follow social media accounts within the context of the National Collegiate Athletic Association Division I Men's Basketball Championship. Changes in the number of followers of participating teams’ Twitter accounts were tracked daily during the tournament as a proxy for fan interest and demand. Independent variables regarding game performance, previous performance, and school performance were analyzed in a regression model. The findings indicated that, during the tournament, fans' social media following behavior was different from the behavior during the regular season games. Further, there were shifts in fans’ social media following behavior in the advanced rounds compared to earlier rounds in a tournament setting. A tournament setting has certain unique attributes that can arouse new consumers' interests, and thus practitioners can tailor their content to appeal to fans, increase follower interaction and reach, and provide better content to maintain the followership gains moving forward, using tournament settings.

Margin of Victory in Tournaments: Structural and Experimental Results

Tournament solutions are standard tools for identifying winners based on pairwise comparisons between competing alternatives. The recently studied notion of margin of victory (MoV) offers a general method for refining the winner set of any given tournament solution, thereby increasing the discriminative power of the solution. In this paper, we reveal a number of structural insights on the MoV by investigating fundamental properties such as monotonicity and consistency with respect to the covering relation. Furthermore, we provide experimental evidence on the extent to which the MoV notion refines winner sets in tournaments generated according to various stochastic models.

Tournament Quasirandomness from Local Counting

A well-known theorem of Chung and Graham states that if h ≥ 4 then a tournament T is quasirandom if and only if T contains each h-vertex tournament the ‘correct number’ of times as a subtournament. In this paper we investigate the relationship between quasirandomness of T and the count of a single h-vertex tournament H in T. We consider two types of counts, the global one and the local one.We first observe that if T has the correct global count of H and h ≥ 7 then quasirandomness of T is only forced if H is transitive. The next natural question when studying quasirandom objects asks whether possessing the correct local counts of H is enough to force quasirandomness of T. A tournament H is said to be locally forcing if it has this property.Variants of the local forcing problem have been studied before in both the graph and hypergraph settings. Perhaps the closest analogue of our problem was considered by Simonovits and Sós who looked at whether having ‘correct counts’ of a fixed graph H as an induced subgraph of G implies G must be quasirandom, in an appropriate sense. They proved that this is indeed the case when H is regular and conjectured that it holds for all H (except the path on 3 vertices). Contrary to the Simonovits-Sós conjecture, in the tournament setting we prove that a constant proportion of all tournaments are not locally forcing. In fact, any locally forcing tournament must itself be strongly quasirandom. On the other hand, unlike the global forcing case, we construct infinite families of non-transitive locally forcing tournaments.

Supplier Collaboration in Collaborative Product Development with Internal Competition

This paper examines a development process in which a firm explores alternative approaches for the development of a single product. To do this, the firm employs competing internal teams, but it also needs to involve key suppliers who are responsible for the development of a key component (or subsystem) via a collaboration with one or several internal teams. We explore how internal competition affects the suppliers' collaborative efforts and the firm's profits, and also how the firm should allocate the internal teams to potential suppliers. Our analysis suggests that this setting is significantly different from standard tournament settings, implying that existing research results should not be simply extrapolated to supply chain settings. For example, unlike in traditional tournaments, we find that increasing supplier competition by increasing the number of teams per supplier can actually increase supplier efforts. We develop a tournament model with two suppliers who differ both in their expertise for the specific type of development project and in their development costs (i.e., their cost-effectiveness). We also identify a critical influential factor: the cost savings resulting from the similarity between different development efforts by the same supplier (with different internally competing teams). If the cost savings are not very high, i.e., if the different development efforts are not very similar, the firm should optimally allocate more teams to the supplier with less relevant expertise or lower cost-effectiveness. Similarly, involving a third supplier is beneficial only if the cost savings are low. While companies make significant efforts to identify the supplier with the best expertise, our results show that allocating more teams to the stronger supplier generally undermines the suppliers' incentives to exert high levels of effort. Hence, allocating the most teams to that supplier might not always improve firm profits.

A Combined Sleep Hygiene and Mindfulness Intervention to Improve Sleep and Well-Being During High-Performance Youth Tennis Tournaments.

To investigate the effects of combined sleep hygiene recommendations and mindfulness on actigraphy-based sleep parameters, perceptual well-being, anxiety, and match outcomes during high-performance junior tennis tournaments. In a randomized crossover design, 17 high-performance junior tennis players completed the baseline, control, and intervention (INT) conditions across 3 separate weeks. The baseline consisted of unassisted, habitual sleep during a regular training week, and the control was unassisted sleep during a tournament week. The players attended a sleep education workshop and completed a nightly sleep hygiene protocol during a tournament week for the INT. Analysis was performed on the weekly means and on the night prior to the first match of the tournament (T-1). Significant differences were observed for increased time in bed, total sleep time, and an earlier bedtime (P < .05) across the INT week. These parameters also significantly improved on T-1 of the INT. A moderate effect size (P > .05, d > 1.00) was evident for decreased worry on T-1 of the INT. Small effect sizes were also evident for improved mood, cognitive anxiety, and sleep rating across the INT week. The match performance outcomes remained unchanged (P > .05). Sleep hygiene INTs increase the sleep duration of high-performance junior tennis players in tournament settings, including the night prior to the tournament's first match. The effects on perceptual well-being and anxiety are unclear, although small trends suggest improved mood, despite no effect on generic match performance outcomes.

Cash, non-cash, or mix? Gender matters! The impact of monetary, non-monetary, and mixed incentives on performance

Standard economic theory asserts that cash incentives are always better than non-cash ones, or at least not worse. This study employs a real effort experiment to analyze the impact of monetary, non-monetary, and a combination of monetary and non-monetary incentives on performance, where non-monetary incentives are defined as tangible incentives with market value. Our overall results suggest that there exists no significant difference in performance in response to monetary, non-monetary, and mixed incentives. However, gender-based differentiation reveals a different picture: the performances of men and women depend upon the type of incentive used. Whereas men’s performance is significantly higher in response to monetary incentives compared to non-monetary ones, women’s performance is significantly higher in response to non-monetary incentives. The gender differences in the effectiveness of monetary and non-monetary incentives do not seem to be triggered by the perceived attractiveness of the non-monetary incentives but rather by the differences between men and women in the feelings of appreciation and perceived performance pressure in a tournament setting. Therefore, our results indicate that gender differences must be considered when implementing incentives.

Entropy of tournament digraphs

The Rényi α-entropy Hα of complete antisymmetric directed graphs (i.e., tournaments) is explored. We optimize Hα when α=2 and 3, and find that as α increases Hα's sensitivity to what we refer to as ‘regularity’ increases as well. A regular tournament on n vertices is one with each vertex having out-degree n−12, but there is a lot of diversity in terms of structure among the regular tournaments; for example, a regular tournament may be such that each vertex's out-set induces a regular tournament (a doubly regular tournament) or a transitive tournament (a rotational tournament). As α increases, on the set of regular tournaments, Hα has maximum value on doubly regular tournaments and minimum value on rotational tournaments. The more ‘regular’, the higher the entropy. We show, however, that among all tournaments on a fixed number of vertices H2 and H3 are maximized by any regular tournament on that number of vertices. We also provide a calculation that is equivalent to the von Neumann entropy, but may be applied to any directed or undirected graph and shows that the von Neumann entropy is a measure of how quickly a random walk on the graph or directed graph settles.

Impact of Contextual Factors on External Load During a Congested-Fixture Tournament in Elite U'18 Basketball Players.

An understanding of basketball physical demands during official matches is fundamental for designing specific training, tactical, and strategic plans as well as recovery methods during congested fixture periods. Such assessments can be performed using wearable indoor time motion tracking systems. The purpose of this study was to analyze the time-motion profile of under 18-years of age (U’18) basketball players and compare their physical demands in relation to team ranking, playing position, match periods and consecutive matches during a 7-day tournament. Relative Distance (RD), percentage of High-Intensity Running (%HIR), Player Load (PL), Acceleration (Acc), Deceleration (Dec), Peak Speed (PSpeed), and Peak Acceleration (PAcc) were recorded from 94 players (13 centers, 47 forwards, and 34 guards) belonging to eight elite teams (age: 17.6 ± 0.8 years; height: 1.91 ± 0.08 m; body mass: 82.5 ± 8.8 kg). WIMU PROTM inertial measurement units with ultra-wide band (UWB) indoor-tracking technology recorded 13 matches during the Adidas Next Generation Tournament Finals in the 2016–2017 season. Paired t-tests and one-way analyses of variance with omega partial squared () and Cohen’s effect sizes (d) were used to analyze for differences between variables. According to team quality, the best teams had lower RD (p = 0.04; d = −0.14). Guards presented higher RD (p < 0.01; = 0.03), PSpeed (p < 0.01; = 0.01) and PAcc (p < 0.01; = 0.02) compared to forwards and centers. The first quarter showed differences with higher RD (p < 0.01; = 0.03), %HIR (p < 0.01; = 0.02), and PL (p < 0.01; = 0.04) compared to all other quarters. The third match of the tournament presented higher demands in RD (p < 0.01; = 0.03), HIR (p < 0.01; = 0.01) and PL (p < 0.01; = 0.02) compared with the first two matches. This study showed that team quality, playing position, match period, and consecutive matches throughout an U’18 basketball tournament influenced the kinematic demands experienced by players during official competition. Therefore, each of these contextual factors should be considered in managing the load and developing individualized strategies for players in tournament settings.

Informational inequity aversion and performance

In labor markets, some individuals have, or believe to have, less data on the determinants of success than others, e.g., due to differential access to technology or role models. We provide experimental evidence on when and how informational differences translate into performance differences. In a laboratory tournament setting, we varied the degree to which individuals were informed about the effort-reward relationship, and whether their competitor received the same or a different amount of information. We find performance is adversely affected only by worse relative, but not absolute, informedness. This suggests that inequity aversion applies not only to outcomes but also to information that helps achieve them, and stresses the importance of inequality in initial information conditions for performance-dependent outcomes.

Informational Inequity Aversion and Performance

In labor markets, some individuals have, or believe to have, less data on the determinants of success than others, e.g., due to differential access to technology or role models. We provide experimental evidence on when and how informational differences translate into performance differences. In a laboratory tournament setting, we varied the degree to which individuals were informed about the effort-reward relationship, and whether their competitor received the same or a different amount of information. We find performance is adversely affected only by worse relative, but not absolute, informedness. This suggests that inequity aversion applies not only to outcomes but also to information that helps achieve them, and stresses the importance of inequality in initial information conditions for performance-dependent outcomes.