Mathematical Analysis of a Soccer Game. Part II: Energy,Spectral, and Correlation Analyses

Following the calculations on the individual and collective behaviors of players and the ball in Part I, further calculations are carried out in Part II concerning (i) the specific kinetic energy of each player, (ii) the power densities of various quantities for different groups of players (forwards, defenders) and for the geometrical centers of the two teams, as well as for the ball, and (iii) the correlations of various quantities among different groups of players and the ball, as well as among the geometrical centers and the ball. Some parameters are compared between the two teams. The relation between these parameters and the potential indicators and all the influencing factors to the final result of a soccer match are discussed. Advantages and limitations of the present approach and some ideas for the future research are outlined.     

Networks metrics and ball possession in professional football

This study aimed to verify the network of interactions resulting from the collective behavior of professional football teams and the influence of ball possession. A dataset of 30 matches of one highly successful team from the Portuguese Premier League, season 2010/2011, was considered. From these 30 matches, 13,958 passes (e.g., 11,127 successfully passes and 2831 unsuccessfully passes) and 7583 collective offensive actions were analysed. The data were analysed using Node XL Template that allows to characterize networks and team activity profiles. The results showed that football players' interactions tended to occur, preferentially, during the offensive phase, wherein the network of contacts was mainly organised in the central and lateral areas of the field. We concluded that the ball possession during a football match endows the team with a larger domain in terms of game actions. Moreover, the results of this study also allow concluding that the ball possession does not significantly influence the final outcome of the game. This study has practical implications for coaches, since it provides a multidimensional analysis of the football match (e.g., networks and ball possession) and offers relevant insights on how creative and organizing individuals might act to orchestrate team strategies. © 2016 Wiley Periodicals, Inc. Complexity 21: 342–354, 2016     

In search of goals: increasing ice hockey’s attractiveness by a sides swap

The popularity and business impact of major sports have been growing globally over time. This paper focuses on ice hockey, specifically the National Hockey League in North America. It reports a striking irregularity in ice hockey’s scoring dynamics relative to comparable sports such as soccer and rugby, namely a scoring surge in the middle section of the game. We explore an explanation for this irregularity related to the convention on the spatial location of the teams’ benches (which are fixed throughout the game) and on-ice sides (which are switched every period). Because a large number of the players’ substitutions occur while the play is in progress, this convention determines the distance forwards and defenders need to travel to make a substitution, and thus indirectly substitution strategies and scoring. We consider two simple operational changes that could increase the number of goals in the NHL by approximately 5 and 10%, respectively, corresponding to roughly 350 and 700 additional goals each season. This would partly offset the current downward scoring trend and thus enhance the game’s attractiveness. The estimated impact of the proposed reforms, one of which is largely costless, is robust across several specifications—using per-minute and per-second scoring data and controlling for various factors, such as bookmakers’ odds.     

Optimizing strategic behaviour in a dynamic setting in professional team sports

This article develops a dynamic game-theoretic model of optimizing strategic behaviour by football (soccer) teams. Teams choose between defensive and attacking formations and between a non-violent and a violent playing style, and can vary these choices continuously throughout each match. Starting from the end of the match and working backwards, the teams’ optimal strategies conditional on the current state of the match are determined by solving a series of two-player non-cooperative subgames. Numerical simulations are used to explore the sensitivity of strategic behaviour to variations in the structural parameters. The analysis demonstrates that the strategic behaviour of football teams can be rationalized in accordance with game-theoretic principles of optimizing strategic behaviour by agents when payoffs are uncertain and interdependent.     

Multidimensional scaling analysis of soccer dynamics

This paper studies the behavior of teams competing within soccer national leagues. The dissimilarities between teams are measured using the match results at each round and that information feeds a multidimensional scaling (MDS) algorithm for visualizing teams’ performance. Data characterizing four European leagues during season 2014–2015 is adopted and processed using three distinct approaches. In the first, one dissimilarity matrix and one MDS map per round are generated. After, Procrustes analysis is applied to linearly transform the MDS charts for maximum superposition and to build one global MDS representation for the whole season. In the second approach, all data is combined into one dissimilarity matrix leading to a single global MDS chart. In the third approach, the results per round are used to generate time series for all teams. Then, the time series are compared, generating a dissimilarity matrix and the corresponding MDS map. In all cases, the points on the maps represent teams state up to a given round. The set of points corresponding to each team forms a locus representative of its performance versus time.     

The computational complexity of the elimination problem in generalized sports competitions

Consider a sports competition among various teams playing against each other in pairs (matches) according to a previously determined schedule. At some stage of the competition one may ask whether a particular team still has a (theoretical) chance to win the competition. The computational complexity of this question depends on the way scores are allocated according to the outcome of a match. For competitions with at most 3 different outcomes of a match the complexity is already known. In practice there are many competitions in which more than 3 outcomes are possible. We determine the complexity of the above problem for competitions with an arbitrary number of different outcomes. Our model also includes competitions that are asymmetric in the sense that away playing teams possibly receive other scores than home playing teams.     

A new points system for soccer leagues: Have expectations been realised?

In the early 1980's the soccer leagues in England and Israel adopted a new points system, by which a team is awarded three points for a win instead of the two points awarded under the old system. The change was intended to improve the quality of the game by encouraging more ‘open’, attacking play, so producing more goals with less games ending in draws, and hence more satisfaction for spectators, leading in turn to more spectators being attracted to the sport. Some of the expected short term effects of the new system are analysed and possible changes in team positions as a result of a different scoring system are considered, empirically and theoretically. It is concluded that expectations have not been realised and that football league administrators gain little from changing points systems.     

The solution of the soccer “dream league” game

On January 5, 1996, Maariv, one of the two leading daily newspapers in Israel, announced “The Dream League” game. Every participant in this game was required to “purchase” from a pool of all the soccer players in the Israeli National League, a team which according to his judgment would be chosen as the best team at the end of the season. Purchasing the players was subject to a given budget and to several other constraints. After the soccer season was over, we were requested by Maariv to find the optimal “Dream Team”.The problem of finding the optimal team is shown to be a generalized version of the well-known knapsack problem. It is formulated as an integer program and solved to optimality by the software NAG. Evidently, the optimal Dream Team is much better (in terms of the total cumulative grade) than the actual winning team chosen by the readers of Maariv. A possible heuristic procedure for solving the game in larger settings is also discussed.     

多无人机协同任务规划问题

研究无人机任务规划问题,从无人机侦查和轰炸两方面入手.首先,运用迭代算法求解出从基地到雷达区域边际上任一目标出入口的最短路径.在此基础上,以无人机在雷达范围内滞留时间最短,以及被探测次数最少为目标,建立多目标最优化模型.通过改进交叉算子的遗传算法找出最优侦查路径.对于轰炸任务,以无人机在雷达范围内滞留时间最短,以及轰炸总时间最短为目标,建立多目标最优化模型.采用改变惯性权重的自适应粒子群算法找出最佳轰炸路线.由于计算时间较长,本文对68个目标进行聚类分析,提出针对轰炸任务的快速算法,相较原轰炸方案,其计算效率提高80%以上.     

Rankings and values for team games

In this article we attack several problems that arise when a group of individuals is organized in several teams with equal number of players in each one (e.g., for company work, in sports leagues, etc). We define a team game as a cooperative game v that can have non-zero values only on coalitions of a given cardinality; it is further shown that, for such games, there is essentially a unique ranking among the players. We also study the way the ranking changes after one or more players retire. Also, we characterize axiomatically different ways of ranking the players that intervene in a cooperative game.     

中国足球甲A联赛的统计分析

本文通过对中国足球甲A联赛三年资料的初步统计分析,得到一些易见的结论;一场球赛的双方进球数服从Poisson分布,甲A联赛总体上存在非常显著的主场效应,但并不是所有球队在每年的联赛中都是这样,只有少数球队才存在统计意义上是显著的主场效应,用计算机模拟给出各名次球队得分的区间估计,并以进球数表示一支球队实力的参数,建立了一种称为特征向量法的预测方法,还依据96联赛成绩,对参赛的12支甲A球队进行了聚类分析.     

Local and global stability analysis of a two prey one predator model with help

In this paper we propose and study a three dimensional continuous time dynamical system modelling a three team consists of two preys and one predator with the assumption that during predation the members of both teams of preys help each other and the rate of predation of both teams are different. In this work we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functional. At the end, numerical simulations are performed to substantiate our analytical findings.     

Ranking of Sports Teams via the AHP

This paper presents an application of the analytic hierarchy approach (AHP) to predict the ranking of the 16 soccer teams of the Israeli National League. Six criteria (attributes) were used for evaluating each team: the team's facility, the team's trainer, the players' level, the team's fans, the previous season's performance and the current performance. For each criterion a matrix of pairwise comparisons between teams was evaluated by a sports expert. The normalized eigenvector of each matrix scales the teams with regard to that criterion. The overall scaling is constructed by weighting the scales of all the criteria. The weight of the criteria are extracted from a judgemental matrix of pairwise comparisons between every two criteria. The normalized eigenvector of this matrix provides the criteria weights. Some comparisons to simpler ranking methods were made in order to validate the model.     

Games of timing with resources of mixed type

The paper considers a game of timing which is closely related to the so-called duels. This is a game connected with the distribution of resources by two players. Each of the players is in possession of some amount of resource to be distributed by him in the time interval [0, 1]. In his behavior, Player 1 is restricted by the necessity of taking all of his resources at a single point, while Player 2 has no restrictions. For the payoff function, defined as for duels, the game is solved; explicit formulas on the value of the game and the optimal strategies for the players are found.     

Measuring the efficiency of NBA teams: additive efficiency decomposition in two-stage DEA

This paper evaluates the efficiency of National Basketball Association (NBA) teams under a two-stage DEA framework. Applying the additive efficiency approach, we decompose overall team efficiency into first-stage wage efficiency and second-stage on-court efficiency and find out the individual endogenous weights for each stage. The empirical results show that NBA teams present a better performance on wage efficiency than for on-court efficiency, as on-court efficiency is influenced by many uncontrollable factors. Moreover, NBA teams, on average, tend to possess a higher weight on the first stage, suggesting that teams’ managers can enhance organizational efficiency more efficaciously by prudentially recruiting players.     

The shift team formation problem in multi-shift manufacturing operations

This paper addresses the problem of assigning operators to teams that work in 1-, 2-, or 3-daily-shift systems. The problem was motivated by, and illustrated with a case situation encountered in Dutch manufacturing industry. The problem addressed forms an extension of cell formation problems which are currently in the phase of addressing labor-related issues in cell design. A generalized goal problem formulation is presented to address multiple, conflicting objectives covering cross-training of workers, ensuring adequate levels of labor flexibility and minimizing labor-related costs. The proposed solution procedure consists of two phases. In the first phase, shift systems, in which applicable machines and the sizes of each shift team are identified. The next phase deals with assignment of operators to various teams and identification of specific cross-training needs for various workers. This phase involves the use of interactive goal programming. The methodology is illustrated by a numerical example derived from the case situation.     

On the development of decision rules for bar quiz handicapping

Some competitions involve teams comprising different numbers of players. For informal games, such as the popular ‘pub quiz’, we argue that teams with fewer players are at a disadvantage. This paper investigates the properties of these games and develops several methods for allocating handicaps within such competitions, so that the competitions may be considered fair, based on a simple Bernoulli model for correctly answering questions and assuming exchangeability of participants. We recommend a natural conjugate prior subjective handicapping rule; with this rule handicaps may be set beforehand given the judgement of the quiz setter regarding the difficulty of the questions. We also describe a posterior rule that provides improved accuracy but is calculated after the quiz is complete. Finally, the paper considers modifications of the proposed rules to cope with multiple-choice questions and progressive quizzes.     

A network DEA assessment of team efficiency in the NBA

In this paper, a Network DEA approach to assess the efficiency of NBA teams is proposed and compared with a black-box (i.e. single-process) DEA approach. Both approaches use a Slack-Based Measure of efficiency (SBM) to evaluate the potential reduction of inputs consumed (team budget) and outputs produced (games won by the team). The study considers the distribution of the budget between first-team players and the rest of the payroll. The proposed network DEA approach consists of five stages, which evaluate the performance of first-team and bench-team players, the offensive and defensive systems and the ability for transforming the points made by itself and by the opponents into wins. It has been applied to the 30 NBA teams for the regular season 2009–2010. The results show that network DEA has more discriminating power and provides more insight than the conventional DEA approach.     

An analysis of Chinese Super League partial results

Some of the history of soccer/world football in China is presented. Then consideration turns to the 2008 Chinese Super League. It has 16 teams. The results from the first half of the season, i.e. 15 rounds, are studied. The response of interest for a specific game is whether the home team won, tied or lost, who the home team was, and who the opponent was. The response is ordinal-valued. A generalized linear model is fit and then, given the remaining fixtures, used to predict the final standings of the season. Other explanatories, such as round number, are considered for inclusion in the model. Simulation is employed to estimate probabilities of interest. This work was supported by US National Science Foundation (Grant No. DMS-0707157)     

Heuristics for the mirrored traveling tournament problem

Professional sports leagues are a major economic activity around the world. Teams and leagues do not want to waste their investments in players and structure in consequence of poor schedules of games. Game scheduling is a difficult task, involving several decision makers, different types of constraints, and multiple objectives to optimize. The traveling tournament problem abstracts certain types of sport timetabling issues, where the objective is to minimize the total distance traveled by the teams. In this work, we tackle the mirrored version of this problem. We first propose a fast and effective constructive algorithm. We also describe a new heuristic based on the combination of the GRASP and iterated local search metaheuristics. A strong neighborhood based on ejection chains is also proposed and leads to significant improvements in solution quality. Very good solutions are obtained for the mirrored problem, sometimes even better than those found by other approximate algorithms for the less constrained non-mirrored version. Computational results are shown for benchmark problems and for a large instance associated with the main division of the 2003 edition of the Brazilian soccer championship, involving 24 teams.