Mathematical Analysis of a Soccer Game. Part I: Individual and Collective Behaviors

Detailed time series analysis of a soccer match is given based on the detailed data of the 2D motions of all 22 players and of the ball for the match. The whole analysis includes two parts. In Part I, the individual and collective behaviors of the players of the two teams as well as the motion of the ball are presented as various time series. Geometrical centers, radii, expansion speeds, possession functions of the two teams are defined and calculated as functions of time. Major ranges of all players as well as of different groups of players (defenders, midfielders, forwards) of the two teams during the entire first half, the attacking phase of team A and the attacking phase of team B are calculated, respectively, showing the structures of the two teams during different phases. Distance coverage of each player and the mean distances covered by different groups of players (defenders, midfielders, forwards) during different phases are calculated. The time portions of possession of the ball by each team and the time portions of different phases are also calculated. In Part II, energy and spectral analysis and various correlations will be derived. The relation between various parameters and potential indicators will be discussed. The major purpose of the present study is to offer some general mathematical tools for the detailed analysis and to reveal some general features of soccer match when the detailed 2D data are available. The results would offer the raw materials for various potential indicators which may eventually be used in the coaching process to enhance the performance and in the prediction of the results of soccer matches.     

When to rush a ‘behind’ in Australian rules football: a dynamic programming approach

In Australian rules football, points are scored when the ball passes over the goal line. Six points are awarded for a goal when the ball passes between the two centre posts, and one point for a ‘behind’, when the ball passes between a centre post and an adjacent outer post. After a behind, the defending team has a free kick from the goal line. It may be worthwhile, particularly in the closing stages of a game, for a defending team voluntarily to concede a behind, by themselves passing the ball between the two outer posts, either to avert the possibility of an imminent goal or to increase the probability of scoring a goal themselves. A dynamic programming model is used to analyse this situation.     

A model of a sudden-death field-goal football game as a sequential duel

A highly simplified ‘football’ game is constructed where individuals can run or kick for a field goal. It represents an attempt to extend the domain of the modeling of duels to a broader category than those previously studied. There are some worthwhile similarities (and important differences) in military, sports and economic tactics which are reasonably well modeled as zero sum games.The theory of duels for military application has been reasonably well developed but there are at least two other activities beyond war in which the zero-sum game duel is worth considering. They are in sports and in the tactics of advertising campaigns or other economic activities where the budgets of A and B are set and the fight is a non-price duel over a market of fixed size. In this paper we concentrate on the extension of dueling to sports.Rather than claim immediate relevance to football as it is played, a highly simplified version of ⌉sudden-death’ scoring is introduced. The basic game is as follows. With equal probability the team toss to see who gets the ball. The team with the ball can either run or try to kick a field goal. The first team to score wins the game.     

足球比赛的新赛制——三败赛

通过对目前采用的几种主要足球赛制的分析与比较 ,总结出目前各种赛制各自的弊端 ,在此基础上提出了一种新的赛制模型——三败赛 .通过运用概率论的知识 ,利用计算机仿真方法对三败赛赛制与原有赛制进行定量分析和比较 ,结果表明三败赛集中了分组赛、淘汰赛的优点 ,并可克服这些赛制的缺欠 ,是一种更为合理的赛制 .     

Determining the best strategy for changing the configuration of a football team

This paper proposes a dynamic programming (DP) approach to find the optimal substitution strategy for a football match, which maximises the probability of winning or the expected number of league points, supported by real data of the English Premier League. We use a Markov process model to evaluate the offensive and defensive strengths of teams by means of maximum likelihood estimators. We develop a DP formulation to derive quantitatively the optimal substitution strategy of a team, in relation to the number required of each type of outfield player. We demonstrate how this approach may help to determine how many of each type of player should start a match and be substituted during a match. We also show how the expected league points would increase if the optimal strategy were followed.     

Decision taking under pressure: Evidence on football manager dismissals in Argentina and their consequences

The paper focuses on dismissals of football managers and whether such decisions appear to contribute to organisational goals or else to be fruitless exercises in scapegoating. Its empirical setting is Argentina where incidence of sackings is more than one per season per club. It models 20 years of football match results and detects a tendency for a change of manager to be followed by deterioration in team performance, with adverse effects concentrated in results of matches played away. The evidence is consistent with decisions being driven by fan and media pressure rather than in realistic hope of improving the position of the club.     

How behaviors spread in dynamic social networks

In this paper, we explore how decentralized local interactions of autonomous agents in a network relate to collective behaviors. Earlier work in this area has modeled social networks with fixed agent relations. We instead focus on dynamic social networks in which agents can rationally adjust their neighborhoods based on their individual interests. We propose a new connection evaluation theory, the Highest Weighted Reward (HWR) rule: agents dynamically choose their neighbors in order to maximize their own utilities based on rewards from previous interactions. We prove that, in the two-action pure coordination game, our system would stabilize to a clustering state in which all relationships in the network are rewarded with an optimal payoff. Our experiments verify this theory and also reveal additional interesting patterns in the network.     

Effect of local information within network layers on the evolution of cooperation in duplex public goods games

Traditional works of public goods game (PGG) are often studied in simplex networks where agents play games through the same type of social interactions. In order to promote cooperation against the defection in PGGs in simplex network environment, many mechanisms have been proposed from different perspectives, such as the volunteering mechanisms, and the punishment and reward approaches. However, due to diverse types of interactions between agents in reality, the study of PGG should also consider the characteristic of multiplexity of networks. Hence, we firstly model the public goods game in the duplex network (for simplification of analysis, the duplex network is considered), in which agents have two types of social interactions, and thus the network is modeled as two network layers. This type of PGG is naturally named as duplex public goods game (D-PGG), in which agents can select one of the network layers to allocate their limited resources. Then for the new game environment (D-PGG), we propose a novel perspective to promote cooperation: degrading the information integrity, i.e., agents get information just from one network layer (local information) rather than from the whole duplex network (global information) in the evolution process. Finally, through theoretical analyses and simulations, we find that if agents imitate based on the local information of the payoff in the evolution, cooperation can be generally promoted; and the extent of promotion depends on both the network structure and the similarity of the network layers.     

随机游动在足球福彩中的应用

通过对一个随机游戏的分析，提出“近似马氏稳态时间”定理并加以证明，而后利用马氏链模型和对策论建立解决方案的最佳模型，并利用此模型预测足球比赛的胜、负的概率。     

Scheduling a non-professional indoor football league: a tabu search based approach

This paper deals with a real-life scheduling problem of a non-professional indoor football league. The goal is to develop a schedule for a time-relaxed, double round-robin tournament which avoids close successions of games involving the same team in a limited period of time. This scheduling problem is interesting, because games are not planned in rounds. Instead, each team provides time slots in which they can play a home game, and time slots in which they cannot play at all. We present an integer programming formulation and a heuristic based on tabu search. The core component of this algorithm consists of solving a transportation problem, which schedules (or reschedules) all home games of a team. Our heuristic generates schedules with a quality comparable to those found with IP solvers, however with considerably less computational effort. These schedules were approved by the league organizers, and used in practice for the seasons 2009–2010 till 2016–2017.

     

Modeling networked systems using the topologically distributed bounded rationality framework

In networked systems research, game theory is increasingly used to model a number of scenarios where distributed decision making takes place in a competitive environment. These scenarios include peer‐to‐peer network formation and routing, computer security level allocation, and TCP congestion control. It has been shown, however, that such modeling has met with limited success in capturing the real‐world behavior of computing systems. One of the main reasons for this drawback is that, whereas classical game theory assumes perfect rationality of players, real world entities in such settings have limited information, and cognitive ability which hinders their decision making. Meanwhile, new bounded rationality models have been proposed in networked game theory which take into account the topology of the network. In this article, we demonstrate that game‐theoretic modeling of computing systems would be much more accurate if a topologically distributed bounded rationality model is used. In particular, we consider (a) link formation on peer‐to‐peer overlay networks (b) assigning security levels to computers in computer networks (c) routing in peer‐to‐peer overlay networks, and show that in each of these scenarios, the accuracy of the modeling improves very significantly when topological models of bounded rationality are applied in the modeling process. Our results indicate that it is possible to use game theory to model competitive scenarios in networked systems in a way that closely reflects real world behavior, topology, and dynamics of such systems. © 2016 Wiley Periodicals, Inc. Complexity 21: 123–137, 2016     

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.     

The collective value: a new solution for games with coalition structures

In this study, we provide a new solution for cooperative games with coalition structures. The collective value of a player is defined as the sum of the equal division of the pure surplus obtained by his coalition from the coalitional bargaining and of his Shapley value for the internal coalition. The weighted Shapley value applied to a game played by coalitions with coalition-size weights is assigned to each coalition, reflecting the size asymmetries among coalitions. We show that the collective value matches exogenous interpretations of coalition structures and provide an axiomatic foundation of this value. A noncooperative mechanism that implements the collective value is also presented.     

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.     

Computer Forecasting of Australian Rules Football for a Daily Newspaper

An exponential smoothing technique operating on the margins of victory was used to predict the results of Australian Rules football matches for a Melbourne daily newspaper from 1981-86 and again for a competitor in 1991-92. An initial ‘quick and dirty’ program used only a factor for team ability and a common home ground advantage to predict winning margins. Probabilities of winning were accumulated to predict a final ladder, with a simulation to predict chances of teams finishing in any position. Changes to the competition forced a more complicated approach, and the current version uses several parameters which allow for ability, team/ground interaction, team interaction, and a tendency for team ability to regress towards the mean between seasons. A power method is used to place greater weight on the errors in closer matches, and errors across the win-lose boundary. While simple methods were used originally, the Hooke and Jeeves method was used in optimizing the parameters of the current model. Both the original model and the improved version performed at the level of expert tipsters.     

Risk assessment for infectious disease and its impact on voluntary vaccination behavior in social networks

Achievement of the herd immunity is essential for preventing the periodic spreading of an infectious disease such as the flu. If vaccination is voluntary, as vaccination coverage approaches the critical level required for herd immunity, there is less incentive for individuals to be vaccinated; this results in an increase in the number of so-called “free-riders” who craftily avoid infection via the herd immunity and avoid paying any cost. We use a framework originating in evolutionary game theory to investigate this type of social dilemma with respect to epidemiology and the decision of whether to be vaccinated. For each individual in a population, the decision on vaccination is associated with how one assesses the risk of infection. In this study, we propose a new risk-assessment model in a vaccination game when an individual updates her strategy, she compares her own payoff to a net payoff obtained by averaging a collective payoff over individuals who adopt the same strategy as that of a randomly selected neighbor. In previous studies of vaccination games, when an individual updates her strategy, she typically compares her payoff to the payoff of a randomly selected neighbor, indicating that the risk for changing her strategy is largely based on the behavior of one other individual, i.e., this is an individual-based risk assessment. However, in our proposed model, risk assessment by any individual is based on the collective success of a strategy and not on the behavior of any one other individual. For strategy adaptation, each individual always takes a survey of the degree of success of a certain strategy that one of her neighbors has adopted, i.e., this is a strategy-based risk assessment. Using computer simulations, we determine how these two different risk-assessment methods affect the spread of an infectious disease over a social network. The proposed model is found to benefit the population, depending on the structure of the social network and cost of vaccination. Our results suggest that individuals (or governments) should understand the structure of their social networks at the regional level, and accordingly, they should adopt an appropriate risk-assessment methodology as per the demands of the situation.     

Evolving networks for social optima in the “weakest link game”

Previous models have applied evolving networks based on node-level “copy and rewire” rules to simple two player games (e.g. the Prisoner’s Dilemma). It was found that such models tended to evolve toward socially optimal behavior. Here we apply a similar technique to a more tricky co-ordination game (the weakest link game) requiring interactions from several players (nodes) that may play several strategies. We define a variant of the game with several equilibria—each offering increasing social benefit. We found that the evolving network functions to select and spread more optimal equilibria while resisting invasion by lower ones. Hence the network acts as a kind of “social ratchet” selecting for increasing social benefit. Such networks have applications in peer-to-peer computing and may have implications for understanding social systems.

A ratings based Poisson model for World Cup soccer simulation

In this paper a method is suggested for predicting the distribution of scores in international soccer matches, treating each team’s goals scored as independent Poisson variables dependent on the Fédération Intemationale de Football Association (FIFA) rating of each team, and the match venue. The results of a Poisson regression to estimate parameters for this model were used to simulate matches played during the 1998 World Cup tournament. For the model to be a more effective predictor, some manual adjustments must be made to the ratings data. The predictions of the model were placed on a web page to create interest in applications of mathematics, and proved popular with the general public.     

On some games of timing of mixed type

The paper considers a zero-sum, two-person game of timing on [0, 1] in which Players 1 and 2 behave as in the so-called discrete-fire duel and non-discrete fire duel, respectively. Player 1 is in possession of one action. Accuracy functions of the players are continuous and nondecreasing, from [0, 1] onto [0, 1]. The game is analysed in three versions and the form of optimal strategies for the players is found.     

Modelling match outcomes and decision support for setting a final innings target in test cricket

^** E-mail: p.a.scarf{at}salford.ac.uk This paper considers the problem of timing the declaration ofthe third innings in test cricket. Data on the outcomes of recenttest matches are analysed in order to develop simple decisionsupport tools. The first of these tools presents match outcomeprobabilities given the position of the match at a potentialdeclaration point. These probabilities are determined usinga multinomial logistic regression model that is fitted to thetest match data. This idea is then extended to consider progresstowards a declaration—match outcome probabilities areconsidered as a function of target aimed for and run-rate. Thedecision tools have been implemented on a spreadsheet and examplecalculations are presented. The modelling described has thepotential for practical use in test matches.