Author(s): David J. T. Sumpter
Football - the most mathematical of sports. From shot statistics and league tables to the geometry of passing and managerial strategy, the modern game is filled with numbers, patterns and shapes. How do we make sense of them? The answer lies in the mathematical models applied in biology, physics and economics. Soccermatics brings football and mathematics together in a mind-bending synthesis, using numbers to help reveal the inner workings of the beautiful game. How is the Barcelona midfield linked geometrically? What's the similarity between an ant colony and Total Football, Dutch style? What can defenders learn from lionesses? How much of a scoreline is pure randomness and how much is skill? How can probability theory make you money at the bookies? Welcome to the world of mathematical modelling, expressed brilliantly by David Sumpter through the prism of football. No matter who you follow - from your local non-league side to the big boys of the Premiership, La Liga, the Bundesliga, Serie A or the MLS - you'll be amazed at what mathematics has to teach us about the world's favourite sport.
Football and mathematics - together, much more than a game
David Sumpter is professor of applied mathematics at the University of Uppsala, Sweden. Originally from London, he completed his doctorate in Mathematics at Manchester, and held academic research positions at both Oxford and Cambridge before heading to Sweden. An incomplete list of the applied maths research projects on which David has worked include pigeons flying in pairs over Oxford; the traffic of Cuban leaf-cutter ants; fish swimming between coral in the Great Barrier Reef; swarms of locusts traveling across the Sahara; the gaze of London commuters; dancing honey bees from Sydney; and the tubular structures built by Japanese slime moulds. In his spare time, he exploits his mathematical expertise in training a successful under-nines football team, Uppsala IF 2005. David is a Liverpool supporter with a lifelong affection for Dunfermline Athletic. collective-behavior.com @djsumpter
Chapter 1: I never predict anything and I never will Paul Gascoigne once pronounced that "I never predict anything and I never will." This beautiful piece of self-contradiction shows us that even when the world is at its most random there is a pattern. I show that the very fact that football goals are unpredictable creates a pattern in the scorelines. Goals turn out to be like bus arrivals, soldiers being kicked by horses, light bulbs blowing, and many other everyday events. We can't predict when a light bulb will go, but we can predict how many we have to change per year. Similarly, over a football season we can forecast how often two-one wins and five-five draws will occur. I go on to look at statistical extremes. How can we measure whether an athlete is truly in a class of his or her own? I answer this question using Lionel Messi and Usain Bolt as examples. I show how to measure their brilliance in terms of how often these extremes occur and how long we will have to wait for another Bolt. I relate extremes in sport to other natural and economic extremes, such as floods and the ups and downs of the stock market. These examples give us a flavour of what statistics can and cannot do, and how we should use them. Chapter 2: How slime moulds invented the Ajax triangle Football formations have evolved over the last 130 years, from a line of attackers running forward with the ball to triangles and diamonds, nets and tiki-taka possession. Millions of years before coaches discovered the best structures for transporting a ball around a field, slime moulds were already using these shapes to transport food. I look at how we can use the geometry of slime and football formations to design efficient rail and road transport networks. I discuss why central planning doesn't always make sense, and how simple interactions can produce highly efficient structures. Chapter 3: Ants play total football I start with a mathematical model of training situations, building on 'piggy in the middle', to look at how players move in response to each other. I then test my predictions on 9 year olds, showing which training drills work best. I go on to look at how humans and other animals interact with each other: how we pass each other in the street; how birds follow each other when they fly; how ants leave and follow chemical trails; and how football players decide to move to the left or right past an opponent. I then use a model to look at the dynamics of teams of birds, ants and football players. I show that ants play 'total football', selflessly doing what needs to be done for the sake of their colony. They swap positions to create a fluid structure, responding to the events as a single unit. The key to their success is that the team is more important than any one individual. Chapter 4: Three points to the bird-brained manager TV pundit Jimmy Hill proposed the change from 2 points for a win up to 3, later claiming that this 'revolutionised football' in the 1990s. I use this example to begin an in-depth look at contests and incentives. I get in to the strategic 'minds' of birds, football managers and cancer cells. Birds fight over worms, managers over points and tumours over the body's resources. I explain why these individuals don't have to be 'intelligent' in order to evolve the winning strategy. Along the way I show why Hill was right, and that 3-point systems provide incentives for the weaker side to play more attacking football. Chapter 5: Power to the people I look at the 'problem' of the evolution of co-operation in biology and economics. If we all follow our own selfish interests how does co-operation evolve? Mathematical biologists have tried to solve this problem, but couldn't agree, causing a massive failure in their own co-operation. I propose a new solution, based on how football managers get teams to establish trust and work together. Using mathematical models of co-operation we can better understand why some groups work well and others fail. These models also allow us to think more broadly about economics and how we plan a stable society, despite conflicting interests. Chapter 6: World in motion Football is now a statistical science. The heat map of player positions and the network of passes, are both part of the overflow of data in football. Creating and understanding these statistics involves new techniques for visualisation. I'll show how these work, focusing on examples like a Suarez heat map or an Arsenal passing network. We have taken this type of analysis a step further in fish, ants and birds, looking at how they respond to each other. Could these methods provide a tool for managers to improve the interactions of their players? The chapter will then take us in to the world of big data and visualisation, covering everything from the human genome to Twitter and Facebook. Chapter 7: So you think you're the best? When Alex Ferguson retired, everyone apart from a few diehard Liverpool fans agreed that he had been the most successful manager in English club football. But does that imply that he was an exceptional talent to start with? In this chapter I describe the preferential attachment model of 'success breeding success'. If an individual is successful one year it increases his or her chance of being successful next. The prediction from this model is a very skewed distribution of success, even when everyone is initially equally talented. Preferential attachment applies just as much to football managers, as it does to entrepreneurs and academics. We can measure the effect in the citations of scientific papers and the distribution of wealth, and see the same pattern in the growth of cities and grouping of animals. Understanding preferential attachment leads us to reassess how we measure the quality of scientists and how we use performance metrics to evaluate people, schools and institutions. Chapter 8: You've only got one song How do you get 50,000 football fans singing the same line? It starts with just one loudmouth who decides it's time to taunt the opposition. A few of his friends join in and a few seconds later the ground is filled with the sound of 'you're not singing ...' or 'who ate all the pies?' The mathematics is straightforward. If one individual infects 9 others, then after the second round of chanting 100 people will be singing, and by the fourth round 10,000 will be involved. This is social contagion, the spreading of an idea from one person through the whole ground. I look at both this start and the eventual end of chanting. Contagion isn't limited to loudmouth football fans. We performed an experiment on an academic audience, showing that when they started and stopped clapping depended on their neighbours. Fashions, the success of songs and videos, even the adoption of cultural values can all be explained in the same way. I study the hype in 2013 around Luis Suarez's possible transfer to Arsenal, and cover a modelling war between Princeton and Facebook about whose influence is diminishing fastest. Chapter 9: The Mexican wave I celebrate the Mexican Wave with a mathematical model of how waves propagate through football grounds, fish schools, bird flocks and singing cicadas. Waves are started by a few enthusiastic individuals and spread rapidly. Simply by standing up as you see your neighbours standing up, together we create a dynamic pattern, pulsating steadily around the stadium. For football fans the waves are a bit of fun, but for fish they are essential for survival. By arranging themselves so that waves of turning spread through the whole group, fish schools can react as one to the attack of predators. In the Australian bush, cicadas try to outcompete their neighbours in song, resulting in a wave a noise spreading through the trees. I go on to look at herd mentality from the starting point of the heavy metal mosh pit. I show how different moshing patterns arise from the same set of simple interactions between the moshers. If concert-goers simply follow each other, move around randomly and crash in to each other, they build swirling circle pits and mosh lanes. These observations tell us much more about the world of human interactions than just the peculiarities of heavy metal. The unexpected and complex patterns that arise when people interact can be captured by mathematical models. These models help us plan how we should evacuate 90,000 football fans from a football ground. They even allow us to plan for the two million people who descend on Mecca each year. Chapter 10: Betting against the masses Betting is becoming less centralised. The old world of men with dogs in a smoky betting shop has been replaced by a new world of micro-second spread-bet online apps and peer-to-peer markets. For experienced punters who think they know better than everyone else, this offers both dangers and opportunities. Making a living betting on the Champions League is difficult, because the volume of betting is high and the crowd is wise. I show how each person entering the betting market makes it harder and harder for speculators to turn a profit. On the other hand, the sheer range of events that we can bet on provides opportunities to identify a winning strategy. But if you think you have got what it takes to bet on League Two football, then make sure you get your programming skills dusted off. Teams of mathematicians work day and night looking for imbalances in the odds on small games. The same algorithms that made them massive profits leading up to the financial crises are now used in the gaming industry. I go on to look at some subtleties in the wisdom of crowds idea, explaining why experts are usually better than groups at specialised tasks. In football, this is illustrated by the mixed fortunes of the crowd-managed Ebbsfleet United. In biology, I look at how fish, pigeons and ants exploit their collective wisdom to make better decisions about where to move. It turns out that a crowd of naive individuals can be essential to making good decisions. Chapter 11: Much more important than that In this final chapter, I go into the philosophy behind my More Than A Game idea. Many mathematicians and physicists believe that mathematics expresses a sort of perfect ideal. They argue, based mainly on a one-liner quote from Einstein, that if a model is simple and elegant then it should be true. When the world doesn't agree with mathematics, there is something wrong with the world. My view is different. Mathematics is much more like football than it is like a perfect ideal. It is a human-constructed way of doing things, which reflects properties of the real world. I show that the pure mathematicians' view is just the 'precise' part of mathematics, and the 'practice' of drawing analogies and building up models is just as important. Only if we think of mathematics in the same way as we analyse football matches can we make proper use of it. And we shouldn't be scared of using our feelings and emotions when deciding which models to use.