THE World Cup is over, which can only mean one thing: the new football season is nearly here. For many fans the early stages of the season are the best. They can still dream of glory for their team before harsh reality takes hold.
Those who don鈥檛 share the passion may scoff at such limitless optimism. But a new statistical analysis suggests that the fans have every reason to dream. You don鈥檛 have to be the best team to win the league. You don鈥檛 have to be among the best to win promotion. And even if you鈥檙e the worst team, the humiliation of relegation to a lower league is not inevitable. The laws of the game contrive to ensure that a little bit of luck can carry you through. It鈥檚 one of football鈥檚 great attractions for fans, and part of the reason why 1.5 billion people tuned in to watch Brazil beat Germany in this year鈥檚 World Cup final.
The role that chance plays in football can be illustrated by a simple thought experiment. Imagine a league in which all the teams are exactly equal. In any given match, each team has an equal chance of winning. What will the table look like when the season is over? It鈥檚 obvious that the teams won鈥檛 all finish with the same number of points. One will be top, another will be bottom, and there will be a spread of points between them, determined entirely by chance.
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But how influential is chance? To make the theoretical league more realistic we need to assign probabilities to each type of result. In professional football approximately one in four matches ends in a draw, so let鈥檚 make the probability of a draw 0.25. This gives each team a 0.375 chance of winning the match. It鈥檚 now possible to 鈥減lay鈥� a full season using these probabilities. As in the English Premiership, we鈥檒l set up a league of 20 teams that play each other twice, with 3 points for a win, 1 for a draw and 0 for a defeat.
A typical simulated season (see 鈥淭urning the tables on the top team鈥�) produces a clear winner with 67 points and a spread between the top and bottom teams of 36 points. This, remember, is with precisely equal teams. In the real Premiership, championship-winning teams average about 80 points and the top-to-bottom spread is about 50 points. Clearly, the difference in abilities of the real teams contributes to the spread of points. However, it is also clear that you can鈥檛 rule out chance as a major factor in the final standings.
But can it decide the winner of the league championship? To find out, let鈥檚 run another experiment. This time take the egalitarian league and add a team that is better than the rest: it still has a probability of 0.25 for a draw, but it is more likely to win its other games than lose them. Let鈥檚 take its probability of winning any given game to be 0.45 and its probability of losing 0.3.
The better team now 鈥減lays鈥� the existing 20 teams and the results are added into the league table. In one simulation, the stronger team finished the season with 67 points. This is better than expected 鈥� the average number of points it can expect with the allocated probabilities is 64. Even so, the stronger team only finished second. A less able but luckier team got 71 points and snatched the league title. Of course, other simulated seasons would give different outcomes, and sometimes the best team would win the league. But for the given probabilities, more often than not it won鈥檛.
What is the underlying reason for this apparent unfairness? It is embodied in the rules of the game. Unlike other team sports such as basketball and rugby, soccer is deliberately designed as a low-scoring game, and this affects the outcome of matches. The fewer the goals, the better chance a weaker team has of winning.
Consider a match in which one team has twice the goal-scoring power of the other. If the game finishes 1-0 鈥� a very common result in professional football 鈥� the probability that the goal was scored by the stronger team is 2/3. In other words, the weaker team has a pretty good chance of winning despite its inherent inferiority.
As the number of goals increases, the weaker team鈥檚 chance of winning diminishes. In a three-goal match, for example, it is 0.25, and in a nine-goal match it has fallen below 0.15. But nine-goal thrillers are very rare in football. In theory, then, football鈥檚 low rate of scoring introduces a substantial amount of randomness into the outcome of matches.
But do the rules produce unexpected results in real life? To find out, we鈥檝e examined matches between the top five and bottom five teams in the Premiership over four seasons. Overall, top-five teams outscored bottom-five teams in these matches by a ratio of 7:3. We can use this ratio of 鈥済oal-scoring power鈥� to calculate expected win rates for the weaker teams, and compare it with the actual win rates over four seasons. Theory turns out to agree with real life: weaker teams are more likely to win 1鈥�0 than 3鈥�2 (see 鈥淯nderdogs come out on top鈥�). Incidentally, when there is an even number of goals, the winning margin has to be at least two goals. This is why the weaker team is less likely to win than when the total is an odd number.
So chance plays a big part in deciding a team鈥檚 league position, and this has its origin in the rules of the game. But there鈥檚 a more difficult question: given the final league table, what is the probability that the league champions were actually the best team?
To find out, we introduce the concept of 鈥減oints ability鈥�. This is the number of points the team would gain if the random effects cancelled each other out. The most probable value of points ability is the number of points the team actually obtained. But given the uncertainty, there will be a probability distribution for the points ability.
To see how this works, look at the top of last season鈥檚 Premiership table. Arsenal won the league with 87 points, followed by Liverpool with 80 points and Manchester United with 77.
How likely was it that Arsenal really was the best team? To find out, we must calculate the probability that Arsenal鈥檚 true points ability was greater than that of all the other teams. Here鈥檚 how it鈥檚 done. First pick a possible value of Arsenal鈥檚 points ability, say 85. Then use the distribution curve to find how probable this is. The next step is to take the distribution curves of all the other teams and calculate the probability that they all had a points ability lower than 85. The two figures multiplied together give you the probability that Arsenal鈥檚 points ability was higher. Repeat this procedure for all the possible values of Arsenal鈥檚 points ability, and add all the results together, and you have the probability that Arsenal was the best team. The answer is 63 per cent. Liverpool鈥檚 probability of being the best is 21 per cent, and Manchester United鈥檚 12 per cent. Two other teams also have a marginal claim that they were the best: fourth-placed Newcastle United at 3 per cent and fifth-placed Leeds United at 1 per cent.
Although this result might give some cheer to Liverpool and Manchester United fans, it will surely be hard for Arsenal supporters to take. There is always a strong feeling that a championship-winning team has proved itself to be the best. But consider this: in over half of Arsenal鈥檚 matches last season, the result would have been altered by just one goal. We can therefore remind ourselves of the crucial moments: the disallowed goals, the missed penalty kicks, the lucky shots, and so on.
On reflection there really is plenty of scope for chance to play its part in the destiny of football teams. We should not regret this 鈥� it is precisely the uncertainty of the outcome which makes the game so exciting.