There’s a dearth of evolution news during the holiday break, so I thought I’d call attention to a oft-discussed problem with the World Cup and a little known paper that suggests a solution. The debate (which, as a football neophyte, I won’t attempt here to resolve or even contribute to) centers on whether the arrangement of World Cup matches is the best way to determine the best team.
To win the World Cup, a team need win only six or more games out of seven against a variety of teams, and to clinch victory it need win only a single game against an opponent it hasn’t previously met in the tournament. This is in strong contrast to American games like basketball or baseball, in which the two final teams are pitted against each other in a series of games, so that you have to win more than one (four in the case of the baseball World Series) to become champion.
There are two interrelated problems here. The first is that the championship is decided with a single game, and the second is that the number of goals that a team must score to win is relatively low—sometimes just one—so a win can reflect luck, or the events of a single day, rather than a persistent and repeated superiority over an opponent. (The American football championship, the Superbowl, is also decided by a single game, but it typically involves several goals and many points.) Can we really be sure that the victor in a single World Cup game is the best national team in the world?
This problem was taken up in 1966 by John Maddox, in an piece he wrote for Nature called, “We wuz robbed” (you can download it by going to this page). If you’re a scientist, you’ll know that Maddox was the plain-spoken and controversial editor of that journal, where he served for two terms (1966-1973 and 1980-1995). He died last year. 1966 was, of course, the only year that Brits ever won World Cup, in a 4-2 final with Germany that featured the only “hat-trick” (three goals by one man; in this case Geoff Hurst) ever performed in a World Cup final.
Maddox fitted the number of goals among all teams in that year’s World Cup to a Poisson distribution. This is a statistical distribution that occurs if there is a constant but very low probability of an event (say, a goal) occurring in a small interval (say, one minute of a game). If the probability is constant, then the distribution of events over a longer interval (say, goals in a 90-minute game) should fit the Poisson. Here from the article is Maddox’s compilation of each team’s World Cup goals, and the expected distribution from under Poisson expectation whose mean is equal to the average number of goals scored by a team (1.234 in this case).
The fit looks pretty damn good, and I confirmed this by doing a chi-square goodness of fit test, which gave the result χ² = 5.73, df = 6, 0.5 > p > 0.4, which isn’t even close to a significant deviation from the Poisson expectation. (I’m told that Mike Whitlock and Dolph Schluter show a similar Poisson distribution of more recent soccer scores in their statistics book The Analysis of Biological Data.)
The fact that the distribution fits so well, as if it were a single team with a fixed probability of scoring goals, led Maddox to say this:
The mere fact that a Poisson distribution can describe so well the distribution of scored by individual teams goes a long way to suggest that the teams were much of a muchness in talent and their scores were independent of each other. From this point of view, the decision that the outcome of a single competition should depend on the outcome of a single game between the two so-called finalists was as much of a farce as a great many West German supporters already know it to have been. If it is assumed that the goal scoring potentiality of the two teams is equally sell described by the Poisson distribution already specified, the chance that the result will be a draw is a mere 0.27. In other words, if two teams are equally matched, the chance that the result will be an active injustice to one of them will be 0.73. By the same token, a team which is slightly less skilled than its opponent can nevertheless expect a one in three chance of winning the deciding match.
Well, I’m not sure I’d consider the loss to an equally-matched team to be an “injustice,” but Maddox has a point. There are not many baseball World Series matches in which the losing team has failed to win a single game, and so we might be wary of saying that a team that wins the World Cup has decisively demonstrated its superiority to all other national teams.
The solution would seem to be making the championshp depend on winning more than one game. Maddox suggests a World-Series-style final of several “replicated’ games, so that
. . the finalists go on playing against each other either until the superiority of one or the other of them is properly established, or until both parties agree to negotiate a draw.
Such a negotiation is of course out of the question, but a series of matches is not. Maddox suggests, tongue in cheek, an alternative:
[R]edesign the parameters of the game of football in such a way that a respectable degree of confidence in the outcome of the competition can be acquired in a reasonable interval of time. If, for example, it were agreed that single cup finals should remain, but that no team should be declared the winner until its score exceeds that of its opponent by three standard deviations of the Poisson distribution, it might be necessary to design the game of football so that it would be practicable for one side to score 100 goals or so within the limits of endurance of the spectators. This implies that the parameter q [the mean score] would have to be much greater than under the present rules. Such a change could easily be brought about, possibly by widening the goalposts or by abolishing goalkeepers.
Nobody’s having that, but why not multiple games in the final? The downside is that this would make the World Cup much longer (especially if multiple games are also held in the earlier stages), and would also eliminate the drama of the championship coming down to a single 90-minute game that the whole world watches. But really, isn’t the World Cup about determining which country’s team is best, a decision that the winning nation can proudly claim for the next four years? Is there anyone here who would defend the present system against one involving multiple games in the final?
Maddox, J. (writing anonymously). 1966. We wuz robbed. Nature 211:670.
h/t: Geoff North