Today is the World Cup final. Soccer is probably the sport in which the underdog has the best chance to upset a better team, as ruefully observed by many sports commentators in the US after coming so very close to beating a much more skilled Belgium side.
In general, we can model the probability that the “better team” wins using some very simple assumptions (and neglecting ties):
(1) If the teams are equally good, there is a 50/50 chance for each to win
(2) As the difference in quality gets bigger, the probability of the better team winning approaches 1
That’s it! It’s had to argue with either axiom. So the chance that a team will win almost certainly looks like a logistic curve:
The Elo chess ranking system is based on this reasoning. But which logistic curve should we use? One that is almost a straight line – meaning that there is a more or less linear correspondence between the quality of teams and the chance of winning – or something more like a step function, in which the better team, even if only slightly better, will virtually always win. Here, different sports may get different answers. In a a very interesting paper, the author argues that the only real difference between sports is how often scoring events happen. It turns out, to a pretty good approximation, scoring events can be modeled as a Poisson process, which means that they occur independently of each other with a fixed average rate (so forget about momentum in sports!) with better teams scoring at a faster rate.
So waiting for a goal to be scored in soccer is a lot like waiting for unstable radioactive nucleus to decay. You know on average about how long you have to wait, but have no idea when this one will go off. (Also, having waited a while does not make it more likely to happen with this kind of distribution.) If it is true that scoring in sports in a Poisson process, then the law of large numbers tell us that events with high scoring rates, like basketball, favor the better team, since lucky deviations have a chance to even themselves out. Contrast this with soccer, in which one mistake, or bad penalty given, can easily swing a match. One solution, of course, is to have a multigame series – as in hockey, baseball, or basketball – which amplifies the advantage for the better team. It is interesting to note that in the NBA, one team, the Miami Heat, has made it to the finals four years in a row [although that streak is in serious jeopardy] and that this year’s finals were a rematch of last year’s. In fact, very long dynasties have occurs in basketball, like the eight consecutive championships from 1959 to 1966 by the Celtics. Basketball favors the better team for two reasons – very high scoring, and long playoff series. In contrast, baseball is much more subject to the rule of chance with shorter playoffs in the early rounds, and low scoring. And soccer, with ultralow scoring and single elimination knockout round, is the best for underdogs.