The knockout stage of the
2010 FIFA World Cup is about to begin in South Africa. At the time of writing, every team has one group stage match remaining, and most teams still have a chance to finish in the top two places in their group and progress to the knockout stage (see the
tournament schedule and
group stage standings).
There are different approaches to ranking world football teams. The most well known is
FIFA's official world rankings, which are derived from points gained and lost in each match according to a heuristic
set of rules that generally reward winning against higher-ranked opponents in more-important tournaments.
A simple alternative with a more statistical basis is an Elo rating system (described in more detail below). A handy property of Elo rating systems is that they directly provide an estimate of the probability that a given team will perform better than another. We can use
Mathematica with that to set up simulations of the knockout stage of the World Cup. This lets us estimate things like the chance of each team winning the tournament. We'll also generate some nice visualizations of the results, such as the following simulated knockout stage (based on the current top two teams in each group):