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A measure of success

I’d like to connect two ideas – the sensitivity of pay to performance and the quality of the England soccer team. I have long argued that money buys success in sport. My evidence is based on running regressions of team win percentages on wage expenditure: the coefficient on wages is usually significant at the 1% level and the percentage of variation explained by wages is substantial- between 30% and 90%, depending on the league and the period of time covered. Most notably, if you average over time the amount of variation explained becomes very high. Then I get asked: if that’s true why have the Yankees had the highest payroll for the last seven years but haven’t won a World Series?

The answer is that sports championships are contests involving repeated trials, which work in favour of a dominant team when outcomes are based on averaging these trials (league or series play), but against it when outcomes are based on elimination. For example, suppose the Yankees buy thesmelves a 60% probability of winning each game they plays (those are pretty good odds). Then if they play a best of five series their probability of winning it is 68%. But if they have to win four series in order to win the Fall Classic, then their probability of success is only 22%. Moreover, even if the process is repeated seven times there is still an 18% probability they will fail (unlikely perhaps, but no so improbable).

Likewise, England is a very successful soccer team in the sense that its win-loss record (treating draws as half a win) is just under 69% over its history, and even allowing for some catch-up in recent years, it is still averaging around the 60-65% mark (in failing to qualify for Euro 2008 its win loss record was 66%). But to win a championship you need to win enough qualifiers, then progress in a group stage, then win quarter-final, semi-final and final. Let us say there are 12 “must win” games in order to win a championship and England has a 65% chance of winning every game. On this basis its chance of winning the whole thing is a pitiful 0.5% and the probability of winning a World Cup or European Championship over the last 40 years was about 10%. That sounds about right to me. In order to give itself a decent chance, England need to achieve a probability of winning of about 75% in every game, and let’s see, that would put it on a par with, um, …Brazil.