By my metrics , the top seven great powers in the world right now are the United States, China, Germany, Japan, Russia, India and Brazil. Your results might vary a bit, but I assume everyone will grant that all these countries would fall into their top 10 list.
According to FIFA, the top seven men's soccer teams in the world are, in order: Brazil, Spain, Portugal, Netherlands, Italy, Germany, and Argentina.
There's not a lot of overlap between those two lists. Indeed, the latter list includes three PIIGS countries plus a few others facing severe debt difficulties. Even if one expands the FIFA list to the top 20, only two more great powers (Russia and the United States) pop up.
Why the lack of correlation? I'd proffer three possible explanations. First, and most important, is culture. What the great powers have in common is possessing proud civilizational identities. While Germany and Brazil might have soccer-mad populations, in the other countries there are other sports -- baseball, hockey , basketball, rugby, and cricket -- that attract more attention and more dollars. The best athletes from most of the great powers don't go into soccer.
Related to this are the skewed industrial policies for sport that some countries pursue. The Washington Post's Keith Richburg looks at why China is ranked 84th in the world, and finds the following:
As in industry, the government picks national "winners" in sports and funnels cash to create champions and win medals. But the support typically goes to individual sports like gymnastics, swimming and diving, and to sports in which Chinese have traditionally excelled, like badminton and table tennis. Soccer teams here are left to look for private sponsorship....
Politics comes into play, several sports journalists and others said, because sports ministry officials, particularly at the local level, would rather invest government money into promising sports prodigies with a quicker guarantee of victory. "It's related to their promotion," said Li Chengpeng, a soccer commentator and author.
Finally, perhaps men's soccer isn't the best metric here. Consider FIFA's ranking for women's soccer: U.S., Germany, Brazil, Sweden, Japan, Norway, North Korea and France. China is 10th and Russia is 15th. The correlation between political power and women's soccer proficiency is much stronger.
The true outlier here is India. Their men's team ranks 133rd, just behind Fiji. Their women's team is somewhat better, just besting Haiti. Even if soccer is not that popular in the subcontinent, it's a country with more than a billion people -- sheer numbers suggest they should field a semi-decent team.
I welcome any South Asian experts to provide some possible answers in the comments.
UPDATE: I should have known that team Passport would be all over this already.
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Daniel W. Drezner is professor of international politics at the Fletcher School of Law and Diplomacy at Tufts University.