Why USC Won't Be No. 1

College's football mess is explained by Arrow's Impossibility Theorem, which dooms any attempt to find a winner from more than two contenders

By Steven H. Wildstrom

Millions of sports fans were confused and frustrated by an inscrutable Bowl Championship Series (BCS) ranking system that denied the University of Southern California's football team a chance to play for the national championship in the Sugar Bowl. After all, USC was the No. 1 team in both the USA Today/ESPN and Associated Press polls.

Football fans tend to blame the complex rating system, particularly the part played by computers. But the real problem lies much deeper: There's no way to devise a system -- other than a series of playoff games -- that can guarantee a fair result. It's not just difficult to get it right, it's a mathematical impossibility, like squaring the circle.

This conclusion was proved in the 1951 doctoral dissertation of Kenneth J. Arrow, who won the 1972 Nobel prize in economics for the work and who is now professor emeritus of economics at Stanford University. The conclusion, Arrow's Impossibility Theorem, remains startling but unchallenged after a half-century. In an election with more than two candidates, no system can guarantee that the choice meets five reasonably straightforward standards of fairness.


  The formal statement of the five criteria of fairness is complex and the mathematical proof much more so, but the rules boil down to common-sense notions. The criterion that the BCS outcome most obviously violates is called the independence of irrelevant alternatives.

The fact that Southern Cal was knocked out of contention partly because the University of Hawaii and Notre Dame were trounced in their final games certainly was a case of the outcome being influenced by irrelevant events, but that's not really what the rule means. Instead, it requires that a candidate who is in first place should not fall out of the lead because of the elimination of any other candidate.

Southern Cal had more first-place votes in both the USA Today/ESPN and AP polls than Lousiana State and Oklahoma combined. But the point totals were sufficiently close that there's no telling how the results would have been scrambled had, say, second-place LSU been removed from contention.


  Independence of irrelevant alternatives is an obvious-enough condition that turns out to be surprisingly difficult to meet in real-life voting systems. The same is true of the other criteria: A) The candidate who receives a majority of votes should always win. B) The candidate who beats all other candidates in head-to-head pairings should win. C) If a candidate leads the first round of a multistage election and all changes in the second stage favor the leader, that candidate should win. D) Finally, no one voter should be able to determine the outcome, that is, no dictators allowed.

The impossibility theorem means neither the computerized rankings nor the coaches and sportswriters who vote in the two polls can guarantee a fair result. And if the individual components don't assure fair outcomes, their combination using an incomprehensible formula certainly doesn't either.

Social-choice theory, as the body of work based on Arrow's work is known, obviously has far more important implications than calling into question the results of a fundamentally silly football ranking system. For example, it suggests that while Democrats may be justified in feeling cheated by George W. Bush's victory in 2000, the fact that Ralph Nader was on the Florida ballot and received more votes than the difference between Bush and Al Gore all but guaranteed an outcome that would be perceived as unfair.


  If we assume that most Nader voters favored Gore over Bush in the end, the elimination of the third-place finisher would have reversed the standing of the top two candidates. This would have made Gore President, but clearly violated the independence of irrelevant alternatives criterion (and others, too).

As for me, I'm a University of Michigan alumnus, and I'm just annoyed that the BCS mess means the Michigan's Wolverines have to play a really angry Southern Cal team in what amounts to a home game in the Rose Bowl. But as Arrow's Impossibility Theorem says, life is unfair.

Wildstrom is Technology & You columnist for BusinessWeek

Edited by Beth Belton

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