Win, Place, and No!

Win, Place, and No!

Win, Place, and No!

How the dismal science applies to your life.
Feb. 6 1999 3:30 AM

Win, Place, and No!

The only truly fair election is the one with one voter.


Sandra Loosemore, CBS SportsLine's figure skating writer, has a mission. She wants to overturn the sport's new scoring system.


The traditional system came under fire after the Great Flip-Flop of 1997 at the European Championships. In the men's free-skating event, five competitors had skated, and the first three places were held by Alexei Urmanov, Viacheslav Zagorodniuk, and Philippe Candeloro, in that order. The final contestant, Andrejs Vlascenko, placed dead last--but in the process he managed to bump Candeloro up to second place, denying Zagorodniuk a silver medal.

(For Loosemore's explanation of how the old scoring system worked, and how it caused the Great Flip-Flop, click here.)

International Skating Union Chairman Ottavio Cinquanta decreed after the championship that the scoring system must be revised to prevent flip-flops. When the new system was announced in July 1998, Cinquanta rejoiced that the problem was solved, so that henceforth "if you are in front of me, you will remain in front of me."

Loosemore has raised several cogent objections to the new system. First, it is more complicated than the old system. Second, contrary to Cinquanta, it won't stop flip-flops. In fact, Loosemore's computer simulations suggest that it will make flip-flops more common, not less.


What should the ISU do? Economist Kenneth Arrow hacked this problem five decades ago with his "Arrow Impossibility Theorem," which proves that no reasonable scoring system can preclude flip-flops. (The word "reasonable" incorporates the ISU's commitment to basing final outcomes only on the order in which the judges have ranked the contestants rather than on their explicit numerical scores.)

When Arrow was laying the foundations for his future Nobel Prize, he was concerned not with ranking figure skaters but with ranking social priorities. But Professor Michael Stob of Calvin College points out that the problems are identical. Judges may disagree about whether Zagorodniuk is better than Candeloro. Voters may disagree about whether a Star Wars defense system is better than a tax cut. In order to name winners, the ISU needs a scoring system to amalgamate the judges' opinions into a single ranking. In order to take political action, the United States needs a social choice mechanism to amalgamate voters' many preferences into a single list of priorities.

It might appear that the United States can afford more ambiguity than the ISU. Members of Congress can get away with cheerfully endorsing multiple conflicting goals and declaring them all "top priorities," whereas at the end of the day, the ISU must declare who gets the gold, who gets the silver, and who gets the bronze. But the fact is that at the end of the day, political decisions do get made, as surely as skating medalists get chosen. The government either cuts taxes or it doesn't. Although the political process is far more complex than anything the ISU might devise, the decisions it arrives at are just as unambiguous.

Arrow set forth two desiderata for any social choice mechanism. First, unanimity, when it occurs, should be respected. If all the judges rate Zagorodniuk ahead of Candeloro, then Zagorodniuk should come in ahead of Candeloro. If 100 percent of the voters prefer Clinton to Dole, then Dole should not be president. This criterion is the easy one. I doubt that any sport, or any polity, has ever violated it.


Arrow's second dictum is that there should be no flip-flops. If A ranks ahead of B, then the entry of a third candidate, C, should not overturn that ranking. Simple majority voting, for example, notoriously violates the no-flip-flop criterion. It's easy to imagine an election where Clinton is initially ahead of Dole, but a third party entry by, say, Jesse Jackson, pushes Dole ahead of Clinton. This can happen even when Jackson runs a distant third, as long as he takes more votes from Clinton than from Dole. Arrow's goal was to eliminate such anomalies with a more sophisticated voting system.

With a bit of elementary mathematics and a lot of keen insight, Arrow was forced to a sobering conclusion: If a "reasonable" voting system is one that respects unanimity and precludes flip-flops, then there are no reasonable voting systems, with one exception--the system that picks one voter and makes him a dictator.

In figure skating, that means the only system that could satisfy Cinquanta is a system with only one judge. (Actually, you could have as many judges as you wanted, as long as you ignored all but one of them.) In politics, it means that if you can't tolerate flip-flops, you'd better start getting comfortable with absolute dictatorship.

Illustration by Robert Neubecker

Is there any way around Arrow's dire conclusion? Well, his proof does leave one loophole. If the number of voters is literally infinite, then the conclusion can be overturned. In that case, there are nondictatorial voting systems that both respect unanimity and eliminate flip-flops. However (and this can be proved mathematically), although such systems exist, none of them can be explicitly described. This observation seems unlikely to be applicable either to figure skating or to constitutional government.

Another avenue is to observe that although every reasonable system allows flip-flops, some allow more flip-flops than others--and we can try to design a system that allows as few as possible. Loosemore argues that the old scoring system is superior in this respect, while Cinquanta asserts the opposite. I haven't studied the issue closely enough to take a firm stand but, based on what I've seen, my money is on Loosemore.

A third approach is to accept flip-flops as part of life and get on with designing a voting system that meets other reasonable criteria. But the people who get excited about this kind of project tend to get carried away. In 1975, the American Mathematical Society wanted to elect a slate of four officers from a list of eight candidates. The rules for counting ballots filled--in the words of the society's secretary--"thirty-three pages of definitions, rules, interpretations, and examples." Having supplied members with a summary of those rules, the secretary felt compelled to append this note: "Please do NOT correspond with the writer about the ambiguities that are unresolved in the above abbreviated description. Instead, refer to Appendices IV and V, where eleven pages of rules appear to leave no ambiguities."

Loosemore might not like Cinquanta's new scoring system, but she can be thankful that it wasn't designed by the American Mathematical Society.