Why Voting Doesn’t Always Give the Best Result

This article on how the International Olympic Committee picks host cities is a great way of demonstrating the problems with voting and Arrow’s Impossibility Theorem.

In selecting a host city, the IOC, acting like a papal conclave, takes a series of votes until a candidate receives a majority. Each of the 100-plus IOC members gets one vote, and after every round the city with the fewest votes is eliminated. In the competition for the 2012 Games, Moscow expired in the first round, New York in the second. Madrid was the top vote-getter in Round 2, but it got the ax in Round 3. London edged out Paris, 54–50, in the final vote.

This violates the independence of irrelevant alternatives part of Arrow’s theorem. That is the rankings of a subset of the possible choices should be the same if we increased possible choices. But this isn’t always the case. Removing the city with the least number of votes can change the outcome each successive round of voting. That is, if Madrid was at the top in the second round it shouldn’t have plunged to the bottom by the third round if the independence of irrelevant alternatives is to hold.

For centuries philosophers, mathematicians, political scientists and economists have searched for the best method of voting. Fifty-eight years ago the economist Kenneth Arrow (later a Nobel laureate) decided to see whether any voting rule could avoid the problems we’ve illustrated. Fix them all at once, he found, and you get–a dictatorship. One voter calls the shots every time. Arrow’s “impossibility theorem” demonstrates that no system of voting always gives the “right” result.

Which I think also goes along well with the quote by Winston Churchill,

Many forms of Government have been tried and will be tried in this world of sin and woe. No one pretends that democracy is perfect or all-wise. Indeed, it has been said that democracy is the worst form of government except all those other forms that have been tried from time to time.

FILED UNDER: Bureaucracy, Democracy, Economics and Business, , ,
Steve Verdon
About Steve Verdon
Steve has a B.A. in Economics from the University of California, Los Angeles and attended graduate school at The George Washington University, leaving school shortly before staring work on his dissertation when his first child was born. He works in the energy industry and prior to that worked at the Bureau of Labor Statistics in the Division of Price Index and Number Research. He joined the staff at OTB in November 2004.


  1. PD Shaw says:

    This seems very similar to the voting patterns used by Lincoln to secure the Republican nomination in 1860. Seward was the top voter getter in the first two ballots, but Lincoln slowly consolidated the votes of candidates that realized they didn’t have a cahnce and the votes of the anybody-but-Seward crowd.

    Political parties ideally aren’t contests for the best candidate, however. They seek the candidate that can best hold the coalition together even if that means everybody’s second favorite choice wins.

  2. Joe says:

    Of course voting never seems completely impartial. Unless a vote is unanimous, there are always going to be disappointed people left in the aftermath who will think something wasn’t fair.

  3. odograph says:

    Why do you think there is a “best” result for Olympic cities?

    (The top few are probably good enough, and you could play rock-paper-scissors for all I care. Though of course I prefer that other countries subsidize the Olympics and not me.)

  4. odograph says:

    BTW, this page gives a rather unsatisfactory integration of Arrow’s Impossibility Theorem with the problems of two party democracy.

    I suspect that the real application is in the cycles of feedback which produce the two party bundles of positions and ideas. We’ve the iterative elimination of items (a moderate VP for McCain perhaps) before the final choice is presented to the full electorate.

  5. Furhead says:

    Did Arrow come up with the term “benevolent dictator”?

  6. Steve Verdon says:


    I don’t know, it would be a good guess, but a quick google search didn’t provide any indications of who coined the term.

  7. An Interested Party says:

    Totally off topic, but here’s possible Verdon post bait…

  8. Steve Verdon says:

    Already seen it. Its kind of pathetic that a police officer has to taser an elderly woman. Frankly, I’d be ashamed.

  9. sam says:

    I dunno why, but this discussion reminded me of a column by Guy Kawasaki, the Mac Evangelist, some years back on why the Mac, which, all things considered, is and was a better machine than the PC, lost out to the PC in the corporate environment: The PC was “good enough” for the tasks required. He didn’t seem to think that price was the fundamental issue, just serviceability to task. You only needed to be good enough, not better.