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.
