Arrow's Impossibility Theorem (idea)

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In 1952, mathematical economist Kenneth J. Arrow proved a rather remarkable theorem.

"There is no consistent method by which a democratic society can make a choice (when voting) that is always fair when that choice must be made from among 3 or more alternatives."

Given that society has a habit of using various voting methods to determine anything from political office, to the location of large events such as the Olympics, to ranking of sports teams, this is a big deal. But it has been proven that using any method of voting, it is possible to get a set of results whereas the winner is in some manner irrational given the votes submitted. Every voting system in existence - and every system that has yet to be invented - has a flaw. This has been proven, mathematically.

Here's a simple example. We'll use the voting method employed in the United States, where each voter only votes for their top choice. There are three candidates, A, B, and C.

The results? A gets 27% of the vote. B gets 33% of the vote. This leaves C with 40% of the vote. Now, C would be declared the winner - this, even though a majority voted AGAINST C.

A better example... suppose we let the voters list the three candidates in their order of preference.

             49% Voters    45% Voters    6% Voters

  1st Choice      C             B            A

  2nd Choice      B             A            B

  3rd Choice      A             C            C 

Now, candiate C obviously won the decision here. However, if you look at the last choice among the voters, the 51% who did not put candidate C as their first choice put him as their LAST choice.

Candidate B would be able to argue, rather convingly, that he is in fact the favorite, as he was not listed as anyone's last choice. Yet he wasn't the winner.

For those with knowledge in this area, here's something more formal.

Let Prefers_i(a,b) mean that person i prefers a to b. Let Prefers be some joint decision procedure that, thus, generates either Prefers(a,b) or Prefers(b,a) for any a, b in some decision set, Set.

Then Arrow's impossibility Theorem says that the following 5 reasonable conditions on the joint preference relation Prefers cannot all be met by any single decision process:

1.Prefers is independent of irrelevant alternatives that is to say, the ordering of any 2 items in Prefers is a function only of their ordering with respect to each other within each of the Prefers_i.
2.Prefers is non-dictatorial that is to say, Prefers is not necessarily identical to Prefers_i for some i.
3.Prefers is pareto-inclusive that is to say, Prefers will rank 2 elements of Set in a particular order if all Prefers_i do.
4.Prefers is transitive.
5.Prefers is a complete ordering on Set.

This theorem was first published in A Difficulty in The Concept of Social Welfare.

_{Sources:
Social Axiom, http://www.geocities.com/TimesSquare/Lair/3936/theme/axiom_social.html
Arrow's Impossibility Theorem, http://www.personal.psu.edu/staff/m/j/mjd1/arrowimpossibilitytheorem.htm}