Week 8: Arrow's Impossibility Theorem

In 1950, American economist and mathematician Kenneth J. Arrow addressed the accuracy of electoral systems that required voters to rank the candidates in order of preference in his book, "Social Choice and Individual Values." He stated that no ranked voting system for more than two candidates can accurately convert a set of ranked preferences into a preference profile for the entire electoral unless it satisfies three specific criteria, which is impossible to do. The following are his criteria:

1. No Dictators (ND): No single voter should have the power to determine the outcome of an election.

2. Independent of Irrelevant Alternatives (IIA): Suppose the electorate as a whole prefers A to B, but some voters change their preference lists. If no voter changes the relative positions of A and B, then A should still be preferred to B when considering the entire electorate.

Ex. If originally all voters (i.e. the electorate) rank the candidates in this order: A, B, C. Suppose some voters change their preference lists to say A, C, B. Candidate A is still ranked higher than Candidate B, so the electorate should still prefer A to B.

3. Pareto Efficiency (PE): If every voter prefers A to B, then it cannot be said that the electorate prefers Candidate B to Candidate A.

Pareto efficiency actually was not a part of Arrow's original theorem; rather, he used the criteria of monotonicity and non-imposition to imply Pareto efficiency. Monotonicity states that if one or more voters change their ranked preferences by putting on candidate higher, then the preference profile for the entire electorate should either be changed by ranking that candidate higher or should not change at all (an individual cannot be ranked lower by having one rating improved). Non-imposition means that every possible preference profile should be possible, i.e. if there are "n" candidates in an election, then every one of the n! lists can be achieved.

So this past week, I've been attempting to prove Arrow's Theorem and various aspects of it, such as how monotonicity and non-imposition together imply Pareto Efficiency. Intuitively, it makes sense, but I'd really like to be able to complete a concrete proof! I will post the proof when I'm finished!

When I first read about Arrow's Impossibility Theorem, I was surprised that a fair, non-manipulable election wasn't possible. What, then, are we striving for, I wondered? And how do we know whether or not a voting method we decide to use is better than the next one over? These questions are quite difficult to tackle, and maybe there isn't an explicit answer.