Back to School and Arrow's Impossibility Theorem

School starts tomorrow, and (hold your applause) I have already done my reading for the week! Basically all the courses I am taking are voting and elections related (hooray!) which makes me more excited to start the semester than I otherwise would be. In doing my reading I was reminded of one of my favorite lessons from last year, Arrow's Impossibility Theorem, and I thought I would share it with you!

Arrow's Theorem basically points out that aggregate voter preferences should be transitive, but they are not. Consider this scenario. We have three voters and three candidates, Barack Obama, Mitt Romney and a chair. The voters prefer the candidates in the following order:

Voter 1: Obama>Romney>Chair
Voter 2: Chair>Obama>Romney
Voter 3: Romney>Chair>Obama

Individual voter preferences are transitive, meaning that if I would rather vote for Obama than Romney and would rather vote for Romney than a chair, then I would rather vote for Obama than a chair as well.

If the chair were not running, Obama would win since 2/3 of the voters prefer Obama to Romney. Similarly if Obama were to get fed up and drop out ,Romney would win since 2/3 of the voters prefer Romney to chair. Since Obama beats Romney and Romney beats chair, it stands to reason that Obama could beat a chair as well, but not so! If Romney were to drop out, the chair would win since 2/3 of the voters prefer the chair to Obama.

Arrow's Theorem is used by proponents of alternative voting systems (for example those where voters weight their preferences) to show that no rank order or majority voting system can accurately reflect the "will of the people."