 So, we might want a voting system to satisfy independence of irrelevant alternatives. But what else do we want? So something we might want is known as monotonicity. Monotonicity means that if more voters support a winning candidate, that candidate should still win. Plurality satisfies monotonicity since more support just means more total votes. The board account also satisfies monotonicity since more support means raising the rank of a candidate and upranking a candidate will increase their point total. But one of the weird things that could happen is instant runoff can fail monotonicity. And one of the disconcerting features of instant runoff is the following. Suppose the preference schedule for an election held using instant runoff is the following. And initially with this preference schedule the vote tallies are going to be A with nine first place votes, B with eight, and C with seven. So C will be eliminated. And so this gives us a revised preference schedule. And now A has 13 votes and B has 11. So A wins. Now the important thing is that A doesn't know that this is what the preference schedule is going to look like. So maybe they sent out some staffers to do some last minute campaigning and they're successful. So suppose some voters who initially ranked B over A over C, here's some last minute campaign ads and decide they like A even more. And so they change their ballot to A over B over C and the new preference schedule becomes. And now the tallies are A with 11 votes, B with six, and C with seven. And so B is eliminated. With B eliminated the new preference schedule becomes. And now the tallies are A with 11 and C with 13. And now C wins. Wait, what? And here's a bizarre result. Even though more people supported A, that support actually caused A to lose the election. Another thing we might want our social choice function to have is neutrality. Neutrality means that renaming the candidates won't change the outcome. For example, using plurality in the preference schedule makes T the winner. Now if we renamed our candidates X, Y, and Z, then in the new preference schedule then Z would win, which meets the neutrality condition because Z is just T renamed. Another feature we might want in a voting system is non-dictatorship. In non-dictatorship no single voter determines the outcome. And it's important to understand that this does not mean that the outcome can't depend on a single voter. Rather what it means is that no single voter's preference is the result in spite of how everyone else votes. Another feature we'd want in a voting system is known as non-imposition. This means that, regardless of the candidate, some preference schedule allows them to win. In 1950 Kenneth J. Arrow, a CUNY graduate, proved the following. It is impossible for a social choice function to satisfy independence of irrelevant alternatives, non-itinicity, non-imposition, neutrality, and non-dictatorship. Arrow actually used a different condition than neutrality, but we could use neutrality instead of Arrow's original condition and prove the same result. Now we should regard Arrow's system with some dismay because it says that the perfect system doesn't exist. However, Arrow viewed his result as a possibility theorem. There are five conditions, you can get any four of them in a voting system. So the four we definitely want are, well, non-dictatorship seems like a good idea, non-imposition, that also seems to be a good idea, neutrality pretty much seems to be an obvious requirement, and that leaves the last two. Now a system that failed monitinicity might be objectionable in principle, but here's the thing about monitinicity, we'd never know if it did, and that's because the only way to know that we failed monitinicity is knowing how voters would have voted. In other words, we had to know that somebody who voted for a candidate wouldn't have voted for that candidate in another universe. And we can say the same thing about systems that fail IIA. If we have an irrelevant alternative affecting the election, we'd never know because it relies on knowing how people would have voted. Still, we can do better than the systems we already have.