 So let's talk about an important result in voting theory called Aero's theorem. And what this is based on is some of the things that we would like to have as a social welfare function or as a voting system. So let's consider what some of those really desirable features are going to be. So we'd like it to be neutral, which is to say that if I rename the candidates, the election result won't change. I'd like the system to have this feature of non-imposition, which means that any candidate has some voting profile where they are the winners. We'd also like the system to have monotonicity, which is that if I have a winning candidate and there is more support for that candidate, if some people are able to retroactively change their votes to support the candidate, that candidate should still win even after this re-ranking. And then finally, we would like the system to have what's called independence of irrelevant alternatives. So if I have one candidate A who outranks another candidate B, then I can only change the social welfare function ranking of these two candidates if at least one voter changes the relative ranking of these two candidates. And perhaps the most important thing we'd like is for this system to have non-dictatorship, which is to say there should not be any single voter who can, by his or herself, completely determine the societal ranking. Nobody should have absolute determination power. It's also convenient if our system is resolute and we're able to resolve ties deterministically. It's worth noting that while our own voting systems tend to be resolute and that we do have some mechanism of resolving ties, that mechanism is often not non-deterministic, but rather is stochastic. It depends on the luck of the draw, literally. Now, there have been a number of recent elections where there is a complete tie in the number of votes for the two candidates and the winner was then decided by drawing cards, rolling dice or something similar. That was a random determination of the final victor. Well, in 1950, Kenneth J. Arrow proved the following result. Suppose I have a resolute social welfare function and I would like it to have neutrality, non-imposition, monotonistic, independent of irrelevant alternatives, and non-dictatorship. Arrow's theorem is pick four. You get to have four of these five conditions, but you cannot have all five. And this actually has a relatively simple proof. So, let's begin with an election profile where some candidate A outranks some of a candidate B. Now, by non-imposition, by a non-imposition assumption, I know that such a profile has to exist. There has to be some profile where A outranks B. And so what I'll do is I've made my decision A outranks B and I'm going to send the voters into one of two rooms depending on whether they individually ranked A over B or whether they ranked B over A. And so I have a bunch of voters and everybody in this room ranked A over B and all the voters in this room ranked B over A. Now, I'm also assuming independence of irrelevant alternatives. And what that means is that these voters can rank any third candidate C any way they want to without changing this result that A outranks B. So, however these voters rank candidate C, this fundamental feature, this fundamental result isn't going to change. And so I'll have some voters change their ranking or set their rankings and I'll rank the voters, I'll have the voters rank them this way. So, some voters will rank A more than B more than C and other voters will rank C higher than A and still higher than B and then all of these voters will change their rank B higher than C higher than A. Now, the important thing here is that all the voters who ranked B higher than A still rank B higher than A, all the voters who ranked A higher than B still rank A higher than B. So, independence of irrelevant alternatives says that if before this voting profile produced A outranking B then however I change the rank of C, I'm still going to have A outranking B. Now, here is an important question. Where does C fall? And so there's two things to remember. The system is resolute so the ties don't occur. So, either A has outranked C or C has outranked A. One of these two things has to have happened. Well, let's take them on a case-by-case basis. Suppose A outranks C. So, A outranks B, A outranks C. Now, what we're going to do is that we're going to take all of these voters who ranked C higher than A to the other room and what that's going to do is I'm going to have one room full of the voters who ranked C higher than A and then the other room is going to be where the voters are who ranked A higher than C. So, because of the way that I've set the voters up, we can just move this ballroom partition over and now I have my two rooms. Everybody here has A outranking C. Everybody here has C outranking A. Now, remember, the system is also neutral and what that means is I can rename the candidate. So, I'll rename BX, I'll rename CY and if I make those changes, neutrality says nothing changes except for the names. So, now A outranks Y, Y outranks A, A outranks X, A outranks Y. Well, independence of irrelevant alternatives means I can focus on two candidates at a time. I don't worry about a third candidate, so I'm just going to focus on A and Y and the actual ranking of X doesn't make a difference. So, I'll ignore X and so here I have an election profile. All of these people have A outranking Y. All of these people have Y outranking A. The election result is that A outranks Y and once again, neutrality means I can rename the candidates. So, now I'm going to rename Y as B. And, well, look at what's happened here. I now have a new election profile where A outranks B, but whereas before I needed all of these people to rank A over B in order for A to win, I only need this group of people to have A outranking V and then A still wins. Well, that was under the assumption that A outranks C. Well, maybe C outranks A. So, let's go back to our original situation of C outranks A. By transitivity, I have C outranks A, A outranks B. So, that means C is going to outrank B. So, this time I'm going to separate the voters into different rooms depending on how they ranked C and B. So, this time I'm going to send everybody who ranked C higher than B. I'm going to leave them in this room. And then everybody else is going to crowd into this other room where B outranks C. And, again, neutrality and independence of irrelevant alternatives will eventually get us to a new election profile where, again, A outranks B. But, again, this time I have a smaller group of voters who need to have A outranking B in order for the social welfare function to return A outranking B. Well, key idea in mathematics, anything we do once, we can do any number of times because I've been able to take a group and send some of them to the other room without changing the election result. I can continue to do that. And I can continue to send people from this room into this room without changing that election result. And, eventually, what's going to happen is there's going to be one person left in the other room. And this person says I'm going to rank A higher than B and that's what is going to happen. Now, because of neutrality, again, this means that they can put A above every other candidate. So whoever the candidate is, A defeats that candidate. And so what that means is that whoever they decide is at the top of the ranking list is going to be at the top of the ranking list. As long as we also have monotonicity. Remember, the peculiarity about monotonicity is it may be possible for some of these people who previously ranked A lower than some candidate. It may be possible for them to raise the rank of A and alter this. Well, if we have monotonicity, that's not possible. The people in here can't lower A's rank by raising it. And so what that means is this one person left in the room is the dictator and they are able to set the social welfare ranking by how they rank the individual candidates.