 This video is going to tell you at a high level probably the most important and surprising thing that we have to tell you about social choice, which is that it's impossible to come up with any social welfare function which is sensible in a particular formal way. So first of all, let's remember what a social welfare function is. A social welfare function is a voting rule where every agent tells me a set of preferences. So it gives me a total ordering over all of the choices that we have to make over all of the candidates. And getting this whole set of different rankings from all of the different agents, the social welfare function gives me back a single ranking, which is the social choice. So think about voting over a set of candidates, everyone has preferences over all the candidates that we're going to vote over, and getting back a single ranking that is the kind of aggregate ranking for all the agents. So what would we want such a ranking to do? What would it look like for such a ranking to do a good job? Well, here's one thing that seems pretty obvious that such a ranking function should do. We'll say that a ranking social welfare function W is Pareto efficient if anytime it's the case that all of the agents agree on the ordering of two outcomes. If everyone thinks that candidate A is better than candidate B, then the social welfare function also agrees about that part of the ordering. So it might say all kinds of other things about candidates C, D, and E, but if everybody thinks that A is better than B, then the social welfare function also is going to tell us that A is better than B. It's going to put A somewhere higher in the ranking than B. So, and that's all we're requiring by Pareto efficiency. We're just saying only in the case when everybody agrees, absolutely everybody agrees, the social welfare function has to rank in the appropriate way. A second thing we might want is what we call independence of irrelevant alternatives. We'll say that a social welfare function is independent of irrelevant alternatives if the way that I rank between A and B only depends on the way that all of the individual agents ranked A and B. So if I want to decide in the social ordering how it is that I compare A to B, I should do that only by looking at how all of the individual agents are relatively ranked A to B. So I shouldn't care about how everyone felt about A and C or about B and D. I should just look at the A, B rankings for everyone and that should tell me how the social welfare function ranks between A and B. Finally, I'll say that a social welfare function has a dictator. If there exists a single agent whose preferences we always just return as the social ordering. So in other words, if I've got N different agents and they always give me some kind of order in and I just say, you know what, I only care about agent three. Everybody else, it just doesn't matter what you tell me. I'm going to throw it in the garbage. I'm not going to think about it at all. I'm just going to give you back agent three and I've decided in advance who agent three is is always going to be the same guy. That's what it means for a social welfare function to have a dictator. Now, it's kind of terrible for a social welfare function to have a dictator because it means that it's not aggregating in any meaningful way. It's just always reporting back what one person thinks. Notice that a social welfare function that does have a dictator trivially satisfies independence of relevant alternatives and Pareto efficiency. Because Pareto efficiency says anytime everybody agrees, the social welfare function has to do the right thing. And when everybody agrees, in particular, the dictator also agreed. So we would get Pareto efficiency for free. Likewise, we would get independence of relevant alternatives for free because we're only going to care about the dictator's A-B rankings in order to decide the A-B ranking. And so we're not looking at anything else from any of the other agents. So we see we can satisfy our two properties from a dictatorial social welfare function, but dictatorial social welfare functions are pretty undesirable to us. Well, here's the punchline. Kenneth Arrow in 1951 famously proved the following theorem. Any social welfare function over three or more outcomes, so candidates in elections that have more than two candidates, that is Pareto efficient and independent of relevant alternatives must be dictatorial. So in other words, these two things that seem very natural that we would want, Pareto efficiency and independence of relevant alternatives, are so constraining on the set of social welfare functions that they imply that we would have to have a dictatorial social welfare function. Since we can't possibly live with a dictatorial social welfare function, what that really means is that we have to give up either Pareto efficiency or independence of irrelevant alternatives. And that's pretty bad news because these are both pretty natural things.