 So in this video, we're going to look at arrows theorem and prove it formally. So to begin, let's establish the notation that we'll use throughout this video. I'll let n be the set of agents. I'll let o be a finite set of outcomes. So these are the candidates that the agents are going to have preferences over. And the really important thing about o is going to be that there are at least three candidates. So arrows theorem is going to require that we have at least three candidates that we're voting over. I'm going to use l to denote the set of all possible strict preference orderings over o. So I'm going to be speaking here only about strict orderings. That means that agents don't have any ties in their preferences. In the end, arrows theorem is going to end up telling us that any desirable social welfare function can't exist, even if preferences are restricted to be strict. So this restriction to strictness is not going to be a limiting thing. It's actually going to be a strengthening thing. We're going to end up seeing that we can't have good social welfare functions even if preferences are strict. I'm going to use this notation here to denote an element of the set l to the n. So in other words, this is going to be a preference profile, a preference ordering one for every agent. And this is, of course, the input that our social welfare function takes. So recall that a social welfare function is a mapping from a preference profile, a set of preferences one for every agent, to a single preference ordering, which is the social welfare ordering. Going on, I'll denote with this symbol a preference and ordering relation here subscripted by a w, the preference ordering that is actually output by the social welfare function. So as I just said, the social welfare function itself outputs an ordering, and this is going to be the ordering that gets output. Now, when the input to w is ambiguous, I might actually have even more complicated notation here saying precisely which ordering it is. So in particular, this notation here is going to mean the preference ordering output by w when given the preference profile here as input. And I'm going to care about this in particular when I change around which preference ordering it is that I talk about. OK, so now let me formally state the conditions that Aero's theorem speaks about. The first is Pareto efficiency. So recall the idea of Pareto efficiency is that when everybody agrees on how two outcomes should be ranked, if it's the case that every agent thinks that A is better than B, then the social welfare function is required also to output that A is better than B. And this is only the case when everybody is unanimous. If even one person feels differently, this doesn't restrict us at all. So formally, I'll say that a social welfare function w is Pareto efficient if for any pair of outcomes O1 and O2 from the set of outcomes, if it's the case that for all agents I, outcome one is preferred to outcome two, then it must be that the social welfare function also chooses O1 preferred to O2. So the next definition we need to set up Aero's impossibility theorem is the idea of independence of irrelevant alternatives. Intuitively, the idea here is that we want to say that the social welfare function should decide the ordering between two outcomes A and B based only on the relative rankings that all of the agents give to these two outcomes. So it should only be allowed to look at whether or not each individual agent puts A above or below B in order to decide whether A should go above or below B in the final ranking. And this should be true for every pair of outcomes. So let's look at how we say this formally. We'll say that a social welfare function w is independent of irrelevant alternatives if for any pair of outcomes O1 and O2 and any pair of preference profiles, I'll call them preff prime and preff double prime, the following condition is true. So let's look at what this condition here says. This says O1 is preferred by agent i in preff prime to O2 if and only if O1 is preferred by agent i in preff double prime to O2. So in other words, this is saying, when I extend it out to the for all i here, this is saying that all of the agents relatively rank O1 and O2 in the same way in both preff prime and preff double prime. In other words, if agent i likes O2 better than O1 in preff prime, it must also be that agent i likes O2 better than O1 in preff double prime. So the condition says that if this is true, then it must be that the social welfare function, when given preff prime as an input, makes the same relative ranking between O1 and O2 as it does when given preff double prime as an input. So again, in words, what we're saying here is that the social welfare function's decision between O1 and O2 in these two different preference profiles has to be the same if the relative ranking that all the agents make between these two things is also the same in both of these two orderings. Finally, I'll define non-dictatorship as follows. I'll say that a social welfare function does not have a dictator. If there does not exist some agent i for whom the following property is true, for all outcomes O1 and O2, it's the case that if agent i likes O1 better than O2, then it must be that the social welfare function also likes O1 better than O2. So in other words, a dictator is somebody who simply determines the social welfare function. It's the social welfare function if it has a dictator is a function that just listens only to one of its inputs, ignores the preferences of everybody else, and simply spits out the one preference ordering that comes from the one dictatorial agent. Having said all of these definitions, we're now ready to formally define Aero's theorem. So what we can say is any social welfare function W that is Pareto efficient and independent of irrelevant alternatives is dictatorial. The way the proof works is as follows. So for the rest of this video, I'm going to give you the proof of this theorem, which I think is a really beautiful proof. Because we're able here to say something about a space of functions, a space of very complicated functions from sets of orderings into orderings without knowing really what function we're talking about. We want to assume simply that our function is P, E, and I, I, A. It's pre-deficient and independent of irrelevant alternatives. And we'll see that that assumption is enough to show that it must be dictatorial. Now, to avoid sweeping this under the rug, let me again remind you that we've made the assumption that the number of outcomes is at least three. That's important for this proof. In a sense, really, it's important because independence of irrelevant alternatives doesn't really mean anything in the case of only two agents. There can't be an irrelevant alternative if there are only two things to be compared. So we're going to need to make use of the fact that there are at least three outcomes, as you'll see in the proof. So let's get started. So we're going to proceed in four steps. We're going to end up proving four intermediate claims, which will eventually get us to the proof that this social welfare function, which we assume is both pre-deficient and independent of irrelevant alternatives, must also be dictatorial. So first of all, here's what I want to claim. It sounds like a strange thing to claim, but it'll get us where we want to go. If every voter ranks some outcome B, either at the very top of his preferences or at the very bottom of his preferences, then it must be that the social welfare function also puts B either at the very top or the very bottom. So let me draw kind of a picture of this. So think of these as being a bunch of preferences from all the different agents. And what I'm saying here is that each agent has to put B either at the top or the bottom. It doesn't matter which, but they have to put it either at the top or at the bottom. And what I want to claim is that the social welfare function then has to do the same thing. It either has to go here at the top or it has to go here at the bottom. So let's see why that would be true. So the way we're going to do this is we'll consider some arbitrary preference profile that satisfies this condition. Every voter rank puts B either at the very top or the very bottom of their preferences. And let's assume for contradiction that the claim that we've just made is not true. Instead, that we don't have B here, but we've got it somewhere in the middle. Well, in that case, there must be some A and some C where A is higher up in the ordering for the social welfare function and C is lower down in the ordering. That's what it means to be in the middle. There has to be something above it and there has to be something below it. So let's call those things A and C. Well, now let's modify our ordering so that every voter moves C just above A in his preference ranking. So all of these rankings, we haven't drawn it, but we have C's and A's in here somewhere because everybody is ranking all of the different order, all of the different outcomes in their orderings. So let's modify whatever this ranking is by constraining where C and A go. So let's imagine that A was in all of these different places and so on. Let's just move it to wherever C was before. Let's just stick it right above A in the ordering and leave everything else unchanged. And let's call that new preference, whoops, it's called a new ranking pref prime. So now we're gonna start using these definitions that we've assumed about the preference ordering. So we know from independence of irrelevant alternatives that in order for this to change or for this to change, these two facts, which we've assumed are true here, the pairwise relationship between A and B and or the pairwise relationship between B and C would have to change for some of the agents here. However, notice that B occupies an extremal position for all of the voters. And that means that C can be moved above A moving from pref to pref prime without changing any of the pairwise relationships between C and B and between A and B. And thus, we can conclude that in the preference profile pref prime, it's also the case that these two things are true, right? We assume that those two things are true in pref for contradiction. And because nothing has changed, we can use the IIA to say that they must also be true in pref prime. And from transitivity, we can, just the fact that we were getting a strict ordering here, we can also conclude, therefore, that A must be ranked above C in the social welfare function. However, look at the ordering that we've actually made here. Recall that we put C above A in every agent's preference order. Wherever A was, we stuck C just above, right? Well, recall that Pareto efficiency says that any time all of the agents feel exactly the same way about something, the social welfare function has to also feel that way. Here, all of the agents think that C is better than A. And so, Pareto efficiency requires that we would have C above A here. But in fact, we have the opposite than A is above C. So that gives us a contradiction. So what does this all tell us? What we've established is what we have up here, that if every voter has some outcome B at an extremal point in his preference list, it must be that it goes into an extremal point of the social welfare function as well. Well, so far, that sounds fine. It doesn't seem like that gets us into big trouble. Let's keep going and see what happens. This is the crucial step now. So now I want to claim that there exists some special voter, and I'm gonna give that voter a name. I'm gonna call him N-Star, who is extremely pivotal in the following sense. I wanna say that he's able to change his vote at some preference profile. So there exists a preference profile where if he changes his vote, he can move a given outcome B from being at the very bottom of the social ranking to the very top. So there exists at least some circumstance in which some person is able to make a really big difference to the social choice. He can move B from being the very least preferred thing to the very most preferred thing. Let's see why that would be true. Well, let's consider a preference profile, Pref, in which every voter ranks B last, and in which the voter's preferences are otherwise completely unconstrained. By Pareto efficiency, it must be the W also rates B last, because every pairwise relationship between B and something else has that something else being preferred to be, and so B has to be at the very bottom of the social ranking. So now I want to basically let voters, so I have some ordering of the voters from one to N, and I'm going to let them successively modify, so let me draw this. So I've got all of these different rankings here from different voters, and I've got some social welfare ranking here. So we start out with everything at the bottom, and we conclude from PE that B must be at the bottom here too. So then I say, let me one at a time take a voter and take B off being at the bottom and put it up at the top, and let me see what happens. When I do this, there has to eventually be a change. It can't be that I do this for everybody and B never changes, because once I've done it absolutely for everybody and I've got B at the top all the way, then Pareto efficiency again is going to constrain me and it's going to say B has to be at the top. So I know there has to be somebody who in making these movements causes something to change at some point. So I'm going to give the name and star to the first voter in this apparently arbitrary ordering on the voters that I've got here, whose change causes something to become different. So I'm going to get the first guy to move B from the bottom to the top and I'm going to look at what happens here. If nothing changes, I'm going to have the second guy do it. I'm going to keep working my way through until at some point B moves, maybe moves somewhere, something different happens. And then I'm going to say, all right, I found my end star. That guy who made a difference is end star. So now let me give some names to preference orderings because I need them for the rest of the proof. So I'm going to denote by pref one, the profile just before end star moves B and I'm going to denote by pref two, the profile just after end star has moved B to the top. And let's note that in pref one, we have B at the bottom in the social welfare ordering. So our social welfare ordering here has B at the bottom. And in pref two, B is in some different place than it was before. And every voter ranks B at either the top or at the bottom, right? So notice here we have Bs at the top for everybody. Here we have Bs at the bottom for everybody. And here again, we have Bs at the top and Bs at the bottom. But what's changed is end star. So here's the one guy who's actually moved something. And by our argument in step one, both of these profiles have B in extremal positions. And so it must be that the social welfare function also puts B in an extremal position. And because we've defined end star as the person who causes B to change, it must therefore be in this profile pref two that B is at the top. And notice that that means now that we've proven the thing we set out to prove that there exists some voter who is able to change his vote at some profile at this profile in order to move some outcome B from the bottom to the top. Okay, let's get to step three. So now I want to reason about this same guy, end star. So recall, end star has special power over outcome B in this very narrow way that there exists this one special profile that we constructed, pref one, where he can move B from the bottom to the top. Well, now I want to reason about this same guy and I want to claim something that all of a sudden seems very strong, that he's a dictator over any pair AC that doesn't involve B. What does that mean? It means that the social welfare function must make every decision between any pair AC where I'm using those variable names just to denote anything that isn't B simply by listening to what end star thinks. So if end star likes C better than A, the social welfare function will as well and it just doesn't matter what anybody else thinks. So that's what we want to claim here. So how does this argument work? Well, so we're going to begin by choosing one of the two elements from the pair and let's call the one that we choose A. So now I'm going to construct a new preference profile, pref three, which differs from pref two in two ways. First of all, I'm going to move A to the very top of end star's preference ordering, leaving everything else unchanged. So I'm going to take A wherever it was here and I'm going to move it up to the top. You'll notice, by the way, in these pictures, if you can read it, some of the variable names are bold and some of the variable names are not bold. The ones that are bold are things we've made explicit assumptions about and the things that are not bold are given for illustrations but we haven't actually assumed. So if you can make it out, you'll see here A is not bold because we haven't assumed anything about where it goes and over here A becomes bold because now we've assumed that it goes to the top of end star's ordering. All right. So anyway, we're making two changes. So our first change is to move A to the top of end star's ordering. And that means that for end star, A is most preferred, or A is preferred to B, which is preferred to C because A and B are both at the top. So wherever C is, it must be further down. And second, we're just going to arbitrarily scramble the relative rankings of A and C for all voters other than end star without changing the position of B, which is always extremal. So we're just going to kind of move around A and C in some kind of arbitrary way, but we're going to make sure that we still leave B in the same place for everybody else that it always was. All right, so that's our preference profile, Prf3. Now notice that in preference profile Prf1, we had that A was preferred socially to B and the reason we can conclude that is that we know that B was actually at the very bottom of the social ordering. So everything was preferred to B. And so A in particular must have been preferred to B. Now, when we compare Prf1 to Prf3, the relative rankings between A and B are the same for everybody because B is in the same extremal position for everybody except for end star who moves B, but we've then changed it to put A above B. And so it's the case that these guys over here have B above A in both cases. These guys have A above B in both cases and end star also has A above B in both cases because here we actually moved A to B above B. So from this fact, we can conclude that in Prf3, it must be that A gets socially preferred to B because by independence of irrelevant alternatives, the social choice between A and B can only depend on their relative rankings by the different agents. And we've just seen those relative rankings are the same in Prf1 and in Prf3. Now we're gonna say something similar comparing Prf2 and Prf3. So in Prf2, we had that B was socially preferred to C because we know that in Prf2, B was at the very top of the social ordering. And so wherever C was, it was somewhere further down. And in a similar way to what we argued before, the relative rankings between B and C are the same in Prf2 and Prf3. Let's see why that is. So we have, for all the agents other than end star, B is in the same extremal position in both cases. So wherever C is, its ordering must be the same. And for end star, things are a little bit different. End star used to have B in the very top position here and here he has B in the second position, but the thing above it is A. So in both cases, wherever C is, it's somewhere downstream. And so again, we see that the, because the relative rankings are the same in both places, it must be that the social ordering again is the same. So here we had that the social ordering says B is higher up than C and so here it must also be that B is higher up than C in the social welfare ranking. So using these two facts that we've just established and transitivity, we can draw the conclusion that the social welfare ordering must give us that A is preferred to C in Prf3. Okay, why do we care about that? Well, let's construct one more preference profile. I promise this is the last one, Prf4, by changing Prf3 in two ways. First of all, let's arbitrarily change the position of B in every voter's ordering while keeping all of the other relative preferences the same. So we're going from Prf3 to Prf4 by taking B and just kind of sticking it somewhere else. And, but we're not going to move around any of the other relative preference orderings. Secondly, we're going to move A to an arbitrary position in N stars ordering with a constraint that A is kept higher than C. So let's look at that. So here we have A in a really special place. Here, we're just going to stick it further down to here. But we're going to make sure that it's still higher than where C is. And what I want you to observe now is that all voters other than N star at this point have completely arbitrary preferences because the only things that we had assumed about those voters ever before was the position of the Bs. Those are the only things that we had bold in those figures because that's all we assumed. All this other stuff here is just completely arbitrary. Furthermore, N star also has arbitrary preferences except for the fact that A is preferred by N star to C. Now, what do we do with prep four? Well, in prep three and prep four, all of the agents have the same relative preferences between A and C, right? So we've moved things all around, but we've done it in a way that still has all of the agents in three and four having the same preferences between A and C. And we know that in prep three, the social welfare function has to rank A higher than C. So by independence of relevant alternatives, it must be that the welfare function also ranks A above C in prep four. But that's pretty interesting because the only thing we've assumed about prep four is that A goes above C for N star. And that means that we've established the thing we wanted to establish, which is that N star is all by himself able to determine the relative ranking between A and C regardless of what everybody else says. So that makes him a dictator over the pair AC. Well, clearly I'm gonna want eventually to say N star is a dictator over everything, not just over AC, but also over pairs AB, pairs that do involve B. So this turns out actually to be very easy to show. So let's consider some third outcome C. So I'm claiming now that N star is a dictator over pairs AB. So let me think about some third outcome C. Well, by the argument that I already made in step two, there's some voter, let's call him N double star, who's extremely pivotal for C, right? So I can make that same argument that I made before where C starts out at the bottom for everyone and the social ordering has it at the bottom. And then I started moving things to the top until something changes and then eventually I can see that it changes there and whoever the guy is who made a change, let's call that N double star. And in principle, maybe this is a different guy, there's nothing in step two that says it has to be the same person. It just says there was nothing special about B that made it the thing that had to move around. It would have worked for anything. So it would work, for example, for C. Well, now I can use the argument in step three to say that this guy that we've just identified N double star is a dictator over any pair alpha beta, which doesn't involve C, right? I'm just basically running through the arguments I had before, but renaming the variables. So of course, AB is an example of such a pair alpha beta, right? So by this argument, there's somebody who's a dictator over the pair AB. So what I want to show is that that person is N star. You know, I might worry that there are different dictators for different things. But notice that I've already observed that N star is able to affect the way that W relatively ranks A and B. For example, we saw this in the movement from the profile pref one to pref two, right? When we had this profile where B was at the bottom, whoops, I'm doing that wrong. Let me do that again. We had this ranking where B was at the top for a couple of people who was at the bottom for N star. It was at the bottom for other people. And then we move it up to the top. We saw that that caused the social welfare ordering to take B and move it from the bottom to the top. And that means that N star is able to affect the relative ranking between A and B because wherever A is, it's somewhere in the middle here in the social ordering. And so the relative ranking between A and B got affected by what N star declared here, by this change that only N star made. And what I've now just argued is that there has to be some dictator over AB pairs. And so if N star was able to make any difference to AB pairs, he has to be that dictator because it's not possible to have both a dictator for AB pairs and a different person who ever makes a difference to the AB ranking. And so that means that this N double star and N star must be the same agent. And that means N star is a dictator over everything. N star is just a full dictator. And that means no matter what social welfare function I have if it's pre-do efficient and if it's independent of irrelevant alternatives it has to be some dictatorial social welfare function which means the only freedom you have in such a social welfare function is the freedom to decide which person is the dictator. And if you have any other social welfare function which isn't dictatorial then it must be that it's either not pre-do efficient or it's not independent of irrelevant alternatives. And that explains why we have voting systems that cause things like the number of votes that Ralph Nader gets to affect who wins the election in Florida. That's a case of not being independent of irrelevant alternatives. And that's usually the axiom that we relax because not being pre-do efficient would be really terrible. That means if everybody all agrees about something sometimes the social welfare function would do something different anyway. That's something that we're usually not willing to tolerate. And so the practical consequence of Ero's theorem is that we end up having social welfare functions that are not independent of irrelevant alternatives.