 Let us now start proving the arrow's impossibility result. So just to remember what arrow's impossibility was. So if we had three or more alternatives in a in the social welfare function domain, so we have a social, Arabian social welfare function f and let's recall what that f was. f was actually a mapping from the ordinal preferences of all these agents. It is taking as input all the ordinal preferences of the agents and popping out one complete ordering over all these alternatives. And if this Arabian social welfare function satisfies this property of weak Pareto and IIA, which we have discussed in the previous module, then it must be dictatorial. So that is the result by arrow. And we will use both these properties weak Pareto and IIA to prove that this result holds. So in order to proceed with the proof, we need the notions of decisiveness. So now what is decisiveness? Informally, you can think of this as a group of individuals, group of agents. If they decide a specific ordering over a pair of alternatives, then that alternative will also come out as the social ordering over those alternatives. So you can imagine that out of this n agents, there is a group of agents who are keeping a specific pair of alternatives, let us say A and B in a specific way. And that ordering is being reflected when you are looking at the final ordering given by this Arabian social welfare function. So that is the intuitive notion of decisiveness. Let us make it formal. So suppose we have this ASWF, which is given by this. And suppose we have a group, a subset of the agents denoted by G. And of course, that is not empty. We are going to call that set to be almost decisive over this alternatives A and B. If the following thing happens, and we are now familiar with this notation, that is whenever this happens, whatever I write within this square brackets, whenever that happens, this implies that this outcome, this is the implication of that happening. So what it is saying, it is saying that for all those agents which are in that group, in that almost decisive group, A is strictly above B. A is strictly preferred than B. And if you are looking at all the agents which are not living in that group, that almost decisive group, then B should be strictly above A. So if this happens, then immediately the conclusion is that in the final outcome of this Arabian social welfare function, A should be strictly above B. So that is what it means. So why is it almost decisive? Because it is almost like saying that if this group decides that A should be above B, strictly above B. So even in the most opposing situation where all the other agents are thinking that B should be strictly above A, still the conclusion among A and B of that group or G is still prevailing. So that is why we call this an almost decisive thing. We will essentially denote this almost decisiveness using this notation that D of G. So this G is essentially the group which is almost decisive. And to denote that this is almost decisive, we use the upper bar. And this is between these two alternatives A and B. Now we make this notion of decisiveness a little more stronger. Now we don't really care whether the other agents are actually in favor of B over A or they are still in favor of A versus B. So you can see that now in the if condition, the second part has actually disappeared. And we have already did this discussion in the previous definition of weak Pareto that whenever in the if condition you have less constraints. So of course if this specific condition is true, then it is actually a superset of this condition. If this condition you can call capital Y and this condition here, this if condition you call X, then Y all the preferences that will satisfy this Y condition will certainly be larger. And you have this smaller set which is X. These conditions will be more strict because not only you have the condition of Y, you also have something more. So therefore the if condition will be smaller. So this Y is a little relaxed condition. And even on that relaxed condition, you are having the same conclusion. So that is the reason when when you are looking at a more relaxed condition, but the conclusion is the same, then that property, that decisiveness property becomes a little stronger than the other condition. So of course, and we are going to denote this decisiveness, the property of decisiveness with this shorthand notation D, G and there is no upper bar anymore. So this is decisive between A and B. This group D, G is decisive over A and B with respect to that Arabian social welfare function F. And it is very natural to conclude that this implication holds. If a group is decisive, then it is almost decisive. Now in order to, so we define this and now we are going to jump into the proof. And before I get into the details of the proof, let us just discuss the overall idea, how we are actually going to prove this result. And it will be, the theorem will be proved in two parts. The first part is essentially what is known as filled expansion lemma. So if we are looking at a group which is decisive over a pair of alternatives, we will actually show that this is almost decisive, not only decisive, it's even weaker, almost decisive. So remember that this, if it is decisive, that's a stronger condition. We are going to assume a little weaker condition. We are going to assume that this is almost decisive over a pair of alternatives. Then the filled expansion lemma says that it is going to be decisive. Now look at this. So this is, we are concluding a much stronger condition, stronger result that it is decisive over all pairs of alternatives, not only between those two things. So that is why it is called the filled expansion, as if you are expanding the idea of decisiveness over all possible alternatives. Now once we have that, then the second part is essentially known as the group contraction lemma. And as the name suggests that if you have a group which is decisive, then this group contraction lemma says that you can look at a strict subset of that group which is also decisive. And now you can already begin to see that why these two parts essentially proves the lemma, proves the theorem. Because if you look at the filled expansion lemma, you know that if you look at the whole set, so if G is actually equal to n, then by weak palette we already know that that is going to be decisive. Because if everyone is preferring a over b by weak Pareto, we already know that the outcome will be, in the final outcome a will also be above b. Now if you now apply filled expansion lemma, then you know that this is, so this is decisive over this pair of alternatives and you can actually replicate that for any pair of alternatives. So it is going to be decisive over all pairs of alternatives. So that is the first part. Now I can repeatedly use the group contraction lemma to reduce the set size, the set of almost decisive individuals until you have only one individual left and that individual is going to be the dictator. So that individual's preference over any pair of alternative is going to be the final outcome. And that essentially is the erosion possibility result. Okay, so that is the overall idea. Let us now go into proving this, each of this lemmas one by one. So the first one is the filled expansion lemma. What does it say? So suppose we have this Arabian social welfare function f that satisfies weak Pareto and IIA. Then for all such alternatives a, b, x and y with the condition that g is essentially a subset of n and it is not empty and certainly a is not equal to b and x is not equal to y. Then what is going to happen is that if this set g is almost decisive between a and b, then it is going to imply that it is going to be decisive over that x and y, where x and y are arbitrary. So we can immediately observe as a pi product of this filled expansion lemma is that whenever we are talking about these two properties weak Pareto and IIA, these two notions of decisiveness are identical because we already know that if something is decisive, then that is going to be almost decisive. Now this is saying the other way around that if you are almost decisive, then you should also be decisive because x can take values like a and y can take values b. Then in this case, almost decisive if a set g is almost decisive over this pair of alternatives a, b, then that is also implying that it is decisive over the same pair of alternatives. So that is just a remark. Now in order to prove the lemma, we will have to consider a bunch of cases and here I am listing down all the cases exhaustively. Let us look at that first and we are going to go over them in a specific order so that it is helping us in the proof. So of course, we will have to look at different conditions on x and y. So the first condition could be that x is neither a nor b and y is also neither a nor b. Then you can look at x being equal to a but y is not equal to a or b and x can be equal to b and y is not equal to a or b. Similarly, the other way around x y can be equal to b or a and x is not equal to any of a or b. And the last two conditions are that x is equal to a and y is equal to b and the other way around x is b and y is a. So this essentially lists down all possible cases. Just that we are going over all these cases in a specific order such that the proof we can use the first two cases in the next set of cases. All right, so let us now focus on case A, case 1 where we are supposed to prove that if it is, if the set G is almost decisive over a and b then it should be decisive over a and y where y is not equal to a or b. Now how, so let me tell you the general strategy, how we are going to do it. We are going to pick some arbitrary R. We will have to show that this is decisive. That means we will have to show for all arbitrary R on which these if condition holds. So remember the definition of decisiveness, the if condition of the decisive part. So you might be wondering what is that if part? So if is nothing but this part. So if this holds, then we will have to show this. So in this context what is that if? So you can pick some arbitrary R on which because we will have to show this decisiveness between a and y, a is strictly preferred over y for all the agents in that group G and then we will have to show that when you are looking at the aggregated outcome there also a should be strictly above y. So this is something that we will have to show and this is what we are given. Now we are going to do it via a different preference profile which we are going to call R prime. And why is that R prime important? Because you see that the property of IIA, the independence of irrelevant alternatives are between two different preference profiles and we are going to pick this R prime in such a way that the relative position of these two alternatives a and y remains the same for all the agents as you have chosen in R. So that we can actually use IIA and we will do construct this R prime in a clever way. So how are we going to construct it? We look at all the G's on which a is above b, b is above y. I do not really care about the other alternatives. They can be anywhere. Just a should be above b, b should be above y. And for all the other cases and this is required because we have to have a above y. What are we doing in the rest of the agents case is that we are going to put b to be strictly above both a and y and between a and y we are going to retain the same ordering for these agents. What was there in R? So you can look at this RI prime. We have kept a and y in such a way. If in the original RI, a was above y, then for that particular agent we will keep a above y. If it was y above a, then we would have kept that. If they were indifferent then we will keep that indifferent. And this is something that we are going to do for all the agents. So the relative position of a and y remains the same for R prime as well as in R. Now, let us first use the first condition that is this is decisive between a and b. So we have, we already know, I mean this is the given condition for case one. So since this is almost decisive, we can use that because here you can see for G a is above b and b is above a for all the agents which are outside G. So then by the definition of this almost decisiveness, we can confirm that a should be strictly above b in this R prime because this is the profile R prime. Fair enough. Now we have the next condition is that we have weak pair it over b and y. And now we can see that b is strictly above y for all the agents. And weak pair it over just says that it is going to be then b should be strictly above y in the final outcome as well. Now we have this a is strictly above b, b is strictly above y for the same outcome f of R prime. Now we can apply transitivity which will say that a is strictly above y. Now that is exactly what we wanted. In this case, we know that R prime has the same relative ordering between a and y for all the agents. And we have kept it in that way. So for all the agents, for all the agents, the relative position between a and y is the same in R prime as well as in R. And we have also shown that in this final outcome of f of R prime a is strictly above y. Now we are just going to use the condition of IIA because what IIA is saying is that if the relative position remains same, then the final outcome between that two alternatives will also remain same. So by that we can conclude that even in R, in f of R, a should be strictly above y. And that is exactly what we wanted to show. So remember this, we need to show that this is the case. And therefore we have proved that this group G is decisive between a and y. Okay, so that's the first case. The second case is that if you have the decisiveness, so the left hand side remains the same, it is almost decisive. The group G is almost decisive between this pair of alternatives A and B. Then we will have to show that X which is neither A nor B and between these two alternatives X and B, G is decisive. So as before, we will have to pick some arbitrary R such that X is strictly preferred over B. So this is the if condition of the decisiveness. So this strict preference between X and B holds for all the agents in G and we don't care about the other agents. We have to show that X is strictly above B even in the final outcome of f of R. And as before, we are going to construct another R prime and in a very similar or complementary way. Now what we have to pick is X is strictly above A, A is strictly above B. And we are going to pick the case that X and B both are more preferred than A for all the agents which are not in G. But again, between X and B, we are going to keep the ordering the same as R. So we have constructed R prime in that way. Now because this is almost decisive between A and B and you can see that here A is above B for all the agents in G and outside G, B is above A. So then by the definition of almost decisiveness, A should be strictly above B in this preference profile R prime, F of R prime. Now using this weak Pareto between X and A because X and A has the same ordering for all the agents. So everybody is actually preferring X over A. So weak Pareto should say that X should be above A. Now we have this again, we are going to use transitivity between these two things. Then we can have that X is more preferred, strictly more preferred in this F R prime than B that we already know by transitivity. Now once we have that and because between X and B, the relative ordering over all these agents are same in R and R prime, then we can use IIA and control that X should be strictly above B even in F of R and that ends the proof. So we have already proved that this G is also decisive for X and B. So that is what we wanted to prove. Okay so now that we know how to prove case one and case two, case three and the rest of the cases will essentially use these conditions in some clever way. I am just going to go over this case three and rest of the things you can just do a reading exercise. It's just I want to skip that because it's just repetitive. So what we have here, so let's go back to case three. So case three was the case where if you have almost decisiveness between A and B, then you are going to have decisiveness between X and Y where none of them are equal to A or B. So how should we do that? So we do it one by one. So first we apply case one where we know that this is going to be, so if you keep the first condition same, the first alternative same, then this almost decisiveness implies the decisiveness here. So between A and B and here A and Y where Y is not equal to A or B that was case one. Now because this is decisive by definition this is going to be almost decisive, almost decisive between the same alternatives. Now you can apply the second case that we have proved. So now the second alternative remains same and the first alternative changes and of course X is neither A nor B. So that will be the conclusion. So using just case one and case two along with the definition of decisiveness we get case three. And similarly we are going to use the rest of the cases, case four, case five, case six and case seven. They are just carefully using the conditions that we have already proved. I am just skipping that. Let us now come to the second part of this theorem which is the group contraction lemma. So as we have already mentioned that if we have this adobe and social welfare function that satisfies week period when IIA and we have a group non-empty group which has at least two or more members because if you have just one member then we are already done. We have proved that that group is decisive which means that that is the dictator. So let us look at only those sets, such groups which are decisive which has two or more members. Then what it says is that there exists a G prime which is a strict subset of G and it is not empty which is also decisive. So you can actually reduce it to a smaller set which is a strict subset and still continues to be decisive. So let us try to prove that. So suppose we essentially partition this group. So we are going to construct because here the condition is that G is already decisive that is given to you. So we can work with any kind of preference profiles. So let us look at this group G1 where the alternative. So remember that in order to do this construction we need at least three alternatives and this is why where the condition of at least three or more alternatives of erosion possibility will be useful. So in G1 you have this ordering of A over B over C for G2 which is the remaining group which is decisive G minus G1 you have this ordering of C over A over B and for all the other agents which are not decisive, I mean which is outside that decisive group the alternative the preferences are B over C over A. So now we can what we can see is that if you look at A and B they are actually above A is above B strictly above B for the entire decisive group. So this is true for all the I is in G and because G is decisive we can conclude that A is strictly preferred even in the final outcome F of R. So let us save this as a as a sub result one. Now what can happen between these two alternatives A and C let us consider that there could be two cases I mean the final outcome can either keep A strictly above C and in that case what we will show is this said G1 will be a decisive set yeah so because we have constructed it in such a way that both G1 and G2 are strict subsets of G we are going to first show that in case one G1 is going to be decisive and in the other case where A is not strictly preferred over C that means C is at least as much as as preferred as A in that case we will show that G2 is going to be decisive and that will end this in this proof. So now we are going to consider so in the first case where A is strictly above C in the final outcome we are considering G1 so what we observe is that A is strictly above C for all the for all the agents in G1 and the other thing is true so C is above A for all the other agents so everybody outside G1 C is above A now so this should remind you some sort of a property so this is almost decisiveness now in order to complete the proof that G1 is decisive what we will have to show almost decisive we will have to show that A is strictly preferred over C in the final outcome okay so we have constructed R and now we are we are going to consider all such R primes where this condition is going to going to hold so A is strictly preferred over C for all the agents in G1 and C is strictly preferred over A for all the agents outside G1 so you can I mean this is not a unique thing you can have multiple preference profiles like R primes let's consider all such R primes where this holds then what we know is by IIA A should be strictly preferred over C as well in those R primes because here in this case one A is strictly preferred over C then this should be the case that for all the other things using IIA that should also hold so notice that IIA is a very powerful condition I mean it is looking at all possible preference profiles and it is connecting with this preference profile so we are getting a handle over all possible preference profiles once we have a specific preference profile and there is a certain kind of an outcome so once that is true then we can actually say that G actually this will be G1 this set G1 is actually almost decisive over this alternatives A and C and now we already know the field expansion lemma so because it is almost decisive between a pair of alternatives it should be decisive over all pairs of alternatives which we just say by the by the name decisive because there is no such pair of alternatives anymore so we can just say it is decisive all right so now in case one we know that G1 is going to be decisive when we have case two which means that this is not true which means C is at least as as good as A in the final ordering F of R then we are going to use this in conjunction with this condition one so you know that F is strictly preferred over B and we also know that C is strictly weakly preferred over A then using these two things using transitivity we can conclude that C is strictly preferred over B so now we can look at G2 and G2 we look we have considered that in a very specific way C is strictly above B and for all the agents outside G G2 B is strictly preferred over C so this is exactly the same kind of if condition that we have for for almost decisiveness and we also know that C is strictly preferred over B under this preference F of R so now we we use we construct all r primes on which this is true just very similar to case one and for all those cases using IIA we can say that G2 is almost decisive between these two pair of ordinates B and C and then we again apply the field expansion lemma to conclude that G2 is going to be decisive so what we have shown is that you can always come up with some strict subsets G1 or G2 of this original set G which was a decisive set and construct preference profiles in such a way such that you have either G1 to be decisive or G2 to be decisive and that concludes the proof