 For a moment, let's forget about the truthful revelation. So the things that we have discussed in the past two modules About truthfulness. Let us set aside that for a moment and just talk about aggregating opinion So players have all the agents have different preferences And in this context, we are only talking about ordinal preferences and not cardinal preferences even though all the results that you prove here will also Follow in the cardinal preferences Can we actually create social preference orders from individual preferences? This is the question that we are going to ask and this is essentially the setup of arrows social welfare function so arrows this Setup is essentially named after Kenneth Joseph arrow who has actually given these results So in this setup, what do we have? So we have a finite set of alternatives the outcomes that we can pick and let's say there are impossible outcomes here And we also have a finite set of n players Now what are the? How are the preferences or the the types of each of these agents defined? Each player has a preference order. This these are their types which is capital ri over a and This ri is is a binary relation. So if you don't know what is a binary relation. It's just a subset so a Binary relation ri is nothing but a subset could be a The whole set also a cross a so you pick an element Let's say a and another element from the same set. Let's say B and you are Using this tuple a and B to denote that a comes before B So what does that mean in in this context of preference? So we are going to write this Notations we will use this notation interchangeably a comma B or a ri B So where we are also making sure that this is the preference order So this means that a is at least as good as B So a is at least as much preferred as B. So you can interpret it in that way Now we are going to also put certain restrictions of certain kind of properties on this preference order ri so here are a bunch of Properties of ri that we are going to assume So the first thing is about completeness. We are going to say that every pair of alternatives a and B that you pick from this set a Either a should be at least as preferred as B or B should be at least as preferred or as a Or the both can happen But it will never be a case that you cannot really compare between two alternatives Even if I give you two alternatives, you will not be able to save whether you prefer this at least as much as the other one Or not that will never happen. That is why it is called completeness You always have some preference ordering among any pair of alternatives in this space a The second property is that of reflexivity which says that If you look at a that is at least as as good as a So you prefer a at least as much as a itself So the third property is about transitivity This is very natural in the context of preference orders Which says that if you prefer a a at least as much as B and be also at least as much as C Then it's very natural to assume that you prefer a at least as much as C. So this conclusion should hold and This three properties we are going to assume all of this although we will make certain small changes whenever we go to Some more restricted preferences as we'll discuss later So set of all preference ordering. So if you look at all possible orders Strict or weak, we'll discuss that later. Let us put all those things in the set R script R On top of that, we are also going to define some ordering to be linear if for every a and b Now you are looking at the same pair of alternatives And you if you have a at least as preferred as be and be also is at least as preferred as a So this is this both condition Both these two things happen together then it must be the case that a should be equal to be So that means that you cannot really have any Indifferences so when can you have a at least as good as be or and be At least as good as a is that when you have preferences like let's say C is your top preference and A and B are at the same level. You you cannot distinguish between these two things in terms of your preferences and D is possibly below So you can say that for these two alternatives a is at least as preferred as be and Also, B is at least as preferred as a Right. So if you if you consider this to be equal to our But now in the case of linear ordering, we are actually ruling out that possibility So it is saying that if both these two things happen then a and B must be equal So there cannot be two different alternatives, which are sitting at the same place Let's say a and C. They cannot sit at the same place The preferences in this case can be see a B D Something like that where you can have a You can clearly say a is strictly preferred over B or B is strictly preferred over a but not both and similar to the weak preferences all linear orderings or the strict preferences are also are denoted by this Notations script of B. So because we now know that what is a strict preference and weak preference we can look at any arbitrary ordering which could be a weak ordering and Decompose it into two parts. So the first part is the asymmetric part Which we are going to denote as PI and there is a symmetric part, which is II So let us look at an example to understand this better So as we have said that RI is nothing but a and then B and C are at the same position So they are sort of equivalent Alternatives for this player I and deep stands below So if you look at if you write this down in the In the form that we have defined. So it is just a binary relation. Then we can actually write it down in the form of This pairs. So we can say that a and Then B and C at the same place and D at the bottom can be written as the collection of all these things Where a is strictly above B a strictly above C Similarly B is strictly above D and C is strictly above B But for B and C we can have both these two things So B is at least as good as C and also C is at least as good as B This is happening because they are actually indifferent in the preference order So now that we have this set this subset of A cross A. So clearly you can see that this is a subset of A cross A So we can write this preference order Into into two parts. So one part is the strict preference order of the linear order Where you have this preferences ABD and ACD, which can be written in the the notation of Binary relation in the following way You have all these things that are shown in red that are the the strict preferences You can collect them together and the differences that you have that can be written in this form and that can be separately Looked at so PI Union UI is actually giving you this R I So that you can say so I mean you can always decompose and this PI and UI are distinct and they are disjoint All right. So that is that is essentially our setup of all the preferences strict preferences weak preferences asymmetric part and the symmetric part Now we are in the position to define. What is a Arofian social welfare function? We have discussed about the social choice function the Previous modules now we are talking about Arabian social welfare function. How is it different from the social choice function? Now, let us look at each of these agents. So each of these players are picking their reference orders from this set R So this is you can say that player 1 is choosing from this script R Player 2 is choosing from script R and so on. So everybody is choosing all possible weak preference orders from from this set of let's say M alternatives and Finally, what you are getting as an output of this function and reveals in social welfare function capital F is also an ordering So you can think of this. So, how can you? Kind of make an intuition about what this social welfare function is imagine that you have Different cuisines different cuisines available as a menu in in the mess in your hostel mess and Finally, you want to have a complete ordering over all the cuisines of which she is collecting together all the preferences of all the residents of that hostel so What is the reason what could be a reason? Because if some of these cuisines are not available, then you can go to the next available cuisine which could be Which is most preferred by all the all by all the agents together all the Residents of that hostel together. So that could be one a way of thinking about social welfare function You have individual preferences all this So you can you can think of this R1 R2 and so on this if there are in in borders in that Hostel or mess and there are We preferences over all of them. So the preferences are given by capital R of I for that player And collectively were trying to come up the total ranking over all these cuisines for the entire society the entire society being that the hostel borders So in this case, we are going to define so it because if of R so by R We are denoting so R is nothing but R1 R2 and so on and this is what it means this is collectively going to be an order and Very naturally as we have defined the asymmetric and the symmetric part Let us denote with F F hat as the asymmetric part of F of R and F bar of R To be the symmetric part of it. Now, let us define two very important definitions. One is of weak Pareto and the strong Pareto So what let us go over this definition slowly So a social adobe and social welfare function F satisfies weak Pareto if for every pair of alternatives a comma b in this set a So this notation this Notation of for the square brackets and this implied part of this definition says that whenever this condition force Whatever we have written here whenever this condition force. This implies that this condition also holds So what what are we saying here? It is saying that if a strictly preferred than B for all the players in this society Then a should also be a strictly preferred than B in the aggregated society So this F of R is nothing but being at the aggregation and F hat means that it is strictly preferred So you can say that let's say we have a society where you have a above B So maybe there are other alternatives in between above or below. I don't really I don't really know Maybe there are a and B just next to each other and there are other alternatives So but the point is everybody is preferring a over B So that is the that is the thing that is happening here So if you give me any such preference order where he is strictly preferred over B for all the agents Then thus if you apply F on top of that Then it must be the case that in the aggregated one a should also be above so this is what this weak Pareto definition means and I should also Tell you that there could be some ours where this condition does not hold. So if the condition So if there exists some are where this if condition is not only so you can think of that if this happens Then immediately you can conclude this. So if the if condition itself is void So there is no such case where he is Strictly preferred over B for by all the agents then we can also assume that this This is implication is vacuously true. So you can you can have a strictly preferred. So we cannot rule out Those kind of F's as not being weak Pareto We will say that is weak Pareto just because that maybe not all the players were having a above B So in that case still it could be weak Pareto We will see more examples to understand this point when we come to that Now what is strong Pareto saying strong Pareto's if condition is a little weaker So you can see here that If this happens then immediately we can say that a is strictly preferred over B And you see that the right-hand side of both this week and strict Pareto are the same The implication is the same Why for the strong Pareto what we are saying is the Arabian social welfare function if Satisfies strong Pareto if for all this Alternatives all the spares of ordinates A and B A is at least as good as B for all the players and there exists some J some player J for which a strictly preferred than B So instead of this being that case so earlier we were having a strictly above B for all the players We are now aligned for Indifferences, so let's say A and B are sitting at the same position. Maybe for most of the players So they are indifferent, but there exists some player some jth player for which a is strictly above B now this What is strong Pareto saying is that if this kind of a preference profile you are given To this function F social welfare function F Then it must be the case that a is strictly preferred over B If that happens, then we will call this F to be Satisfying strong Pareto condition Now you can answer this question think about it that which property essentially implies the other So strong Pareto implies weak Pareto or weak Pareto implies strong Pareto So it might be a little confusing in the in the beginning But you can you can actually write down and try to think through it and you will see that Actually strong Pareto will imply weak Pareto The name itself is saying some hint, but you can you can do it more formally The the intuition is that whenever you have more stricter conditions in the if condition Remember that if you have this kind of a This kind of a situation and you can still say that F is going to Going to give out the the outcome which where a is strictly preferred over B then if you feed this kind of an Preference profile here, of course if it's going to output a to be strictly above B But the the other thing the reverse direction is not true because the for weak Pareto You need the strict preference for all the agents then if you feed this in this Particular weak Pareto definition isn't saying nothing about it. You cannot really guarantee a strictly above B So that is the reason when you have this if conditions in such kind of definitions Which are weaker than the other one? Then you have a actually the property that you are going to satisfy is essentially stronger because the right hand side the implication Is the same for both these cases? All right, so we are going to define this this term. So we say that two preference Ordering so I and I prime agree on a comma B. So these two alternatives a comma B if for agent I if for that agent If a is strictly preferred over B That implies and is implied by that a is also strictly preferred over B in the other preference ordering So if B is strictly above a in PI then it also means and is implied by that B is also above P above a in PI prime and if they are indifferent They implies and is implied by by the same thing in a in the first preference ordering and in the second preference so to denote this agreement we Sometimes we use this notation this shorthand that are I Restricted to a comma B is exactly same as our I prime restricted to a comma B and when all the for all the players this This agreement holds then we are just going to write a restricted to a comma B and our prime is restricted to A comma B So this is this is true for so when this condition holds for all I okay So that is a agreement we will sometimes use this shorthand notation to reduce our writing So the Arofian social choice function satisfies Very important property called independence of irrelevant alternatives. So this IIA stands for independence of irrelevant alternatives Will mention what is what is irrelevant in this case? So what it is saying is that if you pick any pair of alternatives a comma B in this set A Then if you agree if all the players agree on a comma B their their relative orders They can be so there can be situations where a is strictly preferred over B or B is strictly preferred over a for those Corresponding players. So what it means is that it might be the case So in this case what can happen is there could be some players for which a is strictly above B There could be some players which has B strictly above a and there are some players which has a and B at the same position So but the point is that whenever you are going from a from this R to R prime This relative position between a and B are remaining same for those players other the position of other players may change So for instance, there might not be any other alternative Between a and B in this for the first player in the second preference profile. So this could be R In the second preference profile, there could be another R prime in the R prime There could be a C somewhere sitting in between for this first layer, but that does not matter. Those are the Alternatives which are irrelevant alternatives. We are just looking at a comma B and they agree for all the players So if they agree between a and B then what this Independence of irrelevant alternative is saying that if you apply F on R as well as R prime The relative position between a and B in the in the final ordering should also remain unchanged so if In this preference ordering, so when C was not there in between the if a was above B in this In the final outcome. So when you have applied F on that and another preference another Preference ordering where C was there in between in that case also a should be strictly above B if a was B was above a then that Ordering should have maintained And so on so essentially the relative ordering between a and B should not change just because some irrelevant Alternatives have actually changed their position So that is what it means. So let us look at an example a more concrete example to make make this More understandable. So we have R and R prime and What I have carefully picked is that for player one. So these are the preference orders Let's for this example assume. They are strict preference orders, though They are not needed to be a strict preference order So you can see that for player one a is above B. Let us focus our attention only between a and B So a is above B a is above B for player two as well B is above a for player three and B is also above a for player four Now you look at R prime the same ordering is maintained So the position of the other irrelevant alternatives have actually changed So D has gone on top C at the bottom But the related position between a and B for player one player two And so here B is above a and here also B is above a for all these four players have remained unchanged from R to R Prime So these are my candidates. So what it is saying is that if you apply F on on this preference ordering and suppose you have given an outcome where a is Let's say above B There could be some other alternatives elsewhere But a is above B then it must be the case if you apply F on this preference Profile then a should also be above B. I don't really care about the Positions of the other alternatives, but a should be above B. They should be consistent in these two cases Okay, so let us look at some of the very widely used Mechanism very natural Mechanisms by which we can aggregate and come up with a complete ordering over all these agents From this from these individual preferences. So, let's say we assign some scores So let's say score of one to the first position Position score of two for the second position score of three for the third position and score of four for the fourth position The restriction is that every si should this should be at least as much as the second next one And this should be at least as much as the next one and this should so there should be a monotone non-decreasing Relation between all these scores. So this score such as just real numbers Now if you pick a specific alternative, let's say a For all the players you just add the corresponding scores. So this is this is the one This is the reason why this is called the scoring moves and whatever finally you have the score So let's say here if we if we put a S1 so so candidate a is getting two S1's and One is three and one is four. So is three Plus S4. This will be the score for a similarly. You can find out what is the score for B and What are the scores for C and D? so One very special Scoring rule is what is known as plurality where you give only a score of one for the topmost position for all these players And zero for all other places. So in that case a will get a score of two here See we'll get see we'll get a score of one and you will also get a score of one and we will get a score of zero Similarly in this case, we'll get a So B will get a score of two C and D will get score of one each and in particular a will get a score of zero Now Do you see any? Violation or is it satisfying the condition of independence of irrelevant alternatives? Because now what is going to happen is because this Scores are so and so so the the final outcome in this Under this scoring rule. So this scoring rule is nothing but the F in this case So a will be on top because it has score of two and you ordered them accordingly C and D will be the The next position they are indistinguishable because they have the same scope B has a score of zero that lives in the last position Now here what is happening B is on top C and D remains in the second position and A is here So you can see that even though the relative position the relative ranking of all these Players between A and B were remaining the same The outcome has this property that here A is above B while here B is above A So this clearly violates the condition of independence of irrelevant alternatives So this scoring rules essentially does not satisfy the condition of II so now you can begin to see even though these rules are very Very natural and very Easy to think about they do not satisfy this very Important very natural condition that if you if the society did not change its opinion between A and B Why should the outcome change the opinion and that is very difficult to satisfy in particular Based on this IIA and the weak Pareto Condition that we have this and defined earlier Arrow gave its a landmark result which says that if you have at least three alternatives So it will not hold for two alternatives. If you have at least three or more alternatives Then if an adorable social welfare function F satisfies weak Pareto and IIA then it must be Dictatorial so what does dictatorial mean? So dictatorial means you just pick out one specific agent Let's say this and always output its preference order as the output then naturally it is going to satisfy IIA because If A was above B So what is common in these two cases the the same agents have the same relative ordering between those two alternatives, of course if they are the same And if you are using dictatorial, you are just replicating that as an output the the same Ordering relative ordering will satisfy so you can look at this example You can create more examples and see this is happening So in this case the in the dictatorial outcome if this was the dictator A would have been above B and here also A would have been above B because you are going to just replicate this as So dictatorship or the dictatorial Arobe and social welfare function certainly satisfies IIA and Also weak Pareto you can check that but this Result arrows result is essentially saying much stronger thing that if you want to satisfy these two properties IIA and weak Pareto then it must be dictatorial. There is no other mechanism which can satisfy both these two things together So that rules out the scoring rule-based mechanisms and it rules out several other reasonable mechanisms So sometimes this is this mechanism This result is also known as the arrows impossibility result because we don't really need Want dictatorial outcomes. So if you want to have a non dictatorial outcome non dictatorial social Arobe and social welfare function along with the weak Pareto and IIA then it is impossible. That is what it means You