 We have stated the following theorem and SCFF is a strategy proof if and only if it is monotone in our previous module. In this module we are going to prove it formally and one of the reasons of showing this proof is to notice the proof technique and this proof technique will actually be used later as well and in general this principle is quite well utilized in several other parts of mechanism design. So we have to prove an if and only if condition for strategy proofness and monotonicity. So let us first look at the forward direction or the only if direction that is if the social choice function is strategy proof then it must be monotone. So what is the if condition of the monotonicity? So because we will have to show this monotonicity then the if condition must already be pre-existing and now we will have to show that the conclusion of that if condition is also holding. So what was monotonicity? So if we had these two preference profiles P and P prime where in the first preference profile the outcome was A and the case that if you look at the dominated set of A under PI that is weekly increasing in the other preference profile P prime. So this dominated set is going to be weekly increasing for all the agents. Now you can break the transition from this P to P prime into N different stages in each stage only one agent's preference is going to get changed. So what is it? So let us say our first preference was P0 and this was the original preference and the final one N is P prime all the agents have the preference their preferences changed into PI primes. But in P1 only player 1's preference changed from P1 to P2 only player 2's preference has changed and so on. This is how we are going to do it. So in the Kth transition only up to Kth agents preferences have changed and all the other agents K plus 1 to N their preferences are remaining the same as before. Now we claim that what is going to happen is in each of these intermediate stages the outcome is going to be A. So this is the claim that we are going to make and the intuition is very clear. If it does not hold if there exist any intermediate stage where the outcome changes then we can actually create an instance where it is beneficial to manipulate for any of the some of the agents. So we are going to prove it by contradiction. So suppose not suppose there exist some Pk minus 1 and Pk so you are transitioning from Pk minus 1 to Pk such that in Pk minus 1 the outcome was A and in Pk the outcome has changed to P which is not equal to A. So let us assume that. So what is going to happen now one important thing is that because from P to P prime this dominated set has greatly increased so that means whenever we are looking at from Pk minus 1 so this is the preference profile P of K minus 1 and this is Pk that is how we have defined it. So the for player K the relative position of A in the in the preferences has greatly increased so A has gone weekly better off weekly above its current position as it was in Pk minus 1 and we have we are assuming that outcome here was A while outcome here at A is B. Now let us go case by case so there could be three cases so the first case is that in this in this particular preference profile for player K A is more preferred than B and in the new preference profile also A is more preferred than B. Remember we are just comparing between the this two alternatives what is more preferred. Now what you can see is for player K this alternative is worse than A in this preference profile so and it knows that if it reports his preference to be Pk the outcome will be A which is more preferred by that agent even in this particular preference profile Pk prime. So therefore P what our case should misrepresent is whenever its true preference is Pk prime it will report it Pk so that it gets the outcome of A. Similarly the converse thing you can imagine that B is more preferred than A in both these cases then from Pk it knows that the outcome is a worse outcome so it is outputting A while if it misrepresents his preference to Pk prime. It can get an alternative B which is more preferred by the same agent at that preference profile. So in both these cases K missed reports and if the third condition is that B in Pk was above A and it went down so this is the possibility that B was somewhere here so B was above and here it went below. And this three are the three exhaustive cases the fourth condition is cannot happen because A's relative position has weakly bettered off so therefore there cannot be a situation where there was some alternative below A and that went above A in the new Pk. So in this case what happens is that in both these situations either Pk or Pk prime voter K will misreport because if it knows that here the outcome is A if it misreports to Pk prime it gets B which is better here also it is getting a worse outcome in Pk prime and if it misreports to Pk it gets a better outcome. So in both these preference profiles voter K is going to misrepresent. So what we have actually assumed is incorrect because if this contradictory statement is to be true then it is actually contradicting the fact that F is actually breaking the notion of strategy performance. So this must be true that in all the intermediate stages that we have assumed the outcome would be equal to A and therefore even in the final outcome at P prime the outcome should also be A. Okay so now we look at the reverse direction that if these social choice function F is monotone then it must be strategy proof and which is equivalent to saying that if it is not strategy proof then it implies that it is not monotone. We are again going to prove it in a via contradiction so suppose not that means F is not strategy proof yet it is monotone. So let's see what contradiction it leads to. So not strategy proof means that it is manipulable that which means that there exists some player some preference some pair of preferences Pi and Pi prime and some preference profile of the other agents P minus I. Such that this outcome when it manipulates its preference to Pi prime it gets a better outcome than reporting it truthfully under the same truthful preference. So let us assume that this F of Pi prime P minus I is B and F of Pi P minus I this is when they are reporting their preferences truthfully to be equal to A. So hence we what we have is B is more preferred than A. Now what we are going to do is we are going to use a trick that we have done sometime back so we are going to construct carefully a new preference profile where we will show some sort of a contradiction. So let us construct some P double prime such that for all the agents except agent I the preferences are same as the original preference. So for all the other agents we are not changing their preferences for this particular agent Pi what is going to happen is we are putting the top most alternative in this new preference profile to be B and the second top most alternative to be equal to A. So you can imagine that in this Pi prime the top most alternative is B the second top most is A and everything else is remaining the same and this preference profile is nothing but our P double prime. Now you can consider two different transitions so what is happening is from Pi P minus I when you are looking at Pi double prime P minus I. We see that for player I the relative position of A is getting weakly better off because what can happen here clearly we know that in the original preference B was more preferred than A. So in the original preference B was somewhere above and A was somewhere below for agent I so this is only the preference of player I for all the other agents nothing is changing so we don't really need to care about that. Now what we are doing is we are pushing this B on the top and A to the second top so whatever was so the only possibility is that all the alternatives that were except B which was above A is now coming down below A. So that is the reason we can safely write that the dominated set of A under this Pi is weakly getting larger in the new preference profile of Pi prime Pi double prime. Now because we are also assuming that this is monotone so then therefore in this new preference profile the outcome should also be equal to A. So in the previous preference profile the outcome here it was A so therefore it should continue to be A. Now we look at a different transition now we are looking at Pi prime P minus I to Pi double prime P minus I. So what was Pi prime so we know that in Pi prime P minus I the outcome was B and what we have done is again in Pi prime the outcome was B and in this Pi prime we have pushed that alternative to be the top most. And because it is going to the top all the alternatives that was potentially above B in Pi prime that has now gone below B. So you can look at the dominated set of B under this Pi prime that has actually gone weakly increased in the preference Pi double prime. So we can use the same monotonicity condition to conclude that Pi double prime P minus I the social choice outcome at that preference profile would be equal to B. Now these two things are essentially the same the left hand side is the same and B is not equal to A so therefore this actually leads to a contradiction so that actually concludes the proof even on the reverse direction. Alright so that was the proof that is a strategy proofness and monotonicity are one and the same because this is if and only if condition. Now what we are going to do is we will show that monotonicity or strategy proofness because they are equivalent in this case. And you are looking at that condition along with on-to-ness then that will be that will be Pareto efficient. So we already have seen that if we have a set so we have a set of Pareto efficient so let's say this is the set of all Pareto efficient allocations Pareto efficient social choice functions. Then that is contained within the set of unanimous social choice function and that is contained within that on-to. So in this figure the largest set is on-to it is containing the unanimous set of social choice functions which is also containing the set of Pareto efficient social choice functions. Now we are looking at the weakest condition over that social choice function and we are saying that if you want to have on-to-ness plus monotonicity which is same as strategy proofness then you must necessarily be only Pe. So that is what this figure says that we will only hit this particular place and therefore whenever we are looking at only social choice functions that are monotonic or strategy proof Pareto efficiency unanimity or on-to-ness are one and the same. So there is no difference between them. There cannot be a situation a social choice function which is monotonic and on-to but not Pareto efficient. So that is exactly what this lemma is saying that you can shrink this set into only Pareto efficient allocation. So what does that mean? So suppose it is not true so which means that suppose it is monotone and on-to but it is not Pareto efficient. So not Pareto efficient we already know what it means. So it says that you have A and B and a preference profile B such that B is strictly preferred than A by all the agents here by all the agents in set N but the outcome the social choice outcome is equal to A. And if that happens then we can say that this social choice function is not Pareto efficient but we also know that this social choice function is on-to. That means there exists some P prime such that so for every B you can always find that there exists some P prime such that F of P prime is equal to B. So we have selectively chosen this B so this B is the same B which actually Pareto dominates A. Now we are going to do a very similar trick constructing this PI double prime where we have chosen for all the players the topmost position the topmost alternative is B and the second topmost is A and this is true for all the agents here. So this looks like this figure here. So therefore what we can conclude is that this the dominated set of B under this PI prime is which is getting better off PI double prime. And now we also know that this function is monotone. So we know that in PI prime the outcome was B that is by the definition of on-to-ness. So if the outcome was B and your dominated set was actually weekly increasing then by monotonicity we know that the outcome will remain the same. So that's fair enough. So this is let's say point one where the social choice outcome at PI double prime is equal to one. Now you also see that this dominated set for A from PI because in PI what is happening for all the agents B was above A. So it was very similar to the previous example that B was above A and there could be some other alternatives. This was P the original preference. So for player I this was the case for all the other agents B was. So maybe the position of B and A related position could have changed but they were always above B was always above A. So if that was the case in P double prime in this preference profile what we are doing is we are flushing this B and A on to the top. So which means that all the other alternatives are actually falling below A. So A's dominated set is actually getting weekly better off, weekly larger in this P double prime for all the agents. So therefore because in this F of P outcome was A so it must be the case that in F P double prime the outcome should also be equal to A. And that actually has a contradiction. So let's say this is two then one and two are actually in a contradiction because they are the same preference profile but they are leading to two different alternatives. So that actually proves the fact that we cannot have a situation where it is monotone and on to and it is not paired to efficient. So it actually implies that it must be paired to efficient. And as a corollary of this result we can say that if we look at F to be equal to strategy proof which is equivalent to monotone strategy proof and paired to efficient. This implies and is implied by it is strategy proof and unanimous and similarly it is strategy proof and on to. So we will essentially whenever we are in the world with social choice function which is strategy proof on to nice unanimity and paired to efficiency are one and the same. We don't really distinguish between them. So in this setup we have a very landmark result. The most important result in the social choice setup may be one of the first results a very similar to arrows impossibility result. This is known as the Giver-Sotherwood theorem sometimes also called the Giver-Sotherwood impossibility theorem. Suppose we have at least three alternatives and we have the social choice function which is on to and strategy proof. Then it implies and is implied by the fact that F is dictatorial and the result is very much similar. The conclusion is very much similar to arrows impossibility result. But the setup here is of social choice function rather than a social welfare function and the conditions of the two desirable properties that we want are on to nice and strategy proofness. These are slightly different. So but we notice that we actually again come back to the same situation where we had an impossibility. So in certain textbooks you will see the same statement for Giver-Sotherwood with instead of on to nice it is replaced with paired to efficiency or unanimity. They mean the same because because of this corollary here that they all these things when you are talking about strategy proofness they are equivalent.