 we have reached the last module of this course and in this module we are going to discuss few of those things which we haven't got the time to discuss in detail in the course. But the discussion here will be a little less rigorous than we have done for the other cases, but it is important to know these results and if you are interested in a later course of mechanism design, you can actually learn more about these properties. So we have discussed efficiency to a very large extent of this second part, the mechanism design part of this course and we have also seen when and where it can be made budget balanced and in particular we have looked at auction settings where we try to make the auction budget balanced and we failed. So here we are going to see that there is a inherent tension between efficiency and budget balance and how can we actually get to that result is what we are going to discuss next. This is a result due to green and la forme. So when we are talking about gross class, we essentially try to find out the efficient allocation which is nothing but the sum, the allocation that maximizes the sum of the values of all these agents. In this first result we are going to tell you how gross mechanism is unique for the efficiency. So the result says this was done by green and la forme and independently by homestown and both of them have shown that if the type space is sufficiently rich and I am not going to explain further what is sufficiently rich, you can imagine that all possible kind of valuations are feasible as we have discussed in the case of Robert's result, Robert's theorem. So in that sufficiently rich class of type spaces, every efficient and DSIC mechanism is a gross mechanism. This is a very strong characterization result that if you want efficiency and dominance rate incentive compatibility then you must be in the gross class. Now we are going to give the proof sketch only for two alternatives for multiple, more than two alternatives. This proof can be actually extended and that is exactly what this homestrom's result do. This presentation is essentially from homestrom paper itself. So what we are going to do is we are looking at the welfare under these two alternatives. So let us say we are looking at the sum of the valuations of all the players at this alternative A and that in alternative B. And we know that efficiency means that whenever this TI, the sum over TI A is larger or at least as much as TI B, then A will be chosen, breaking the tie in favor of A. Okay, so now what we are going to do to understand why this efficiency is very much tied to the gross payment rule. Let us first look at what happens when we fix all the types of all the other agents except agent I. So for T minus I, for both these alternatives A and B, we are going to fix those values. We are only going to change TI of A. So this value here, even for player I, even for the alternative B or the allocation B, we are also going to hold its valuation to be fixed. Now what happens is that if you now change this TI of A, you can actually choose TI A to be sufficiently negative such that this inequality does not hold anymore. So in that case, B will start becoming the winner and after a certain value, threshold value of TI A, which we are going to call as TI star of A, the outcome will start becoming A because that after that threshold, this inequality starts becoming true. Okay, so let us consider that threshold TI star A. We know that this is if you are thinking in terms of a line, TI star A is a threshold below which if you and this is just TI A. So TI A is being varied between over a real line and below this threshold it is going to be the outcome is going to be B and above this the outcome is going to be A. This is because we have actually fixed all the other valuations, even the valuation for that agent for the other alternative B and for all the other agents at both alternatives A and B, the valuations are fixed. Fair enough. So now we are going to look at two different situations. So a TI A, which can be slightly above this. So let us say somewhere here, which is TI star of A plus epsilon. If TI A is so, then we can actually apply the condition of DSIC. So DSIC says what? So it says that if you reveal your valuation truthfully, then the utility that you are going to get is going to be at least as much as if you misreport to something else. And we know that if you misreport and the outcome does not change, then nothing changes. So you cannot really be better off. The only way you can change your payoff is by changing the outcome. So suppose you misreport to something which is living somewhere here, so that the outcome becomes B. And in that case, the utility will be this right hand side. And because of DSIC, this inequality should hold. And now I would also like to mention one point that we have said long ago, that if you are looking at only DSIC mechanisms, then by changing the valuation, the valuation report, if you cannot change the outcome, then the payment should also remain the same. So we have said that it is only going to be dependent on the outcome of the mechanism and not on the type. Because if you could change the, if you could change your valuation report, which does not change your allocation, but it changes the payment, then there is a possibility of misreport manipulation. So that mechanism will never be DSIC. So I had asked you to prove this, maybe you can go back to that exercise again and see that whenever you are changing, so your payment as long as the outcome remains the same as A, it is going to be the same payment. So we can, without loss of generality, just mention the payment with respect to the allocation itself. So this is one observation. And this inequality is due to DSIC. So this is when your true type is slightly above TI A star. So the outcome, if you reveal it truthfully, the outcome becomes A. So now consider another situation where the true type, let us say TI of A, the same TI of A is actually falling below this. So the true type is somewhere here, TI star A minus delta. So here if you reveal your type truthfully, then the outcome becomes B. And because in B, your type is the same as TI of B, which we haven't changed. So in that case, the valuation becomes TIB if he reports truthfully. And the payment would be PI of B. So this utility is the utility when this agent I is being truthful. And the right hand side is suppose it reveals something else. It misreports to somewhere here, some value report which is here. And thereby the outcome changes to A. But the point is that its true type for A is actually TI A star minus delta. So this is the, this is the valuation that this agent gets when the outcome A is chosen. And it makes a payment of PIA. By DSIC, this inequality should also get satisfied. Now what you can see here is that if you look at this quantity, TI star A minus PIA and TIB minus PIB, all that is different in these two inequalities is that they are flipping their size of the inequality. And epsilon and delta are subtracted, added or subtracted into this two cases. Now because epsilon and delta are arbitrarily, we can actually take the limiting value of this epsilon and delta. And what we can get is that this inequality will become an equality. Because here if you let delta go to zero, then you have an inequality in the reverse direction than the inequality here. So you get this equality and this equality will be very crucial. So we will be using it very carefully. We also know that TI star is the threshold of the efficient outcome. So what does that mean? So that means that this is that point where if you take the sum of all the agent's values at that outcome A, it must be exactly equal to the sum of the values of all the other agents. So because TI star, we are actually holding all these types to be exactly same. And TI star A was the point beyond which, if it is more than that, if your type is type of player I at alternative at allocation A goes beyond this point, then of course this part will become larger than the other part. And therefore the outcome will start becoming the efficient outcome will start becoming A. And if it is smaller than that, then this becomes larger. So therefore the outcome will be B. So this inequality should get satisfied and this is by definition of TI star A. Now from one and two, what we are going to do is we are looking, we are interested in looking at what is the difference between PI A and PI B. You can do this reorganization and find out that this is going to be exactly equal to summation of TJBs where J is not equal to I and some minus summation over TJA where J is not equal to I. And this should remind you something very similar that we have done in the gross payment class. So gross payments were such that this inequality should get satisfied only when you have this kind of a structure in the payment which has one component which might not depend on TI at all. And the second component which depends on all these agents. So essentially we are looking at PI of X. So any outcome X, X can be either A or B. And the second term will be the sum of the valuations of all the agents at that outcome. And the first term will not depend on type of player I at all. And if you think carefully this is exactly the gross payment. So this is not the proof, but it is an intuition how you can actually go about and think about how the payment would look like when we have efficient outcomes, efficient and DSIC mechanisms. So for two players certainly the mechanism is going to be gross payment. I mean the payment will be in the gross class of payments. And this can be even extended for more than two alternatives, two allocations. The second result, so now we know that if you want DSIC and efficiency, then you better be in the class gross. The Green Elephant gives another result which says that no gross mechanism is budget balanced. So you cannot find any gross mechanism which falls in this class and is budget balanced. And therefore you can actually conclude the fact that if you are looking for efficient and budget balanced mechanism, then that set is null. So you cannot find anything. So this is an impossibility result. So you are not going to get into the proof of this. This is a little elaborate, but even then I have given the proof sketch. So this notes will be available so you can take a look at it. So yeah, so finally we can have this, summarizing these two results, we can have this corollary. If the valuation space is sufficiently rich, then no efficient mechanism can be both DSIC and budget balance. That is the conclusion that we can make. Now how should we get around this impossibility result? So one of the early attempts to bypass this result is to weaken the condition of DSIC. So DSIC is a much stronger condition, but you can also look at something called a BIC mechanism and we have seen that before. So allocation still becomes the remains the efficient one, but the payment in that setting, now that we have a prior distribution over all these types, we can actually have, we can take the expectation with respect to t minus i given the agent knows its own TI. So this is an x-intering expectation of over this prior. So the mechanism designer in this case is actually doing this expectation and charging this much amount of payment to each of these agents. So delta i Ti is the payment that it is charging to that player. As soon as it reports Ti, it is going to recharge this amount and that is going to be nothing but the look at the same summation of all the valuation of all the other agents except agent i at that optimal value and that optimal allocation, but you are also taking the expectation with respect to t minus i. So this function even though the original sum was a function of both Ti and t minus i because this is now being expected over t minus i. So this ceases to become a function of t minus i. So this finally becomes just a function of Ti. So the allocation as before remains the efficient allocation, but the payment with this definition of delta i Ti is modified in the following way. It is summing all these delta j's tj. So we have used this delta i Ti just to define how we are going to calculate the payment under this new mechanism. We are going to sum over all the agents this value of delta j tj for all the agents except agent i and take a product with 1 over n minus 1. So we will see why or what is the purpose of keeping this multiplier of 1 by n minus 1 and subtracting now delta i Ti. This is also quite similar to the VCG mechanism, but the only difference is that if this is expectation with respect to t minus i for each of these delta j's. And for that reason sometimes this DAGVA is also called, this mechanism is called the DAGVA mechanism. So this mechanism is also called, sometimes called the expected VCG mechanism. And the name DAGVA comes from the names of its inventors. So this payment implements the efficient allocation, this efficient allocation rule in Bayes Nash Equilibrium. So what do we mean by that? So we will have to look at the utility of each of these players and take the expectation with respect to t minus i given Ti. So we know that this is the payoff, this is the evaluation of this agent i when the allocation is A star t. So the allocation remains the same, it is still the same efficient allocation. Payment is replaced by this DAGVA mechanism and we are now taking the expectation with respect to t minus i given Ti. So this is the x entering utility of player i. So we can just expand this out and this is quite straightforward, you can do this exercise. So what happens is that this term, so we have this term separately handled and this delta i Ti comes in the beginning. And we notice that this delta i Ti term was nothing but the expectation of t minus i given Ti for all the agents except agent i. And now we are going to sum that with this Ti A star t and together they will be constituting the sum over all agents including agent i. And that becomes the first term here. And the second term, the point to observe here is that this term is no longer a function of Ti. So here everything is, so in the first expression there is Ti term there but for the second expression there is no Ti. So what we can say here is that this term, so this first expression because A star t is the maximizing the arg max of this sum. So this is going to be larger than any other outcome. So if we look at the same summation over all these djs and if we replace this with any other alternative then this quantity, this term itself even before this expectation is going to be larger than that. And in particular we are going to pick it very carefully, we are going to pick it for the same efficient allocation when this agent is misreporting to Ti prime. And this is the very similar idea that we have done even for the VcG mechanism. If you do that this inequality is going to get satisfied and because this is not dependent on Ti we can again refactor all these terms and go to something like this expression here. So where we have this Ti da gva when agent i is misreporting to Ti prime and this term is nothing but the utility of that, the valuation of that player when it is reporting Ti prime. So it shows that it is in expectation, so in a Bayesian Nash equilibrium reporting its type truthfully is the best response. The important part here is that the payment of this da gva is chosen in such a way that if we take the sum over all these things, so remember what was the term here 1 over n by 1 and then this term here and then the second term was minus the delta i Ti. And if we take the summation over all these agents, in this summation we are excluding one specific delta i term and when we are taking the sum over all those cases the whole sum will have n minus 1 copies of each of this delta i. So we can write, so this term can be actually written, so this double summation can be written as a single summation multiplied by n minus 1 and this n minus 1 was the reason why we have had this multiplier 1 by n minus 1, they cancel out and the second term is simply the equal to this part of this first term. And after we have taken the summation over and the payments over all these players, these two terms essentially cancel out and it becomes exactly equal to 0. And that was the reason what we have, why this Pi of da gva was chosen in such a form so that it becomes budget balance by design. So we can conclude that this da gva mechanism is efficient by design, we have proven that it is BIC and now we have also shown that it is budget balance. So this is one mechanism which satisfies these three properties even though it is at a compromise of the incentive compatibility property. So what about the participation guarantee, the individual rationality constraint, one can create examples to show that da gva is not interim individually rational. And in particular there is a impossibility result due to Meyerson and Sutherwood which says that in a bilateral trade even the simplest form of the mechanism where two types of agents, only a seller and a buyer, no mechanism can be simultaneously BIC, efficient, IIR and budget balance. So this essentially seals the problem that we cannot really go any further. So we cannot really weaken this mechanism any further unless we change some of these properties or we go for some other kind of approaches, let's say approximation or something. All right, so that essentially concludes the discussion of mechanism design which is in the scope of this course. So to summarize and kind of get a feel of all the results that we have discussed so far, let me give you a very broad pictorial overview of the whole domain, the space of mechanisms that we have discussed. So on the left hand side of this figure we have looked at the valuation or the type space or the domain as we have said. We are talking about the social choice functions, its domain can be unrestricted, can be restricted and so on. And then we on the right hand side we look at the type of DSIC mechanisms which we can, which we have characterized through several results. So the first is the largest class, so unrestricted, this is the class that takes care of all other subclasses, the subdomains. So if it is unrestricted, then by the Gibberish Otherweight result we know that the outcome is going to be the mechanism that can satisfy DSIC is dictatorial. So of course we are also looking at some other properties but at this point we are not so much audit. The broader result is that in the unrestricted domain to satisfy dominant strategy incentive compatibility we will only have the dictatorial mechanism. Now let us look at the next domain restriction which was single-picked preferences and along with anonymity and on-to-ness we have seen that this mechanism, this subclass can have non-dictatorial cases and which falls under this class of mechanism called median border. In fact this is if and only if so both directions are true and that is the result due to MULA. Similarly if we look at another subclass which is the task sharing domain we have also discussed that anonymity and path to efficiency uniquely identifies that the mechanism is going to be uniform rule and that is also if and only if result and this is a result due to SPRUMO. Now we have after that we have actually moved into the domain of quasi-linear preferences. This is a preference domain where the we are also talking about transfers, these are mechanism designed with transfers and the utility is in the quasi-linear form. So if we have unrestricted quasi-linear preferences then we have these results due to robbers which says that the DSIC mechanisms must necessarily be affine maximizers and this is a unidirectional thing unlike SPRUMO or MULA which implies this in both directions, this is only in one direction so it's a necessary result. So the second condition that when we actually look at the efficient outcome under this quasi-linear preferences then we have seen that it has to be according to the results due to green and la forme it has to be sitting in this gross class of payments. So the mechanism has to be the gross class of mechanisms and on top of that within that efficiency class if we also want to ensure budget balance then again by the same result of green la forme by the same paper of green la forme we see that the set is going to be empty. So there is no such mechanism which can be simultaneously efficient budget balance and also DSIC. Okay so that was all about the DSIC mechanisms that we have discussed several results. Let us now move on to the BIC mechanisms and there we have also looked at the similar situation of quasi-linear preferences and this is mechanisms with transfers and if we have efficiency and budget balance here it is possible unless unlike the previous case and this is a sufficient condition that if you use DAGVM mechanism then we can actually satisfy this there is no characterization of BIC mechanisms so characterizing BIC mechanisms is much more difficult we have only done that for the single object allocation but apart from that there is no no such general purpose characterization results. Due to myersons and southern weight we know that if we also want to ensure efficiency budget balance and interim individual rationality then that set will be empty so we cannot really find any mechanism which satisfies all these three properties together along with BIC. And for the single object allocation so which is a very special case of the quasi-linear domain if we want to satisfy DSIC mechanisms we have seen this result due to myersons so all these results are essentially due to myersons the DSIC characterization says that the allocation rule should be non-decreasing and the payment should be given by the myersons payment formula. Similarly for BIC mechanisms a very similar thing works the allocation rule has to be non-decreasing in expectation now and an equivalent myerson payment rule can be designed. Now if I look for BIC as well as optimal mechanism which is maximizing the revenue then it has to it leads to a mechanism which is a reserve price auction so we have the same auction as the second price auction but with the reserve price and that happens I mean the reserve price has to be said based on the based on the priors that we have but the point is that we can actually design a mechanism and that mechanism happens to be very simple a second price option with reserve price and that maximizes the revenue for the auction alright so that gives you a kind of a panoramic view of all the things that we have discussed in the mechanism design part the first part of this course was more about developing the foundational stuff on game theory so hope you have liked this course