 the Meyerson's characterization result now and we also know what kind of what is the type of mechanism which are truthful in single object allocation domain. Let us now look at some of the examples of such mechanisms which are truthful and false in this class. So, the first two examples are quite standard. So, constant allocation and dictatorial we know that those are monotone allocations, but they are not super interesting because they are very trivial mechanisms. The first interesting mechanism is that of second price auction. We have already seen it in various contexts. Let us see why it is falling in the same class and how we can actually pictorially represent the second price auction in the context of Meyerson. So, let us look at the payment formula in this case. So, we know that the first part is let us say it is assumed to be equal to 0. Then the second price auction we know that this is not a randomized allocation, it is a deterministic allocation. You give the object to the highest valued agent. If that agent is a high highest value then it gets this object with probability 1 and all other agents get it with probability 0 and it pays the second highest bid. So, let us look at that agent which has the highest value in this case. So, imagine agent i is the highest valued agent, then this function is nothing but a step function. We already have seen this diagram before that this is the allocation. So, this is the fi of ti t minus i and where i is the highest valued agent. So, the ti value is the highest. So, we can look at the second highest value and denote that with t minus i 2. So, this is the max over all the other agent's valuations of their types. So, as long as it is the type of agent i is below this value it is not going to win that object and after that it is going to win that object with probability 1 and exactly at that point the allocation can be anything. So, that could be multiple agents who can be probabilistically given this object. That will not have any impact in terms of the payment that it makes. So, you can look at this. So, this is going to be exactly 1 for player 1. So, this is going to be ti. So, essentially this is nothing but this whole rectangle here. Now, if you look at this quantity here this is nothing but this green shaded region. So, if you subtract that out payment is nothing but this red part here. And therefore, the utility is this the ti which is nothing but the whole area of this rectangle minus that payment which is going coming back again to this blue this green part here. So, and as we can see that the payment is just this part which is nothing but the 1. So, what is this area under this red region? It is the second highest bit multiplied by 1. So, therefore, this is the second highest bit. So, we know that this payment is nothing but the second highest bit. So, this is the second price option and it can be very well classified inside this Myerson class of mechanisms with the appropriate values of fi and ti. Now, let us look at the efficient allocation with the reserve price. So, we have seen this earlier that the second price auction in the second price auction on top of asking the agent for the second highest bit as their payment one can also set a reserve price and fix something like the maximum value of t minus i2 and that reserve price i. So, what happens in that case is this threshold gets replaced by the max of these two quantities t minus i2 and that reserve price here. And there also we can see that the allocation rule is monotone and therefore, the corresponding payment will be just this this point. So, max of these two things and the argument is exactly similar to the second price option. So, we will not go over it once again. So, this efficient allocation with reserve price is also a non-decreasing allocation rule and therefore, you can design this Myerson mechanism which will be truthful. But now, let us look at some not so common allocation rules. These are something which we have not seen before and we unless we knew the Myerson mechanism or the Myerson's characterization, perhaps we would not have thought about this kind of rules to be truthful. So, suppose the allocation has three components. The first component is whether to decide that if that agent that object is to be sold at all or not. So, a0 is the decision where it is not sold, a1 is the decision where it is sold to agent1 and a2 is sold to player2 and there are only two players in this case. So, now given the type profile t1 comma t2, the seller first computes this corresponding number which is a maximum between three numbers. So, the first thing is some arbitrary constant let us say 2 and the second and third are t1 square. So, the type of player1 squared and t2 the type of player2 cubed and select it will select a0 a1 or a2 depending on which of this is the maximum. So, if this is this maximum, so if this turns out to be the maximum then the mechanism will not sell this object at all. If this happens to be the maximum then it will give the object to player1 and in this case it is going to give it to agent2. This is a very strange mechanism but what we are going to argue is that here also this mechanism, this allocation rule is monotone and therefore you can design the corresponding the payment formula which will make sure that this mechanism is implementable in dominant strategies. So, why is this monotone? So, you can see that you can break the tie in whichever order and suppose there exists already a tie breaking rule. So, the player1 will get this object if it is t1 crosses this threshold of max of 2 times t2 square t2 cube. So, essentially square root of that. So, if this happens, so it is essentially spelling out the condition under which this becomes the maximum. So, if that becomes the maximum then player1 will get that object. So, in the it will have a very similar allocation function it will also be a step function and this threshold will be given by different numbers something like this. This is for player1. If this threshold becomes this then it will after that it will start getting that object before that it will not get that object. Similarly, for player2 we will get this object where this threshold is being replaced by this. So, this will then that will be the allocation rule for player2. Actually, this should be t1 and this should be t2. There is no t3 anywhere. Okay. So, that is the condition. So, what we can see is this allocation rule for both this both these players is a monotone allocation rule and therefore, we can equivalently compute the corresponding payments using the Meyerson's formula and we can make this mechanism terminally strategic incentive compatible. Okay. So, let us now look at the property of individual rationality. We have seen this property earlier but in the context of single object allocation this will have certain implications on the on the Meyerson's result that we have shown in the previous module. So, the mechanism is exposed individually rational and we emphasize this term exposed which means that even after all the agents have revealed their types participating is weakly preferred. So, this is the meaning of exposed even after all agents have revealed their types. So, even after observing everyone else's type it will be beneficial for you to participate. So, the definition is fairly simple your expected utility is going to be non-negative for every type of your own and the types of the other players and this should hold for every player. So, what is the so what are the implications of individual rationality on the Meyerson's characterization result that we have seen. So, here is the lemma which says that formally. So, a single object allocation in the single object allocation setting if we have a DSIC mechanism f comma p then it is going to be individually rational if and only if this the constant that we had in the in that integral formula is going to be non-positive and on top of that if you also want to ensure no subsidy condition then this P i T i t minus i has to be non-negative for all these all these players then this constant quantity is going to be exactly equal to 0. So, this is this is nothing but the condition of no subsidy that is the payment given by each of these players has to be non-negative then this quantity has to be exactly equal to 0. So, let us look at why this is true and in fact the proof is really very simple and straightforward because this is a individually rational we already know that this mechanism is DSIC. So, we can without loss of generality assume that the Meyerson's payment formula holds and the payment formula so because we will have to ensure that this is going to be non-negative for every valuation. So, what is the definition of individual rationality this inequality should hold for every T i and T minus i and we know this from the Meyerson's result this the payment formula that we have to satisfy we just replace those values with T i to be equal to 0. So, then the first part actually goes away it is going to be 0 and also those two integral parts. So, P i had P i T i T minus i had the first term was just the 0 and P minus i, but the second part has T i F i T i T minus i this also goes to 0 when you have plug in T i to be equal to 0 and integral from 0 to T i F of x T minus i d x that will also go to 0 if we plug T i to be equal to 0. So, essentially in this part there will be only this term left and because it has to be non-negative we have this P i has to be non-positive P i 0 of T minus i has to be non-positive. So, conversely if you have this to be non-positive then we will have to show that this also satisfies the individual rationality constraint and we can do that just by writing down the utility expression explicitly we have this T i F T i T minus i and then expand out the the payment formula as we have here. We notice that these two terms get cancelled out and now this quantity is nothing but this is a probability distribution. So, therefore all these terms will be non-negative and we are taking the integrals of this part will be non-negative as well and because this term is this P i 0 is non-positive so negative of that will be non-negative. So, the whole term will be non-negative as well. So, therefore we have proved that this mechanism with this condition of on P i 0 is going to be individually rational. Okay, so in part 2 we are going to prove on top of individual rationality you also want to ensure no subsidy. So, all the payments has to be non-negative and if that has to happen for all T i's you can as well put that T i to be equal to 0 and that has non-negativity now and by individual rationality we already know that this has to be non-positive. So, the only possibility is that this is exactly equal to 0. So, and the converse thing is always true that if we have this to be exactly equal to 0 then it is very straight forward to see that this is going to be individually rational by the very condition that this is exactly equal to 0 and it is also not going to be not going to have any no subsidy condition because it is not going to give any subsidy because because of the fact that the allocation policy is monotone. So, we can again draw something like that that the picture that we have shown last time the payment is always going to be non-negative. Okay, so that with that let us now go into some mechanisms which are non-vickery options. So, so far we have discussed only vickery options that you have a threshold and only one agent who is crossing that threshold is the sole winner. So, this is something like a mechanism which is a deterministic mechanism but our characterization result of Myerson does not only apply to deterministic mechanisms it also applies to non-deterministic or randomized mechanisms. So, let us look at some of those examples some randomized mechanisms which is also truthful under this setting of single object allocation. And we are going to focus mostly on budget balance that is the payment that has been made by all the players if that if we take their sum that should be very close to 0 if not exactly equal to 0. Okay, so the first mechanism that we are going to define is that the object belongs to the highest bidder again this is a deterministic mechanism but the payment is such that everyone is compensated some amount here also the allocation is very straightforward the highest bidder gets the whole object and all the other agents get zero allocation. So, their payment if you look at the Myerson's formula only the constant term remains everything else goes away. So, but for player one there is something which is left over so we will calculate that. So, let us assume that so in this mechanism those constant values are decided in the following way the constant value for player one and that for player two is nothing but minus 1 1 nth of the third highest bid. So, without loss of generality we are assuming that this t1 is greater than t2 t2 is greater than t3 and so on they are ordered in a decreasing order of their of their types. Now everyone else apart from this highest and the second highest bidder receives 1 nth of the second highest bid. So, which means that this is going to be t i 0 comma t i t minus i that is going to be minus 1 by n of the second highest in the rest of the set tj j is not equal to i. So, in this case this is just going to be t2 because that is the second highest bid. So, now we can use that constant things and the fact that because this is dominant strategy incentive compatible the Myerson's payment formula will will hold. First thing to notice that this kind of a mechanism is a monotone allocation rule. So, you are always giving it to the highest bidder. So, the highest bidder the threshold that when it crosses the threshold of the second highest bid it is going to be the winner before that it is 0 and therefore, this is a monotone allocation rule. So, the the payment. So, this is the that constant part. So, p0 p1 0 comma 2 minus 1 this is the this is that expression which we have already defined. Now, the player 1 is going to be the winner. So, therefore, this this is t1 multiplied by 1 and the last term of that integral formula is integral from 0 to t1 if 1 x or t minus 1 dx and this player does not win until t2. So, therefore, this integral will be nothing but t1 minus t2 after t2 it starts winning with probability 1 and therefore, when you sum all of them up t1 cancels out from both sides from here and here and what we are left with is this. So, similarly player 2 pays just this amount because the rest of the part is 0 because it does not get that object all the other agents are getting minus 1 by n times t2. So, this is according to the definition. Now, if you look at the the sum of all these payments you can do this calculation there will be m minus 2 all other agents what you will have is 2 by n times t2 minus t3 and as n increases you can see that this excess amount of money that is going to be taken away by the auctioneer goes very much close to 0. So, this is a deterministic mechanism that also redistributes the money but it redistributes ensuring that this is truthful. Let us look at a non-deterministic mechanism a randomized mechanism which allocates the object with probability 1 minus 1 by n to the highest bidder and with probability 1 by n to the second highest bidder. So, this is also a monotone allocation rule you can convince yourself. So, what is this constants? So, constant is nothing but minus 1 by n we can pick the constants in whichever way we want this mechanism picks it as minus 1 by n times the second highest bid in the rest of the agents. So, for for instance this will be t3 for player 1 this will be t2 t3 for both player 1 and player 2 and t2 for all the other players. So, so we can we can do this calculation so we can see that so when player 1 so again we are assuming that the value the type of player 1 is the highest and it keeps on decreasing with increasing index of this players. So, the first constant so this is the p0 the constant of this integral and so with probability this the player becomes the winner. So, this is t1 times f1 and f1 is nothing but 1 minus 1 by n and then we have the the integral so this this entire part is the integral from 0 to t1 ff1 of x comma t minus 1 dx. Now, we know that between 0 to t3 the player does not win from t3 to t2 this is the second highest bidder. So, therefore, it has a probability of winning which is 1 by n so between t2 and t3 and from t2 onwards to t1 this is the highest bidder. So, it will have this probability of winning which is 1 minus 1 by n this is between t1 and t2. So, if you do the integral divide the integral into these parts you will get this all these terms and if you sum them up you will get this quantity here. Similarly, you can compute the second highest bidder a second highest bidder will only have up to this point here it is not winning here it is winning with probability 1 by n and when you take the sum you will see that this is going to be exactly equal to 0 and for all other players this the the payment is going to be just the constant quantity because they never win. So, their the only constant quantity is their their payment and because this is negative it means that they are getting paid. So, player 1 is actually paying player 2 is paying nothing even though it is getting some share of this allocation the all the other players are not getting anything but they are getting some payment. So, they are being compensated with some payment but the interesting thing is now if you sum all these payments together it will be exactly equal to 0. So, this mechanism is certainly strategy the this mechanism is certainly strategy proof because of the property that we have used this is the this is following in the Myerson's class the allocation is monotone and the payment is following this payment formula but additionally it is also ensuring budget balance. What it is losing because of this allocation rule it is not efficient and we have seen this before the allocation if that becomes efficient then it is not going to be budget balanced which is the problem with VCG that that problem persists even in this situation we have found a mechanism which is which is redistributing the money. So, someone is paying and someone is getting paid and the allocation is such that this is monotone and it is strategy proof but it is not efficient yet it is budget balanced. So, since our objective is budget balance we have seen a mechanism which is budget balance and also calls in the Myerson's class.