 Now, we can extend the idea of that optimal mechanism design to multiple agents in the previous module we did it for only one agent. So if it is BIC and IIR and also maximizes the revenue then we are going to call that an optimal mechanism in this context. So the previous results that we have seen so far this BIC and IIR actually reduces the fact that all these FIs the probability of getting allocated this object are non-decreasing in expectation that we have seen this is the characterization result. And then also PITI has a very specific formula and IIR enforces that this constant of that integral formula must be equal to Z. So therefore the expected payment made by this agent I is nothing but this payment multiplied by the GITI. So what's the probability that this TI is taking a value of lowercase TI. So this expression and this expected payment of player I we are going to sum it over all players and then try to maximize it subject to the conditions like FIs are all non-decreasing in expectation. So we are going to do a very similar exercise like the one agent case and to simplify the expression. So we are going to do this substitution of this PITI to the integral formula and the steps of this transformation of the order interchange of the integrals and all the steps are exactly the same. Finally we get to this point where we replace that first term which is TI minus 1 over GITI by lowercase GITI. This is the same term which we are going to call the virtual valuation of player I. So we will be using this term quite often. So this is not the exact valuation because the exact valuation is TI, we are subtracting out something which is dependent on the prior of that type and that is why we are calling this a virtual valuation. So some different sort of valuation and we are trying to maximize that with respect to these two terms which are having a we are taking the product and integrating over it. So the first term is the probability, so its own prior over this TI and the second term is the expected allocation of this agent. And we know just by expanding this alpha ITI this is exactly what it means. Now we can take this term and this term together and call that the whole vector. So it is a joint probability even though it is a it can be decomposed into the products because all these TI's are independent but we can use a shorthand notation of G of t which means the joint probability distribution over this whole type profile. And this FITI is exactly the type of player I and similarly because this was only with respect to the type. So this 0 to BI is nothing but T of I and here the integral was over all t minus I. So the integral is over the whole type profile t. So this is just a shorthand notation for multiple integrals. So there are in integrals here they are compressed into one. So we have this part which is the virtual valuation of player I multiplied by the probability of that agent getting allocated and then G of t. So now what we see that this is a common term for all the players this and that and this part is essentially dependent for that player. So we can actually write the total revenue generated by all the players by taking the summation over all these agents and all that we have is we can interchange because this part as we said is independent of any player. So we can shift this summation inside and write this to be the expression that we were trying to maximize. So sum over the FIs multiplied by WIs and that summed over all the players. So therefore our optimal mechanism design problem reduces to maximizing this subject to the condition that this FIs has to satisfy the condition of NDE. It has to be non-decreasing in expectation. That is what is ensuring that this is BIC. Now if we just forget about this condition of NDE for FI and we just try to maximize this. We have a summation which we are trying to maximize and we know that this FI TIs will be so if we take the sum over all Is that is exactly equal to 1. So we are actually trying to maximize the convex combination of this WIs. So it's a very natural thing and we have seen this very early in this course that it is best to give probability of 1 to the highest value in this convex sum. So once we are trying to maximize the convex combination of multiple numbers, the one that will maximize is when we are putting the entire mass on the maximum value. So and that is the kind of very elementary solution to this problem. So we can give this FI T to be equal to 1 for that agent whose WITI value is the maximum and 0 otherwise and if WITI is less than 0 for all agents then we do not allocate that object at all. We keep it unsold. But there is a problem with this part alone because now we can actually construct situations under which based on whatever conditions that this WITI satisfies, this FI T might not be non-decreasing in expectation. And therefore we cannot really solve it in an unconstrained manner. And if you want to take a look at what is the example, you can take a look at the original paper by Meyerson, the optimal auction design paper. Essentially the idea is the example is such that the following condition is violated. So remember that even in the single agent case we assume some regularity conditions on this WIs. So in that case the W was monotone non-decreasing and this GIs were actually satisfying the condition of monotone hazard rate which actually gave rise to the fact that WI was monotone increasing and that was helping us in making this constraint optimization problem into an unconstrained optimization problem. So we are going to assume a very similar thing in this context. We are going to call this virtual valuation WI to be regular if that condition holds precisely. That is you have this WI satisfying this monotone increasingness condition. So if SI is less than TI then WI of SI should also be less than WI of TI. So if that monotonicity condition is satisfied then we can actually solve this problem and there is no issue with this example one. So this point one that we have made you can allocate the object to the agent whose virtual valuation is the maximum and keep it unsolved if everyone's virtual valuation is negative. And you can notice that the condition that we have imposed earlier the monotone hazard rate condition if we impose the monotone hazard rate condition for each player then definitely this WI TI is going to be monotone increasing. But monotone hazard rate for ensuring monotone hazard rate for every agent is much more demanding than the condition of regularity. So essentially there are examples which you can find in this Meyerson's original paper which says that if you have monotone hazard rate condition of course you are going to satisfy regularity but regularity is weaker. So you can have regular virtual valuations which might not satisfy monotone hazard rate condition. So we are actually imposing a weaker condition in terms of regularity and that is exactly what we need. Suppose we assume that every agent's valuations are regular, virtual valuations are regular then the allocation rule of the optimal mechanism is the same as the solution of the unconstrained problem. And the proof sketch is very similar, I mean the solution is as given in this equation one what we have discussed before you give the object to the highest virtual valuation agent and charging the payment which is given by that payment integral formula. And this optimal allocation rule also happens to satisfy this non-decreasingness property. So even though we are just interested in finding a mechanism, we are just trying to ensure the FIs which were satisfying non-decreasing in expectation, the solution that we got from this optimization problem when we are maximizing the revenue and it happens to be non-decreasing. So not only BIC, this mechanism is going to be DSIC and that exactly is the observation that we are going to make, this is non-decreasing and NDE. We will see in the next module that what implication does it give. We have actually solved the optimal mechanism design problem for selling one single object, single indivisible object and we have found the allocation and the corresponding payment integral according to my essence formula.