 Now we are going to address a very important question from the point of view of the option here, how to maximize the revenue. So far we have only discussed about the truthfulness, but now we not only want to reveal the information truthfully, but we also want to maximize the payment that has been made by all these agents. Now, when we are talking about the revenue optimality, and that is what we mean by optimal options, we have to assume some sort of a prior over the types or the valuations of the agents. And why is that, if we try to do a non-prior, try to maximize the revenue from a non-prior setting, we will have to consider something like a worst case revenue. And the worst case revenue might not be that good, it can be very, very less compared to what is possible to earn in practice. And the reason being that when you are selling an object, when the auctioneer is selling an object, it might also have some information about the valuations of the buyers from their previous interactions. If there is an object which is being sold over time, or similar kind of objects being sold in an auction over time, then it has some sort of a prior that what the valuation distribution is. And accordingly, we can define the expected revenue that you can earn from this mechanism. And that is the reason we will now shift to the Bayesian setting, and we will redefine these notions of incentive compatibility and individual rationality in the Bayesian setting. So Bayesian incentive compatibility, we have already discussed very early in this mechanism design segment. So it says that if we, so now we are going to define BIC in the context of single object allocation. Now assuming that the type set are intervals between 0 and BI for agent I, and the common prior is defined on top of that type profile. So it's a Cartesian product over all the types of all these agents. And let us assume that this lower case G denotes the density of that common prior distribution which is a CDF. Now with this notation G minus I, S minus I given S I, we are going to define, we are going to denote the conditional distribution over S minus I, the types of the type vector of the all other agents given agent I's type which is S I. So what we are assuming that agent I knows its own type, this is something like an interim state where it knows its own type, but it does not know the types of the others. It only can predict or have a belief over that which is probabilistic. Similarly, lower case G minus I is just the density. And all these things S minus I given S I, this is derived from the original G using the Bayes rule as before. Now what we are going to say is that every mechanism, so the mechanism is again mapping on the type profile into an allocation which could be a probabilistic allocation. And this payments, the actual payments that has been made after the realization of the types, this induces an expected allocation. So when the agent I does not know the types of the other players, it can only predict what will be the allocation in an expected way and we will define that shortly. So it can define the expected allocation and the expected payment according to those common prior capital G and its density is lower case G. And those payment, this expected allocation and expected payment rule, we are going to define by this alpha and pi, the vector pi. So what is this alpha? So alpha again has components for each of these players. So this is the ith component of that allocation. But now notice that it is no longer a function of S minus I because we are going to take the expectation with respect to S minus I and therefore it is only function of two things, what its true type is and what is its reported type is. So now it is going to be defined in the following way. So look at this part which was there in the previous setup also when we were talking about this randomized allocation. But the randomized allocation was in the prior free setting. So it was defined for the entire type profiling when SI, so agent I sees its type which is SI and the type of the other players is S minus I. This was the probability of allocation of that agent, of that object to that agent. Now on top of that what we are going to do because agent I does not deterministically know what S minus I is, it is expecting that with respect to that belief which is S minus I given TI. So TI is its true type. So the belief will depend only on the true type while it might report something different. So based on its report the mechanism will work but it will always evaluate that mechanism according to its true type and that is why there is this difference. So notice that this one is the same one which it has reported and this one, this type is the same one that is its true type. And at this point we are also, I would like to make one comment that we have seen deterministic mechanisms where the allocation is exactly giving the object deterministically to one agent or to some other agent. We have seen in the previous setup in the module on Myersons mechanism that you can also give randomized allocations. Now that is after everybody has reported, so this randomization is happening on the side of the mechanism designer. So mechanism designer may toss a coin or do some randomization and give the object probabilistically to one of these agents. So this is one level of randomization that happens at the mechanism designer level. But the probability or the prior based thing, so this randomization is actually happening at the agent's place. So the agent is creating its belief or has a belief based on their g-i, the common prior and it is taking the expectation with respect to that belief. So that is another randomization how this s-i's are getting generated in agent i's own mind. And we are taking the product of these two different probabilities and then taking the expectation over this s-i's. So this is what I wanted to keep in mind because whenever we are talking about the single object allocation, particularly when we are talking about Bayesian incentive compatibility or some guarantees in the Bayesian world and also our mechanism is also randomized. Then we have these two levels of expectation. The first expectation is coming from the agent side, which is the belief over the types of the other players. And the second level of randomization is coming from the mechanism designer side, how it is allocating the object to the different players. So something to keep in mind. So similarly we can define what is the expected payment for this player i. Again s-i is the reported type and t-i is the true type. As before we are going to take the expectation with respect to this posterior distribution. So you can call this a posterior given its own type. It is generating the belief over the types of the other agents, taking the expectation with respect to the actual payment which is given by the mechanism. Because it is reporting s-i, the mechanism will decide the payment according to that. And assuming that the other players are reporting s-i in truthful manner. So here the probability can only be defined in terms of the true valuation profile, not what they do if they misreport. So if that happens then this is the expected payment for player i. And therefore the expected utility will be just that you are, so when agent i is reporting this type truthfully let us say, so s-i is actually equal to t-i. In that setting you can write it as t-i multiplied by that expected probability of this allocation minus the expected payment. So this is sort of the interim utility that this agent i is getting. Now we are in the right position to define what is valuation incentive compatibility. As before the inequality that when it is reporting its type truthfully the expected payoff or expected utility that they are getting is at least as much as when it is misreporting to some s-i. And this inequality should get satisfied for all s-i and t-i in capital T-i. Note that there is no s-i anymore because that has already been taken expectation over. And as before we are going to say that this allocation rule is Bayesian implementable. If there exists some payment such that this f comma p is b-i-c, note that we are going to call that this mechanism, this f comma p mechanism to be Bayesian incentive compatible. Of course implicitly we are assuming that there exists a common prior g and we are taking the expectation of this allocations and the payments according to that g. So in this setup we are going to assume the independence of all these priors of this the distribution over these types. And also then we are going to do the characterization of b-i-c mechanisms very similar to what we have done for the d-s-i-c mechanisms and we will see a very similar result which is also a result due to Meyerson but now for the b-i-c mechanisms. So what we are going to assume is that this the joint distribution of all the types of the agents is decomposable in a product form and therefore I mean this is essentially just saying that all these types are generated independently from each other. Now we are going to use for simplicity whenever agent i is reporting its type truthfully. So this ti is the same as the type that is its true type that alpha of ti is the shorthand for that notation instead of writing it twice. So now an allocation rule is going to be non-decreasing in the expectation. So remember that the earlier we were saying it is non-decreasing and we were just using the fact that fi of ti t-i is at least as much of fi of si t-i and whenever ti is greater than si. So we define this as non-decreasingness. Now we are going to apply the expectation with respect to the prior and we are going to define that it is going to be non-decreasing in expectation if for every si which is strictly less than ti instead of fi now the expected allocation is non-decreasing. So that is a non-decreasing in expectation quite straightforward definition. We will note that this rules that are non-decreasing that we have defined in the previous case so all those rules so all the characterization results that we did and all the examples that we have given earlier which were non-decreasing they are always non-decreasing in expectation because they are non-decreasing for every ti-i so of course if you take expectation with respect to that they are also going to be following the non-decreasingness in expectation but now what can happen is we can possibly have more rules and we will see examples of that more rules that are non-decreasing in expectation and that opens up a lot of many other mechanisms which on which we can actually try to maximize our revenue. So let us first discuss the characterization result the result which is very similar to the Myerson's characterization of DSIC mechanism so here we are trying to do the characterization of the BIC mechanisms so you are going to say that this mechanism the result goes as follows this mechanism f comma p in the independent prior setting so that is what we are looking at is BIC if and only if two conditions hold and those conditions are very much analogous to the DSIC condition f is non-decreasing in expectation now and also this payment formula satisfies this integral formula which is very similar so you are the expected payment at type ti again notice that there is no t minus i anymore because they have all been expected over the expected payment is has three components first one is the constant component the second one is alpha ti times alpha of ti so you see f or fi has all been replaced with alpha i ti and similarly for the last component which is the integral part and this should hold for all ti in capital ti so this is this is the as you can see is a Bayesian version of the earlier theorem but the proof can follow in very similar line so we are not going to get into that proof you can use the ideas of the same proof and just have to take the expectation at appropriate places and redo the proof I leave that as an exercise now since we said that there are more NDE rules than the NDE rules so let us look at one example where it is NDE but not NDE so let us assume that there are two players and each of them have five possible types their typeset has five possibilities and they can be represented in the form of a matrix so the five different types for player one is given here and the five possible types are given here so each of this blocks are actually representing each type profile and now we can so this matrix is showing that under which conditions player one is getting allocated so they are represented with one the the slots that are blank that the agent two is getting allocated the object now how can we see that whether alpha one T one so let us look at alpha one T one the expected allocation for player one under this T one you can see when T one is having this value then the object is getting allocated to player two under all possible circumstances so if you are actually looking at the corresponding allocation with probability one it is going to play at two and with probability zero it is going to go to player one so therefore this expected allocation you can do this calculation the expected allocation for player one when T one is equal to this this value here is going to be zero similarly it is zero here as well but here it has a probability of one over five because there are five possibilities in in one of those cases it is getting an allocation in other four cases it is getting allocated to T two so if you take the expectation over that it will be one by five similarly if the T one is this then it is two by five and at this value it is three by five so you can see that as T one is increasing because T one is increasing in this direction the alpha one T one is also monotonically non decreasing similarly you can look at the alpha two T two so because here and now you will have to look at count the number of blank boxes because we are considering alpha two that is allocation probability of player two so this will be three by five in this case it increases then it goes to three by five again here it is four by five because there are now four blank boxes here also it is four by five and finally it is one because it is getting allocated to player two all together so for both these players they are monotone but you can already begin to see that it is not monotone for F one T one T two or F two T one T two so you can see that if T two was having this particular value here then the allocation so if we fix T two at this point then player one is not getting the allocation here then it is getting the allocation here but again not getting the allocation here so this violates the monotonicity condition so you can also create an example for T two where it is violating that but the point is that this is this allocation rule is not non decreasing so in the Bayesian setting we can also define the analog of the individual rationality property so we are going to say that a mechanism is in the interim individually rational so this is something that happens after agent I has observed its own type but it is still speculating what is the type of the other player so the expected utility or expected payoff of this agent is always going to be non-negative at that intering level for all types of this player so if it is deciding whether to participate after watching its own type it should always participate because that is that is the sort of belief system is telling it okay so now we can actually extend the result of that Myerson characterization for BIC mechanisms to BIC and IIR mechanism so you can see that the mechanism F comma P is BIC and IIR if and only if the first two conditions are coming directly from the characterization of BIC and the third condition is telling you that if the constant part in this integral should be non positive this is something that we have also seen in the individual rationality condition when we discuss the prior phrase setting so here the this constant is going to be non positive so we will not go again to prove this very formally but we can give the pointers that this conditions one and two that we have said is coming directly from the BIC mechanism so this is an if and only if condition so they are equivalent to BIC so all that we need to prove is to show that IIR along with this conditions one and two are equivalent to three and that is not very difficult so in the forward direction so we want to show that IIR along with one and two are equivalent to three so we first show that if we assume IIR with one and two this implies three so how can we do that we just need to apply the the condition of IIR so what is IIR so this this one and we are going to plug the value of TI to be equal to zero so therefore this term this term here vanishes and all that we have is minus pi i is zero is going to be non negative which means this pi i is zero should be non positive that is very straightforward we did the similar exercise in the previous case now the other direction that if if we assume condition three that is if we assume this is true and also this one and two this inequality is true so then whether we should have a condition of IIR so how can we do that we can just write the the utility of player i and if we if we now substitute the value of pi i according to that integral formula what we what is left so because this term will get cancelled out what we will be left with is minus of pi i is zero and this integral part and we know that this is a probability distribution so therefore it is going to be non negative the integral will be non negative and pi i we already know that this is non positive so minus of that will be non negative as well and that will be greater than equal to zero so this is certainly an IIR so all right so we have actually proved this am I as well so we will use this results appropriately when we are trying to find the optimal mechanism which maximizes the revenue