 Let us now look at how slots are allocated in position options and how they are priced So as we discussed before the value of an agent I is a product of three components The first component is its value. So how much it values that particular click is independent of which position that click came from it is once the User is in the website of the advertiser. That is the value that it gets Now the the probability of getting a click had two components The first one was the first one was the position component and the second one was the Component of the of the user that is the quality component and now this three things together was is the valuation the total valuation expected valuation of this agent and from the point of view of the of the Search engine or the publisher of this ad This is row. I component is estimated and What is AI AI is essentially nothing but the allocated slot to agent I So therefore the the complete allocation is given by this vector a a which gives the the corresponding slot So a 1 to a n of the different slots that has been allocated to these different Now because we are actually looking at the efficient allocations We are going to pick the the allocation which maximizes the sum of the valuations of these agents So this is the same as the VCG mechanism as we have done before Now the claim is that if you are doing this kind of a allocation The efficient allocation it is going to be efficient if and only if it is ranked by expected revenue mechanism. So We have already seen the expected revenue expected revenue is going to be nothing but this this total quantity because this is the probability of getting a click and whenever that click happens this agent is going to give this value and if you are Sorting them with respect to that Expected revenue Then you are going to get an efficient allocation and the proof is fairly simple I am not going to do it in the most rigorous manner But the idea is that all that we need to do whenever the agents report their valuation is is to multiply that with that reported theta and if that if you do the sorted allocation so the first slot goes to the agent who has the highest why had times by theta i value and Give it the first slot then overall when you are taking the sum of all these things multiplied by the corresponding Position effect the position probability Of of that slot that is definitely going to be the maximum and the reason is very very simple all these p is have a Monotone relationship when you are going from the position one to position M and If you have a bunch of numbers, which are why had times theta i how should we Place them we should start with the highest value highest value going to the highest position any other allocation of this This advertisers to those positions will automatically be suboptimal than that. So that's the that's the allocation we are sorting all these agents with respect to their product y hat times theta i and Starting to allocate them the positions Sequentially so that is going to be the efficient allocation and This is because this is a sorting problem. I mean it's already computationally tractable So now the allocation decision is done We need the payments to make it DSIC and quite naturally because the allocation is efficient The natural candidate is going to be the VCG mechanism and this is something which is used in the designing the Adoptions on Facebook Now what we are going to discuss in this module is essentially a simplified version of that of that mechanism certainly This mechanism this kind of a thing that we are going to discuss here is not The same as the actual implementation in Facebook or other social media web pages But the principle remains the the same that how we are going to price them and how The prices per click should be will be the same in both this in all these cases So let us consider the VCG in position option. So what What is given in this case each of these agents are asked for their bids that how much they are true values are so theta Is are essentially for their advertisements and this bidders reveal or the advertisers reveal this be be eyes and be eyes nothing but the theta I had as we have Used the notations elsewhere Now we can say that the this numbers of these bidders So the the bidder numbers are unimportant all that we matter all that matters is essentially how we are going to Sort them so without loss of generality. Let us assume that They are numbered in in the way Such that they are sorted according to their values of why Row I had times be Row one had times be one is going to be the first position. So the the top most position The second agent has the second highest value of row two Had times be be two and so on So the allocation is essentially the efficient allocation is going to be the allocation which allocates These agents in this order. So now what is going what we are going to define is this in a minus I start this is something that will be requiring whenever we are defining the VCG mechanism If you are removing a specific agent, let's say agent I then all the agents that were after that agent I That is from I plus one to N Then now start getting once better slot because the agents were sorted according to their Row I had times be I So if agent I is not present now all the other agents starting from I plus one to N They will now get allocated once what about because that is now empty So we can write down the the PI VCG the payment under this VCG mechanism in the following way So as before we have all this J not equal to I so that means So imagine in this in this allocation List so suppose this was the position of agent I Now we are considering when agent I is present versus when it is not present But we are only summing the valuations of all the agents except agent I so therefore it will not consider this agent's valuation at all If you only look at the agents above it and also the agents that are allocated slots below it, right? So now if we look at the the allocation per agent I is not present that means that all the the Agents which were allocated after I they actually move one position above So therefore we can start so in this to so you take the subtraction of of that quantity With the quantity where the allocation is the same but now agent I is also present So therefore this agents this agents do not move up. They remain where they are So for all the agents that are above I they actually cancel out from both these sides So we don't really need to look at them. So we'll have to start from agent I Will go up to in minus one. So this is the this is the place until pitch agents have some Some value so we I mean this is just for a notational purpose because we are using this notation of J plus 1 So for the jth position or the ith position starting from that ith position The the next set of agents who were one step later in the actual ordering of these agents They now start getting this the slot jth slot Similarly when agent I is present then j j plus 1th agent is essentially getting the jth plus 1th slot So there is no change in shift in their position. So this is this is the two terms So all the other terms I essentially cancelled out now we can compress these two Summations into one by writing this this difference of the corresponding position effect multiplied by their Their valuation their expected valuation and of course This is for all the agents up to in minus one the last agent in in this list Will always have a zero payment because if that agent is removed the allocation of any of the other agents do not change So therefore both these things will essentially cancel out and they will have only see Now this is the portal expected payment So you can think of this as if the after observing that they have been clicked or not Now these click is essentially a probabilistic event. So therefore what the Search engine or the publisher can charge is for a specific click It can it can charge just by dividing with the probability of that So PI times rho i hat is essentially the Probability of getting a click for this agent I which is at the ith position and If that that quantity is in the denominator and the payment that we have just discussed is in the numerator That is the the payment that this agent makes for every click and that is how the BCG mechanism is actually implemented in the context of sponsor search options