 Now, we are going to discuss one of the most celebrated mechanisms of mechanism design with transfers and this is known as the VCG mechanism named after the inventors Picri, Clark and Groves. So, this is a mechanism which is also known as the Clark's pivotal mechanism. This belongs to the class of Groves class of mechanisms. So, how is it belonging to the Groves class? If you look at this specific HI theta minus I function, remember that we had all the mechanisms in the class of Groves essentially satisfies the allocative efficiency property that is all of them are actually trying to maximize the sum of the valuation of all the agents by picking their allocation. The only difference in different mechanisms in the Groves class is in their payments and that is distinguished by the choice of the HI theta minus I. The payments were HI theta minus I minus sum over the valuation of all the agents at that efficient allocation of theta I theta minus I and when agent J is type is theta J and this is summed over all the J which is not equal to I and this will be the payment of agent I. Now, based on different HI theta minus I, you can get different mechanisms and different payment rules and in the case of the Clark's pivotal rule or the VCG mechanism, this is going to be the max over all the agents' valuations, the max over the sum of the valuations of all the agents except agent I. So one another way of interpreting this is as if the player I is now removed from the whole system, it is not participating in the mechanism anymore and we are trying to maximize the sum of the valuation of all the agents except agent I and what would have been the maximum value that is given by this HI of theta minus I. So therefore the payment under the VCG and we are going to denote that as PI VCG of theta I theta minus I is given by this following expression. We have the maximization with respect to and alternative which is maximizing the sum of the valuation except agent I and the subtraction part, the second part of the expression remains the same, it is the same efficient allocation. So notice that here agent I is actually present and it is choosing that alternative. So the mechanism is choosing the alternative or the allocation in presence of agent I but we are just looking at the sum of the valuation of all the agents except agent I. So this difference is essentially the VCG payment. Now one thing we can immediately notice here is that here we are looking at the sum of the valuations except agent I in both these cases and in the first expression we are trying to pick the alternative that maximizes the sum of the valuation, pick the allocation which maximizes the sum of the valuation. So no matter whichever other alternative you pick, in particular even if that is the efficient including agent I. Now when we are looking at the agents, all the agents except agent I, this is going to be suboptimal. So this will be the maxima and this expression will be less than or equal to that value. So we can conclude that this expression, the payment here is actually non-negative that is given by this expression here, payment under VCG is always going to be non-negative which means that it is no subsidy. Remember the definition of no subsidy that we mentioned in the last module and because it is no subsidy that is asking. So it is not paying any of these agents, they can either make some positive payments or they can make zero payments but it is never giving, the mechanism is never giving any subsidy to any of these agents. This is automatically going to imply that it is also no deficit because if you take the sum of all these payments they are also going to be non-negative. Now let us come back to the interpretation of this payment rule. So what is happening here is that you are looking at the sum of the value of the other agents when agent I is not present and then subtracting it out from the same sum of the players of sum of the value of the other players when agent I is present. So this is one way of interpreting this thing and what this mechanism is doing is you can in some sense think of as if the if you are if you are not present then other players who are possibly getting some better evaluation because of your presence they are getting something less than what they could have got if agent I was not present. So this difference is essentially the marginal damage that this agent I is creating on the other agents and this mechanism is just asking that agent to compensate that. So that is one way of actually interpreting. If you ever forget the actual expression this is the way you can actually reconstruct the the payment formula that is the interpretation of the payment part but you can also interpret the utility under this BCG mechanism in which case we are just taking the sum the valuation of that agent at this efficient allocation and subtract out the payment then if you reorganize the terms you can find out that the first term will be the sum of the valuation of all the agents including agent I at that efficient allocation and back because this is the the social welfare maximizing allocation this term is going to be the maximum social welfare in presence of and this second term if you just look at it it is maximizing the utility in the world when agent I is not present. So it is the maximum social welfare in absence of agent I. So the utility that each agent gets is the marginal contribution of I to the social welfare. Let us discuss a few examples to understand the how the BCG mechanism works. So the first thing is the very old example of single object allocation and here the type is the value of that object for that player if that object gets allocated to that agent so theta I in that case is nothing but the value of that agent and if that agent does not get this item then its value is 0. So that is the setting. Now let us look at the expression of the BCG payment and try to understand what it means so what will be the BCG payment for this kind of a setting. So as before the allocation because BCG mechanism is nothing but a mechanism in the class of gross class of mechanisms the allocation rule is the efficient allocation rule allocatively efficient. So the object because this is single and indivisible object it goes to that agent who values it the most. So when all the agents are reporting their types whoever has the highest type the item goes to that agent. Now if we look at the payment that this particular agent makes we are first looking at if that agent was not present what would have been the sum of the value of all the other agents. Well now if that agent is not present then it will go the object will go to the second highest bidder because that is the maximum bidder maximum valued agent in the rest of the game. So this sum of the valuation is going to be the the value of the second highest bidder second highest valued agent and when agent i is present if we look at the agent is that agent i is getting these objects so assuming that agent i is the winner that agent is getting the object so the valuation of all the other agents is essentially going to be 0. So therefore this term for that winner is going to be 0 and this term is going to be the second highest bid. So if we go back to the previous example where the there are four agents which we discussed in the previous module they had these values and their allocation was so the efficient allocation would have been to give the item to this agent 1 and that agent will pay mean the payment and the VCG payment will be 9 which is the second highest valuation in this case. And all the other agents will pay 0 because from the same expression we can see that if agent i was not present so suppose now we are considering another agent i which is not the winner. So for that if that agent was not present nothing is going to change still in this summation the highest valued agent will have that object and its valuation will remain the same. So for instance for the second agent here this maximization with respect to all the other agents except that agent is going to be 10 and even if when this agent was present this this valuation the sum of the valuation of all the other agents was 10 because when it was present the efficient outcome has actually given the object to this agent the sum of the valuation of that agent was 10 and everybody else as well 0. So you are both these terms are essentially the same for all the losing bidders so the losing agents who are not getting that object so therefore their valuation their payment under VCG is going to be 0. Now since we have called this mechanism a pivotal mechanism let us ask this question that what is pivotal in this VCG payment. Let us look at the this example to understand this point here on the on the rows we have different players A B and C and suppose there are three alternatives three allocations that are possible either it is a football ground which is going to be built or it can be a library or a museum so this is the example where the the city planner or the municipal corporation is deciding whether to build any of this and each of these agents have a certain value associated with each of these outcomes so for instance player A does not get any value if the football ground is built it gets a high value if library is built and moderate value if museum is built for player B something different happens so what would be the outcome in this case so you know that the outcome under this VCG mechanism is going to be the efficient allocation so we are going to take the sum of each of these columns some of the valuations of each of these columns and whichever maximizes that will we are going to pick that and it's easy to see that if you pick museum then the sum of the valuation is maximum which is 150 for everything else it is going to be smaller. Now let us look at so that is the outcome so the museum is the the allocation decision what will be the VCG payment so in order to find the VCG payment for player A we will first have to remove agent A from the system and see what would have been the optimal value or the sum of the maximum sum of the value for all the other agents so if agent A is removed then we are left with these numbers only and we see that in that case building football is the one that maximizes the sum of the values so 105 will be the sum of this two numbers the valuation of all the other agents which is giving this number and when agent A was present at that situation the outcome was museum and that was giving a valuation the sum of the valuation to be 100 for the other two agents so this difference between these two things is essentially the payment that agent A is supposed to pay under this VCG mechanism similarly if you do the similar exercise for player B so then you are going to remove that player B you can see that in that case library would have been the outcome so 1.2 would have been the sum of the values of all the other agents when agent B is not present and when it is present again this will be 50 and 50 so that is the sum of the valuation of all the other agents when agent B is present so this difference is essentially the payment that it is making now when you come back come to this third agent that is agent C here something interesting happens so you see that whenever agent C was is removed the outcome does not change so the maximum value is still going to be 100 for the for the other two players and that is going to be the maximum none of the other things can actually overshoot that value so therefore the and when agent C is present then also the outcome is museum so the sum of the values of the other agents are actually remaining the same so we are going to say that this agent C is a non-pivotal agent so what is pivotal whenever you remove that agent from the system the outcome changes if that happens then we are going to call that player is a pivotal player because because of its presence the outcome was something different and because of its absence the outcome changes but C is not like that even if it is not there the outcome does not change so this mechanism is essentially not charging any money to the non-pivotal agents so that is why it is called the pivotal mechanism so let us look at another example of combinatorial allocation so we are in this case selling multiple objects so suppose there are two objects one and two so either it can be so none of these objects could be sold if only one can be allocated and two can be allocated separately or the bundle of one and two can be allocated together now the agents might not have a kind of linear valuation that is it is might not be additive so it's a value for one so let's say when can that happen I mean the sum of these values the the valuation of the bundle is not exactly equal to the sum of the valuation of those individual items in that bundle this can happen if there are complementary objects so let's say you have a shoe pair of shoes one is left foot of that shoe and the other one is the right foot if you combine them together you have a high value but individually they have very low value so similarly there could be situations where this might not be the the valuation maybe smaller than the sum and you can you can think about this this kind of situations that is why it is called the combinatorial valuations agents can have different values for different combinations of these objects so let us assume that this table is essentially showing that what is the value for agent one and two for this different bundles and very naturally when nothing is allocated the valuation is zero so here and later on we are going to use some shorthand notation because this is an independent private value whenever we have this valuation of A this is the the alternative of the allocation A under this type theta i this valuation will sometimes be written in a shorthand with this theta i of A which means the same thing it's a value of that agent when the allocation A is chosen okay so under this setup the efficient allocation you can you can verify this because you can give this item to agent two and this item to agent one and that will maximize the sum of the valuation of all the agents this is the efficient allocation let's say that allocation is called the A star allocation now what is what is VCG payment in this case so what we have to do to compute the first part is that we'll have to remove agent one but if agent one is removed then the best thing that you can do is to give the entire bundle to agent two which gives you this valuation of 14 and when agent A is agent one is present what is the valuation that the other agent gets under that allocation under that efficient allocation that is going to be nine so this difference is essentially the payment that agent one is supposed to make and you can see that because it's a true valuation for its own allocation is going to be six so six minus five is going to be the payoff so you can also calculate the payoff separately so similarly when you remove agent two from the system then the best thing is to give the entire bundle to agent one and that is going to be the the sum of the valuation of all the other agents minus when it is present what is the valuation that the that agent one is getting that is going to be six here and that difference is essentially the six which is going to be the VCG payment for this agent two and similarly you can calculate what is the payoff and the utility of that player