 Now, that we know what is a VCG mechanism, let us now discuss some of its properties in the context of combinatorial auctions. Just to remind you the combinatorial auction is those kind of auction where every agent has some valuation on a combination of this object. So there are let us say m indivisible objects in this context and every agent has a valuation for a subset of these objects. So we can denote by this set omega the set of all possible subsets of m which is nothing but the power set of m. So we will also use the term bundle to denote a subset of these objects. And now what the theta i or the type in this context it is defined in the following way. So you take this any of the subset from this set omega and theta i will actually map that subset into the set of real numbers. So one can say that the so if you have a specific subset s of these objects theta i of s is essentially telling you how much this agent values that particular subset of these objects. Fair enough. Now in this context even though we the mapping it is mapping to the whole of real line will only focus on the non-negative part of the real line. So which means that this this types of the valuation of this agents are always non-negative. This is also sometimes referred to as the objects that are goods that is it is always going to give you non-negative valuation it can give you a zero valuation in which case you don't really want that object. But it can can never give you negative valuation. So contrast this with some objects like bags I mean the name comes very similarly like goods if that object if you if allocated to you let's say some task that is allocated to you that task is essentially costly. So therefore you don't have a positive valuation for it rather you have a negative value for it. So those kind of settings we are excluding from our discussion of combinatorial options. So in this setup the allocation of this allocation in the combinatorial option is nothing but partition of these objects and how is this partition defined. So it's a partition of those m objects into n plus 1 bins. So each of these bins from 1 to n and actually corresponding to which bundle is goes to which agent and the 0th bundle is nothing but the that set of objects which are unallocated. And all this AI is an agent is a disjoint. So therefore if an object goes to a specific agent it does it cannot go to some other agent. And of course when we take the union of all this sets all these components of this allocation and this should certainly be equal to m that is all objects are exactly in one of this partition. So let us assume that a is the set of all such allocations. So all such allocations which does this partition of these objects into n plus 1 bins. So we are also going to assume if a specification does not get any object that is its allocated partition is the empty set then its valuation is going to be 0. So this is quite natural. And in addition we are also going to assume that these valuations are selfish. What does that mean? So there is no dependency across multiple allocations. So if agent i gets an allocation of AI its valuation only depends on AI. What can be a different situation? So perhaps agent i and agent j are in some sense interrelated. They might be good friends and then the valuation that one agent gets might not just depend on its own allocation but it also depends on the other agent's allocation. So we are excluding that. So we are just focusing on selfish valuation that your type only your valuation only depends on your allocation. Okay so in this context where we are looking at this allocation of goods the VCG payment of our agent that gets no object in the efficient allocation is 0. So we know that the VCG mechanism always picks the efficient allocation that is the allocation that maximizes the sum of these valuations of all the agents. But suppose in that efficient allocation an agent gets no object so its allocation set. So let us say this is agent i whose allocation in this optimal in this efficient allocation is empty. In that case the VCG payment will also be equal to 0 that is the claim that we are making. So how should we prove it? So we first look at so because this is just allocation of goods one can actually look at two different scenarios. So first scenario is when all the agents are present because we will need that in defining the VCG payment. So where all the agents are present and we are picking that allocation which maximizes the sum of the value. So notice that because these are independent private values we can just rewrite that as the theta G of A itself. We can use this notation instead of writing it in the value form. Similarly we can also look at the optimal the efficient allocation excluding agent i. Suppose agent i was not present in that set up what will be that allocation which maximizes the sum of the values of all the other agents. Now notice that what we are doing here is that we are looking at the same allocation set. So which means that we still have this n plus one length vector so an allocation is nothing but a n plus one length vector. But now that agent i is not present we can deterministically say that this ai is going to be empty. So in all these allocations when we are talking about a minus i star in all these allocations that ith agents allocation is going to be null. So that is another way of representing this and this is possible because we are talking about the allocation of goods I mean just an allocation of certain objects. If you look at some of the textbooks you will find that this kind of a set up can be put under a very much more general setting after removing a specific agent. The choice set is essentially is monotonically decreasing. So that property that more general property is known as the choice set monotonicity. But without going into it we are just focusing our attention to the combinatorial allocation of goods and that is in that setting we can very easily see that we can actually define this allocations in the following in the same way. So the allocations still fall in the same set A yet we are going to denote the allocation excluding agent i just by putting that ith component of that allocation to be always equal to null. Now what we have already seen is that the VCG payment is always going to be non-negative and that is something that we have noticed we have argued in the previous module. Now we are going to show the other direction. So we will show that this BIVCG is also going to be non-positive in the in this specific case where the allocation of agent i is null in the efficient allocation. So how should we do that? So let us write down the PIVCG again. Now we know that this is going to be the sum of the valuations of all the agents when agent i is not present and here we are looking at the sum of all the other agents when agent i is present and we are picking the efficient allocation. The only thing is that in that efficient allocation agent i gets nothing. So we know because of the way we have defined this a in this world without agent i we can define the type because in this world of a minus i star the ith component of this allocation is always going to be null and therefore theta i of null is going to be zero. So we are going to add this term to this expression on the right hand side and also going to subtract out this theta i ai star which is the valuation of agent i in the efficient allocation and by definition this is going to be equal to zero. So now what we get here is when we add this term to this term then we get the sum over all agents including agent i for this a minus i star and the same thing when a star over a star. Now this quantity if we look carefully this is the sum of the valuation of all the agents including agent i and a star by definition maximizes this term. So for any other allocation that you can look at this is this sum is always going to be larger or at least as much as the valuation the sum of the valuation in the other allocation. So you can clearly write that this is non positive and that essentially proves our result. So we have already shown that the PIVCG is non negative and now we are showing that the same PIVCG for this particular agent i who got nothing allocated in the efficient allocation will be non positive. So therefore it must be equal to zero. So that is what we wanted to do alright. So let us now look at a different property called the individual rationality property. So intuitively it means that the agent should be voluntarily participating in this mechanism if they participate then they should get some non negative utility under no circumstances they should get the agent should get any negative payoff. So the mechanism formally we can define it in the following way that this mechanism is individually rational if you look at the valuation minus the payment under that mechanism for this reported types of theta this should be non negative for every theta. So any type profile that you look at it should be non negative and it should also be true for all agents i in n. Now again we can make this make a similar claim that if if the allocation if we are looking only of about allocation of goods the combinatorial allocation of goods then VCG mechanism is definitely going to be individually rational. So again you might find a very analogous result in some text or some other courses which talks about that those kind of monotonicity the choice set monotonicity and also condition that there is no negative externality which we actually ensure by looking only at the goods. So VCG mechanism satisfies individual rationality under those general conditions but here we are for our simplicity we are looking at a very specific set which satisfies those two properties. So what we have to prove here is that the utility for every agent is non negative that is how individual rationality is defined. So first look at so pick some arbitrary agent i and look at its utility and then expand out the payment function the payment formula. Now what we will get we will get a very similar term where the first term is essentially the sum of the values of all the agents at that at that efficient allocation. And in the second term we have all the sum all the agents the sum of all the agents except agent i in that optimal allocation excluding agent i. So now we are what we are going to do we are going to do a very similar trick we are going to add and subtract this quantity theta i a minus i star as before a minus i star is nothing but that allocation it is still living in that set a with the ith component being identically equal to now. So if we look at that so we know that this is going to be exactly equal to 0. So what advantage it gives us is that we can now club these two terms together and we can say that that is nothing but the sum of all the agents the valuation sum of the valuations of all the agents including agent i but under this different allocation which is a minus i star. Now by definition we know that this this difference what we have from here is non negative and this term the second term is always going to be non negative as well. So actually this will be exactly equal to 0. So together this term is always going to be non negative so therefore because we have chosen this agent i to be arbitrary this is definitely going to be true this non negativity is going to be true for all agents in the player set. So that shows that under these conditions PCG mechanism is actually individually rational.