 We've seen in the past that the VCG mechanism gives rise to dominant strategies and efficient allocations. We know that in general it doesn't give rise to individual rationality or budget balance. In this video, we'll see that under some pretty mild additional assumptions, it is possible for us to get both of these things. Now, remember that individual rationality means that it makes sense for agents to participate in a mechanism. If they had a choice between participating and not participating, they would be happy to participate. And recall that budget balance means that it doesn't cost money to run the mechanism, that the mechanism either keeps completely breaks even all the time, which we've already seen in the previous video is impossible for VCG, or at least is weakly budget balance at least turns a profit or breaks even. What we'll see in this video is that we can have both of these two properties. So, let me start with individual rationality. To get individual rationality, I'm going to need two different assumptions to be true of the environment. The first is called choice set monotonicity. And this assumption says that for all agents, the set of outcomes that are achievable without that agent present is a weak subset of the set of outcomes that are possible when that agent is present. So, in other words, when I remove an agent from the mechanism, as of course I have to do inside the payment evaluation for VCG, the mechanism set of choices that are available to it weakly goes down. So, in particular, no new choices become possible when somebody gets removed. The second assumption that I'll make is called no negative externalities. And I'll say that we have no negative externalities if for all agents and all choices that can be made without that agent, the agent's own valuation for each of these choices is non-negative. So, in other words, when you get dropped from the mechanism, it isn't possible for the mechanism to choose something that actually causes you pain. It might choose something that you like less well than what it chooses when you are present, but it can't choose something for which you have negative utility. Let's look at two examples of realistic scenarios that satisfy both of these properties to give you a sense of why these properties are reasonable to assume. So, the first is the road-building referendum problem. So, consider this problem where we want to have a vote between agents to decide whether or not to build a public good like a road. So, the set of choices has nothing to do with the number of agents because the two choices are that either the road gets built or the road doesn't get built. Remember that the payments come from the payment function. Those aren't the choices. And so, the choice set is monotonic. It's monotonic in the weak sense that it just doesn't change with the number of agents, but remember that we allowed for a weak subset. Secondly, let's assume that no agent negatively values the project. So, some agents might like it better for the road not to be built than for the road to be built, but let's say that none of the agents experiences negative utility in any situation. That still seems like a reasonable model of this problem, and that would give us no negative externalities. Okay, well, we saw an example previously where choice of monotonicity arose in a pretty trivial way because the set of choices just didn't change. Let's look at a richer example where the set of choices does change. So, here let's think about kind of a simplified stock market in which we have two different kinds of agents. Some of them have a single unit of a stock that they want to sell, so they have one share of Apple stock and they want to sell it. We have another group of agents who want to buy one share of Apple stock, and we have some number of agents on both sides of the market. Now, the choices that the mechanism can make here are different ways of pairing together buyers and sellers. So, every buyer can interact with only one seller, every seller can interact with only one buyer, and nobody has to interact. So, the pairing where nobody gets paired is valid, the pairing where just two people get paired is valid, and the pairing where as many people as possible get paired is valid. So, we have these various different ways of establishing trades that might happen in this market. Now, think about what happens if I add a new agent to the market. None of the pairings that worked when that agent wasn't present go away when I add a new person. They're all still possible to do because I can always just set that new person aside. But I can also have that new person participate in trades that weren't possible without that person there. And thus, you can see here the choice set monitonicity is satisfied in a more interesting way because new options become available when I add an agent, but no old options get ruled out. And it's natural in this kind of a setting to assume that agents have zero utility both for other people trading with each other and for other people not trading at all. And that being the case, the agent has zero utility for everything that can happen when they're not present and so there are no negative externalities. So, now we're ready to consider the main result that we want to establish for this part of the video which is that these two assumptions that we've just made of choice set monitonicity and no negative externalities are sufficient to make the VCG mechanism ex post individual rational. Now let's remember what ex post individual rational means. As I said at the beginning of the video, individual rationality means that agents always have a weekly positive utility for participating in the mechanism and ex post means that this is true regardless of what valuations any of the agents have. So this is true for every realization evaluations that the agent himself and also all of the other agents might have. So this is the strongest kind of individual rationality. So this is the most desirable property about individual rationality that we would hope to prove about VCG and that makes this an encouraging result. So here's how we prove it. We begin by saying that all of the agents truthfully declare their valuations in equilibrium because we want to think about what happens in equilibrium and then we can write this expression for agent eyes expected utility for participating in the mechanism. Well, he gets his value for the allocation that actually happens. So this is the choice that VCG actually makes given that everybody reports their values truthfully and he gets his value for that choice. And then he pays the VCG payment function that we all know and love. So because I've assumed that everybody reports truthfully I don't have any V hats anywhere so I can collect terms and particularly I can roll this into one of the sums so I hear get a sum over all of the agents minus a sum over all of the agents except for I. Now it comes to the part where I use choice that monogynicity. So remember that this expression X of V is the outcome that VCG chose which means that it maximizes social welfare. And by choice that monogynicity this other outcome X of V minus I was also one of the available choices when we when we picked this instead. And so what that means is there must have been at least as much social welfare arising under this choice as under this choice. If not the optimization would have just picked this one. And so writing that as an equation the sum of agents values for this choice is at least as big as the sum of agents values for this choice. I'm just moving that equation up here and now I can also use no negative externalities. Let me just write the definition of no negative externalities. So the value that I has for the choice that gets made when he's not present is greater than or equal to zero. And what that means is that if I change this expression here to J not equal to I then I've just taken out a non negative number here which means that this whole thing got smaller. And that means the inequality still holds. And so I have this expression here. And that's exactly what I wanted to prove because remember the agents utility is this expression. And so I want to ensure that this number is bigger than this number. If it is then this whole thing is positive and I have individual rationality. And that's exactly what we just find here that this in fact is positive or non negative. And so we do obtain individual rationality. Okay, so much for individual rationality. Now I want to talk about budget balance. So here's the property that we need turns out one property is enough for showing that VCG is weekly budget balanced. And we call this no single agent effect. So an environment exhibits no single agent effect if it's the case that for all agents I all valuations of the agents other than I that are possible and all choices X that maximize the social welfare. There exists another choice X prime, which is feasible without I which means it's possible for us to pick it, even if I has been dropped. And for which all of the agents other than I get more social welfare than under the original allocation. So in other words, if I drop an agent I, and then I pick some other choice instead without I, everybody other than I is at least as happy with the new choice as with the old choice. So the welfare of agents other than I is weekly increased when we drop by that might sound a little confusing. It's actually a really natural property. Here's an example. Think about a single sided auction where, for example, I have one good for sale. The choices in this case are all of the different agents I could give the good to. And if I drop an agent, one of two things is possible. Either that agent wasn't winning the auction before, in which case the social welfare maximizing outcome remains the same after the agent has been dropped and nothing changes. And so then I get equality in this expression, or that is the agent who was winning before. And then I pick a different winner. Once that agent has been dropped, I'll get less social welfare overall, which is why I was picking a different agent in the first place. But when I consider only agents other than I, they're all going to be happier when I has been dropped than when I was present. Because one of them gets to win after I has dropped. And in the case where I was present, one of them wasn't winning. So you can see it's kind of actually a pretty natural property. No single agent effect. So it turns out that this is all we need to prove that VCG is weekly budget balanced. And indeed, it's a pretty direct proof. So as before, let's assume truth telling and equilibrium. And now what we want to think about is the sum of transfers that agents make to the center. So the sum of all of the payments. And I want to show that the sum of all of these payments is greater than or equal to zero. That's going to mean weak budget balance. So once again, this is the VCG payment function. And now I'm just summing the payment function across all of the agents because I'm interested in the aggregate payment. From the no single agent effect condition, I know that for all of the agents other than I, the value they have for a world for the choice that gets made when I is dropped. The social welfare maximizing choice for them that gets made when I is dropped is at least as big as the sum overall of their utilities for the choice that gets made when I is not dropped. That's just directly what no single agent effect means. And you can see that these are exactly the terms that I have in the VCG payment function. And so the result follows directly. This blue term is bigger than the green term, which means that this expression up here is just positive for every I or non negative for every I. And so that gives us what we want, which is that this whole sum is greater than or equal to zero. Well, I have one last bit of good news to give you. Here's a theorem from Krishna and Perry, which says something else about the revenue in VCG, which is even more encouraging than what we just saw. It says, consider any Bayesian game setting in which VCG is exposed individually rational. And the claim is that in such as case VCG collects at least as much revenue as any other efficient and X interim individually rational mechanism. Now, here are a couple of things to kind of walk through about what what is being claimed here. First of all, you might wonder about this little preamble. Didn't we already talk about what happens when VCG is exposed individual rational? Well, not really, because what we looked at before was two conditions that are sufficient for VCG to be individual rational. We didn't claim that there aren't other things that might be true about the setting that would also result in VCG being individual rational. So what this theorem says is don't worry about how you got there. Think about any setting in which VCG is exposed individual rational. It'll include what we talked about before, but it'll include other settings too. And what it says is VCG collects at least as much revenue as any other efficient mechanism. I think to notice here is even if that mechanism only uses Bayes Nash equilibrium, we're not assuming that this other mechanism is dominant strategy efficient. So this other mechanism gets to be drawn from a much bigger pool, the dominant strategy mechanisms and the Bayes Nash equilibrium mechanisms. And another sense in which it's drawn from a bigger pool is that we're only going to require of this competitor mechanism that it's X interim individual rational. Whereas we're requiring a VCG that it's X post individual rational. So this is a X interim individual rationality is a weaker thing to require, which means there's a broader set of mechanisms that would satisfy it, which means that we're making a stronger statement when we compare VCG to this broad set. So a way of understanding this result is that VCG is as budget balanced as any efficient mechanism can be. So I'm not saying here necessarily that the revenue is greater than or equal to zero. Sometimes it will be and sometimes not. But in cases where the revenue is still non is still negative. What this result says is if VCG was X post individual rational, there's no other mechanism which is efficient and which gets more revenue. So VCG is always getting as close to budget balance as efficient mechanisms can as long as VCG remains individually rational.