Let us look at an example of VCG in action, of the Victory-Clarke-Groves mechanism in action. And it'll be a routing, a transportation problem. So here is a directed network of links, each link have the cost of the length of it if you wish and the goal is to find a shortest path from A to F. And obviously looking at it, be quite obvious that the shortest path is this one. So this is the path that would be selected by a traveler who wishes to get there, from A to F most quickly. Now, according to VCG, how much would the owner of the link AF pay? Well, intuitively he's not part of the, or the path, so he shouldn't have to pay or receive anything. Let's see what VCG says. So the shortest path with AC's declaration of what their cost is, is five right. And so the cost to all the agents is that shortest path totaling five. Had AC not been in the picture, and not declared anything therefore, what would the total cost to the agent have been? Well, it would have been the very same path, so the shortest path. And so the cost would have been 5 also and so, the amount that AC pays, if you wish, is the difference between the cost to the agents within the picture and the cost without in the picture which of course is 0. And this is true for all the other edges that don't participate in the shortest path. All these edges. Let's draw them blue. These edges not participating in the shortest path will neither pay nor receive anything according to VCG. So that's the easy part. What about edges that do participate in the shortest path? So let's look for example at one of them, at AB. Well, with the shortest path as we know is this one. And how much would all agents other than AB itself, what is the cost on the agents on this sort of path? Well, it's 1 + 1, is the cross of two. Had AB not been in the picture, what would have happened? Well the shortest path would have been this one over here. So this wouldn't be the shortest path of a cost of 2+3+1. And so the cost to the agent AB is the difference between the cost without them in the picture, which would be -6 And the cost with them in the picture, which is -2. That is the cost to all the other agents and the difference is -4. So the cost to AB is -4, in other words AB will get a payment of 4. Making, if you wish, a profit of 1 because the cost is 3, you'll get a payment of 4 and you'll have a benefit of 1. So this is for AB. What about BE, for example? Well it's the same sort of analysis. BE the cost to all other agents without BE in the picture is -6. Its still, this would have been the shorter route had BE not been in the picture and cost would be -6. Now the cost to the agents with B in the picture, to the agent other than BE is 4. Why is that? It's 1 + 3. And so the cost to, this is the -4 over here, so the neck of it is that the cost imposed by VCG on the link BE is -2, in other words, they would get a payment of 2, making, again, a profit of one if you wish. What about EF? Same sort of calculation, although the outcome would be a little different because in that case they will getting a payment of 3. Why is that? Well without them In the picture this would no longer be the shortest path. So the shortest path would be a cost of 7, whether it's this one or this one, in both cases the cost of the first path will be 7, so the total cost to the population, without them in the picture, would be 7. With them, the cost is again the 1 + 3, it's a 4, and so they get the difference of 3. So the payment would be 3 and therefore the net profit, if you wish, it would be 3- 1, namely 2. So you could ask why are, why is it so different. You look at all these links somehow their net payoff is different. Is that fair and fairness is in the eye of the beholder. But the rationale for it is that they have different market power. The amount of social value they bring is different because without them, the outcome would be different to the population and that's why they have a different sort of payments and different cost structures here. That's an example of DCG in operation.