We will now walk through an example or a whole process of generating as many counter intuitive examples as we like, based on Sen's impossibility result. So Sen proposed another list of four axioms. We've seen the first two. Each input is complete and transitive. The output is completely intransitive. We've seen Pareto. If every input is prefers A over B, then the output should also say A is better than B. The fourth one, is again the problematic one. It says there should be at least two decisive voters. What is one decisive voter? It says that, A decisive voter is one where they have the voter has at least, One paired comparison where she holds the absolute power. If she thinks A is better then B then you made everyone else thinks B is better than A, the output should say A better than B. Now of course that is a very strong power and if there's only one such person with these powers then that is a dictator. So there needs to be at least two decisive voters for the system to be a real voting system. But still, that gives two or more voters a very strong power, To rule over the others. And it turns out that this strong power would make voting systems impossible. There is no voting system that can satisfy all four axioms. It's impossibility result here. And what happens that the decisive voter impose a strong externality on all the other voters. And therefore this blocks axiom two. Just like IIA in Aeros system blocks XM2 and possibly leading to simplic output even though all inputs are transitive. And we'll illustrate this through a way to generate as many examples as you want. And here is how those examples can be generated. Lets say we've got three voters and five candidates. Okay, Voters one, two, three, candidates A, B, C, D, E. Lets look at these pair wise comparison AB, BC, CD, DE and EA. To start constructing counter examples, we can just say that each of the three voters have the identical and cyclic input which is A better than B, better than C, better than D, better than E, which is in term better than A. This clarity is a cyclic input and therefore should not be allowed, but we're not done yet and this is intermediate stuff, and two, three of the same. Now let's say, voter A is the decisive voter on the A,B pairwise comparison. Voter two is the decisive voter on the C,D pairwise. And voter three is the pair decisive voter on the E,A pairwise comparison. We denote that, that fact by these three red phrases. . And now, we want to say that, we would like to make these three inputs transitive. Okay, not cyclic anymore and yet the output will be cyclic. Okay, clearly right now the output is cyclic. So we want output to be cyclic, and yet input not to be cyclic. Because our goal is to generate counter intuitive examples. And indeed we can do that. We just simply need to look at the pair wise comparison where voter one has the decisive power. And then swap the AB for voters two and three. Okay, similarly, where voter two has the size and power swap C,D for voters one and three, and where voter three has size and power swap that pair as comparison for voters one and two. Because now if you look at each of the three voter's input list, it is transitive. Okay, this says, for example, the first one, A, A, better than B, B, better than C, D also better than C, D is better than E, and A is better than E. That is not sick leave. That satisfy all the transitivity that you need. Similarly for the other two voters two and three and therefore now the three inputs are indeed transitive and yet because we're only flipped where there is a decisive voter. So the output remains transitive. Still A, Better than B, B better than C. C better than D because voter two is vote is the decisive vote. D better than E and E better than A because voter three is vote is decisive vote. We just flipped the non decisive voters, so that all inputs are transitive and yet output is still transitive is circulate, because A better than B than C than D than E in turn than A That is a cycle from A back to E Now we got more than one decisive voters, we follow the Pareto principal our input our cyclic and yet our transitive and yet output is cyclic. In fact, if you remember all the way back to lecture one, when we talk about Prisoner's Dilemma, when we introduced gain to barter competition. That is a special case of Sen's counterintuitive examples too. Because we can view, that game in the following light. The two prisoners again, one and two. The numerical values doesn't matter in this illustration. They each got two choice: not confess/confess, not confess/confess. Okay, prisoner one can, be the decisive voter between confess or not confess, between the two columns. Okay, And prisoner one would say, you know what? I think confess is better than not confess in any case. So, he says B is better than A, and D is better than C. Similarly, prisoner two would say that confess better than no confess. So C is better than A and D is better than B So out of the four possible candidates, A, B, C, D, describing the two by two configuration of this game. One prisoner says B better than A, D better than C. The other prisoner says C better than A, D better than B. Since they control these actions, they are the decisive voter for those two pair wise comparisons respectively. And both agree that A is better than D. Okay, If both decide not to confess, that's better than both confess. And now, here comes the problem. The dilemma is now reflected in this cyclic output of the four candidates voting result. Because A's better than D both agree. And prisoner two says, D is better than B. And prisoner one says B is better than A. And they are the two decisive voters for these two. And therefore you form a cycle. A better than D better than B better than A. Similarly, you have another cycle the other way around. A better than D, better than C, better than A. And that is why you see prisoner's dilemma, because the two prisoners is each a decisive voter between two actions. And together with a common sense choice, let these two cycles, and there's, therefore no transitive result that satisfy both prisoners needs. So, now we have come to the end of this lecture. This is relatively a short lecture, about we are going to have two long lectures coming up in the next two lectures seven and eight The key message here is that, Wikipedia success depends on the positive narrow affect as well as good faith collaboration. The forming consensus collaboration can be modeled by either bargaining or voting We spend a little bit of time on bargaining. . And for voting, we saw that, it may not satisfy some intuitive conditions as it compresses many rank order lists into a single one such as IIA, which is the flawed intuition. And it is an important principle that, a voting can indeed be made sensible. And as a universal ride that provide the bridge between individuals and aggregate and effect a means to provide between check and balances against absolute power, and it forms the foundation of consent from the governed to the government. So this is an important principle that we got to just a little bit of time to cover and we will move on now to influence power models in the next lecture.