0:01

We will now walk through an example or a whole process of generating as many

Â counter intuitive examples as we like, based on Sen's impossibility result.

Â So Sen proposed another list of four axioms.

Â We've seen the first two. Each input is complete and transitive.

Â The output is completely intransitive. We've seen Pareto. If every input is

Â prefers A over B, then the output should also say A is better than B.

Â The fourth one, is again the problematic one.

Â It says there should be at least two decisive voters.

Â What is one decisive voter? It says that,

Â A decisive voter is one where they have the voter has at least,

Â One paired comparison where she holds the absolute power.

Â If she thinks A is better then B then you made everyone else thinks B is better than

Â A, the output should say A better than B. Now of course that is a very strong power

Â and if there's only one such person with these powers then that is a dictator.

Â So there needs to be at least two decisive voters for the system to be a real voting

Â system. But still, that gives two or more voters a

Â very strong power, To rule over the others.

Â And it turns out that this strong power would make voting systems impossible.

Â There is no voting system that can satisfy all four axioms.

Â It's impossibility result here. And what happens that the decisive voter

Â impose a strong externality on all the other voters.

Â And therefore this blocks axiom two. Just like IIA in Aeros system blocks XM2

Â and possibly leading to simplic output even though all inputs are transitive.

Â And we'll illustrate this through a way to generate as many examples as you want.

Â And here is how those examples can be generated.

Â Lets say we've got three voters and five candidates.

Â Okay, Voters one, two, three, candidates A, B,

Â C, D, E. Lets look at these pair wise comparison

Â AB, BC, CD, DE and EA. To start constructing counter examples, we

Â can just say that each of the three voters have the identical and cyclic input which

Â is A better than B, better than C, better than D, better than E, which is in term

Â better than A. This clarity is a cyclic input and

Â therefore should not be allowed, but we're not done yet and this is intermediate

Â stuff, and two, three of the same. Now let's say, voter A is the decisive

Â voter on the A,B pairwise comparison. Voter two is the decisive voter on the C,D

Â pairwise. And voter three is the pair decisive voter

Â on the E,A pairwise comparison. We denote that, that fact by these three

Â red phrases. .

Â 3:03

And now, we want to say that, we would like to make these three inputs

Â transitive. Okay, not cyclic anymore and yet the

Â output will be cyclic. Okay, clearly right now the output is

Â cyclic. So we want output to be cyclic, and yet

Â input not to be cyclic. Because our goal is to generate counter

Â intuitive examples. And indeed we can do that.

Â We just simply need to look at the pair wise comparison where voter one has the

Â decisive power. And then swap the AB for voters two and

Â three. Okay, similarly, where voter two has the

Â size and power swap C,D for voters one and three, and where voter three has size and

Â power swap that pair as comparison for voters one and two.

Â Because now if you look at each of the three voter's input list, it is

Â transitive. Okay, this says, for example, the first

Â one, A, A, better than B, B, better than C, D also better than C, D is better than

Â E, and A is better than E. That is not sick leave.

Â That satisfy all the transitivity that you need.

Â Similarly for the other two voters two and three and therefore now the three inputs

Â are indeed transitive and yet because we're only flipped where there is a

Â decisive voter. So the output remains transitive. Still A,

Â 4:42

Better than B, B better than C. C better than D because voter two is vote

Â is the decisive vote. D better than E and E better than A

Â because voter three is vote is decisive vote.

Â We just flipped the non decisive voters, so that all inputs are transitive and yet

Â output is still transitive is circulate, because A better than B than C than D than

Â E in turn than A That is a cycle from A back to E Now we got more than one

Â decisive voters, we follow the Pareto principal our input our cyclic and yet our

Â transitive and yet output is cyclic. In fact, if you remember all the way back

Â to lecture one, when we talk about Prisoner's Dilemma, when we introduced

Â gain to barter competition. That is a special case of Sen's

Â counterintuitive examples too. Because we can view, that game in the

Â following light. The two prisoners again, one and two.

Â The numerical values doesn't matter in this illustration.

Â They each got two choice: not confess/confess, not confess/confess.

Â 5:58

Okay, prisoner one can, be the decisive voter between confess or not confess,

Â between the two columns. Okay, And prisoner one would say, you know what?

Â I think confess is better than not confess in any case.

Â So, he says B is better than A, and D is better than C.

Â Similarly, prisoner two would say that confess better than no confess.

Â So C is better than A and D is better than B So out of the four possible candidates,

Â A, B, C, D, describing the two by two configuration of this game.

Â One prisoner says B better than A, D better than C.

Â The other prisoner says C better than A, D better than B.

Â Since they control these actions, they are the decisive voter for those two pair wise

Â comparisons respectively. And both agree that A is better than D.

Â Okay, If both decide not to confess, that's

Â better than both confess. And now, here comes the problem.

Â The dilemma is now reflected in this cyclic output of the four candidates

Â voting result. Because A's better than D both agree.

Â And prisoner two says, D is better than B. And prisoner one says B is better than A.

Â And they are the two decisive voters for these two.

Â And therefore you form a cycle. A better than D better than B better than

Â A. Similarly, you have another cycle the

Â other way around. A better than D, better than C, better

Â than A. And that is why you see prisoner's

Â dilemma, because the two prisoners is each a decisive voter between two actions.

Â And together with a common sense choice, let these two cycles, and there's,

Â therefore no transitive result that satisfy both prisoners needs.

Â 8:02

So, now we have come to the end of this lecture. This is relatively a short

Â lecture, about we are going to have two long lectures coming up in the next two

Â lectures seven and eight The key message here is that, Wikipedia success depends on

Â the positive narrow affect as well as good faith collaboration.

Â The forming consensus collaboration can be modeled by either bargaining or voting We

Â spend a little bit of time on bargaining. .

Â And for voting, we saw that, it may not satisfy some intuitive conditions as it

Â compresses many rank order lists into a single one such as IIA, which is the

Â flawed intuition. And it is an important principle that, a

Â voting can indeed be made sensible. And as a universal ride that provide the bridge

Â between individuals and aggregate and effect a means to provide between check

Â and balances against absolute power, and it forms the foundation of consent from

Â the governed to the government. So this is an important principle that we

Â got to just a little bit of time to cover and we will move on now to influence power

Â models in the next lecture.

Â