0:01

It is not easy to model Rough Consensus formation.

Â One possibility is through the model of bargaining.

Â In a bargaining model, the contributor would need to reach a compromise or

Â otherwise there will be no agreement. But each contributor's utility function

Â and the default position, in case of no agreement, need to reflect the goodwill

Â typically observed in the Wiki collaboration.

Â We'll soon spend about five minutes, very briefly, to look at how bargaining can be

Â modeled to reflect both the desire to cooperate, but also some competition in

Â their different viewpoints. The other possibility is through voting

Â theory. Actually in Wikipedia, voting is very

Â rarely done, of course voting is used to select committee members on a regular

Â basis and sometimes voting is explicitly carried out.

Â But most of the time, a voting is not done explicitly but implicitly.

Â 1:09

And you can think of each committee member as a person with a partially ordered

Â preferences among the choices presented in front of the committee.

Â And everybody got some preferences in certain order of possible outcome of that

Â discussion. I say partially ordered, because for some

Â possible outcome you just don't care about the order.

Â 1:33

So, in a round table, or oh, each editor may have a slightly different preference

Â order. Eventually, they have to agree to a common

Â one. And there will be a threshold to say that,

Â if this final agreement is too different from my own preference list,

Â Then I will veto. And in the world of forming rough

Â consensus, Basically, everybody got a certain vetoing

Â power. But you have to model the partial

Â ordering, You have to model the distance between one

Â order and some other partial order. And then you have to model the threshold

Â before an editor stand up and decide to veto and thereby preventing a consensus

Â from forming. None of these modeling staffs is actually

Â mature in the study of Wikipedia group dynamics.

Â Nor is the bargaining model mature in the study of this topic.

Â So, you're going to see a huge gap between the theory of bargaining and voting on the

Â one hand and the actual practice, Of rough consensus formation in Wikipedia

Â or in many other dynamics of a group in trying to reach consensus.

Â The gap is very big and this is a word of caution that this lecture as well as the

Â next two lectures, seven and eight. Six, seven, eight.

Â From by far, the largest the gaps between theory and practice, between the question

Â I want to answer and the unambiguous language we have at our disposal.

Â By far the largest in this course. You probably can add up the practice theory

Â gap of all the other lectures and multiply by ten. A so if there's a way to quantify

Â the distance and still be smaller than the gap you'll be up serving today and the

Â next two lectures. But that also means that there might be

Â ways for you to make the contribution in closing this gap a little bit.

Â 3:53

networks and irrational behavior and psychological process in people's minds

Â and therefore is certainly not easy thing to model.

Â But having said that, let's, let's very quickly go through the bargaining part.

Â This is a kind of model that started by Nash as another part of his Princeton PhD

Â dissertation based on an axiomatic approach.

Â We will not have time to cover this in this video.

Â We will push that to the advanced material part of the lecture.

Â We will spend a few minutes, however, just on a more intuitive process driven type of

Â model in the eighties developed by Reubenstien, for interactive offer.

Â So what is this? Suppose you've got,

Â 4:45

The following problem. Got Alice and Bob who like to divide $1.00.

Â Okay? And Alice would propose to Bob, how about I keep 80 cents, you get twenty

Â cents. And Bob will say, no, I refuse the order

Â and I would like to propose I take 90%, you take ten%l And this offer process will

Â go back and forth until one person says, alright, I take your offer.

Â 5:27

Okay. We'll later, perhaps all the way to

Â Lecture twenty talk about other issues related to fairness of cutting a cake or

Â dividing a dollar. So both of them would like to reach a

Â consensus and therefore get something, but they clearly have also competing

Â interests. And you think, hold on a second,

Â This process can go on forever, right? So suppose you've got time slots with

Â certain duration, [unknown] for time slot. And the first user can provide a number as

Â one between zero and one for the first user.

Â And x2, which is 1-x1, cuz we're talking about $one to be divided to the other

Â user. And the other user can either take it or

Â reject and propose something different. But this process can keep on going

Â forever. So, there must be some kind of friction,

Â some reason for people to say, I got to stop and take the offer and one could be

Â the price for disagreeing, being, time. Okay. You really would like to conclude

Â the deal and reach a consensus. So, I'm going to say each of these two

Â users, the pay off function or utility function, use of i equals X of i. That is,

Â how much you get. Times and exponential function, E to the minus ri kT,

Â T is the duration of each time slot. K is what time slot are you talking about right

Â now. Is it the tenth or the hundredth time slot

Â and ri is a bargaining power parameter. We'll see its impact momentarily. And this

Â holds true for both the first and the second user.

Â 7:27

In other words, your pay-off depends on how much you get, but also, it drops

Â exponentially fast as these iterations keeps on going.

Â The exponent will vary depending on your bargaining power relative to the other

Â user. If you take this model,

Â Then skipping [unknown], It is intuitively clear that, if waiting

Â for the next round on the negotiation they give you, give me the same pay-off, except

Â this round's offer, then I might as well accept the offer. Okay?

Â So carrying that intuition a little further, you can see that,

Â 8:32

From the first user's point of view. And this is from the second user point of

Â view. Okay? Or, I should say this is from the

Â second user point of view. This is from the first user point of view.

Â If what I am getting right now basically equals to what I might be getting next

Â round, if you will take my next transfer offer, then I might as well just take your

Â current offer. And flipping the rows of persons and

Â users, you get the equation. If so, then,

Â 9:10

This pair of points x1, star x2 star constitute an equilibrium.

Â So this is a rough sketch. It's a hand-wavy way to establish a basic

Â intuition. Now, it turns out that today there's a

Â unique solution to the above pair of equations, which is x1 star x2 stars

Â equals the following, one minus e to the minus r two time slot duration and about

Â one minus e to the minus r one plus r two T. And equilibrium resource to the second

Â user is to follow them. Now, in order to gain some intuition out

Â of these two equations we're going to take the extreme case where T shrinks to zero.

Â This becomes a very efficient bargaining offer and counter offer process.

Â And I highlight, this is approaching zero not that it's, identical to zero.

Â And as it approaches zero, we have a nice approximation e to the rT becomes, like,

Â one minus rT when T becomes very, very small.

Â Then we can simplify these expressions and arrive at the following equilibrium

Â result, which is very intuitive. X1 = r2 over one+ r2.

Â And x2 = r1 over r1 + r2. In other words the [inaudible]

Â equilibrium, kind of resource that you get for the two users is basically dependent

Â on the relative bargaining power of the other user.

Â The denominator is the same r1 plus r2. Okay. It's the sum of the bargaining

Â power, but if the second user have a lot of bargaining power, that means it's decay

Â of utility. Remember, it's exponential e to the minus

Â r2 kT. This is a very small number.

Â That means that decay of my utility as time goes on,

Â 11:31

Decays slowly, whereas if r2 is a big number, then it decays faster.

Â So, smaller r2 means slower decay means stronger ability to wait out the

Â negotiation and therefore a stronger bargaining power.

Â 11:49

And indeed, if this a smaller R2, that translates into a smaller allocation of

Â the resource to the first user because the second user has a stronger hand.

Â Conversely, the first user, it has a smaller r1 relative to r1 + r2.

Â That means she has more ability to wait out the negotiation and therefore stronger

Â hand that implies that the second user's resource allocation will be smaller.

Â So, smaller ir means stronger hand for that user and therefore, less resource

Â allocation for the other user. This is very intuitive result describing

Â the equillibria of this basic Rubinstein model of interactive offer for bargaining.

Â Of course, this is bargaining about $one. And where do we put the line to divide it

Â between two people? How do we go from here to understanding

Â Wikipedia's discussion page? There is no mature mathematical models of.

Â Results. If you are interested, you might want to go through some process of the

Â talking page and history page or even run an experiment on certain Wikipedia article

Â with those contributors and see if similar model might be applicable.

Â