It is not easy to model Rough Consensus formation. One possibility is through the model of bargaining. In a bargaining model, the contributor would need to reach a compromise or otherwise there will be no agreement. But each contributor's utility function and the default position, in case of no agreement, need to reflect the goodwill typically observed in the Wiki collaboration. We'll soon spend about five minutes, very briefly, to look at how bargaining can be modeled to reflect both the desire to cooperate, but also some competition in their different viewpoints. The other possibility is through voting theory. Actually in Wikipedia, voting is very rarely done, of course voting is used to select committee members on a regular basis and sometimes voting is explicitly carried out. But most of the time, a voting is not done explicitly but implicitly. And you can think of each committee member as a person with a partially ordered preferences among the choices presented in front of the committee. And everybody got some preferences in certain order of possible outcome of that discussion. I say partially ordered, because for some possible outcome you just don't care about the order. So, in a round table, or oh, each editor may have a slightly different preference order. Eventually, they have to agree to a common one. And there will be a threshold to say that, if this final agreement is too different from my own preference list, Then I will veto. And in the world of forming rough consensus, Basically, everybody got a certain vetoing power. But you have to model the partial ordering, You have to model the distance between one order and some other partial order. And then you have to model the threshold before an editor stand up and decide to veto and thereby preventing a consensus from forming. None of these modeling staffs is actually mature in the study of Wikipedia group dynamics. Nor is the bargaining model mature in the study of this topic. So, you're going to see a huge gap between the theory of bargaining and voting on the one hand and the actual practice, Of rough consensus formation in Wikipedia or in many other dynamics of a group in trying to reach consensus. The gap is very big and this is a word of caution that this lecture as well as the next two lectures, seven and eight. Six, seven, eight. From by far, the largest the gaps between theory and practice, between the question I want to answer and the unambiguous language we have at our disposal. By far the largest in this course. You probably can add up the practice theory gap of all the other lectures and multiply by ten. A so if there's a way to quantify the distance and still be smaller than the gap you'll be up serving today and the next two lectures. But that also means that there might be ways for you to make the contribution in closing this gap a little bit. networks and irrational behavior and psychological process in people's minds and therefore is certainly not easy thing to model. But having said that, let's, let's very quickly go through the bargaining part. This is a kind of model that started by Nash as another part of his Princeton PhD dissertation based on an axiomatic approach. We will not have time to cover this in this video. We will push that to the advanced material part of the lecture. We will spend a few minutes, however, just on a more intuitive process driven type of model in the eighties developed by Reubenstien, for interactive offer. So what is this? Suppose you've got, The following problem. Got Alice and Bob who like to divide $1.00. Okay? And Alice would propose to Bob, how about I keep 80 cents, you get twenty cents. And Bob will say, no, I refuse the order and I would like to propose I take 90%, you take ten%l And this offer process will go back and forth until one person says, alright, I take your offer. Okay. We'll later, perhaps all the way to Lecture twenty talk about other issues related to fairness of cutting a cake or dividing a dollar. So both of them would like to reach a consensus and therefore get something, but they clearly have also competing interests. And you think, hold on a second, This process can go on forever, right? So suppose you've got time slots with certain duration, [unknown] for time slot. And the first user can provide a number as one between zero and one for the first user. And x2, which is 1-x1, cuz we're talking about $one to be divided to the other user. And the other user can either take it or reject and propose something different. But this process can keep on going forever. So, there must be some kind of friction, some reason for people to say, I got to stop and take the offer and one could be the price for disagreeing, being, time. Okay. You really would like to conclude the deal and reach a consensus. So, I'm going to say each of these two users, the pay off function or utility function, use of i equals X of i. That is, how much you get. Times and exponential function, E to the minus ri kT, T is the duration of each time slot. K is what time slot are you talking about right now. Is it the tenth or the hundredth time slot and ri is a bargaining power parameter. We'll see its impact momentarily. And this holds true for both the first and the second user. In other words, your pay-off depends on how much you get, but also, it drops exponentially fast as these iterations keeps on going. The exponent will vary depending on your bargaining power relative to the other user. If you take this model, Then skipping [unknown], It is intuitively clear that, if waiting for the next round on the negotiation they give you, give me the same pay-off, except this round's offer, then I might as well accept the offer. Okay? So carrying that intuition a little further, you can see that, The following two equations, if simultaneously satisfied for the point x1 star and x2 star, these two scalars would constitute some kind of equilibrium, From the first user's point of view. And this is from the second user point of view. Okay? Or, I should say this is from the second user point of view. This is from the first user point of view. If what I am getting right now basically equals to what I might be getting next round, if you will take my next transfer offer, then I might as well just take your current offer. And flipping the rows of persons and users, you get the equation. If so, then, This pair of points x1, star x2 star constitute an equilibrium. So this is a rough sketch. It's a hand-wavy way to establish a basic intuition. Now, it turns out that today there's a unique solution to the above pair of equations, which is x1 star x2 stars equals the following, one minus e to the minus r two time slot duration and about one minus e to the minus r one plus r two T. And equilibrium resource to the second user is to follow them. Now, in order to gain some intuition out of these two equations we're going to take the extreme case where T shrinks to zero. This becomes a very efficient bargaining offer and counter offer process. And I highlight, this is approaching zero not that it's, identical to zero. And as it approaches zero, we have a nice approximation e to the rT becomes, like, one minus rT when T becomes very, very small. Then we can simplify these expressions and arrive at the following equilibrium result, which is very intuitive. X1 = r2 over one+ r2. And x2 = r1 over r1 + r2. In other words the [inaudible] equilibrium, kind of resource that you get for the two users is basically dependent on the relative bargaining power of the other user. The denominator is the same r1 plus r2. Okay. It's the sum of the bargaining power, but if the second user have a lot of bargaining power, that means it's decay of utility. Remember, it's exponential e to the minus r2 kT. This is a very small number. That means that decay of my utility as time goes on, Decays slowly, whereas if r2 is a big number, then it decays faster. So, smaller r2 means slower decay means stronger ability to wait out the negotiation and therefore a stronger bargaining power. And indeed, if this a smaller R2, that translates into a smaller allocation of the resource to the first user because the second user has a stronger hand. Conversely, the first user, it has a smaller r1 relative to r1 + r2. That means she has more ability to wait out the negotiation and therefore stronger hand that implies that the second user's resource allocation will be smaller. So, smaller ir means stronger hand for that user and therefore, less resource allocation for the other user. This is very intuitive result describing the equillibria of this basic Rubinstein model of interactive offer for bargaining. Of course, this is bargaining about $one. And where do we put the line to divide it between two people? How do we go from here to understanding Wikipedia's discussion page? There is no mature mathematical models of. Results. If you are interested, you might want to go through some process of the talking page and history page or even run an experiment on certain Wikipedia article with those contributors and see if similar model might be applicable.