0:33

so let's have a look at these things and try and understand a little bit about

Â them. So, the first one is your game on

Â networks, also known as a graphical game where we're choosing now levels of

Â actions. We could have it in zero one, we could

Â have the action, it could be xi and zero to infinity,

Â So some people are taking some level of action and this we'll look at as work

Â from Yann Bramoulle and Rachel Kranton. And the game that they're thinking about

Â is one where people are choosing some level of information to acquire about

Â something. Okay, so you're expending some effort and

Â this effort is learning about something and how much time do I spend learning

Â about this. So it might be learning about you know,

Â in, in California we vote on things in terms of propositions.

Â How much time do I spend learning about this, and then maybe my friends can just

Â ask me and free ride on the fact that I do the research or maybe I free ride on

Â them and so forth, okay? So this, this structure that they look at

Â is a classical public goods structure. So the payoff for any given individual is

Â going to look as follows. So what are you getting?

Â You're getting some f, some function of your own action, xi, plus the action of

Â your neighbors. So you've got sum of j in your

Â neighborhood. And what those j people are doing.

Â Okay, so that's the benefit you get. You get some benefit from the overall

Â level of public goods as being and generally this is going to be a concave

Â function, so initially there's high benefits to this, but it is going to tail

Â off. with the level.

Â So, as we think about the total level that's being provided.

Â The total level of the sum of x's as it goes up, we're going to think of this as

Â being a concave function. So, eventually, you know, more and more

Â learning, it's just not worth the information.

Â So. Information, a little bit of information

Â is really valuable, a little more is a little more valuable, eventually, it's

Â just, you know, I'm, it's overload, it's not worth anything to me.

Â 2:44

Okay, so we've got a concave f, and each person pays a per-unit cost of their

Â effort. Okay?

Â And, and then we pool our information so I get to learn my information plus

Â everything that my neighbors learned. And we pool that together.

Â That's the value I get. And then I get a nice concave function of

Â that. And then I pay some cost per unit of, of

Â effort that I have. So that's a very simple, structure.

Â And, once we have that kind of structure. We can let,

Â So let's assume that this thing's nicely differentiable.

Â We can take a derivative of it. We can also look at the maximizer of

Â this. So suppose that I just look at the

Â function where I look at the total effort that's provided and look at the total

Â cost, okay. And I want to look at a, a situation

Â where I, I maximize this thing with respect to x.

Â So, maximizing, let's suppose I want to, choose the x which maximizes f of x minus

Â c of x. Well, maximizing that I'm going to set f

Â prime of x minus c is equal to 0 or f prime is equal to c.

Â So, lets let x star be the solution, that if a given individual was choosing how

Â much total effort should be provided. they, they and they had to pay the cost

Â of the whole effort they would choose x star.

Â Okay. So we'll call x star this level.

Â Okay. So, what's true about x star?

Â if you look at any pure strategy Nash Equilibrium, now let's suppose that each

Â person chooses the x's, and they look at what their neighbors are doing.

Â in all pure strategy Nash equilibria, the total amount that any individual is

Â getting in their neighborhood has to be at least x star, okay.

Â And why is that true? Suppose that the amount that was being

Â provided in somebody's neighborhood was less than x star.

Â Right, so say x star over 2. Well, if given that this is a concave

Â function, x star is the point at which this derivative is exactly equal to c.

Â So, when we look at where f prime is, that's equal to c.

Â The slope is exactly equal to the cost. So, if you're seeing less action, then

Â you would still have, you would have f prime.

Â Be greater than c. And so that means a little bit more

Â action by one person, so if I add a little bit ex-, extra x, the gain I'm

Â going to get in terms of my f is going to have a, a more rapid gain that what I'm

Â going to lose in terms of cost. So if anybody is seeing less than x star

Â in their neighborhood... They would like to increase xi.

Â So if the total, if this sum, if that was less than x star, then you should

Â increase xi and so that could not be a Nash Equilibrium, I'd want to, I'd

Â strictly want to increase my activity because I want to push us up to this

Â level where the trade off begins to be just equal to the cost.

Â 6:06

So what that tells us is that, in a Nash Equilibrium, if a, it has to be that

Â everybody is getting at least x star. And, if I'm getting more than x star, it

Â could be that there's lots of people that happen to be providing public goods for

Â me. So my neighborhood is really full of

Â people providing lots of public goods. If we're in a setting where I've got

Â extra activity going on in my neighborhood.

Â Then it should be that I'm not providing any action.

Â And why's that? Because if we've gone too far, then by

Â decreasing my action, if I'm providing positive action, what I, what I get in

Â terms of the decrease, so again this f prime, it's a situation where by

Â decreasing what I'm doing, I'm actually, so if we're in a, a situation where I've

Â gone too far, so this is the x star where things are exactly equal.

Â If we've gone too far, then what I would lose by decreasing my action.

Â 7:09

Would be less in terms of the slope than the c.

Â So there we'd be in a situation where if prime is less than c, so actually

Â decreasing my action would save me cost and the loss and benefits would be less

Â than that. So it has to be that if, if we're over

Â then I'm providing nothing. Okay, and that is possible in some of

Â these situations that if I have lots of friends providing public good, I'm

Â getting too much. I just, I, I free ride completely, okay.

Â So, that's a basic structure and so what that means is that then equilibria are

Â going to boil into situations where we can, we can talk about two different

Â kinds. one is where we're, we're in a situation

Â where there's things that are distributed where everybody is providing some action,

Â and the, the total actions are just balancing exactly up to this x stars.

Â Everybody's seeing exactly x star in their behavior, in their neighborhood.

Â Everybody's acting enough to just balance these things.

Â And then there's specialized equilibria, where some people take actions and other

Â don't, and where basically the people who take actions are going to, are just

Â going to specialize and produce the public good, just like in a best shot

Â public goods game, and then, those people around them are going to free-ride

Â completely, okay? And so to get a feeling for this, let's

Â have a look. So let's suppose that we'll just

Â normalize x star to be 1. Let's look at you know, here's a few

Â different equilibria on different networks.

Â Here's a situation where we've got a specialized equilibrium.

Â These people provide the full action, their neighbors free ride.

Â Everybody gets exactly x star in their neigbhorhood.

Â Nobody wants to increase their action. in fact, sorry, some, some people.

Â These individuals get more than x star. They get two times.

Â So they're happy. They're getting double the provision of

Â public goods. They certainly don't want to contribute

Â anything. Okay.

Â And these people are getting exactly one. They're providing it because their

Â neighbors are not providing anything. Here's another equilibrium.

Â Everybody provides 1/3. And that balances because we're in a

Â situation where each person is seeing exactly x star in their neighborhood, so

Â nobody wants to increase or decrease. Right?

Â So this is a distributed one, where we've got an exact distribution.

Â Okay. And this one is the, is similar to the

Â first one, but just with roles reversed, okay?

Â So you could generally, you know, if, if we had disconnected individuals, they'd

Â each have to provide one, there's nobody else to, to free ride on.

Â you could have situations like this where the isolated person.

Â Their friends provide they don't have any friends so they provide the public good

Â here. You could have either person provide it.

Â you could also have an equilibrium like this where they do a half, a half, right?

Â Or you could do one where you've got 3 4ths and 1 4th.

Â So you could have a whole series of different equilibria and then in each of

Â those cases the other person would be providing one.

Â you could have a situations here that look like the best shoppable goods.

Â You could have ones with these distributions, you could.

Â So these are whole series of different equilibria, some of them are specialized.

Â Where certain people end up doing one, other people end up doing zero, and then

Â there's other ones where distributed, where you've got some actions by

Â different players but their adding up to one, okay?

Â Okay. So, we can begin to see the connection

Â between this, and the best shot public goods game that we saw before.

Â if you look at the specialized equilibria here.

Â Those are going to look like maximal independent sets.

Â Right? So you have the people, some people just

Â provide the full public goods. Their neighbors free-ride.

Â And no two nodes that are, connected can both be providing this, the public goods.

Â So you end up with every node in n is either in this set of s that provides the

Â public goods or linked to somebody. And proposition by BramoullÃ© and Kranton,

Â then, is the set of specialized Nash Equilibria here are profiles such that

Â the maximal independent set is, a maximal independent set is, is exactly equal to

Â the set of specialists people providing the public good.

Â Okay, so that's just a straightforward extension of what we saw before in, in

Â the public goods game. the best shot public goods.

Â But now we're actually allowing people to take multiple actions and we see that

Â it's a specialized equilibria correspond to that.

Â Okay. so what's interesting in, in terms of the

Â Bramoulle and Kranton analysis is they show that these the specialized

Â equilibria are actually ones that have a nice property.

Â And, it's a sort of a stability property. So, imagine what you do is you start with

Â some strategies, set of strategies that people are playing.

Â So let's look at our equilibria, and ask which ones are sort of stable to

Â perturbations. And what we'll do is that we're going to

Â perturb this by changing everybody's actions by a little epsilon.

Â And the the epsilon can be positive or negative so we could, you know, move,

Â some, some people we bounce some people up so they provide a little more

Â information, other people down so they provide a little less information.

Â 12:40

So we start at some point, which might be an equilibrium.

Â We want to ask is that stable? We perturb it a little bit, and then what

Â we do is we look at okay, what's the best response?

Â So if everybody was choosing their maximal action, given what everyone else

Â is doing under this perturbation, what do we get?

Â Okay, so we, we go from x zero to x one and then, you know, x one goes to x two

Â and so forth. And what we would like is that if you

Â perturb things by some small epsilons, if you always converge back to x, then we'll

Â say that the equilibrium is stable. Okay.

Â So that's the notion of stability. and let's first think of, of a dyad, so

Â we've got two individuals, these two individuals, and let's suppose that, you

Â know, person 1 is doing something say like a 1 3rd and person 2 doing 2 3rds.

Â that's not going to be stable in the sense that if we perturb this a little

Â bit, so suppose we move this down to 1 3rd, minus epsilon and this up to 2 3rds

Â plus epsilon, well that in fact is also an equilibrium.

Â So if we did that perturbation it wouldn't go back to the original one, it

Â would actually stay at this one, okay? So in a, in a dyad, no matter what

Â numbers you put in there, you know, we could put in zero and one, and, and have

Â this go to 0 plus epsilon, 1 minus epsilon so no matter what numbers you put

Â in, as you perturb this, it would, it would stay there.

Â So as long as x one is less than or equal to x two.

Â Add a little bit to x one. And subtract a little bit from x two.

Â You would stay there. So you're in a situation where you end up

Â not having, any movement in terms of, of the perturbation.

Â nothing's stable. Okay,

Â But when we look more generally. the equilibria are, the only stable

Â equilibria aren't necessarily going to be specialist equilibria.

Â And they're going to be such that every non-specialist has exactly, at least two

Â specialists in his or her neighborhood. So, for instance, if we look at three

Â nodes, and again, our x star is one. This thing is stable.

Â 15:02

These three are not stable. Okay, and let's just go through and try

Â to understand why is this stable? What's the logic, here?

Â So, let's suppose that we perturb this a little bit.

Â So for instance we take a little bit of epsilon away, so we take this person down

Â to 1 minus epsilon. And we bump this person up to epsilon.

Â And let's, let's just keep this person constant for now.

Â if we do that, and now we look at the best responses to this.

Â Well, this person has already a neighbor providing at least one.

Â They're still at 2 minus epsilon in their neighborhood.

Â They're way bigger than the x star of one.

Â They're getting too much action. They can get rid of their epsilon and go

Â back to zero. So, they're going to tend to go right

Â back to zero. Once they go back to zero, this person's

Â going to go back to one, and we end up going back to the, to the situation that

Â we have. So here the fact that I'm already getting

Â too much means even if you perturb that I'm going to want to go right back to

Â zero. So the people that are in the

Â neighborhood of these specialists are going to stay there if they have at least

Â two specialists in their neighborhood they're going to go back to zeroes.

Â And that means that specialists are going to be forced back to one because

Â all their neighbors are providing zeroes. Whereas this one is unstable, and it's

Â unstable for the reason the dyad was, right?

Â We could make this a 1 minus epsilon and you know then, move this guy to epsilon,

Â this guy to epsilon. Once that happens, well then these people

Â are happy but this person's going to want to decrease more and, and it would

Â actually move eventually towards this equillibrium.

Â you know we could change this by moving this to minus epsilon, add an epsilon

Â over here. that would still be an equilibrium and so

Â forth. So, so the only ones that turn out to be

Â stable, turn out to be the ones that every, that, that every specialist.

Â Has specialized equilibria and such that every non-specialist has at least two

Â specialists in their neighborhood. Okay, so you can go through a sketch of

Â the proof of this basically the stability of such equilibria for, for perturbations

Â of this for the non-specialist, the best response they go right back to zero, and

Â you converge right back. For any other equilibrium, there is

Â somebody providing goods as a non-specialist and then you perturb that

Â agent up and the other neighbors down just the way we did, and basically you

Â can, you knoq work through that kind of logic, so the idea of the proof is much

Â similar to what we just worked through in terms of the examples, and you can deduce

Â this proposition in that manner. Okay, so what says is that is that these

Â specialized equilibria are somehow special in this model.

Â So you get, they're stable in the sense that if you do perturbations then you're

Â going to get pushing people towards that. Let me just sort of make one comment on

Â this, and sort of an interesting contrast.

Â another way we could think about, you know, so here we've got a game with

Â multiple equilibria. There's lots of different equilibria in

Â this game. And, you know, part of the reason that,

Â that one's interested in applying the stability concepts is just to find out

Â are some of these more naturally equilibria than others?

Â And so if we perturb them and we get back to them, we say, well, that seems to be a

Â more natural one than another one. another way we can think about stability

Â is instead to think, think about pairwise stability, right?

Â So suppose your links are costly, and if I'm specializing, and I've got a bunch of

Â friends who aren't specializing, I'm providing all the good.

Â They're not doing anything. I would start dropping links to the

Â non-specialists. Right?

Â So in some sense in that world, what you would have is the only reason people

Â would be willing to, to maintain links would be if their friends were also

Â providing some of the local public goods. So I normally want to keep these

Â friendships if they're actually providing some value.

Â And so there, what you're going to get is the non specialized equilibria are

Â going to be the only, stable ones. So, what this says is that, you know?

Â It depends very much on what kind of stability notion you put in, as to what

Â you might select out. And whether you think about perturbing

Â the, the network, or perturbing the actions.

Â You could get very different kinds of conclusions.

Â And so, you know, ultimately I think what this says is there's still a lot we need

Â to understand about these kinds of games. they are tractable games.

Â They have some interesting features. Some of them tie back to some of the

Â things like the best-shot public goods games and so forth and they, they can be

Â you know, interesting to analyze further. Okay.

Â So, so that's one example of these kinds of games.

Â if we begin to introduce heterogeneity to these kinds of things that's going to

Â change the nature of equilibria. We could begin to enrich things that way.

Â There's a lot of enrichments we can make in these kinds of games.

Â And what we are going to do next is look at another class of games.

Â And this other class of games is going to be one where we allow for strategic

Â complements and there the games again are going to be nicely behaved in the sense

Â it will be much easier to get a full characterization of what the equilibria

Â look like and, and it'll be nice we'll have some nice relationships to the

Â network structure and that'll allow us then to.

Â