Okay, folks, now we're going to talk a little bit about games where we have richer action spaces. So people can take some level of action, beyond 0-1. And I want to talk about just two examples of this that give some illustration of these things. And one is going to be basically a game of substitutes that's going to have a lot of relationships to the. best shot public goods game that we looked at before, and another game is going to be one where there's games of complements and that'll be a different one. And it'll tie back to network structure more easily. so let's have a look at these things and try and understand a little bit about them. So, the first one is your game on networks, also known as a graphical game where we're choosing now levels of actions. We could have it in zero one, we could have the action, it could be xi and zero to infinity, So some people are taking some level of action and this we'll look at as work from Yann Bramoulle and Rachel Kranton. And the game that they're thinking about is one where people are choosing some level of information to acquire about something. Okay, so you're expending some effort and this effort is learning about something and how much time do I spend learning about this. So it might be learning about you know, in, in California we vote on things in terms of propositions. How much time do I spend learning about this, and then maybe my friends can just ask me and free ride on the fact that I do the research or maybe I free ride on them and so forth, okay? So this, this structure that they look at is a classical public goods structure. So the payoff for any given individual is going to look as follows. So what are you getting? You're getting some f, some function of your own action, xi, plus the action of your neighbors. So you've got sum of j in your neighborhood. And what those j people are doing. Okay, so that's the benefit you get. You get some benefit from the overall level of public goods as being and generally this is going to be a concave function, so initially there's high benefits to this, but it is going to tail off. with the level. So, as we think about the total level that's being provided. The total level of the sum of x's as it goes up, we're going to think of this as being a concave function. So, eventually, you know, more and more learning, it's just not worth the information. So. Information, a little bit of information is really valuable, a little more is a little more valuable, eventually, it's just, you know, I'm, it's overload, it's not worth anything to me. Okay, so we've got a concave f, and each person pays a per-unit cost of their effort. Okay? And, and then we pool our information so I get to learn my information plus everything that my neighbors learned. And we pool that together. That's the value I get. And then I get a nice concave function of that. And then I pay some cost per unit of, of effort that I have. So that's a very simple, structure. And, once we have that kind of structure. We can let, So let's assume that this thing's nicely differentiable. We can take a derivative of it. We can also look at the maximizer of this. So suppose that I just look at the function where I look at the total effort that's provided and look at the total cost, okay. And I want to look at a, a situation where I, I maximize this thing with respect to x. So, maximizing, let's suppose I want to, choose the x which maximizes f of x minus c of x. Well, maximizing that I'm going to set f prime of x minus c is equal to 0 or f prime is equal to c. So, lets let x star be the solution, that if a given individual was choosing how much total effort should be provided. they, they and they had to pay the cost of the whole effort they would choose x star. Okay. So we'll call x star this level. Okay. So, what's true about x star? if you look at any pure strategy Nash Equilibrium, now let's suppose that each person chooses the x's, and they look at what their neighbors are doing. in all pure strategy Nash equilibria, the total amount that any individual is getting in their neighborhood has to be at least x star, okay. And why is that true? Suppose that the amount that was being provided in somebody's neighborhood was less than x star. Right, so say x star over 2. Well, if given that this is a concave function, x star is the point at which this derivative is exactly equal to c. So, when we look at where f prime is, that's equal to c. The slope is exactly equal to the cost. So, if you're seeing less action, then you would still have, you would have f prime. Be greater than c. And so that means a little bit more action by one person, so if I add a little bit ex-, extra x, the gain I'm going to get in terms of my f is going to have a, a more rapid gain that what I'm going to lose in terms of cost. So if anybody is seeing less than x star in their neighborhood... They would like to increase xi. So if the total, if this sum, if that was less than x star, then you should increase xi and so that could not be a Nash Equilibrium, I'd want to, I'd strictly want to increase my activity because I want to push us up to this level where the trade off begins to be just equal to the cost. So what that tells us is that, in a Nash Equilibrium, if a, it has to be that everybody is getting at least x star. And, if I'm getting more than x star, it could be that there's lots of people that happen to be providing public goods for me. So my neighborhood is really full of people providing lots of public goods. If we're in a setting where I've got extra activity going on in my neighborhood. Then it should be that I'm not providing any action. And why's that? Because if we've gone too far, then by decreasing my action, if I'm providing positive action, what I, what I get in terms of the decrease, so again this f prime, it's a situation where by decreasing what I'm doing, I'm actually, so if we're in a, a situation where I've gone too far, so this is the x star where things are exactly equal. If we've gone too far, then what I would lose by decreasing my action. Would be less in terms of the slope than the c. So there we'd be in a situation where if prime is less than c, so actually decreasing my action would save me cost and the loss and benefits would be less than that. So it has to be that if, if we're over then I'm providing nothing. Okay, and that is possible in some of these situations that if I have lots of friends providing public good, I'm getting too much. I just, I, I free ride completely, okay. So, that's a basic structure and so what that means is that then equilibria are going to boil into situations where we can, we can talk about two different kinds. one is where we're, we're in a situation where there's things that are distributed where everybody is providing some action, and the, the total actions are just balancing exactly up to this x stars. Everybody's seeing exactly x star in their behavior, in their neighborhood. Everybody's acting enough to just balance these things. And then there's specialized equilibria, where some people take actions and other don't, and where basically the people who take actions are going to, are just going to specialize and produce the public good, just like in a best shot public goods game, and then, those people around them are going to free-ride completely, okay? And so to get a feeling for this, let's have a look. So let's suppose that we'll just normalize x star to be 1. Let's look at you know, here's a few different equilibria on different networks. Here's a situation where we've got a specialized equilibrium. These people provide the full action, their neighbors free ride. Everybody gets exactly x star in their neigbhorhood. Nobody wants to increase their action. in fact, sorry, some, some people. These individuals get more than x star. They get two times. So they're happy. They're getting double the provision of public goods. They certainly don't want to contribute anything. Okay. And these people are getting exactly one. They're providing it because their neighbors are not providing anything. Here's another equilibrium. Everybody provides 1/3. And that balances because we're in a situation where each person is seeing exactly x star in their neighborhood, so nobody wants to increase or decrease. Right? So this is a distributed one, where we've got an exact distribution. Okay. And this one is the, is similar to the first one, but just with roles reversed, okay? So you could generally, you know, if, if we had disconnected individuals, they'd each have to provide one, there's nobody else to, to free ride on. you could have situations like this where the isolated person. Their friends provide they don't have any friends so they provide the public good here. You could have either person provide it. you could also have an equilibrium like this where they do a half, a half, right? Or you could do one where you've got 3 4ths and 1 4th. So you could have a whole series of different equilibria and then in each of those cases the other person would be providing one. you could have a situations here that look like the best shoppable goods. You could have ones with these distributions, you could. So these are whole series of different equilibria, some of them are specialized. Where certain people end up doing one, other people end up doing zero, and then there's other ones where distributed, where you've got some actions by different players but their adding up to one, okay? Okay. So, we can begin to see the connection between this, and the best shot public goods game that we saw before. if you look at the specialized equilibria here. Those are going to look like maximal independent sets. Right? So you have the people, some people just provide the full public goods. Their neighbors free-ride. And no two nodes that are, connected can both be providing this, the public goods. So you end up with every node in n is either in this set of s that provides the public goods or linked to somebody. And proposition by Bramoullé and Kranton, then, is the set of specialized Nash Equilibria here are profiles such that the maximal independent set is, a maximal independent set is, is exactly equal to the set of specialists people providing the public good. Okay, so that's just a straightforward extension of what we saw before in, in the public goods game. the best shot public goods. But now we're actually allowing people to take multiple actions and we see that it's a specialized equilibria correspond to that. Okay. so what's interesting in, in terms of the Bramoulle and Kranton analysis is they show that these the specialized equilibria are actually ones that have a nice property. And, it's a sort of a stability property. So, imagine what you do is you start with some strategies, set of strategies that people are playing. So let's look at our equilibria, and ask which ones are sort of stable to perturbations. And what we'll do is that we're going to perturb this by changing everybody's actions by a little epsilon. And the the epsilon can be positive or negative so we could, you know, move, some, some people we bounce some people up so they provide a little more information, other people down so they provide a little less information. So we start at some point, which might be an equilibrium. We want to ask is that stable? We perturb it a little bit, and then what we do is we look at okay, what's the best response? So if everybody was choosing their maximal action, given what everyone else is doing under this perturbation, what do we get? Okay, so we, we go from x zero to x one and then, you know, x one goes to x two and so forth. And what we would like is that if you perturb things by some small epsilons, if you always converge back to x, then we'll say that the equilibrium is stable. Okay. So that's the notion of stability. and let's first think of, of a dyad, so we've got two individuals, these two individuals, and let's suppose that, you know, person 1 is doing something say like a 1 3rd and person 2 doing 2 3rds. that's not going to be stable in the sense that if we perturb this a little bit, so suppose we move this down to 1 3rd, minus epsilon and this up to 2 3rds plus epsilon, well that in fact is also an equilibrium. So if we did that perturbation it wouldn't go back to the original one, it would actually stay at this one, okay? So in a, in a dyad, no matter what numbers you put in there, you know, we could put in zero and one, and, and have this go to 0 plus epsilon, 1 minus epsilon so no matter what numbers you put in, as you perturb this, it would, it would stay there. So as long as x one is less than or equal to x two. Add a little bit to x one. And subtract a little bit from x two. You would stay there. So you're in a situation where you end up not having, any movement in terms of, of the perturbation. nothing's stable. Okay, But when we look more generally. the equilibria are, the only stable equilibria aren't necessarily going to be specialist equilibria. And they're going to be such that every non-specialist has exactly, at least two specialists in his or her neighborhood. So, for instance, if we look at three nodes, and again, our x star is one. This thing is stable. These three are not stable. Okay, and let's just go through and try to understand why is this stable? What's the logic, here? So, let's suppose that we perturb this a little bit. So for instance we take a little bit of epsilon away, so we take this person down to 1 minus epsilon. And we bump this person up to epsilon. And let's, let's just keep this person constant for now. if we do that, and now we look at the best responses to this. Well, this person has already a neighbor providing at least one. They're still at 2 minus epsilon in their neighborhood. They're way bigger than the x star of one. They're getting too much action. They can get rid of their epsilon and go back to zero. So, they're going to tend to go right back to zero. Once they go back to zero, this person's going to go back to one, and we end up going back to the, to the situation that we have. So here the fact that I'm already getting too much means even if you perturb that I'm going to want to go right back to zero. So the people that are in the neighborhood of these specialists are going to stay there if they have at least two specialists in their neighborhood they're going to go back to zeroes. And that means that specialists are going to be forced back to one because all their neighbors are providing zeroes. Whereas this one is unstable, and it's unstable for the reason the dyad was, right? We could make this a 1 minus epsilon and you know then, move this guy to epsilon, this guy to epsilon. Once that happens, well then these people are happy but this person's going to want to decrease more and, and it would actually move eventually towards this equillibrium. you know we could change this by moving this to minus epsilon, add an epsilon over here. that would still be an equilibrium and so forth. So, so the only ones that turn out to be stable, turn out to be the ones that every, that, that every specialist. Has specialized equilibria and such that every non-specialist has at least two specialists in their neighborhood. Okay, so you can go through a sketch of the proof of this basically the stability of such equilibria for, for perturbations of this for the non-specialist, the best response they go right back to zero, and you converge right back. For any other equilibrium, there is somebody providing goods as a non-specialist and then you perturb that agent up and the other neighbors down just the way we did, and basically you can, you knoq work through that kind of logic, so the idea of the proof is much similar to what we just worked through in terms of the examples, and you can deduce this proposition in that manner. Okay, so what says is that is that these specialized equilibria are somehow special in this model. So you get, they're stable in the sense that if you do perturbations then you're going to get pushing people towards that. Let me just sort of make one comment on this, and sort of an interesting contrast. another way we could think about, you know, so here we've got a game with multiple equilibria. There's lots of different equilibria in this game. And, you know, part of the reason that, that one's interested in applying the stability concepts is just to find out are some of these more naturally equilibria than others? And so if we perturb them and we get back to them, we say, well, that seems to be a more natural one than another one. another way we can think about stability is instead to think, think about pairwise stability, right? So suppose your links are costly, and if I'm specializing, and I've got a bunch of friends who aren't specializing, I'm providing all the good. They're not doing anything. I would start dropping links to the non-specialists. Right? So in some sense in that world, what you would have is the only reason people would be willing to, to maintain links would be if their friends were also providing some of the local public goods. So I normally want to keep these friendships if they're actually providing some value. And so there, what you're going to get is the non specialized equilibria are going to be the only, stable ones. So, what this says is that, you know? It depends very much on what kind of stability notion you put in, as to what you might select out. And whether you think about perturbing the, the network, or perturbing the actions. You could get very different kinds of conclusions. And so, you know, ultimately I think what this says is there's still a lot we need to understand about these kinds of games. they are tractable games. They have some interesting features. Some of them tie back to some of the things like the best-shot public goods games and so forth and they, they can be you know, interesting to analyze further. Okay. So, so that's one example of these kinds of games. if we begin to introduce heterogeneity to these kinds of things that's going to change the nature of equilibria. We could begin to enrich things that way. There's a lot of enrichments we can make in these kinds of games. And what we are going to do next is look at another class of games. And this other class of games is going to be one where we allow for strategic complements and there the games again are going to be nicely behaved in the sense it will be much easier to get a full characterization of what the equilibria look like and, and it'll be nice we'll have some nice relationships to the network structure and that'll allow us then to.
