Okay, so we've got some basic ideas behind strategic formation. And now, we're going to talk about modeling how we measure efficiency and also how we can model people's decisions form and delete links. And so, in terms of modelling incentives and you know, just sort of keeping track of these things, let me just first make a very simple point. so let's think of a world where we need consensus to form links. So, two individuals have to agree to be friends, and I can't form a friendship with somebody else if they don't agree to be friends with me. Now one way people might think of doing, of modeling this would be as just modeling it as a game. And the simplest possible game you can imagine is everybody just announces who they want to be friends with. And then if both people, if two people both announce each other, then we form a friendship between them, and if they don't both announce each other, then we don't form a friendship. Okay. Well, Nash equilibrium would be a situation where nobody wants to change their list of announcements given the set of announcements of other individuals. And so, l-, let's just talk through why that is doesn't really work so well as a m-, model of, of network formation in these kinds of settings. And imagine that we're in a setting where they're just two individuals. And if they're separate, they get a value of 0, and if they're connected, they get a value of 1. Okay, and now we have a game where they beat, they simultaneously announce whether they're willing to form their relationship. well both of these are Nash equilibria. it in this case it's a situation where if, if I don't think the other person's going to, going to announce me, that doesn't matter what I do, I can't get the friendship anyway, so I might as well not announce it. So, there's an equilibrium where neither person announces the other, because they can't you know, add or really change things. There is also an equilibrium where both of them announce the friendship and then it firms. And so there is two nash equilibria and in this case that's somewhat problematic as a model here. Because this is the simplest possible model and it predicts anything it could happen in terms of linking to forming or not forming. And yet, any reasonable, communication among these individuals should lead to this link for me. Now there are ways of dealing with this in terms of the Game Theory. We can put in stronger solutions concepts and trying to do things that way, so instead of looking at Nash equilibrium we can allow for slight errors by players and see what happens. Or, but there are other examples that aren't quite as simple as this one which give other solution concepts trouble. So what we're going to do is instead of using off the shelf non cooperative gain theory, we're going to model incentives using a very simple concept which we'll call pairwise stability... 'Kay? So the idea here is, is simple. What we'll do is, is we'll look at a network and we'll say that it is stable. In particular we'll say that it's Pairwise Stable. If nobody who's involved in some particular relationship would gain by deleting that relationship. Okay. So in a situation where it takes mutual consent to form a link, either person can get rid of the link. So if one person decides they don't want to go and be friends with somebody else they sever that relationship. So one person can delete a relationship and it takes two to form one. So what we, what we do here is we have a situation where no single agent is going to gain from deleting the link, and no two agents both gain from adding a link with at least one strictly gaining. So we can worry about indifferences, but the idea is that if two people both gain weekly, and somebody gains strictly, then a relationship should form. Beneficial relationships should be pursued when available, and ones that aren't beneficial should not be pursued and should be deleted. Very simple concept in that already is what we'll begin to, to put a lot of structure on networks. So in terms of notation, pairwise stability is, is defined as follows. So we'll say that the network, we'll say that g is pairwise stable, so this is pairwise stability of a network g. It's stable if the utility of i for any link that they're involved in is greater than or equal to the, what they would get from deleting the link. Okay, if it was less they should have deleted it. So for g to be stable it should be that they get at least as much as they would get from deleting any of the links they're in. Nobody gains from severing the link. And if somebody really wants a new link, if somebody wants a new link it should be that the other person didn't want it. Otherwise it should be added, ok? So for this to be stable and and not to be subject to further changes, it should that if somebody wants to add it, their partner doesn't want to add it. So if some like i j is not in g. This, if i g is not in g then it should be that if one person wants it the other person doesn't want it. So it could be that neither person wants it, but it can't be that they're both happy to have this link and it not to be in there. Okay. So this is a very weak concept. Why is it weak? Well, it's only looking at pairs of individuals. It's only looking at one link at a time. And it just makes sure that there's no single link that would be better deleted. And no link that's not present that would be better to add. Okay. But often this already, is fairly powerful. So sort of the minimal set of requirements that we might think of in terms of stability. It'll often began to narrow things down. Now, there's all kinds of other variations. So this is this concept came out of the paper with Ashford Lowenski. so the Ashford Lowenski 96 paper, people have looked at a lot of other variations on these kinds of things, we'll talk about some of them but, you know this, this will give us some basics to work with. Okay. So, so now when we go back to that example we had before both are Nash equilibria, but this is the only pairwise stable[UNKNOWN], right. So both people would gain by adding this pairwise ability just says that this is the only stable network. Okay. So let's take a look at this in action. So let's look at a slightly richer example and we'll walkabout where these numbers come from a little bit later. But let's suppose that we have a situation where everybody's symmetric, if if nobody's connected, they all get payoffs of zero. So, we'll just normalize payoffs with no connections to zero. Let's suppose that if you form a relationship with one other person, you get a value of three each. So, if two people form a dyad, they get a value of three. So, if both sets of people formed relationships and we have three to four people and we have two relationships, and everybody gets a three. if, if we add a link to this network where these two individuals now decide to form a link together, then their payoffs go up so now they have two relationships each. They get a marginal benefit, a bit little more, they get 3.25. But lets suppose these people are jealous, they don't like their friends to have new friends. So this is different then the connections model this is a situation where now I'm, I'm losing time with my friend because now their spending more time with somebody else so they get a value of two. now these people if they connect to each other, they get more value. But then these people are losing value because now their friends are spending time with other individuals. So we can think of this, this will come out of a collaboration network where if people I'm collaborating with are collaborating with other people, then that means we spend less time together, I get less value... So this is one where we've got negative impact of other people forming new relationships. And so you can go through and, and have different paths here, and here, you know, when these people now form a relationship, their value goes from 2.5 up to 2.78. But these people are, go down from 2.5 to 2, so they, they're losing more time[INAUDIBLE] and so forth. And then these people form a relationship. They go up from the 2 to 2.3. And so forth. Okay, so this is a very simple setting. And what we see in this setting. In terms of the, value. the arrows represent. Moves from one network to another network. Which would be improving, or it would sort of means that this, this network is not stable because the individuals here would gain by adding a link, and then this one's not stable, and this one's not stable, right? So each one of these is pointing to a new one, and we end up with the only pairwise stable network for this set of payoffs. You, given all the permutations of these things you're going to end up with everyone connected and everyone getting 2.33, okay. So, that's the pairwise stable network in this set. Okay. Now the interesting thing is well, they, they're getting, they're worse off than they would have been had they stopped here. The difficulty is, this is not stable in the sense of individual incentives. People who have incentives can move on from there. So let's talk about that a little bit in more detail. So let's talk about he efficency, and contrast that with the individual incentives. Ok so pair wise stability handles individual incentives. Now lets talk about evaluating overall welfare. So one notion that comes out of economics due to[UNKNOWN] and the late 19th century is known as Pareto efficiency. And what does Pareto efficiency mean? It says that a network is Pareto efficient if there is not some other network for which everybody is at least as well off, and somebody, some of the individuals are strictly better off. Okay? So there's not something that one can do which is unambiguously better for everybody. Nobody suffers and some people are made better off. So if something is not Pareto efficient then society really has better options. Just unambiguously better options. If something is Pareto efficient then it means that if somebody gains by move, by some change, somebody else loses. Okay? So Pareto efficiency is a weak notion of efficiency. There can be lots of pareto efficient outcomes but it it does rule some things out. So it's going to rule out things which are just unambiguously bad and you can do better by. Okay? Now when we look at a stronger notion. Instead of just keeping track of well some people are just better off or is everybody better off or not. Sometimes we have choices to make, that some people are going to be better off, and some people are going to be worse off. we could talk about just a, a stronger notion of efficiency, which we'll refer to as efficiency, or we could refer to as strong efficiency. If G is a maximizer of the overall sum of, of payoff. Okay. So you know, this would be Pareto if, if you allow it for people just to move utility back and forth. You can always make, you know, if you make everybody if you make the some better off then, then you could make everybody better off by, by making appropriate transfers. But more generally this is just going to be a notion which is known as Utilitarianism. [SOUND] Which means that you care just about the total utility or some weighted, in this case the equal weights on all the individuals, of the utility in this society. Okay so this is utilitarianism with equal weights on everybody, I just care about total utility and I, I, I actually don't care you know, some people might gain or lose but if overall it's better then I want to go with that. Okay. So this is a stronger notion that will narrow things down a little bit more. So if we look back at the picture we looked at before, pairwise stability was moving us to this complete network... If we look at the overall maximizer, the overall maximizer would be here. so this is an efficient network and it's also pareto efficient. this one is pareto efficient. There's no other network which makes everybody better off. This one is a better in terms of the overall sum. Right this one gives us a higher sum than this one does. But these people, some people go down and other people go up. Okay, so this is the overall efficiency and it's Pareto efficient. This one is Pareto Efficient, but not overall efficient in terms of, of maximizing total sum of utilities. you know, this one is not Pareto Efficient or Efficient. this is better for everybody and, and you know, similar here, the, these are. So, here already in this example, we see that what society would like to do in terms of picking something which maximizes over your all utility Or even something which is[UNKNOWN] efficient. They can end up things, which are worse, in the sense that everybody is worse off than what would happen if the society could oppose the network. And part of it is due to the fact that individuals aren't accounting for the harm that they can afflict on others when they make their decision. Right, so they're, they're are selfishly, maximizing their own payoffs and not accounting for what that does to other individuals in the society. And that's not unusual in, in a lot of studies and once we start looking at individual incentives. they're going to be misaligned. And especially in network contexts, where what individuals do. and what benefits they get depends on the full structure of the network and what other people were doing. It's not going to be unusual that we're going to find some conflict between what is individuals are going to do and what society would like them to do. What's going to be interesting is, is trying to figure out when this happens, and to what extent it happens, and why it happens, and. Whether interventions can help or not. so there's going to be a whole series of questions but one of the basic themes once we start looking at strategic network formation is there's payoffs involved. And individuals are going to be forming the relationships that they find beneficial, but what's good for them is not necessarily good for the overall society because their actions have... Implications for other people that they are not necessarily taking into account when choosing those actions. Okay, we'll look into this a little more detail now. We'll come back and look. say the connection model, and some other models. To try and analyze what are the efficient networks, what are the Pareto stable networks. Do we see conflict and so forth. And that'll be our next topic.