Okay, so we've got some basic definitions and ideas behind us in terms of

understanding equilibria in network games.

And now we can look at a little more structure.

And what I want to do is, is, talk a little bit about when it is that, that

there's multiple different actions that can be sustained in a, a, given network.

So when is that it's possible that some people adopt a new technology and other

people don't? Or that some people are, are becoming

educated, other people are not, and so forth.

So when is it that we actually can sustain multiple actions, even when we've

got a lot of homogeneity in the society. Even when anybody has the same

preferences and so forth, we still end up with different people taking different

actions based on their position in the network, okay.

So this is just sort of an interesting question conceptually to understand when

this can happen. And, so lets take a look at it.

And what we're going to do is, is look at a paper by Steven Morris a cord, a simple

coordination game. And this is going to be a game where you

care only about the fraction of your neighbors taking a different action.

So you prefer to take action one if a fraction Q or more of your neighbors take

action one. So suppose that Q is a, a half, then if

you just want to match the majority of your friends.

So if the majority of your friends take action one, you want to do that.

If the majority of your friends take action zero, then you prefer to take

action zero. Okay?

So this is a game of, of strategic compliments.

And a very simple one where everybody just cares about the fraction.

So everybody's threshold is just a fraction of their degree.

It's the same fraction. But we could have Q be a half, it could

be two thirds, or maybe you need two thirds of your neighbors to take this

action before, you know, this new technology, before you're willing to

adopt it and so forth, okay? So a sim, a very simple coordination

game. And let me say a little bit about the

background of this game. the game where it's actually a half is

also what's known as the majority game. And this is a game which has been studied

quite a bit in the statistical physics literature.

And has some background in the, physics and, and, agent based, literatures.

And, you know, part of the reason is that, that, there's certain kinds of

particles. Where the particles might be sitting in

some sort of lattice structure. And the particles react to what other

particles are doing. So, if other particles end up in one

state, then they end up trying to match the state or they could end up going in

opposite directions, but in certain situations they'll flip into be in a

certain state if, if more of their the other, so as more of their neighboring

particles become excited, they become excited, for instance.

And depending on what that threshold is, then that ends up having a percolation so

that you can end up having this move through different kinds of of materials.

And so that's been an area of study in physics.

And this actually has a nice interesting relationship to these kinds of games on

networks, where an, a given node cares about what its, its neighbors are doing,

and would like to match actions to the neighbors.

And in this case, we have the simple Q which describes what's the fraction that,

have to take action one before I want to take action one, okay?

Okay. So let's, let's think about what

equilibria look like in this game, so we're going to look at pure strategy,

Nash equilibria in this type of game. And, let's let S be the subset.

So we've got these N agents, one through N, [NOISE] they are connected in some

network, right? So there's some network describing which

people are connected to which other ones, and so forth.

And what we want to do is we want to color them so that some of them take

action one. And we'll let s be the set of individuals

that take action one, okay? So what can we say about an equilibrium

in this game? Where S is the set of people who take

action one. Well, it's going to have to be that every

person in S has a fraction of at least Q of its neighbors in S.

Okay. So the only way that they're going to

want to take action at one, is if at least Q of their neighbors are in S.

Right, so, so that just follows directly out of the fact that you only want to

take action one here if at least q of your neighbors do.

And it has to be that everybody not in S, doesn't want to take action one.

So it has to be that everybody who's not in that set has to have a fraction of at

least one minus Q of their neighbors outside of S, so that fewer than Q of

their neighbors are in S. More than 1 minus Q of their neighbors

have to be outside, okay? So, for any, for, for, the set S to be

the group that take action one, for that to be an equilibrium, these two things

have to be true, okay? And basically that characterizes a set of

equilibria. So S is going to be in equilibrium, if, a

pure strategy equilibrium, if and only if this is true in this game.

All right? Okay.

So now a definition which is actually an interesting definition in terms of a

network what, what's known as cohesion, following Stephen Morris' definition.

and we'll say that a group S, some group of nodes S, is R cohesive.

Where R is going to be something R is going to be some number in zero to one.

So we've got some number in zero one. And we'll see that S is r-cohesive if

when we look at everybody in the set, right?

So look at all of the people in S. And look at what fraction of their

neighbors are in S. So here's how many neighbors they have,

here's how many of their neighbors are both neighbors and in S.