Okay, well now let's suppose there's two of us, person one and person two.

I'd like the other person to experiment. If I think p's less than a half, why

don't I let them try it, and I'll sit by and just choose A?

And if, if they experiment and play with B for a while and it pays off well, then

I can switch to B. But I don't have to pay the cost of the

experimentation. I want a free ride.

'Kay? Now that becomes a game, which is

actually going to have a fairly complicated equilibrium, especially when

you start putting that game in a network with all kinds of players.

And we begin to have the players connected to other players, and now we

look at this, this simultaneous decision of, who's going to choose B in this

period? Who's chose it in the last period?

What are our beliefs? What do I think everybody else's belief

is, and so forth. So when you, when you get to, to overall

looking at this game, the game becomes very complicated.

both because of the strategic aspects, and because of the, the, Bayesian

inference. And so, now you can begin to see why it

might be that in fact when we put humans in the laboratory and, and ask them to

play games, or to make these kinds of choices, that they might not behave in a

fully Bayesian manner. and its just, just complicated to do.

It's, it's hard to even write the model down and solve it.

Okay so in fact, the way that let me just say a little bit about how this is solved

in terms of the Bala and Goyal approach. So what they did is assume that players

are not going to be strategic about this, and each person is just going to choose

things which maximize their own payoff and, and not worry about the gaming

aspect of it. And secondly it, I'm not going to infer

things from the fact that other people are making different kinds of choices.

I'm just going to, to keep track of what have I seen in terms of my histories of

As and Bs, okay. So I just, I just keep track of my, what,

whatever I've seen through myself and my neighbors.

And I'll just keep track of what are the relevant payoffs, and important, most

importantly, how many times have I seen B payoff two, how many times have I seen it

payoff zero? And then I can update on what I think p

is, just based on, on those observations. And I'll ignore everything else, and I

won't do the complicated updating, I won't game things.

I'm just going to look at, at the twos and zeros, and decide whether or not I

want to switch from A to B or B to A. Okay, so let's look at that.

Okay, so what's a proposition you can prove then fairly directly?

the first thing you can show is, let's suppose p is not exactly a half, where

I'd being different between choosing an action.

Then with probability one, there's a unique time, or there'll be a time, a

random time, sorry. such that all agents in a given component

play just one action and play the same action from that time onward, okay.

So bascially what's, what's happening is that as long as p's not exactly a half,

and we would be sort of indifferent we're basically going to eventually all end up

choosing the same action. And we'll just lock in on some action

and, and play that forever after at some time, okay.

So, so, sometime we'll all eventually converge, and, and play the same action

forever. Okay, so that's the nature of the

proposition. So let's talk through the intuition and,

and basic proof behind this, why is this true.

and I'm just going to sort of sketch out the proof, it's, it's fairly easy to, to

fill in the details here. So let's suppose that, that this weren't

true, right? So if, if it wasn't true, then basically

somebody's going to be having to switch back and forth infinitely many times,

otherwise we'll eventually converge. So somebody's gotta be going back and

forth between A and B infinitely often. And then, in particular somebody let's

suppose we just have one component, and this just works, you know, regardless of

which component you're looking at. So, let's suppose somebody is playing B

infinitely often. Okay, so they, if we don't converge,

somebody's gotta be playing B infinitely often, okay.

Now we can use the law of large numbers. So law of large numbers is going to tell

us that if somebody plays B infinitely many times, then they're going to come to

get an arbitrarily accurate estimate of what p is.

So with the probability going to one in time, they will, their belief will

converge to p. And so, what does that mean?

Well, in order for them to keep playing B, if their belief about p is becoming

arbitrarily accurate, then it must be that p is converging to bigger than a

half, otherwise they would stop, right? So over time, they're good, they're good

Bayesians, they know how accurate their belief is.

They would either converge to p above half or not, or below half, because it's

not allowed to be exactly half under our assumption.

If it's above half, then they'll keep playing it.

If it's below half, then eventually they would stop playing it.

Because now they'd be arbitrarily accurately convinced that it's, that it's

not good. So if it's not good, they should learn

that, and they'd stop playing it. If it is good, they'll keep playing it.

Okay, so it must be that if they do play it infinitely often, then it's gotta be

the case that they're converging to the good belief.

Otherwise they would've stopped. Okay?

So, now this means that, that they have to be converging to the true belief or,

or with probability one that the, the true p has to be bigger than a half.

so then, everybody who sees this person is actually going to see this sequence

played. They're going to also see B played

infinitely often. They're also going to have to converge to

the belief that P is bigger than a half. And so they should all start playing B,

right? So if this person is, is learning that B

is, is good, then their neighbors are all going to have to converge.

Then these people are all going to see B infinitely often and converge to p.

Their neighbors are going to have to converge, and so forth.

So the neighbors of agent must play, then all agents must have to play B.