So, we've been talking about strategic network formation.

And we talked a little bit about the variations of positive and negative

externality models. And, and why we might end up having

inefficiencies in terms of the networks that form.

And now I want to talk a little bit about.

the possibility of transfers, so subsidies, payments across different

individuals. And try to understand how that might

rectify things, and in terms of, you know, what we've seen, we saw this

conflict, you know we can talk about all the difference kinds of ways of modeling.

And fitting such things, so here we're just going to talk a little bit about the

transfers at this point. And then try and understand what we see

there. There's a lot more that could be said on

this subject, but I want you to give you some feeling for it and at least a little

bit of a basic understanding. Okay, so what do we mean by transfers?

so here we mean sort of outside intervention, somebody taxing or

subsidizing relationships, say a government supporting R and D

relationships. it could also be due to the fact that

there's bargaining among the individuals involved.

So, somebody says look, you know, it's worthwhile to form this link.

I'm going to pay something to you to help you form this, favors exchanged among

friends and so forth. So, the idea is that whatever those

utilities numbers we're dealing with some of that could be moved from one node to

another. Either by some outside entity saying I'm

going to tax some people and, and subsidize others or by individuals

bargaining and say look, I'll give you something if you're willing to do it.

And, and certainly when, when countries form alliances, there can be payments

made either explicitly or implicitly. In terms of the arrangements to make sure

that these things are in both people interest's in forming relationships.

Okay, so lets have a look at this in detail and so what we can think of in now

we're changing the base utility to the utility plus some transfer.

Where this could be either a positive or negative number depending on whether

somebody is making net payments or getting that receipts as a function of

the network. So, for instance it could be that the

peripheral players say look, it's really beneficial to be connected in this star,

I'm willing to do favors to the center. And then the center is willing to

maintain these relationships because they get value form the other players.

so if we just sort of thinking about transfers.

One possibility in terms of, let's go back to the inefficiency that we had in

the co-authorship setting. So, remember the co-authorship model that

we just talked about in our last video. We've got a situation now where the

problem was that people wanted to over connect.

And we could imagine that in this situation one possibility is the

government says okay, we're going to tax people who form extra links and then move

that to to the other players. So, the raw paths you know from, from

just forming just one relationship was three now if people form this extra

relationship here, their payoffs went up to 2.35, and these people went down to 2.

One possibility is now what we do is we actually charge these individuals, so we

charge them a 0.625 each and then pay that to these individuals.

Right, so we tax people saying look, if you're going to form relationships,

you're going to have to pay something for that and then we reallocate those taxes.

So, instead of these people getting the 2s, now they get 2.625.

Okay, so those particular taxes and subsidies, are a way of equilibrating the

payments in this model, right? And so now, when we look at the

incentives, what happens is the individuals no longer have an incentive

to form this extra link. Because they have to pay the tax

involved, and so this now becomes pairwise stable, if they include the cost

that they're going to to have to pay in terms of taxes and so forth.

And so what we've done is we've aligned the interest of the individuals.

Now everybody sees the, an equal fraction of the value of this network, and these

people say, oh, that's not a good idea. I'm better off in this network and so

they don't form this extra link and this turns out to be pairwise stable.

So that's a situation where, you know, equalizing the payoffs by, by proper tax

and subsidy and reallocating that. Now everybody gets an equal payoff.

And they have incentives to form the right relationships.

Okay, so one possibility is, is we just, you know, tax and subsidize in a

completely Egalitarian way. So, set the transfers that are being made

to any individual. What we do is, we just look at the total

value of the network. And average that across all individuals,

and if they were getting less than the actual amount, then they're going to get

a positive transfer. If they were getting more than the

average amount, then they'll have a negative transfer.

So, we're just going to adjust the transfers to move everybody back to the

center. So, now everybody, in terms of their net

utility when we account for the transfers is just the average overall utility,

okay? Now everybody in the society has exactly

the same incentives as a utilitarian planner would have, because now

everybody's utility is just one nth of the total utility in a society.

So now, the utility anybody gets is, is exactly proportional to the efficiency of

the network, so, ones that are more efficient everybody gets more value.

Ones that are less efficient, everybody gets less value.

Now the most efficient, so directly out of this we're going to get the, get the

overall efficient network is going to be pairwise stable.

Right? So now we, we've solved that problem of

efficiency being pairwise stable, by just equilibrating things and making sure that

everybody is an equal sharer in the pie. Okay, that's wonderful.

it works well, but it, it, it could involve a lot of transfers.

It could involve a lot of spreading money around.

And in particular or spreading utility around, it could involve.

making transfers that are going to violate some fairly basic conditions.