0:00

Okay, so we're back and we're talking about strategic network formation.

Â And now we're going to have a look at the connections model,

Â the symmetric version of the connections model and

Â see what is efficient in that setting and then what's pair y stable.

Â So, just to remind you of this model,

Â we had this benefit parameter delta somewhere between 0 and 1.

Â That gave the direct benefit and then this decays as you get further away.

Â So, in terms of overall value, the utility that a given

Â individual gets is represented then, by this utility form, where you sum

Â over different individuals and you look at the path length to those individuals.

Â 0:43

And then you might have a delta, which depends on those

Â personal connection and then you raise that to that power.

Â So if you are not connected at all you can make this infinite and then this will

Â indicate a zero and then you subtract off costs for maintaining links.

Â Okay and then the simplest version of this, in terms of the symmetric version,

Â will get rid of the subscripts and all these things and

Â everybody will bear the same types of costs and have the same types of benefits.

Â So, when we looked at that then for

Â different networks we have utilities that each individual gets.

Â 1:30

Okay, let me just say one thing before we go on.

Â So, one nice thing about this type of approach is that once we've

Â specified utilities and welfare calculations for each individual.

Â Then, we end up having the ability to evaluate networks and

Â say which ones are good and which ones are bad,

Â directly by doing calculations in terms of the welfare properties.

Â And this is something that's very different from the random network.

Â So, is a network formed by preferential attachment better than a network that

Â is formulated at random?

Â Well, we have no idea because we didn't have any specification of what

Â that did for the individuals involved.

Â And so here, we've got some idea that there are preferences and

Â where you can value friendships at a certain level.

Â We value indirect friendships at a certain level, and once we've done that,

Â now we can assign values to these things and then make welfare evaluations.

Â So, it's not only that by doing this we get predictions about which ones are going

Â to form, but we also can evaluate them and so that's an important aspect of this.

Â And one other thing to say in terms of pair y stability and

Â this idea of forming networks strategically.

Â It's not necessarily true that each individual has to be Machiavellian or,

Â calculating about all their friendships or I want to form this friendship

Â because it's beneficial and that one because it's not.

Â It's more just that what's necessary is that individuals will tend

Â to form things which are beneficial and

Â when things aren't working out, they tend to get rid of them.

Â So as long as their pressures in those directions.

Â Then we can talk about dynamics, and so forth afterwards.

Â But that'll give some push towards these kinds of networks, and

Â these are ones that are going to be stable,

Â in the sense that nobody then has an incentive to move away from these.

Â So they'll be rest points of different processes, where people don't have to

Â be so calculating, but at least pushed in the right directions, and

Â eventually reach these kind of networks.

Â 3:35

Okay, so let's have a look now at this particular setting.

Â And let's look at the ones, the efficient networks, so

Â the ones that are maximizing total utility in this setting.

Â And they break into three different categories.

Â So, we're going to deal with situations where there's very low cost to links.

Â Situations where there's a medium cost to links and

Â then situations where there's a high cost to links, so the cost is above some level.

Â And basically for

Â very low cost of links, the complete network is the unique efficient network.

Â And that's very intuitive.

Â Links are so cheap and

Â in particular, when c is less than delta- delta squared, that's going to be

Â the situation where it's more beneficial to have a direct relationship, right?

Â This is the value for

Â direct relationship compared to an indirect relationship of distance 2.

Â So shortening anything of distance 2 or even further.

Â 4:28

The gain in changing that,

Â is bigger than the loss in terms of cost of adding a link.

Â So adding a link is always going to be beneficial, and

Â the complete network is going to be uniquely efficient in that setting.

Â And it will have to worry about externalities,

Â but that's going to be the conclusion.

Â In the middle range, then, star networks are going to be uniquely efficient.

Â So, the only architecture which is going to be efficient, is going to be have

Â somebody in the center and then everybody else have that one link to that person.

Â And that's going to be the thing that maximizes total sum of utilizes necessity,

Â and the only thing which maximizes the total sum of utilities.

Â So that does it and its the only type of architecture that does it.

Â And then once costs are so

Â high it just makes sense that nobody should connect any links

Â are just too expensive it doesn't make sense to have anybody talking to anybody.

Â So for very high costs the empty network is uniquely efficient, okay?

Â So the meat of understanding this is really going to be understanding

Â this middle part because the other extremes are going to be fairly,

Â if links are so cheap you might as well just add them all.

Â If links are so expensive, it doesn't make sense to add any.

Â And in the middle, what's really the insight here is that you want

Â to have the star networks form.

Â Now, a couple of things to say is that this, is a fairly stark characterization.

Â It's actually going to be true for a set of models beyond what is true here.

Â As long as you get a higher benefit from a direct connection, and a lower benefit

Â from an indirect connection, and a lower benefit from things.

Â There won't be anything special about it being delta, delta squared, delta cubed.

Â It just has to be some value, some value for two distance, a value for

Â three distance.

Â And you'll see that everything we say in all the proofs and so

Â forth, would go through if you just separated,

Â put in, substituted something else for the deltas, and so forth, okay?

Â So it's not special to the functional form.

Â 6:30

Okay, so let's first try and

Â understand stars before we go into a formal proof of this.

Â So letÂ´s think about, we start with one relationship that gives us 2 delta- 2C,

Â and we think about adding a second one.

Â So, there's two different ways we can add this second relationship.

Â One, is that we could connect the person to somebody who's already connected, or

Â they could form a new relationship.

Â If they form a new relationship, then weÃ¨ve got four people connected,

Â four benefits, four costs.

Â Four minus delta four c.

Â If we connect this person here, each one of these 2 still gets a benefit so

Â we'd still end up with 4 deltas and 4c's.

Â 7:15

So here, what's important is the indirect benefits that flow through the network

Â generate extra value.

Â And so connecting in this way it

Â gives us a higher value than connecting in this separate way, okay?

Â So that's the start of it.

Â Now when we think, let's connect this third person in somehow,

Â we want to connect them as well.

Â Well, we could connect them to say, one of the agents,

Â one of these peripheral agents or we can connect them back to the centre.

Â And again, we're going to have 6 values and

Â 6 costs coming from the direct links because there's 3 links in either network.

Â And the question is why is it better to do it in the star form?

Â Well, in a star form, all these indirect connections now are at a distance too.

Â Whereas in this one some of the indirect connections are at a distance three.

Â 8:06

And so by doing it in a star form, we end up with a higher value for

Â all the indirect connections.

Â We get 6 delta squared as opposed to 4 deltas and 2 delta cubes.

Â These longer distance relationships are worth less, they're worth a delta cubed.

Â Which is less than a delta squared and so we get higher value by

Â having more direct connections which come through a star.

Â So those indirect connections are shorter and more valuable that way, okay?

Â And then, if you think about adding a fourth person in.

Â Well, if you add them in directly to the center, again,

Â more value from the indirect connections, than if we added to the periphery.

Â Where now a lot of these indirect things are going to be of distance three,

Â as opposed to all of distance two here in the other network.

Â So the stars are coming out because they're the most efficient way to connect

Â people with a given number of links with the least distance between them.

Â So it's a very efficient way to do connections.

Â 9:08

When is it that you want to keep connecting?

Â So, let's suppose we've got the four individuals and we keep them in a star.

Â Or else we could add extra links in, to connect these two people.

Â Well, what we get here is now, before here when

Â between these people, there was a distance 2 now there's a distance 1.

Â So what we've done is we've moved some of the distance 2 things, over to direct

Â connections, but we've paid a couple of extra costs of maintaining links to do so.

Â So we have more relationships, but more benefits because of that.

Â And so the question is, when is it that this gain in

Â having shorter distances outweighs the cost?

Â And it's precisely 1 delta minus delta squared is bigger than c, right?

Â When is it better to have direct relationship than an indirect

Â relationship?

Â So delta minus delta squared is bigger than c,

Â then you don't want to stop at a star, you want to just keep adding links and

Â shortening all those indirect relationships.

Â So that gives you the idea of when you'd want the complete network

Â versus instead stopping at a star.

Â So a star is the most efficient way if you wanted to use the minimal number of links.

Â And then there's a question of whether you want to keep adding links.

Â And that'll happen if the links are cheap enough and

Â the benefits outweigh the costs.

Â 10:30

Okay, so that's just a quick peek at it.

Â And the next thing I'll do is actually go through a form of proof of this.

Â You can skip that if you like, or you can look through the details.

Â So the formal proof is very straightforward.

Â But we'll just go through verifying that the star is actually the most efficient

Â way to do this, do a formal proof of this proposition.

Â And then we're going to come back and look at the pair y stable networks and trying

Â to understand the difference between the networks that are most efficient and

Â the ones that are pair y stable.

Â