Hello there. So we're going to talk now a little bit
about dynamic strategic network formation.
So add some dynamics to the process instead of just a static look at what's
stable. And looking at these dynamic processes we
can think of, of you know adding dynamics for different reasons.
you know again as I sort of mentioned earlier in the course we don't
necessarily want to just add enriched models because that adds realism.
That's not a good reason to enrich your model because it complicates the model
and we want things to be as simple as possible.
So we only want to add it if it's going to give us something that we didn't
get before. And here in particular what it's going to
do is, is begin to give us some predictions of which networks might form
when there might be multiple ones which are stable.
And you know, another possibility is that by doing this you could begin to capture
forward looking behaviour. Where people were sort of asking, well if
I do this, then what's going to happen further down the line.
We're going to start by just really focussing in on this fact that it's going
to refine static, models. And there's three different approaches to
deal with dynamics. We can think of dynamics where they are
myopic and error-prone and, and nature's just marching along.
And we're thinking about an evolution of a system or we can think of very forward
looking calculating types. And we're going to look at a fairly
myopic version of a model right now, but a fairly simple one.
So the idea here is that we're going to look at a dynamic process where people
can form links over time. And, even if there are multiple pairwise
stable networks and some of them happen to be efficient, we might not reach
those. And this process we'll look at was first
proposed by Allison Watson in 2001 and the process is really the simplest one
you can think of in terms of adding a dynamic.
Nature just finds a link, uniformly at random picks a link, and then it, it's,
then that link is added if adding that link to the current network would benefit
both individuals involved. At least one of them strictly benefiting,
and if the link is already in the network.
Then it will be deleted if it would, if deleting it would help either of those
individuals involved. So, it's basically like the pairwise
stability concept but instead we're just going to look a link at a time.
So, we start at some network randomly pick a link and then if it's already
there, we think about deleting it. If it's not there we think about adding
and then, just continue randomly picking links and, and so forth.
So now we've got a nice dynamic process, it's going to march along and then we can
ask where will it, where, where will it end.
So, first thing we can say is that any resting point, so if, if, if this process
ever stops at some network and never moves from that network.
It must be that whichever links are recognized nobody wants to add a new one
and nobody wants to delete an existing one.
Therefore it must be pairwise stable. So this process is going to identify
pairwise stable networks. It's going to come to rest at pairwise
stable networks. And so the proposition that Allison Watts
showed is an interesting one. Where, let's suppose we consider the
connections model, where c is less than delta, so it actually makes sense just to
form individual links to people you're not connected with.
But c is bigger than delta minus delta squared.
So, it stars are going to be efficient networks and a star would be pairwise
stable. So, if you actually had the center of a
star, where this sort of low to medium cost range where stars are going to be
efficient, and pairwise stable. And a point that she made is that if we
look at this dynamic process as end grows the probability that this process
actually stops at a star goes to a zero. So, even though a star would be one of
the pairwise stable networks, the chance that a process of fairly of natural.
Dynamic process is actually going to reach one of those efficient networks is
going to zero. So most likely you're going to end up
with an inefficient network. And, and let me just go through the basic
ideas here and then we can go through a short proof of this.
So the ideas of, are fairly straight forward.
So let's imagine we started at an empty network and we just start building in
these things up. So, we started an empty network and you
know, two people are, are identified. And they can form a link let's call them
one and two so that the first two people who have a chance to form a link.
Well, we're in a situation where c is less than delta, so they're going to form
that link, it's beneficial. Now another person comes along, we
recognize another link, now there's different possibilities in this, in this
setting. one is that, that somehow the link
involves two individuals other than one and two.
And those people would want to form that link if, if we happened to find three and
four they'd want to form a link because that link's not theirs and they're myopic
they think this is good, it's beneficial. They would form that link.
Already, we're on a on a path that's not going to lead to a star.
and question is we have to be careful to find out if that three, four ever be
deleted and maybe they'd form a new link to one and so forth.
But at this point the the we're on a bad tragectory we could also think of a
situation instead. Where maybe, some individuals recognized
along with the link to one or two. Okay.
So we do happen. So we do happen to, to go on a good
trajectory towards the star. And once that happens then as new links
come in, it's quite possible that the new links being recognized are not directly
to either one or two, but to somebody else.
In which case you know, we can end up having things move out in ways that are
not going to lead to a star. Okay, so the, the, the fact that people
are biopic and not necessarily thinking, oh we have to get to a star, that's the
best thing for us. Instead adding links when they're
valuable means that this thing could arise in a way that's going to look very
different from the star. And, the, you know, we would have to have
basically an order where the ordering of the links that are recognized would have
to just happen to be exactly in a configuration of a star for a star to
arise. And a that these are the links that are
recognized and not any other ones before we get to finishing it is going to zero.
So that's the basic idea behind this proposition of balance and watts.
