Well see, suppose you tell me there is a vector x let's just say x,

Okay? So vector allocates some good resources

x1, x2 up to xn among the end users competing against each other.

And I say, that this is a particular function, vector is that alpha fair

allocation, vector if here's the definition, okay.

If for any other feasible vector y or such y.

If I look at the difference for each user, Between getting something from x factor

versus getting from the y allocation. And then I normalize it by Exxon through

the power alpha. There's where the alpha comes in.

That's some kind of alpha parameterized normalization of the difference in

drifting away from x allocation to y allocation.

And then I look at the joint impact, the collective impact to all the users in the

system summing over r. If this summation which is a single

scaler, is negative, Okay?

Less than, equal to zero then I say, this vector x is alpha-fair vector.

Now, that's a mouthful for definition. Let's process it again.

We say a vector x is alpha-fair if for all any other feasible allocation y, you can't

give me an unfeasible allocation and ask me to compare, okay.

Any other feasible allocation y, if I look at the collective.

Alpha profit must normalize the drift from x2 such vector Y, is always a bad thing.

Okay, the sum is negative, then, I say this vector indeed is alpha fair

allocation. All right, now you can disagree with the

definition, but it turns out to be a useful definition and therefore a good

definition. So, what about our utility function?

This is a vector not a function. Well it turns out that without proving

this, we'll just state the, the result, it says, if you maximize the following

parameterized utility function parameterized by alpha, so I'm writing

alpha in the subscript for this utility function as a function of x.

If you maximize this function, what does it look like?

What is a branching definition? It looks like it's simply, x to the power

one minus alpha, over one minus alpha. If alpha is not one, okay.

If alpha is one, this denominator is not defined, but you can use Lapatel's rule,

and see that when alpha is one, this actually is log of x,

Okay? Now see the familiar and important special case of log utility.

So, if this is the definition of your utility function,

Then maximizing such utility function will lead to an answer of the optimizer of x

star that satisfies the definition of alpha fair allocation.

That's why we call this the alpha fair utility function.

Another name for it is called Isoelastic Utility Function.

Okay. Why isoelastoc? It gets, gets back to our

notion in the last slide, Okay?

In a homework problem I think you easily verify that if you take this definition of

u of x, you will see the resulting demand function,

Okay? And then that implies the elasticity.

It is actually one over alpha. In particular, when alpha is one for large

utility, the corresponding demand function is just one over p and the corresponding

normalized demand elasticity is just one over alpha.

That is just one. In that case, one alpha is one.

Okay. But in general, for other non-one alphas

is one over alpha s eta. That means the elasticity of demand is all

independent of everything else, okay? Except the shape.

This problem to alpha. That's why its called isoelastic.

Later, we'll see many good uses of isoelastic or alpha fair utility function.

Including the following special case. When alpha is zero, what do we have?

That's simple. We're just looking at the allocation

itself. It's just of sum allocation for example,

some rate in maximization. When alpha is one, that's a log utility

function. It turns out that at least two of what's called a proportional fare.

If you look at this definition here, when alpha is one it means that your

normalization is simply xi itself, you're not trying to skew the shape by any means.

And, whether you like the definition or not, you can see why this is called alpha

proportional fair'cause proportional back to normalized by X.

So, when alpha is one, you have this special case called proportional fairness.

Now, when alpha's really big, for example, alpha approach infinity, then it turns out

that it approaches what's called a max-min fair.

I will later talk more about fairness, so let's leave it just like that for the

moment. And conclude this part of the video module

with simple application of this utility model,

Okay? Whether it comes form focus group study or

fairness or demand elasticity. Apply this simple utility model to talk

about a very important phenomenon called the Tragedy of Commons.

Now, this was first formally mentioned by Lloyd a long while back, 1833.

And then codified and popularized by Harding's article in 1968.

Now this article had other kinds of provocative and somewhat controversial

arguments. Tragedy of Commons is only one small part

of it but we have seen a liberal use of this term, tragedy of Commons, quite a few

times. In lecture one, when we talk about power

control, interference. That's the trade of commerce in the air.

When we were in lecture two okay, transferred commerce in terms of second

prize correctly internalizing the externality imposed on other option

competitors. In lecture, I think six, okay, voting

theory, okay. The K degree of externality imposed on

other voters and the need for something like voter count.

And later, in chapter lecture, I think, fourteen for TCP, fifteen for P2P and

eighteen for WiFi. Okay?

In all of these cases for tech networking discussion we will see a different

manifestation of Tragedy of Commons but the reason it was called Tragedy of

Commons was because it was a, a thought experiment, okay. Just like this binary

number experiment in lecture seven. It says that suppose that you've got a

piece of common grassland as the commons, okay, and there are different farmers and

each farmer says okay, should I get another cow or not?

Okay, here's a cow. Well, it looks like a, a mouse, but let's

say it's a cow, right. So, should I get a cow now?

Well, if I get a cow, the cow's going to need to eat grass, , okay.

But it, you know, the cost of eating the grass is like if there are n farmers, is

like one over n, proportional to that. But the benefits to me of the milk and

maybe selling the cow for its meat later down the road is one, one unit, okay.