This used to measure the amount of uncertainty but also can be used to
measure the amount of variance. Well, speaking of variance, how about the
standard deviation, or may be the second order moment of this vector,
okay? Normalized.
And that's called the Jain's Index in certain networking literature,
okay? A lot of these try to normalize this so
that you going to map whatever vector into a number in the range between zero
and one, where one is most fair, okay?
Now, we have seen another related but subtly different approach where we
provide an objective functions and optimization problem, and then try to
understand what is the resulting optimizer's property.
For example, alpha fair utility function, okay? Utility parameterized by the
parameter alpha between zero and infinity says that I'm going to look at [SOUND]
one minus alpha. This function, if alpha is not one and by
L'Hopital's Rule this function if alpha is one,
okay? If I maximize a summation of these utility functions, I got a utility
function for the entire vector. And the resulting maximizer, x star,
satisfy the so called alpha fair notion. Including proportional fair that we
talked about, one alpha is one. Including max/min fair
that says, you cannot make one user's rate higher or resource higher without
making someone whose already getting a smaller amount of resource get even
smaller resource, okay? And that's called the max/min fair
which happens as alpha becomes very large, close to infinity,
okay? These two are special cases. As you can see already, these two
different approaches, a normalized diversity index and an alpha fair
objective function are actually different,
right? For example, the treatment of efficiency
is different. For alpha fair utility maximization,
efficiency is implicit and bad in it. But still in there and normalize the
index is not effected by the magnitude often,
okay? So, one, one the vector is the same as
100, 100th vector. Can these two approaches be unified?
And can more approaches be discovered? In fact, how many more approaches are
there? Well, let's try something that we saw
back in Lecture 6 when we talked about the axiomatic construction of a voting
theory and of bargaining theory. Back then, we looked at the axiomatics a
treatment of errors impossibility theorem, as well as
the axiomatic treatment of bargaining by Nash, in the advance material part of the
lecture, okay? So, in one case, a set of axioms
that led to impossibility result and another case led to a unique possibility
result. So, what kind of axioms are we talking
about here? We will see that in the advance material
of this lecture, the axiom of continuity. Very briefly it just says that, the
function that maps, a given vector of allocation in to some scalar.
The fairness value, okay?
Should be a continuous function, okay?
Of so called homogeneity that says, if I scale this x by say a factor of five,
five pi vector x, it should give the same as if I was
looking at x. In fact, doesn't matter if it was a five
or any positive constant. So, scale does not matter.
Another way to call that kind of function is called homogeneous function,
okay? So, inefficiency is automatically taken
out of consideration for the moment. And then, of saturation.
This is a tenical axiom where we will skip this on to the advanced material of
partition that says, you can grow the population size and the notion of
fairness still remaining well defined. And finally, of starvation that says,
the allocation of half, half between two users equal allocation should be no less
fair than the allocation of zero, one, which stops one of the two users,
okay? Notice this is not saying that you put allocations should by axiom be viewed
as most fair. It simply says, it is not less fair than
starvation, okay? So, it'ss a very weak statement,
therefore a very strong axiom. It turns out that if you believe these
five axioms, then skipping to the relation we will be able to derive a
unique family of functions F. And that's a vector of allocation to a
scaler representing the fairness value. And this scalar will allow us to do two
things. A, allow us to do quantitative
comparison, okay?
Between two vectors, which is more fair. And the second is scale.
Not only gives you a order comparison, but also provides a numerical scale to
talk about how much fair is one allocation with respect to the other.
So, it is our job now to go through that unique family of fairness function
constructed based just on these axioms of the function.