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RICHARD DE NEUFVILLE: So
the consequences, then,

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of these axioms is that if
you accept these axioms,

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you can construct a valuation
function that is meaningful.

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Now, it is only meaningful
in terms of ranking.

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That is, I can say given
a set of alternatives,

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I can talk about
the first, second,

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and third in terms of which
is my ranking for them.

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But I can't say with
any meaningfulness

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that one is twice as
valuable as the second,

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or the second is one and a half
times a third, and so forth.

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Cardinal distances don't
particularly count.

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In the same way, any
function of this sort,

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there are many that
there are monotonic.

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A monotonic transformation is
an equivalent value function.

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For example, if I take
this function of these two

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variables, and I say I have X
squared, X1 squared times X2,

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and I take logarithms
of it, this second one

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here will have rank combinations
of X1 and X2, these two

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qualities, in the same
order as this other one.

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But the numbers
can't be interpreted

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as being meaningful,
in terms of one

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being twice as good
as the other because

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of the monotonic transformation.

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So it's good for ranking.

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As you can appreciate, that
limits the kind of usage

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you can make of a
valuation function.

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So I think you're ready, Indra,
for another pause and poll.

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So if you are, please launch it.

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Well, it seems like there's
a clear special interest

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in the monotonicity
one, which is OK.

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So let's stop the poll now.

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I got the message.

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And I'm open to
discussion of it.

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So somebody would like to pose
a question about something

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that they would
like clarification

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on, or more the
discussion on, or what

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have you, please go ahead.

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INDRA: Do you have
a specific question,

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or would you just
like the professor

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to go over the principle again?

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Yeah, so Stuart has a question.

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What is a monotonic transform?

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RICHARD DE NEUFVILLE:
A monotonic transform

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is something like taking a log,
or multiplying everything by 3,

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or dividing, or taking a square
root-- something that does not

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change the ordering
that's associated with it.

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And there's a large
class of them.

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Let me put it that way.

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So the point to be taken is that
if you have a valuation that

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is X1 squared times X2, you
have numbers coming out,

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but those numbers don't
particularly mean anything.

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If you subject it to any
one of these transformations

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like a square
root, or logarithm,

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or whatever that will transform
that original function,

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you have something
that is equally

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going to give you
the correct ranking.

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And so there's just lots
of different possibilities.

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So to say that one measure
of X1, X2, X1 squared X2

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is 50 and the other one is 30,
the comparison of two things

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doesn't say that one is
67% better than the other,

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or that 20 difference
makes any sense.

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It's that number
can be manipulated.

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That's the point of that, is
there's no one way of doing it.

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That is the point.

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