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RICHARD DE NEUFVILLE:
And so what

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are the conditions
for a value function?

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That is, that the construct
of a value function,

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one that is legitimate in size,
meaningful and unambiguous,

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depends upon three axioms,
three assumptions, if you wish,

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that you might have.

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And I'm going to
go through them,

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but by going through
them, we also

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recognize when it is not
possible to have a value

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function.

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And I want to emphasize
that, and this

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addresses a very important
aspect in terms of optimization

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in particular, but in
terms of valuation.

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There's a large
branch of literature

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which simply does not
recognize this reality,

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and I want to draw
your attention to it.

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So there are three axioms.

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The first one is known as
Completeness or Complete

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Preorder, and it basically
says that people, in fact,

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have preferences
over all choices.

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That in no way means
that you've thought

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about all the possible
choices, but that you are

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capable of making a preference.

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There isn't some thing which
you could not decide about.

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And deciding about it permits to
say that I'm equally attracted

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to having chocolate or
strawberry ice cream,

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that you don't have to
necessarily prefer one

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to the other, but you
can have a valuation.

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And it's OK to say a dish of
strawberry or chocolate ice

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cream has the same value for me.

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It doesn't mean you have
to prefer one to the other,

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but you can make choices.

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That is for a logical one,
that's a necessary assumption.

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Otherwise, it doesn't mean--

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if you can't make
preferences, you

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can't have a function
over what happens.

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The second one is transitivity.

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That is if thing x1 is
preferred to thing x2,

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and thing x2 is
preferred to thing x3,

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then the preference
is transitive,

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that the x1 is better than
x2, which is better than x3.

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Therefore, x1 is
preferred to x3.

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Now, it's assumed
true for individuals,

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and it is true that depending
on how you phrase it,

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there are various ways
you can trick people

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into thinking differently about
it, depending on what else--

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the context you put in,
their known examples of it.

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But this is a
fairly standard one.

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If you're a strawberry
ice cream fan, and you

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prefer it to chocolate,
and you really hate--

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I don't know--
marshmallow ice cream,

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then it's presumed that you
are preferring strawberry

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to marshmallow also.

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So this is a normal thing.

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The trick here that I want
to bring to your mind,

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and we'll explore
it in a moment,

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is that this fairly
obvious choice

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of A being preferred to B being
preferred to C, therefore A

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is preferred to C, that is not
necessarily true for groups.

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So whereas we can have
a meaningful value

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function for individuals,
having it for groups

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is not very plausible.

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That is, and when I talk
about groups, I mean,

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we're talking about
a board of directors.

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We're talking about
a design team.

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We're talking about
a city council.

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We're talking about
almost anything

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that you can imagine
where you have

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individuals or factions
or business groups talking

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about things.

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Right now, I'm dealing
with a company in Morocco,

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and as with others,
getting decisions

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made is a tug of war
between the finance people,

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the marketing people, the
industrial people each

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have a different set of
values as what makes sense

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from their point of view.

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And they do not have a
consistent value function

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for decisions about
what to spend money on

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or how to run the company.

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But that's the transitivity one.

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Yes?

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AUDIENCE: Yeah, so I just had
a comment on transitivity.

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So these are some
of Savage's axioms,

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I'm assuming, for
subjective expected utility.

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But the transitivity one,
there was this experiment that

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was done, and it's called
the Ellsberg paradox, that

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shows that under
ambiguity, transitivity

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doesn't always
hold because people

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will want to choose the
non-ambiguous option usually.

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RICHARD DE NEUFVILLE: Yes.

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Yes, point well taken.

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And there are various
ways that, as you say,

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under various
ambiguous situations,

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and there's also
the Allais Paradox,

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and there's a cluster of
psychological experiments

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where by asking a question
that context is something else,

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you can get people to
focus on different aspects.

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So you can-- there's
a range of exceptions.

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But in general, this
is an acceptable one.

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And my point here is not
so much to say that it is--

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there aren't exceptions.

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It is that in this case,
it's relatively OK,

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if you had to think
about your own decisions.

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But when we get to groups,
it is definitely not OK.

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That's the point I want to
emphasize in this context.

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So well done from an
advanced doctoral student

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who studied this, and
you're absolutely correct.

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But for the-- let's--

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yeah, let me just
leave it at that.

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Well said.

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I appreciate your comment.

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It's excellent.

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Good going.

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So the third one is sometimes
called the Archimedean

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Principle or the
Monotonicity, but it basically

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says that if we are looking
between a range, which

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is the thing up top between
the best of something

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or the most of something
and the worst or something,

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for anything in between,
that the value of the thing

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in between is a weighting
function-- that is,

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a sum proportion such that the
value of the thing in between

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is a certain
proportion of the best

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and a complimentary
proportion of the worst.

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It basically says that the
valuation may be going up

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all the time, or it may be
going down all the time,

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but it's not going up and then
down or up and down and up.

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That is, it is
basically monotonic

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so that you can think about
something being better

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than something and some
else thing being better.

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If you don't have that,
you can't be transitive.

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So the question is then, well,
OK, is this a reasonable thing?

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Because if you want a consistent
way to value something--

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so if you're saying I'm saving
lives or making money, that

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more and more is always
better or less, less, less

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is always better, that there
isn't a point where you say,

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well, it's good to have so much.

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But then if I have more
money, that's really bad.

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Or if I've saved a lot of
lives, but if I save more,

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it's really bad or worse.

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So is it a reasonable
assumption?

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I mean, it's a
necessary assumption

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for a consistent value
function, but it is reasonable.

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So no, it's not always true,
is the reason I bring it up.

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And let's continue on here.

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Let's look at some
example cases.

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If I have sound
from a speaker, I

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want it loud enough
so I can hear it.

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But at some point, if
the sound got too loud,

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maybe it's just because
I'm older than the youngest

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people might like it really
booming, but at some point,

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it's just too loud for me.

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I can't do it.

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Do I like salt in my food or
sugar in my coffee or whatever?

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Well, more is better up to a
point, until more is worse.

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So the point being
this valuation

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can be very complicated.

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Now, we get around this
in many cases by saying,

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say to the salt on
the food, we can say,

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all right, I don't
mind having more salt.

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It's always better to have more
salt on the table, I suppose.

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But I don't have to use it.

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So having it available
is not the same thing

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as having to use it.

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So in that case,
you can assume--

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you can think
through the problem

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as saying, well,
at a certain point,

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I don't benefit anymore
from it, so it levels off,

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but it's not worse
to have it available.

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So there are ways around
this particular thing.

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But what I want
to emphasize here,

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that there is this
Monotonicity assumption,

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and it doesn't always
apply unless you work

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your way around it in some way.

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