When we talked about influence diagram we
included in the influence diagram nodes
that represent the agents' utility
function. And those utility functions we
said indicate an agents' preferences
regarding the state of the world or
different aspects of the state of the
world. What are these utility functions
and where did they come from? So utility
functions are necessary for our ability to
compare, complex scenarios that involve
uncertainty or risk. It's not difficult
for a person to say, that they prefer an
outcome where they get four million to one
where they prefer three million. But it's
not quite as easy to encode a more
complicated, preference that allows us to
compare the utility of these two
lotteries, as they're called. Where the
one on the left gives the agent probabil,
gave the agent $four million with
probability 0.2, and this, the one on the
right gives the agent $three million with
probability 0.25. Which of those lotteries
do we prefer if we had to make that
decision? It turns out that the way to
formalize the decisions making process of
an agent in this type of scenario is by
ascribing a numerical utility to these
different outcomes, to the outcome of four
million. To the outcome 3,000,000 and to
the outcome of $zero. And then we can use
the principle of maximum expected utility
to decide between these two different
lotteries. Specifically we can then
compare 0.2 times the utility of the
outcome 4,000,000 plus. 0.8 to the
authority of the up come zero dollars
versus the converse which is 0.25 versus
the authority, expected authority for the
second lottery to 0.25 time to the
authority of three million dollars plus
0.075 and [inaudible] zero dollars. And if
you compare this to expressions and this
side weather we prefer the one in the
right, the one in the left, the one in the
right or, or there are equally budget in
our view. Now. It might be natural to
assume that utilities should be linear in
the amount of pay off that we get, so that
$five, is preferred about half as much as
$ten. It turns out that that's not the
case for most people and one example of
that is this decision making situation
over here, where on the left the agent has
the option of getting four million dollars
with probability of 0.8, and on the right
they have the option of getting three
million dollars with certainty. Most
people tend to prefer the lottery on the
right, but if one computes they expect
payoff of. These two different lotteries.
We can see that the expected payoff over
here is, is zero, four million times. 0.8
which is 3.2 million. Whereas, on the
right, we have an unexpected payoff of
3,000,000. So, the expected payoff on this
side is higher and nevertheless, people
prefer the lottery on the right. Another
example, very famous example of this type
of, of this type of preference is what
called the Saint Peter's birth paradox.
Saint Peter's birth paradox is, is an
imaginary game that one can play, where a
fair coin is tossed repeatedly until it
comes up head for the first time and if it
comes up head for the first time on the
nth toss you get to [inaudible] nth
dollars. So what's the expected pay off in
this case. Well the probability that is
comes up heads on the first toss is half
and then you get $two. The probability
that is comes up heads for the first as a
quarter and. The payoff here is $four.
Eight, third tosses eight times eight,
times eight dollars and it's easy to see
that the expected payoff over here is
infinite. So in principle people might be
willing to pay any amount to pay, play
this game because they expect the payoff
is bigger than any amount that they would
be paying to play but the fact is that,
for most people. The value of playing this
game. Is approximately $two, which is a
strong indication that their preferences
are not linear in the amount of money that
they earn. So, let's try and quantify that
using this notion, which is called a
utility curve. The utility curve, in this
case, [inaudible] the X axis with the
dollar amount that you get. And on the Y
axis, the utility that an agent describes
to that. And now, let's compare a few
different scenarios here. So first, let's
compare, let's look at the utility of
getting $500. So if we go up from 500 to
the utility car, we consider the utility
of this outcome is going to fall over
here. So, this is going to be the utility
of $500. But now, let's look at, at the
decision, at situation that involves some
risk. So let's look at a set of lotteries
where I get $zero with probability one
minus p, and a $1,000 with probability p.
Because of the linearity of expected
utility, all of these lotteries are going
to sit on this line over here where
depending of the value of p I have a
different weighted combination between
getting the utility of $zero and the
utility of $1000. So for high values of p
close to one we would be sitting on this
side of the curve and otherwise for
example for low values of p we would be
sitting close to here. Specifically what
happens if we look at the probability p
equals 0.5? Well, in that case, we would
have this point on the curve, over here.
Now the important thing to notice is that
the utility of this point where I get
$1,000 probability 50 percent and $zero
probability 50%, that utility in this
example is considerably lower than the
utility of $500. So I prefer. To get the
$500 for certain, which is what most
people will take. Now if we look at. What
the lottery is worth. That is the risky
version. We can see that, that sits over
here and might for example be
corresponding to getting $400 with
certainty. So that $400 is called the
certain equivalent of this lottery over
here. That is, it's the amount that you'd
be willing to trade for this lottery in
terms of getting that money for certain.
The difference. Between. These two
numbers, the expected reward and the
utility of, of that lottery is called the
insurance premium or the risk premium. And
it's called that because that's where
insurance companies make their money.
Because of a person's willingness to take
less money with certainty over a more
risky proposition. So we can see that this
kind of a curve that has this shape, this
concave. Shape, is, is representing a risk
profile which is risk averse. That is, a
person is willing to pay for taking less
risk. Other profiles would, of this, of
this curve, would represent, different
behaviors. So, for example, if the utility
was linear in, in the reward, that would
be a behavior that was called risk
neutral. Conversely if we had a curve that
looked like this, which is a convex
function. That would be risk-seeking.
[sound] And risk seeking behavior occurs
for example in Las Vegas where one is
willing, or in other gambling situations,
where one is willing to actually take a
lost in term of the expected reward for
the small chance of getting a really high
payoff. Now it turns out that people often
have utility curve that looks like the
[inaudible] so if the x axis is the amount
of money that we get and we are
[inaudible] play. It's the zero point over
here, which is at one's current. State.
And we ask, how much do you prefer to earn
money, and how much do you prefer to lose
money? What's your utility for these
different, changes to one's state? We can
see that, one's preferences for earning
money typically exhibit a form of
diminishing returns. [sound]. Which gives
us the, this concave. Utility curve,
which, suggests risk, averse behavior, in
the sense that we would prefer a certain
amount of, we would prefer to get money
with certainty relative to the, expected,
relative to the payoff equivalent,
uncertain lottery. Now on the negative
side of the spectrum many people exhibit
some kind of behavior that is actually
more risk seeking, which means that many
people would prefer a small probability of
a large loss. Relative to a small loss
that you get with certainty, and, that's a
that's a behavior that one often sees.
More importantly in this region of the
space which is close to one's current
state the behavior is often risk neutral
this is small. Small losses or small gains
on the order of a small number of, of
dollars. And of course, it depends on
one's, one's baseline. Are often something
that you don't really don't care about,
having the uncertainty, and the expect-,
expected payoff is often very close to
the, to the utility of the, expected
payoff. Now one final important
observation regarding utility functions is
that one?s utility often depends on many,
many things not just on the monetary gain.
So in all of the attributes that effects
the preferences must be integrated into a
single utility function. This is something
that many people find very painful because
it forces us to do things like put human
life for the loss of human life on the
same scale as, as monetary gain. The point
is even if we don't do this explicitly,
even if we decline to put human life for
example on the same scale as monetary
gain, the fact though our decisions are
indicating that we're making those
decisions. So for example, when an airline
chooses not to run maintenance on the
airplane every single. Time that the air
plane lands that's a financial decision
because that would be too costly but at
the same time it also definitely increases
the chance of loss of human life because
of, because of an accident. Now, it's not
just big companies that make these
decisions. We make these decisions
ourselves. So we don't change the tires on
our cars every month, or every week.
Because that would be too costly. But,
clearly, having better tires is something
that is likely to increase our chances of
surviving an accident or a skid. So these
tradeoffs are ones that we make all the
time, whether we recognize it or not. And
so it's important when we think about a
decision making situation, to list, for
ourselves, all of the different things
that could affect our decision, money,
time pleasure. And many, many other
attributes and think about how we could
bring them together into a single utility
function. Specifically, in the context of
human life, people have spent a lot of
time thinking about how to bring in human
life into one's utility function. And what
turns out to be the wrong strategy, in
terms of reflecting people's preferences,
is to have a utility for the monolithic
event of someone's death. And that turns
out to be a very difficult thing to
contemplate. What, what seems like a
better strategy, in general is this notion
of a micro [inaudible], which is a one in
a million? Chance of death. And so one
puts the risk explicitly into the utility
function. And, and so, what is a one in a
million chance of death worth? Well, back
in 1980, so a while ago, people did, this,
this study. And it turns out that a micro
[inaudible] was worth approximately twenty
of, $twenty in 1980 dollars. And. So of
course you can account for inflation but
it's not a huge amount of money. [sound].
And that turns out to be a much better way
of ranking people's utility for outcomes
that involve risk to human life and asking
about the utility of, of death. The
second, way, that people use a medical
decision, making situations specifically
for accounting for human life, is this
notion of [inaudible], or a quality
adjusted life here. So, each quality
adjusted life here, which is a year
adjusted for one's quality of life, has a
certain, utility associated with it, which
allows it to be compared. With other
aspects that affect our utility in a
decision making situation. One example
from a real world situation, is in this
context of prenatal diagnosis. Where,
researchers did extensive work in
eliciting utility functions that involve
prenatal testing. So the relevant
variables in this scenario include, the,
whether the, baby is going to end up with
some kind of genetic disorder.
Specifically, the one that they focused on
was Down's syndrome. But at the same time,
there's other aspects that affect one's
utility. So For example, V. Pain of.
Testing for down syndrome is one aspect.
The comfort of knowledge that you know
what, what you're going, what's going to
happen is something that also turns out,
out to contribute to one's utility
function. Prenatal testing runs the risk
of the loss of the fetus and that is also
clearly a component of one's utility
function. And at. At the same time, the
potential for future pregnancy. That is
whether there will be a future pregnancy
or not, is another component of one's
utility function. So if we think about the
space here, the utility function depends
in a complicated way of a large number On,
on these five variables and this is fairly
high dimensional space over which to
illicit to illicit utilities. Fortunately
it turns out that many people have a lot
of structure in your utility function. And
specifically they can breakdown the
totality function as a sum of [inaudible],
just as we had in the context of
[inaudible] diagram. And for many people
that the composition looks like the
utility of the testing. The, a separate
component for the utility of the peace of
mind of knowledge. And then we have. These
two pair wise utility terms, the first of
which is a term that depends
simultaneously on Down's syndrome, and on
the loss of the fetus. And the second is,
the utility that depends on the loss of
the fetus, and the potential for future
pregnancy. So people's utility function,
for many people, decomposes in this way.
Which it turns out we could actually.
Think about it as a graphical model that
has singleton terms as well as these pair
wise terms over here. And, that allows us
to considerably reduce the number of terms
that we need to list in order to get a
usable utility function. [sound] So to
summarize our utility function is what we
can use to determine preferences about
decisions that involve risk or
uncertainty. In order to define or elicit
the utility function, we generally need to
consider multiple factors all of which
affect our utility. In most cases, the
relationship between these different
factors, the. Between say money and the
utility or micromores and the utility.
This relationship is usually a nonlinear
one. And the shape of the utility curve
determines one's attitudes towards risk.
Finally the actual utility function is
usually a multi-attribute utility that
integrates all these different factors,
and it often helps to decompose this
utility function into tractable pieces,
often as a sum of these pieces, which
allows us to make this [inaudible] problem
much more manageable.