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When we talked about influence diagram we
included in the influence diagram nodes

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that represent the agents' utility
function. And those utility functions we

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said indicate an agents' preferences
regarding the state of the world or

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different aspects of the state of the
world. What are these utility functions

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and where did they come from? So utility
functions are necessary for our ability to

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compare, complex scenarios that involve
uncertainty or risk. It's not difficult

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for a person to say, that they prefer an
outcome where they get four million to one

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where they prefer three million. But it's
not quite as easy to encode a more

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complicated, preference that allows us to
compare the utility of these two

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lotteries, as they're called. Where the
one on the left gives the agent probabil,

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gave the agent $four million with
probability 0.2, and this, the one on the

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right gives the agent $three million with
probability 0.25. Which of those lotteries

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do we prefer if we had to make that
decision? It turns out that the way to

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formalize the decisions making process of
an agent in this type of scenario is by

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ascribing a numerical utility to these
different outcomes, to the outcome of four

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million. To the outcome 3,000,000 and to
the outcome of $zero. And then we can use

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the principle of maximum expected utility
to decide between these two different

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lotteries. Specifically we can then
compare 0.2 times the utility of the

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outcome 4,000,000 plus. 0.8 to the
authority of the up come zero dollars

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versus the converse which is 0.25 versus
the authority, expected authority for the

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second lottery to 0.25 time to the
authority of three million dollars plus

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0.075 and [inaudible] zero dollars. And if
you compare this to expressions and this

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side weather we prefer the one in the
right, the one in the left, the one in the

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right or, or there are equally budget in
our view. Now. It might be natural to

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assume that utilities should be linear in
the amount of pay off that we get, so that

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$five, is preferred about half as much as
$ten. It turns out that that's not the

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case for most people and one example of
that is this decision making situation

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over here, where on the left the agent has
the option of getting four million dollars

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with probability of 0.8, and on the right
they have the option of getting three

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million dollars with certainty. Most
people tend to prefer the lottery on the

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right, but if one computes they expect
payoff of. These two different lotteries.

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We can see that the expected payoff over
here is, is zero, four million times. 0.8

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which is 3.2 million. Whereas, on the
right, we have an unexpected payoff of

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3,000,000. So, the expected payoff on this
side is higher and nevertheless, people

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prefer the lottery on the right. Another
example, very famous example of this type

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of, of this type of preference is what
called the Saint Peter's birth paradox.

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Saint Peter's birth paradox is, is an
imaginary game that one can play, where a

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fair coin is tossed repeatedly until it
comes up head for the first time and if it

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comes up head for the first time on the
nth toss you get to [inaudible] nth

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dollars. So what's the expected pay off in
this case. Well the probability that is

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comes up heads on the first toss is half
and then you get $two. The probability

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that is comes up heads for the first as a
quarter and. The payoff here is $four.

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Eight, third tosses eight times eight,
times eight dollars and it's easy to see

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that the expected payoff over here is
infinite. So in principle people might be

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willing to pay any amount to pay, play
this game because they expect the payoff

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is bigger than any amount that they would
be paying to play but the fact is that,

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for most people. The value of playing this
game. Is approximately $two, which is a

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strong indication that their preferences
are not linear in the amount of money that

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they earn. So, let's try and quantify that
using this notion, which is called a

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utility curve. The utility curve, in this
case, [inaudible] the X axis with the

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dollar amount that you get. And on the Y
axis, the utility that an agent describes

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to that. And now, let's compare a few
different scenarios here. So first, let's

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compare, let's look at the utility of
getting $500. So if we go up from 500 to

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the utility car, we consider the utility
of this outcome is going to fall over

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here. So, this is going to be the utility
of $500. But now, let's look at, at the

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decision, at situation that involves some
risk. So let's look at a set of lotteries

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where I get $zero with probability one
minus p, and a $1,000 with probability p.

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Because of the linearity of expected
utility, all of these lotteries are going

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to sit on this line over here where
depending of the value of p I have a

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different weighted combination between
getting the utility of $zero and the

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utility of $1000. So for high values of p
close to one we would be sitting on this

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side of the curve and otherwise for
example for low values of p we would be

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sitting close to here. Specifically what
happens if we look at the probability p

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equals 0.5? Well, in that case, we would
have this point on the curve, over here.

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Now the important thing to notice is that
the utility of this point where I get

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$1,000 probability 50 percent and $zero
probability 50%, that utility in this

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example is considerably lower than the
utility of $500. So I prefer. To get the

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$500 for certain, which is what most
people will take. Now if we look at. What

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the lottery is worth. That is the risky
version. We can see that, that sits over

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here and might for example be
corresponding to getting $400 with

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certainty. So that $400 is called the
certain equivalent of this lottery over

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here. That is, it's the amount that you'd
be willing to trade for this lottery in

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terms of getting that money for certain.
The difference. Between. These two

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numbers, the expected reward and the
utility of, of that lottery is called the

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insurance premium or the risk premium. And
it's called that because that's where

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insurance companies make their money.
Because of a person's willingness to take

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less money with certainty over a more
risky proposition. So we can see that this

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kind of a curve that has this shape, this
concave. Shape, is, is representing a risk

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profile which is risk averse. That is, a
person is willing to pay for taking less

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risk. Other profiles would, of this, of
this curve, would represent, different

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behaviors. So, for example, if the utility
was linear in, in the reward, that would

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be a behavior that was called risk
neutral. Conversely if we had a curve that

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looked like this, which is a convex
function. That would be risk-seeking.

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[sound] And risk seeking behavior occurs
for example in Las Vegas where one is

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willing, or in other gambling situations,
where one is willing to actually take a

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lost in term of the expected reward for
the small chance of getting a really high

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payoff. Now it turns out that people often
have utility curve that looks like the

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[inaudible] so if the x axis is the amount
of money that we get and we are

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[inaudible] play. It's the zero point over
here, which is at one's current. State.

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And we ask, how much do you prefer to earn
money, and how much do you prefer to lose

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money? What's your utility for these
different, changes to one's state? We can

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see that, one's preferences for earning
money typically exhibit a form of

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diminishing returns. [sound]. Which gives
us the, this concave. Utility curve,

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which, suggests risk, averse behavior, in
the sense that we would prefer a certain

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amount of, we would prefer to get money
with certainty relative to the, expected,

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relative to the payoff equivalent,
uncertain lottery. Now on the negative

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side of the spectrum many people exhibit
some kind of behavior that is actually

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more risk seeking, which means that many
people would prefer a small probability of

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a large loss. Relative to a small loss
that you get with certainty, and, that's a

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that's a behavior that one often sees.
More importantly in this region of the

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space which is close to one's current
state the behavior is often risk neutral

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this is small. Small losses or small gains
on the order of a small number of, of

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dollars. And of course, it depends on
one's, one's baseline. Are often something

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that you don't really don't care about,
having the uncertainty, and the expect-,

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expected payoff is often very close to
the, to the utility of the, expected

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payoff. Now one final important
observation regarding utility functions is

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that one?s utility often depends on many,
many things not just on the monetary gain.

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So in all of the attributes that effects
the preferences must be integrated into a

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single utility function. This is something
that many people find very painful because

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it forces us to do things like put human
life for the loss of human life on the

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same scale as, as monetary gain. The point
is even if we don't do this explicitly,

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even if we decline to put human life for
example on the same scale as monetary

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gain, the fact though our decisions are
indicating that we're making those

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decisions. So for example, when an airline
chooses not to run maintenance on the

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airplane every single. Time that the air
plane lands that's a financial decision

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because that would be too costly but at
the same time it also definitely increases

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the chance of loss of human life because
of, because of an accident. Now, it's not

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just big companies that make these
decisions. We make these decisions

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ourselves. So we don't change the tires on
our cars every month, or every week.

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Because that would be too costly. But,
clearly, having better tires is something

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that is likely to increase our chances of
surviving an accident or a skid. So these

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tradeoffs are ones that we make all the
time, whether we recognize it or not. And

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so it's important when we think about a
decision making situation, to list, for

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ourselves, all of the different things
that could affect our decision, money,

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time pleasure. And many, many other
attributes and think about how we could

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bring them together into a single utility
function. Specifically, in the context of

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human life, people have spent a lot of
time thinking about how to bring in human

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life into one's utility function. And what
turns out to be the wrong strategy, in

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terms of reflecting people's preferences,
is to have a utility for the monolithic

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event of someone's death. And that turns
out to be a very difficult thing to

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contemplate. What, what seems like a
better strategy, in general is this notion

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of a micro [inaudible], which is a one in
a million? Chance of death. And so one

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puts the risk explicitly into the utility
function. And, and so, what is a one in a

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million chance of death worth? Well, back
in 1980, so a while ago, people did, this,

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this study. And it turns out that a micro
[inaudible] was worth approximately twenty

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of, $twenty in 1980 dollars. And. So of
course you can account for inflation but

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it's not a huge amount of money. [sound].
And that turns out to be a much better way

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of ranking people's utility for outcomes
that involve risk to human life and asking

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about the utility of, of death. The
second, way, that people use a medical

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decision, making situations specifically
for accounting for human life, is this

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notion of [inaudible], or a quality
adjusted life here. So, each quality

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adjusted life here, which is a year
adjusted for one's quality of life, has a

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certain, utility associated with it, which
allows it to be compared. With other

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aspects that affect our utility in a
decision making situation. One example

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from a real world situation, is in this
context of prenatal diagnosis. Where,

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researchers did extensive work in
eliciting utility functions that involve

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prenatal testing. So the relevant
variables in this scenario include, the,

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whether the, baby is going to end up with
some kind of genetic disorder.

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Specifically, the one that they focused on
was Down's syndrome. But at the same time,

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there's other aspects that affect one's
utility. So For example, V. Pain of.

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Testing for down syndrome is one aspect.
The comfort of knowledge that you know

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what, what you're going, what's going to
happen is something that also turns out,

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out to contribute to one's utility
function. Prenatal testing runs the risk

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of the loss of the fetus and that is also
clearly a component of one's utility

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function. And at. At the same time, the
potential for future pregnancy. That is

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whether there will be a future pregnancy
or not, is another component of one's

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utility function. So if we think about the
space here, the utility function depends

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in a complicated way of a large number On,
on these five variables and this is fairly

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high dimensional space over which to
illicit to illicit utilities. Fortunately

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it turns out that many people have a lot
of structure in your utility function. And

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specifically they can breakdown the
totality function as a sum of [inaudible],

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just as we had in the context of
[inaudible] diagram. And for many people

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that the composition looks like the
utility of the testing. The, a separate

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component for the utility of the peace of
mind of knowledge. And then we have. These

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two pair wise utility terms, the first of
which is a term that depends

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simultaneously on Down's syndrome, and on
the loss of the fetus. And the second is,

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the utility that depends on the loss of
the fetus, and the potential for future

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pregnancy. So people's utility function,
for many people, decomposes in this way.

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Which it turns out we could actually.
Think about it as a graphical model that

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has singleton terms as well as these pair
wise terms over here. And, that allows us

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to considerably reduce the number of terms
that we need to list in order to get a

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usable utility function. [sound] So to
summarize our utility function is what we

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can use to determine preferences about
decisions that involve risk or

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uncertainty. In order to define or elicit
the utility function, we generally need to

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consider multiple factors all of which
affect our utility. In most cases, the

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relationship between these different
factors, the. Between say money and the

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utility or micromores and the utility.
This relationship is usually a nonlinear

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one. And the shape of the utility curve
determines one's attitudes towards risk.

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Finally the actual utility function is
usually a multi-attribute utility that

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integrates all these different factors,
and it often helps to decompose this

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utility function into tractable pieces,
often as a sum of these pieces, which

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allows us to make this [inaudible] problem
much more manageable.