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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, gave the agent $4 million with probability 0.2 and this the one on the

right gives the agents $3 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 4 million to the outcome 3 million and to the outcome

of $0 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 million plus.

0.8 to the utility of the outcome $0 versus the converse which is 0.25, ver,

versus the utility, expected utility for the second lottery which is 0.25 times

the utility of $3 million + 0.75 * the utility of $0.

And we can compare these two expressions and decide whethere we prefer the one on

the right, the one, the one on the left, the one on the right or, or they're

equally good in our view. Now,

it might be natural to assume that utilities should be linear in the amount

of payoff that we get so that $5 is preferred about half as much as $10.

It turns out that it's not actually 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 4 million dollars with a probability of 0.8 and on the

right they have the option of getting $3 million with certainty.

Most people tend to prefer the lottery on the right, but if one computes the

expected payoff of these two different lotteries we can see that the expected

payoff over here is is 4 million * 0.8 which is 3.2 million.

Where as on the right we have an expected payoff of 3 million 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 this type of preference is what's called the St.

Petersburg Paradox. St Petersburg Paradox is an imaginary

game that one can play where a fair coin is tossed repeatedly until it comes up

heads for the first time. And if it comes up heads for the first

time on the N-th cost you get 2^N dollars.

So what's the expected pay off in this case? Well the probability that it comes

up heads ont eh first toss is half and then you get $2. The probability that it

comes up heads for the first time is a, a quarter and the pay off here is $4.

A, a pro, third tosses A time eight, times $8 and, it's easy to see that, the

expected pay-off over here is infinite. So in principal people might, be willing

to pay any amount to pay this, to play this game, because their expected pay-off

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

fact is, that for most people. The value of playing this game is

approximately $2 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 the utility

curve. The utility curve in this case has, as

the x axis, 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, let's look at the utility of getting $500.

So if we go up from 500 to the utility curve, we can see that the utility of

this outcome is going to fall over here. So this is going to be the utility of

$500. But now lets look at a decision a

situation that involves some risk so lets look at a set of lotteries where I get

zero dollars with probability one minus P and a thousand dollars with probability

P. Because of the linearity of expected

utility all these lotteries are going to sit on this line over here, where

depending on the value of P, I have a different weighted combination between

getting the utility of $0 and utility of $1000.

So, for high values of p1, = 1. we'll be sitting on this side of the

curve and otherwise for example, for low values of P, we will 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 $1000 probability 50% 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 would say.

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 certainty

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 persons 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

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That would be risk seeking. .

And our 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

loss in terms of the expected reward for the small chance of a getting a really

high pay-off. .

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

following. So if the X axis is the amount of money

that we get and we arbitrarily raise the zero point over here which is at once

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 ones preferences for earning money typically exhibit a form of

diminishing returns. Which give us this concave utility curve

which suggests risk adverse 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, that's a behaviour that one often sees.

More importantly, in this region of the space, which is close to one's current

state the behaviors often risk neutral. That 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 base line.

are often something that you don't really care about having the uncertainty.

And the expected pay-off is often very close to the, to the utility of the

expected pay-off. 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 effect

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, umm, put human life or the loss of human life on the same scale 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, de facto 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 airplane 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 trade offs are ones that we make all the time whether we recognize it

or not. And so its important when we think about

a decision making situation to list out 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 human life, into ones

utility function, and what turns out to be the wrong strategy, in terms of

reflecting peoples preferences, is to have the 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 micromort,

which is a one in a million chance of death.

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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 micromort was

worth approximately $20 of, $20 in 1980 dollars.

And, so of course you can account for

inflation but it's not a huge amount of money.

And that turns out to be a much better way of ranking people's utility for

outcomes that involve risk to human life than asking about the utility of death.

The second way that people use a medical decision making situations specifically

for accounting for human life is this notion of equality, or equality adjusted

life year. So each quality adjusted life year, 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 effect our

utility in the 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, relevant variables in this scenario include the, whether the baby is going to

end up with some kind of genetic disorder.

And specifically, the one they focused on was down syndrome.

But at the same time, there's other aspects that effect one's utility so,

for example, the pain of testing for Down's 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 the result of contributes towards 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 complicated way on a large number on these five

variables, and this is fairly high dimensional space over which to elicit to

elicit utilities. Fortunately it turns out that many people

have a lot of structure in their utility function and specifically they can break

down the utility function as a sum of sub utilities just as we had in the context

of the influence diagram and for many people that decomposition 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 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 can actually think about 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. 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 a 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, or

micromorts and the utility. This relationship is usually a non-linear

one and the shape of the utility curve determines one's attitude towards risk.

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

integrates all of 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 elicitation

problem much more manageable.