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So, utility functions are necessary for our ability to compare complex scenarios

Â that involve uncertainty or risk. It's not difficult for a person to say

Â that they prefer an outcome where they get four million to one where they prefer

Â three million. But it's not quite as easy to encode a

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

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

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

Â right gives the agents $3 million with probability 0.25.

Â Which of those lotteries do we prefer if we had to make that decision.

Â It turns out that the way to formalize the decisions making process of an agent

Â in this type of scenario is by ascribing a numerical utility to these different

Â outcomes to the outcome of 4 million to the outcome 3 million and to the outcome

Â of $0 and then we can use the principle of maximum expected utility to decide

Â between these two different lotteries. Specifically, we can then compare 0.2

Â times the utility of the outcome 4 million plus.

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

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

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

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

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

Â equally good in our view. Now,

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

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

Â It turns out that it's not actually the case for most people and one example of

Â that is this decision making situation over here where on the left the agent has

Â the option of getting 4 million dollars with a probability of 0.8 and on the

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

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

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

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

Â Where as on the right we have an expected payoff of 3 million so the expected

Â payoff on this side is higher and nevertheless people prefer the lottery on

Â the right. Another example, very famous example of

Â this type, of this type of preference is what's called the St.

Â Petersburg Paradox. St Petersburg Paradox is an imaginary

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

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

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

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

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

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

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

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

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

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

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

Â approximately $2 which is a strong indication that their preferences are not

Â linear in the amount of money that they earn.

Â So, let's try and quantify that, using this notion which is called the utility

Â curve. The utility curve in this case has, as

Â the x axis, the dollar amount that you get.

Â And on the y axis, the utility that an agent describes to that.

Â And now let's compare a few different, scenarios here.

Â So first let's, let's look at the utility of getting $500.

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

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

Â $500. But now lets look at a decision a

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

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

Â P. Because of the linearity of expected

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

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

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

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

Â curve and otherwise for example, for low values of P, we will be sitting close to

Â here. Specifically, what happens if we look at

Â the probability P equals 0.5? Well, in that case, we would have.

Â This point on the curve over here. Now the important thing to notice is that

Â the utility of this point where I get $1000 probability 50% and $zero

Â probability 50%. That utility in this example is

Â considerably lower than the utility of $500.

Â So, I prefer to get the $500 for certain which is what most people would say.

Â Now if we look at what the lottery is worth, that is, the risky version, we can

Â see that, that sits over here and might for example be corresponding to getting

Â $400 with certainty. So that $400 is called the certainty

Â equivalent of this lottery over here. That is, it's the amount that you'd be

Â willing to trade for this lottery in terms of getting that money for certain.

Â The difference. Between these two numbers.

Â The expected reward and the utility of of that lottery is called, the insurance

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

Â insurance companies make their money. Because, of a persons willingness to take

Â less money with certainty over a more risky proposition.

Â So we can see that this kind of a curve that has this shape, this concave shape

Â is, is representing a risk profile which is risk averse.

Â That is a person is willing to pay for taking less risk.

Â Other profiles would of this, of this curve would represent different

Â behaviors. So for example if the utility was linear

Â in in the reward, that would be a behavior that was called risk neutral.

Â Conversely if we had a curve that looked like this, which is a convex function

Â 8:18

That would be risk seeking. .

Â And our risk seeking behavior occurs for example in Las Vegas, where one is

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

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

Â high pay-off. .

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

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

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

Â current state. And we ask how much do you prefer to earn

Â money, and how much do you prefer to lose money?

Â What's your utility for these different changes to one's state?

Â We can see that ones preferences for earning money typically exhibit a form of

Â diminishing returns. Which give us this concave utility curve

Â which suggests risk adverse behavior in the sense that we would prefer a certain

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

Â relative to the payoff equivalent uncertain lottery.

Â Now, on the negative side of the spectrum many people exhibit some kind of behavior

Â that is actually more risk seeking. Which means that many people would prefer

Â a small probability of a large loss. Relative to a small loss, that you get

Â with certainty and that's a that's a, that's a behaviour that one often sees.

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

Â state the behaviors often risk neutral. That is small small losses, or small

Â gains on the order of a small number of, of dollars.

Â And of course, it depends on one's one's base line.

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

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

Â expected pay-off. Now one final important observation

Â regarding utility functions is that one's utility often depends on many, many

Â things, not just on the monetary gain. So, in all of the attributes that effect

Â the preferences must be integrated into a single utility function.

Â This is something that many people find very painful because it forces us to do

Â things like, umm, put human life or the loss of human life on the same scale as

Â monetary gain. The point is even if we don't do this

Â explicitly, even if we decline to put human life for example on the same scale

Â as monetary gain, de facto our decisions are indicating that we're making those

Â decisions. So for example when an airline chooses

Â not to run maintenance on the airplane, every single.

Â Time that the airplane lands, that's a financial decision, because that would be

Â too costly. But at the same time, it also definitely

Â increases the chance of loss of human life because of, because of an accident.

Â Now, it's not just, big companies that make these decisions.

Â We make these decisions ourselves so we don't change the tires on our cars every

Â month, or every week, because that would be too costly.

Â But, clearly, having better tires is something that is likely to increase our

Â chances of surviving an accident or a skid.

Â So, these trade offs are ones that we make all the time whether we recognize it

Â or not. And so its important when we think about

Â a decision making situation to list out for ourselves all of the different things

Â that could affect our decision, money, time, pleasure and many, many other

Â attributes and think about how we could bring them together into a single utility

Â function. Specifically in the context of human life

Â people have spent a lot of time thinking about how to bring human life, into ones

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

Â reflecting peoples preferences, is to have the utility for the monolithic event

Â of someone's death, and that turns out to be a very difficult thing to contemplate.

Â What what seems like a better strategy in general is this notion of a micromort,

Â which is a one in a million chance of death.

Â 13:12

And so one puts the risk explicitly into the utility function.

Â And, and so, what is a one in a million chance of death worth?

Â Well, back in 1980. So, a while ago,

Â People did, this, this study. And it turns out that a micromort was

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

Â And, so of course you can account for

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

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

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

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

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

Â life year. So each quality adjusted life year, which

Â is a year adjusted for one's quality of life, has a certain utility associated

Â with it which allows it to be compared with other aspects that effect our

Â utility in the decision making situation. One example from a real world situation

Â is in this context of prenatal diagnosis. Where researchers did extensive work in

Â eliciting utility functions that involve prenatal testing.

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

Â end up with some kind of genetic disorder.

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

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

Â for example, the pain of testing for Down's syndrome is one aspect.

Â The comfort of knowledge that you know what, what you're going, what's going to

Â happen, is something that also the result of contributes towards utility function.

Â Prenatal testing runs the risk of the loss of the fetus.

Â And that is also clearly a component of one's utility function.

Â And at at the same time, the potential for future pregnancy.

Â That is whether there will be a future pregnancy or not is another component of

Â one's utility function. So, if we think about the space here, the

Â utility function depends in complicated way on a large number on these five

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

Â elicit utilities. Fortunately it turns out that many people

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

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

Â of the influence diagram and for many people that decomposition looks like the

Â utility of the testing. The a separate component for the utility

Â of the peace of mind of knowledge, and then we have.

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

Â simultaneously on Down's syndrome and the loss of the fetus.

Â And the second is, the utility that depends on the loss of the fetus and the

Â potential for future pregnancy. So people's utility function for many

Â people, decomposes in this way, which, it turns out, we can actually think about as

Â a graphical model that has singleton terms, as well as these, pair-wise terms

Â over here. And that allows us to considerably reduce

Â the number of terms that we need to list in order to get a usable utility

Â function. So, to summarize our utility function is

Â what we can use to determine preferences about decisions that involve risk or

Â uncertainty. in order to define or elicit a utility

Â function, we generally need to consider multiple factors all of which affect our

Â utility. In most cases, the relationship between

Â these different factors, the. Between say money and the utility or, or

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

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

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

Â integrates all of these different factors.

Â And it often helps to decompose this utility function into tractable pieces,

Â often as a sum of these pieces which allows us to make this elicitation

Â problem much more manageable.

Â