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