The Standard Model: Expected Utility

Problem Set Lesley is posted. It is also known, less affectionately, as Problem Set 1.

You may be asking yourself what is going on with the problem set names. For extra credit, you tell me. That is all I will give you: it is a puzzle, it is a riddle, and it is fun. You will not work it out today — you will probably need a few more problem sets in hand before the pattern shows itself. Your only hint is that they are things that come in six, because there are six problem sets. If you do not care for activities of this kind, ignore me entirely. Nothing depends on it, and I concede that some of the fun here is for me rather than for you.

Problem Set 0 stays optional; it is background material on optimization. Lesley is the first real one, and it is due Tuesday, September 15. By the end of today you will be equipped to do a good deal of it. Start early.

We are trying to stay low-tech in here. If you prefer taking notes on a computer, go ahead — you do you. But it is my experience that things are captured differently in different modalities, and that writing by hand does something typing does not.

Today especially: take notes, ask questions, interrupt me. Nearly everything today is a definition you will have to hold precisely, and the moment to sort out an imprecision is the moment it appears, not three weeks later.

I was asked at the start of class whether there is other material you can use to help you — something to read alongside the course. That is a hard question to answer, for the following reason. Is there anything out there that will directly explain the material in this class? No. You could go and read graduate textbooks. It is torture, and it is much, much, much harder than simply doing the class.

So here is what I actually recommend, and I will repeat it a gazillion times over the term: make up problems yourself. That sounds crazy. It is very easy, I promise, and it comes in levels.

The first level is to take a problem that already exists and change the numbers. The second is to change the functions — here we have logs, so make them square roots, and see what survives the swap. Those are iterations on an idea, and they are worth more than re-reading. The third level, the real one, is to ask yourself why the question exists before you ask me. I wrote everything for a purpose, as I mentioned last time. Reconstructing that purpose is a way of engaging with the material that pays off well beyond the problem in front of you.

The last thing you can lean on is this page, and the ones like it. The whole of the first lecture has now been transcribed and reworked and posted, images and all. You can ask questions and leave comments in the margins. For the welcome lecture there is admittedly not much to say. But once we are into actual material, read what was said — or my variation on it. It will not be literally what came out of my mouth; I swear too much and I am too sassy, so it gets trimmed. It keeps the zhuzh.

Today we begin the first major topic, choice under uncertainty, which runs for roughly the first third of the course. There is an outline at the front of the deck. It is there for reference rather than for me to talk through — if you are ever wondering whether something is in a given lecture, glance at the first couple of slides and you will be able to work it out.

I will also, throughout the term, give you what I think of as meta-lectures: ideas that matter in this course and also outside it. The whole course has that feature. The intention is that we come out of it better thinkers, not merely better students. Your life does not begin and end in this class, whatever your instructors may have implied over the years. So the question worth keeping in view is: what are the portable tools here — the things we can carry out of this room and use profitably somewhere else?

I will tell you right now that behavioral economics is unlikely to come up in your job, although it is likelier than most subfields. Thinking is going to come up.

A key part of thinking clearly and precisely is defining your ideas. It is part of good economic modeling, and it is part of a good life. A great many of the debates we get into as a society come from two people not talking about the same object, and it gets genuinely gnarly when nobody has bothered to say what they mean.

Health outcomes are a good example, because they are fuzzy. What constitutes sickness, health, good health, bad health? We can all agree that a person with cancer is sick. Is a person with a bad diet unhealthy, if a bad diet is all they have? It is not so clear. And if we want to think about policy — which, as economists, we do — we had better have an idea of what our outcomes are and how we are defining them. Sickness and health is the easy case, where we can at least reach for natural heuristic outcomes correlated with what we mean. With psychological objects it gets slippery.

Where people get most hung up in this first stretch of the course is not listening to my definitions. You have to take my definitions, which will appear in bold on the screen, exactly as they are written.

Here is why I am going to insist on that. Take risk. We all have a feel for it — raise your hand if you think you are risk averse. Now: what do you think that means? I am not going to make you answer, since that is an uncomfortable question to be handed immediately after putting your hand up. But you had some definition in your head. You either raised your hand or you considered it and did not.

Whatever that definition was, it has a tight relationship to the one I am going to give you in a few minutes, and it is not the same, because yours is a little loose. What most of you were probably thinking is something along the lines of I do not like risk. Spoiler: nobody does. Nobody at all likes risk. It is almost definitional, almost circular. And it is not what the definition says.

So your task, in this course and when communicating with anybody about anything, is to say what you mean. It is perfectly fine to announce that you are going to call some particular thing risk. Lawyers do this compulsively — the beginning of every legal document is a long list of definitions, precisely because they do not want to be tripped up later. They will write: for the purposes of this document, a house is any rectangular object with a door. That is a silly definition of a house. But inside that document, every rectangular object with a door is a house. That is how definitions work.

Our first definition is the gentlest one in the course. This is a warm-up, and I will flag it for you when things get harder.

A lottery — or a gamble, or a risky prospect — is a set of possible outcomes and a probability of each outcome occurring.

When I define something with three names like that, it means I intend to use those words interchangeably. Gamble, risky prospect, lottery: synonyms, from here on.

These can be boring casino-shaped things. If you walk up to a roulette table and put $10 on black, you have just purchased a lottery. A roulette wheel has 38 slots, 18 of them black. With probability 18/38 you receive $20 — your ten back, plus ten more — and with probability 20/38 you receive nothing, because your ten is gone.

They need not be casino-shaped, though. When you buy a car you are buying a lottery. Illustratively: with probability 1/2 it is a car you love, call that high value; with probability 2/5 it is a car that adequately serves your needs, call that low value; and with probability 1/10 it is a lemon, worthless. Note a key feature of probabilities, and a classic easy way to make a mistake: they have to sum to one. Note also that the outcomes here are not monetary. Nothing in the definition says they must be.

And we can go abstract, because notation is convenient: you get outcome $x_1$ with probability $p_1$, outcome $x_2$ with probability $p_2$, outcome $x_3$ with probability $p_3$, where again the three probabilities sum to one. This is the theme from the first lecture — I teach using mathematics — and it is not that hard. I simply got lazy and did not want to write outcome one, so I wrote $x_1$. Sometimes I will change the letter. Why? To keep your brain nimble. Not to confuse you.

Writing lotteries out in paragraphs is annoying, so we have shorthands.

You can draw a probability tree. Suppose I have to flip a coin heads twice in a row to pay you $20, and otherwise you get nothing; the tree branches, and you can read the branches off. Perfectly coherent, and you can see immediately why it is an inconvenient way to write lotteries down for any length of time.

You can write a vector of outcome–probability pairs. Here is our roulette bet: Payoff = ( $20, 18/38 ; $0, 20/38 ). And the car: Car Value = ( high, 1/2 ; low, 2/5 ; worthless, 1/10 ).

I am using language very intentionally here. You see the words vector and pairs and something in your brain goes flat. Do not let it. They are just words. All I have done is write down a sentence.

And now notice something critical, because it will come back today more than once. That object is not a mathematical object. Why not? It has a semicolon in it. You cannot do mathematics to semicolons — not that I am aware of. It is a shorthand sentence, nothing more. I use the semicolons deliberately, so that I myself do not become confused, start dividing things, and go off multiplying pieces of a sentence together.

Finally, you can write a lottery as a piecewise equation — the payoff equals $20 with probability 18/38, and $0 with probability 20/38. These will come up from time to time, and they will not bite you. It is the easiest sentence in the world to read.

Our object of study is lotteries. Our natural question is how people choose among them. The entire first section of this course is that question.

There are immediate theories available. The one on the left, for example. That is a theory of choice: always take the left-hand option. It is complete, it is applicable in every circumstance, and it has a coherent account of a short-lived human’s life. It would probably get you killed, certainly at intersections.

We want something more descriptive, so let us develop some models of behavior — the classic ones first, and then a look at where they fall down.

Before that, one habit I want you to build, because it will save you repeatedly. When in doubt, map the question back into your head as: what is the person actually doing? I am never going to ask you esoteric mathematical questions. The example I was scribbling on the board while you were walking in, the one involving the derivative of a log function, is not a question I would ever ask. The kind of question I actually ask is Question 2 on the problem set: a person is choosing between a risky investment and a safe investment, and has to put some mix into each. That is what the person is doing. So your answer had better be some mix of those two things.

Our first theory is expected value theory, an old, old idea with a hilariously circular-sounding statement: people choose the lottery with the largest expected value.

Which means nothing until we say what an expected value is. So here is the first real definition of the course — a mathematical one.

The expected value of a lottery $x = (x_1, p_1 ; \ldots ; x_n, p_n)$ is

An expected value is an intrinsic property of a lottery. If that word is unfamiliar in this use: intrinsic means inextricable. You cannot remove it from the lottery. If a lottery is numeric — if its outcomes are numbers — then it has an expected value, much as objects have color, or quantum particles have spin. It is simply a fact about the object, and it does not reduce any further.

The lottery itself I will often write as a boldface $x$. On the board, where I cannot make a boldface anything, I will write a little script $\ell$. This one has $n$ lumps — $n$ particular outcomes, indexed 1 through $n$ — where outcome $x_1$ happens with probability $p_1$, outcome $x_2$ with probability $p_2$, and so on through the dot-dot-dot, where we are yada-yada-ing in the Seinfeld sense. The expected value is the product of each outcome and its probability, all summed up.

Many people find that this is where their brain starts asking why it is in this class. Take a breath. Let us write our abstract lottery with only three parts:

and then the big summation sign just means we take each of these products and add them together:

Why write it with the sigma at all, if this is all it means? Two reasons. First, you already know that notation. Somebody taught it to you; it is in your head somewhere, and I am dredging it up. Second, this gets long. If there are 25 outcomes I am writing all afternoon, and I am already crowded on the board.

Notice, immediately, that non-numeric lotteries do not have well-defined expected values. If I flip a coin and give you a shoe on heads and no shoe on tails, the expected value is half a shoe. Kind of. I suppose. That is a bizarre object and I am not going to put you in that circumstance. We can and will define choice over non-numeric lotteries, quite a lot — there are ways of turning them into usable objects — but not with this tool.

So: our fifty-fifty gamble over winning $10 or winning nothing has expected value $10 \times \tfrac{1}{2} + 0 \times \tfrac{1}{2} = 5$. Calculating expected values should be something you can do in your head, when the numbers are kind.

And the theory applies immediately. It sits at the top of the screen: people choose the option with the largest expected value. So if I offer this person $3 for sure, against that gamble, they take the gamble. Note that $3 for sure is itself a lottery — we can write it as ( $3, 1 ), and its expected value is the product of 3 and 1, which is 3. Five is bigger than three. Done.

Now for the first behavioral insight of the course, from 1713, and by a distance the oldest thing we will look at.

Nicolas Bernoulli — of the Bernoullis, a prolific mathematical family; Bernoulli variables, which take the value zero or one, are named for them — proposed the following bet. I want you to actually think about this one.

I am going to flip a coin, and keep flipping until it comes up heads. Then I pay you, as a function of how many times we flipped. If I flip heads immediately, you get $2. One tails and then a heads, $4. Two tails and then heads, $8. Three tails, $16. And so on.

I am good for it, incidentally. I will genuinely pay out up to a million dollars, and I do not lie to you in this class. You would have to be quite lucky. I once gave somebody $1,024 on this bet — nine tails in a row and then a head — which is the $1,024 I mentioned last class and promised would make sense today.

You can see immediately that this is a valuable lottery. So I am not going to give it away. I am going to sell it, to the highest bidder, right now — you have to pay me for the right to flip the coin.

We needed a coin, and one arrived from the third row: a Wisconsin state quarter, cow and cheese wheel on the obverse, Washington on the front. Kind of on the nose there, Wisconsin.

I started the bidding at $4, because $2 is obvious — you are guaranteed at least $2, and everybody agreed on that mathematics. Hands went up at four, five, six, seven, eight, nine, ten, eleven, twelve. Twelve going once. Somebody said twenty-four, conditional on being allowed to flip the coin themselves, which I declined; I know how to manipulate a coin, and in any case it has to hit the table. So it sold for twelve.

I flipped. Heads, first try. Two dollars. Our buyer is in the hole ten.

That looks a little inexplicable — I did just take ten dollars off a student in front of everyone — so ask yourself the question that matters: how much were you willing to pay for that bet? Because every single bid in this room was under twelve dollars, and that means none of you is following expected value theory.

Here is why. The probability of heads on the first flip is one half, and it pays two dollars; one half times two is one dollar. The probability of a tails and then a heads is one quarter, and it pays four; one quarter times four is one dollar. The probability of two tails and then heads is one eighth, and it pays eight. Every rung of this ladder halves the probability and doubles the payment, so every rung contributes exactly one dollar.

An infinite sum of ones is infinite. Truncated at my ability to pay — a million dollars — it comes to about twenty-one dollars,¹ which is still more than anybody here was willing to bid. So an expected value maximizer should have paid a fair bit. You are not expected value maximizers.

And think about the genuinely unbounded version, where I simply keep flipping forever. Then you are adding another dollar for the branch in which I flip 399,000 tails in a row and then a head. That can happen. It is statistically not going to happen, but it can, and infinite things are weird like that.

So we have discovered that expected value theory is not a very good description of this room.

The other Bernoulli, Daniel, wrote in 1738:

That is a very geeky way of saying something you learned in ECON 201: marginal utility is decreasing.

Think about it through the coin flip. Winning $2, $5, $10 is great. The difference between $500,000 and $1,000,000, though it is a lot of money, is shrinking in terms of each marginal dollar. The value of one more dollar when you are about to win five hundred thousand is less than the value of one more dollar when you are about to win one.

Bernoulli’s direct implication is that our choice object should not be that mathematical property all lotteries have. Every numeric lottery has an expected value, and that is a fact about it — but it is not the thing people ought to be choosing over. Because people exhibit diminishing marginal utility of money, a more reasonable description is that just because a gamble is worth infinite money, which this one mathematically is, does not mean it is infinitely valuable to a would-be gambler.

And there is a corollary, which I will formalize shortly: maximizing utility, when a person has diminishing marginal utility of money, means that person will turn down a great many risks.

I asked earlier who was risk averse, and promised to define it. Here is the slipperiest definition in this stretch of the course, and I am going to repeat it several times to be sure it lands.

A person is risk averse if, for any risky lottery $x$, she prefers to have $EV(x)$ for sure instead of lottery $x$.

I will say it again. A person is risk averse if, for any lottery, she would rather have that lottery’s expected value than play the lottery out.

That is the only definition. Nothing more, nothing less.

Somebody asked whether there is a measure of how risk averse a person is — whether we can say something about a person who would also accept a discount. That is a natural question and we will get to it in a few slides. For now we have only this first binary classifier, and two natural cousins.

A person is risk neutral if, for any risky lottery $x$, she is indifferent between $EV(x)$ for sure and lottery $x$. Risk neutral is the tie: exactly on the line.

A person is risk loving if, for any risky lottery $x$, she prefers lottery $x$ to $EV(x)$ for sure.

Keep our handy coin flip up on the board: $10 with probability one half, $0 with probability one half.

I offer Alice $4. Alice is risk averse. Does she take the $4 or the lottery — or do we need more information?

More information. Here is how we get there. First we remind ourselves that the expected value of the lottery is 5. Then we remind ourselves how the natural numbers work: five is a bigger number than four. I told you Alice is risk averse, so if I offered her $5 she would take it. That is the definition, literally on the screen in front of you. This is what it means to use a definition precisely.

But I am not offering her $5. I am offering $4. Maybe Alice is only a little risk averse. Maybe she would take $4.99, and $4 is too steep a discount — she is not that risk averse. We need to know how risk averse Alice is.

Question two, and it is always Bob; it is always Alice and Bob. Bob is risk loving, and I offer him $6.

Hands went up for the lottery, then for the six dollars, then for needing more information. Winner, winner: more information. Why? Because six dollars is bigger than five, by the way the natural numbers work. We just did this.

Bob is risk loving, so he would rather have the lottery than $5 — that is exactly what the definition says. I am offering him $6. Six is a bigger number than five. So we need to know how risk loving Bob is.

Here is a clarifying habit, and a genuinely good way to explore a definition on your own time: ask about the limit cases. For this lottery, a few natural ones are $4.99, $5.01, one penny, and $9.99.

You might think these are silly. They are illustrative.

Let me do the penny. Alice is risk averse. Part of your intuition — that risk aversion means, in some sense, not liking risk — is not contradicted by the definition. It is merely incomplete and imprecise. What we know is that Alice would rather have $5 than this lottery. Now I offer her a penny. And, well — maybe she hates risk astronomically. She could still take the penny. It is bigger than zero.

That feels counterintuitive, and you need to stuff your feelings down and use the definition. Everybody in this room understands these words. There is genuinely nothing confusing about them; the confusion lives in the application, not the vocabulary. Which is exactly why we have to be judicious.

Work $5.01 and $9.99 yourself. They are quick, and quick is the point.

Evidence suggests that people writ large are risk averse. Daniel Bernoulli’s observation about diminishing marginal utility has risk aversion as a natural corollary, as we will see in a moment. And your own intuition, for most of you, says people are risk averse.

Except not all the time. Except when it comes to flipping coins for small amounts of money when you are bored. There is entertainment value in that, and you are allowed to be entertained by things.

But very few people would take the following bet: take your current net worth, double it and add a dollar, against losing everything. That is a positive expected value bet, and Sam Bankman-Fried described and endorsed exactly this bet.

Statistically, an expected value maximizer takes it. As the head of a large company, or as a human being, you would be mad to. Why? Because half the time you have nothing. Half the time you have squandered all of your clients’ money and you end up in prison. Cautionary tales.

A defense of the position was offered from the room, and it deserves three answers. One: it is genuinely not rational, and we will describe the many ways in which it is not. Two: he does not know what he means by that word — and we will all know what it means by the end of this course. Three: you may hold this as a religion if you like. It is a religion that occasionally leaves you broke.

Now we are armed with Bernoulli’s observation, our intuition, and several years of economics classes. That lets us build a more satisfying theory of behavior: people choose the option with the largest expected utility.

The move is that we cannot merely take the monetary outcomes in a lottery — the tens and the zeros — at face value. We have to transform them by something that describes how much $10 matters to me, or to you.

The expected utility of a lottery $x = (x_1, p_1 ; \ldots ; x_n, p_n)$ is

Put a small star next to that in your notes, because it is not exactly right. It will be corrected within the hour. There are treats for those who come to class; this is not much of a treat.

The recipe is simple enough: take each outcome, transform it by the utility function — whatever that function may be, let that go for the moment — multiply by the probability of that outcome, and sum.

So where the expected value of our coin flip is 5, its expected utility is

and I apologize in advance that $u$ and $v$ are hard to tell apart in my handwriting. I will do my best.

Now note something. I have told you nothing whatsoever about this function. It is a mathematical function. You cannot pull the numbers out from inside it — they are inside the function, and I have not told you what the function is. Avoid the temptation.

So let us ask some questions. Could an expected utility maximizer choose $100 with probability one half and $0 with probability one half, over $100 with certainty?

Hands for yes, hands for no, hands for confused. Interesting.

What is happening for those who are confused is that something that feels like common sense is colliding with the fact that I did not tell you the function. What if this person hates money? I have not even said that $u$ is increasing. It is a $u$ in parentheses.

So much of an undergraduate education is brain off in this particular sense. Turn it on here. Sure — an ascetic monk who hates things, which is what all of our characters are until further notice, says no, no, I do not want a hundred; zero is much better, zero is clean, zero is pure, absence is everything. Could that person choose the fifty-fifty over $150 for sure? Sure, same reason. Over $200? Sure, same reason.

If we put no restriction on the utility function, expected utility theory can explain any individual choice.

It is not, however, vacuous — and of course we are going to put restrictions on, and they will match your intuitions. But as a piece of training, notice what you brought into the room with you. You did not say the utility function out loud, but part of you was thinking function go up. We want to acknowledge that assumption. In most circumstances it is perfectly benign, and we will carry it through most of this class.

Somebody asked whether the utility function could simply be the expected value. No — and the reason is worth being precise about, because it is a type error. The utility function $u$ takes a real number and returns another real number. The expected value operator takes the outcomes and their probabilities and returns a single number. Those are different sorts of objects.

You have all taken an economics course before, presumably — anybody who has not? Sometimes I get physics students in here. I think they hear I am a nerd. They tend to be good students. So you have seen this function, and probably this one, and variations on those themes: lots of functions that qualitatively look like that. The sad little teapot.

Let me state the situation plainly. Expected utility theory can explain a choice if there exists some function such that a maximizer holding that utility function would make that choice. But even with no restrictions whatever on the utility function, combinations of choices can violate the theory.

Here is how that works. I need the board for this, so the screen goes off.

First, notation. That squiggly greater-than sign, ≻, is not a greater-than sign. It means chooses, or prefers. So this whole object is a sentence — it cannot be mathematics, because there is a semicolon running around inside it.

Suppose a person makes these two choices:

( $100, 1/2 ; $0, 1/2 ) ≻ ( $200, 1/2 ; $0, 1/2 )

( $200, 1/3 ; $500, 2/3 ) ≻ ( $100, 1/3 ; $500, 2/3 )

Call the first pair $A$ and $B$, and the second pair $B'$ and $A'$.

Step one: this is not math. We turn it into math using the definition, and we do it mechanically, without thinking about what it means. The first choice says $EU(A) > EU(B)$, and now that is a real greater-than sign. Now we are cooking. Apply the definition of expected utility to each side:

Step two: this is mathematics now, and I have the same thing on both sides of an inequality, so I can strike it out. Away goes the $\tfrac{1}{2}u(0)$. And the halves, while we are at it, because why not. The first choice reveals

Fine. We have learned that this function is decreasing, at least between those two points. Now take the second choice, $EU(A') < EU(B')$. Here everybody is tempted to go fast. Do not go fast. Write it down; it takes two seconds.

The $\tfrac{2}{3}u(500)$ sits on both sides, so it goes. The thirds go. And notice what is left: literally the same two expressions as before, with the inequality flipped.

Both of those cannot be true. So even without any restriction at all on the utility function, patterns of choices can lead to violations of the theory.

Notice the technique, because it is the technique you will use on the problem set and on exams: brain off, then brain on. Brain off — take the definition and apply it to the choice scenario, mechanically, without interpretation. Brain on — remember your high-school algebra and cancel what is common. That is generally how this works, and you will get plenty of practice.

Of course we are going to put restrictions on the utility function. Thus endeth the broad lesson about the person who hates money; we will not worry about that person when applying expected utility theory.

The natural restriction is that more is better — that $u$ is increasing in its argument $x$. The line goes up.

If we are willing to assume that, we get a nice result. But we need an intermediate one first, and it is, by intention, the most tedious definition in the entire course. On day two. It is meant to make us think.

Lottery $x$ dominates lottery $y$ — equivalently, lottery $y$ is dominated by lottery $x$ — if for every amount $z$, the probability of getting at least $z$ in lottery $x$ is at least as large as in lottery $y$, and strictly larger for at least one $z$.

I will say it again, because it is a mouthful. Lottery $x$ dominates lottery $y$ if for every amount $z$, the probability of getting at least $z$ in $x$ is at least as large as in $y$, and strictly larger for at least one $z$.

Now: are there any words in that sentence you do not understand? No. That is rather the point. And to answer the question that came from the room — $z$ is not utility and it is not money in any special sense. $z$ is just a number. Every amount means every number.

So let us practice.

First pair. $A$ = ( $200, 1/2 ; $0, 1/2 ) and $B$ = ( $100, 1/2 ; $0, 1/2 ). Does $A$ dominate $B$?

There is an intuitive answer here and you have to shove it down deep. The definition is right there. What we have to do is think about every amount $z$.

Oh no — there are infinitely many numbers, and I will be here all day. We can use tricks, except we do not know what the tricks are yet, having met this definition five minutes ago. So let us simply start picking.

Pick $z = -15$. That is a number. What is the probability of getting at least $-15$ in $A$? One: your outcomes are $0 and $200 and both clear it. In $B$? Also one.

Minus fifteen was a poor place to start. Good job, Bushong. But now we can use our heads: any number between $-15$ and $0$ gives the same answer, so there is no point walking through them one at a time. Pick $z = 1$ instead.

Probability of getting at least 1 in $A$? One half — it is no longer one, because zero is not bigger than one. In $B$? Also one half, by the same logic.

Still tied. That is fine. It is still possible that $A$ dominates $B$; we simply have to keep going, because we need to find one $z$ where the probability is strictly larger.

So try $z = 101$. Many of you have already intuited this number. Probability of getting at least 101 in $A$? One half. In $B$? Zero.

There it is. $A$ dominates $B$. And notice there is no way to shortcut this entirely, though we can shortcut a little with our brains, as I just did out loud. The whole exercise took about three minutes.

Second pair. $A$ = ( $200, 1/2 ; $0, 1/2 ) and $B$ = ( $150, 1/2 ; $50, 1/2 ). Does $A$ dominate $B$?

Choose $z = 49$. Probability of getting at least 49 in $A$? One half. In $B$? One. So $A$ cannot dominate $B$: the definition demands at least as large for every $z$, and it has just failed at $z = 49$.

Does $B$ dominate $A$? We can more or less see that it does not, but here is how to show it formally. Pick $z = 151$. Probability of getting at least 151 in $A$? One half. In $B$? Zero.

So neither of these dominates the other. That is a permitted outcome. Dominance is a partial ranking, not a complete one.

A third pair, which I leave to you: ( $200, 1/3 ; $150, 1/3 ; $75, 1/3 ) against ( $150, 1/2 ; $75, 1/2 ). Same procedure, and it is perfectly coherent to mix up the probabilities like this. Then, as an exercise of exactly the kind that appears on the problem set, imagine the lotteries were something like $10 with probability $p_1$ — leaving one of the probabilities symbolic — and ask: for what probabilities does lottery $A$ dominate lottery $B$? A perfectly coherent question. You just have to move things around.

And why did we introduce this at all? For this result:

Under expected utility, with the assumption that more is better, a person will never choose a dominated lottery.

That is fun. Your intuition — the one many of you had twenty minutes ago, that nobody is going to choose that over that — was right. What we have dredged out is that your intuition was leaning on a hidden assumption, namely that more is better. Which is a pretty benign assumption, I agree. But it was lurking, and the purpose of an exercise like this is to notice it, name it, and then say that in most circumstances it is perfectly fine.

To square the circle — to bring risk aversion, risk neutrality and risk seeking into all of this — we have to say something more about the utility function than that it goes up. We have to describe it in a bit of detail.

Here is a very helpful fact. The mathematical proof of it is genuinely quite difficult. The statement is not.

Under expected utility theory, a person is risk averse if and only if her utility function is concave.

Concave is the sad little teacup: it goes up, and it flattens as it goes. If the phrase if and only if is unfamiliar, it means the two ideas are synonyms. A concave utility function is risk aversion. Risk aversion is a concave utility function. They are the same thing. And in mathematics, concavity corresponds to the second derivative being negative.

And now the other two, which are going to be obvious. A person is risk neutral if and only if her utility function is linear. A person is risk loving if and only if her utility function is convex — going up, and bending the other way. This is a careful hand-motion day.

So, in this class: if I say that Sarah is a risk-averse expected utility maximizer, you may assume Sarah has a concave utility function. I have just told you that. I told it to you in different words. And if instead I tell you that Sarah has a concave utility function and behaves according to expected utility theory, you may assume she is risk averse. You have not actually assumed anything extra. The two statements are synonymous.

And when I say something like Sarah behaves according to expected utility theory and is risk averse, I am implicitly assuming that more is better.

I promised you, twenty minutes ago, that the expression I wrote down for expected utility was not exactly right. It is not.

Expected utility theory actually operates over final wealth states. I have been doing things a bit sloppily. The proper way to do this is to incorporate however much money a person has at the time, and we use a symbol for that: $w$, the person’s wealth — $w$ for banana.

So if we take a lottery $x$ with outcome $x_1$ with probability $p_1$, $x_2$ with probability $p_2$, and so on and so on, then the proper definition is:

To evaluate the prospect, take the probability of each outcome and multiply it by the utility of that outcome incorporating the wealth you began with. Sometimes I will say outcomes and sometimes income, because these are changes in wealth, and changes in wealth are usually called income. Do not forget to be a human being in this class.

Going back to our favourite example, the fifty-fifty coin flip over $10 and nothing, the actual expected utility is

where I write the $+\,0$ for completeness, though of course you could leave it off.

Why does any of this matter? Remind yourself of the history lesson we started with — Daniel Bernoulli, and sensations inversely proportional to the goods already possessed. Your wealth, and the word may conjure particular pictures but I mean simply the sum of your material goods, shapes the way a bit of income lands on you. Elon Musk does not care about ten dollars. Of course not. Why would he? He has a trillion. A million dollars is not cool. You know what is cool? A trillion dollars.²

Here is where we will conclude today, and it is a real application.

Suppose you have $1,000. Sorry, poor students. And suppose there is a 10% chance that you suffer a loss of $250.

Before I write it down, a note about examples. Throughout this class I will use a great many examples that look like this one, and they are silly in their magnitudes. Of course they are. The spirit of this one is not silly at all — it is absolutely a real-world situation. We could incorporate people’s actual wealth and the myriad chances of loss they face from health risks and everything else, and we would get a very nasty expression on the screen and precisely the same qualitative lesson. It would just take forever.

So, in the simplified version, the lottery you face is

( $1000, 0.9 ; $750, 0.1 )

Notice that I wrapped the loss directly into the wealth level rather than carrying it around separately. It helps.

Now: insurance is not free. You have to buy it. The insurance agent says, I will fully insure you if you pay me some amount $p$. You can think of $p$ as the price — and for those of you who have not bought insurance, which is probably most of you, the price you pay to have your money or your health insured is called a premium. Hence $p$.

If you buy the insurance, you face the lottery

( $1000 − p, 1 )

That is a degenerate lottery — one outcome, with probability one. As I promised earlier, those are perfectly coherent and they break nothing. And why $1000 - p$? Because you have to pay. When you pay for things, it is negative.

So what is your willingness to pay for full insurance? That is a term of art we will use a few times. It just means: what is the amount of money you would be willing to hand over, right on the line?

Turn the sentences on the board into mathematics the same way we did earlier. Call the uninsured lottery $A$ and the insured one $B$, and ask what makes

When in doubt, start doing stuff. The expected utility of $A$ is $0.9\,u(1000) + 0.1\,u(750)$, and we set that equal to $1 \times u(1000 - p)$, where I write the 1 purely for completeness:

and the $p$ that satisfies this we call $p^{*}$, the special price that makes the two sides equal.

Many of you will now want to keep going — to pull things out of the function, to solve for something. You cannot. It is a function, and I have not told you which one. You are stuck, and being stuck right here is the correct place to be. The answer to the question is that $p^{*}$ is the price at which you are indifferent between buying insurance and facing the risk uninsured. That little expression is a sentence, and that is the sentence it says. How one would actually solve for it depends on what the utility function is — which is exactly where we are going next.

You are, believe it or not, equipped to do a fair bit of Problem Set Lesley already. So look at it. Just start there. When in doubt, start, and see what happens. You can do a good deal of it; your mileage will vary.

Office hours begin now-ish — now-ish being however long it takes me to walk back to my office. Otherwise, I will see you Tuesday.