Prospect Theory

There is a smattering of you whose work I will look at in this stack and go, oh, God, I don’t know who’s who. And I am not going to call you to my office. You are adults. You know what you need to know or not know. Any other comments, questions, thoughts? Again, problem set two, code name Martin, is due in 10 days, essentially. This is the shortest gap, also. It is just a calendar quirk of this year’s particular calendar. It has to do with the way Labor Day fell. It is very boring, but sorry. But again, you need not actually turn it in that day if you do not like. Depending on your degree of procrastination and your self-control, you can turn it in at the exam. Solutions will be publicly posted a week from Thursday.

Why did the name change? You tell me. Again, it is a puzzle. It is fun for somebody or nobody. When you know the answer to the riddle, you will be able to tell the remaining problem set names, and you will know why they are in the order that they are. So that is the entirety of the clue you are going to get.

Other questions, comments, thoughts before we begin? Again, don’t suffer. Don’t grind on things by yourself in your room at 11 p.m. That is a bad strategy for success in this class. A good strategy is, when in doubt, just start doing something. So start this problem set as soon as you are done with this class, if you have time, and spend an hour and figure out: is this going to be something you can kind of work through, or are you hopelessly lost? And if you are hopelessly lost, it is diagnostic. It is not zero information. It is that I did not do a good enough job of explaining something, or that somebody else in your academic history did a poor job of explaining something and confused you in some way, which is fine. And I am unlikely to be able to fix that in this class, standing up here in front of 50 of you, but I am much more likely to be able to fix that, no guarantees, if you let me know.

Finally, as a reminder, if office hours just never work for you, if you are like, I can never make the time after class, then let me know. I will try to make an office hour for you. And before the exams I will make a bunch of office hours. But that is not an excuse to procrastinate or exhibit a lack of self-control.

Let us zoom in, then out, and in, then out, and then get nauseated, and then do that a couple more times. We first developed a descriptive model of human behavior of the form of expected value maximization. This is a person who looks at each lottery, does a little arithmetic in their head, determines which has the higher expected value, and chooses that. This is defined for any lottery over numerical outcomes. Then we said, that may not be great. Let us try one in which we imbue the person with a function that changes the objective numbers in the lottery, the outcomes, to a transformed number that we sometimes call utility. Utility is a meaningless thing. It is like widgets, an economics term that we all made up. But it is meaningful in the sense that the comparisons tell you what a person will do, which is an actual thing that we can observe. If the expected utility, defined as the probability of each outcome multiplied by the utility of that outcome, including the wealth, all summed up, if the expected utility of A is greater than B, then a person will choose A over B.

And then we said, well, that theory has some flaws. Some of the flaws are kind of contrived. It took very clever people some time to come up with things like the Allais paradox, where we mix things in a certain way and get people to flop their choices. Or the Ellsberg paradox, where we show that maybe people do not have a coherent notion of probability; it varies when you describe things ambiguously. We said, well, one of the central challenges is not any of that. It is the question of what we are trying to describe with this mapping from expected value to expected utility.

We are trying to describe the fact that people are risk averse. Everybody in here, show of hands: could you recite the definition of risk aversion, more or less as it was on the slide? Hands not up. Flashcards, baby. Whatever your thing is, it is like 15 words long, but it needs to come out of a holster, or whatever non-violent thing you pull out of a notebook. It should come off the dome easily. So we wanted to explain that people exhibit risk aversion. That is, that people would rather have (definition time) the expected value of a lottery for sure, rather than play out that lottery. So we transformed everything through this function that has a curve. And we noted that a concave utility function is synonymous with risk aversion. They are one and the same, under expected utility theory. But that concavity, although it seems to buy us everything we want, in fact has this problem: it requires too much concavity, in a sense. Thursday’s lecture illustrated what we call this calibration idea, that you need too much concavity to get the risk aversion we see every day. So we said, look, we are going to need a new theory. And that is where our friends Danny and Amos come in, in the form of prospect theory.

I am also going to discuss a little bit of one of the problems in class, either next week or maybe on Thursday, I cannot remember. So if you are a person who fancies yourself good at mathematics, do problem seven first. If you want to get good at math, do this before I do it in class. I am trying not to look at any of you. You know who you are. If you are like, ah, I’m good, dog: I will show you tricks. We are going to have to use a little bit of log rules, and exponentiating both sides, and moving stuff around, and cross-multiplying. It is a bunch of high school algebra stuff. You know how to do it. If, again, you are trying to go further in your academic future, do this on your own.

So today we discuss the main behavioral theory in this section, which is called prospect theory. I introduced this briefly on Thursday last week. I used the example of Moe from The Simpsons getting pecked by a bird. The notion is going to be that contrast, rather than absolutes, carries value.

In expected utility theory we can write down the entire theory in a very shorthand way, and we can write down an expected value maximizer the same way. Where we are going to get to today is something like distorting probabilities, and distorting something that looks like a utility function, and going back to changes rather than absolutes as the argument of that function.

Here are Danny and Amos, looking very chill. Amos Tversky died in the mid-90s. Danny Kahneman won the Nobel in 2002. He died two years ago; I think it was two years ago. They jointly proposed prospect theory, and they chose the name for its intentionally vague meaning. What does this mean? It means nothing. It meant nothing until they defined it. They offered that choice under uncertainty, the object of this section of the class, has two phases, in some sense. The first phase is an editing phase, and I will define what that means in a moment. The second is an evaluation phase: how do I think about these things? I will walk through these bit by bit.

The editing phase, or the editing stage, whatever you want to call it, says that if I give you a bunch of text and describe some choice scenario, your brain is going to subconsciously or consciously try to organize and formulate that into some other, more manageable thing. You cannot choose over paragraphs. It is meaningless, in some sense. And along the way, some choice problem, either in the real world, where the choice problem is something visual, say (most of the information that goes into our brains is visual), or a description on paper (visual in some sense, I guess), is transformed in your head into lotteries.

This is a very loose description, of course. It sort of has to be. For instance, they propose that one thing you need to do is code outcomes as gains and losses relative to some reference point. Most of the time this is kind of obvious. If I tell you I am going to flip a coin, and you get plus $10 if it is heads and plus $0 if it is tails, or lose $10 if it is tails, then that coding step is immediate. If anything, the step of incorporating the wealth is the less natural one. Many of you may have found that when exploring problems. But this is a thing that needs to happen, and in other environments it is less obvious. When deciding over health care plans, when deciding whether to engage in some risky activity. You fill in the blanks. Gross. Don’t fill in the blanks.

Another thing they offer as part of the editing stage is cancellation. This is wrong. It is wrong in a mathematics sense. I have described this before: the notion that if I say, suppose there is a three-quarter chance you do not face a lottery, and then if you do face a lottery, you face the following lottery, you cannot, or you ought not, ignore that three-quarter chance. I mean, you can do whatever you like, and people will, but it is not the correct probabilities, at least. And they say, well, this is just a natural consequence of editing. A person edits this down in their head, and it leads to this heuristic.

They also offer a basic simplification heuristic, things like rounding off probabilities and stuff like that. You might notice from even the nature of my description that I am waving my hands around a lot, and that is because this notion is loose. I mean that in a mathematics sense. There is no mathematics on the slide at all, in fact. And I am basically not going to talk about the editing stage other than the slide. It is not because it is not real. It is definitely real. It is because it depends too much on the specifics in a given environment. When do people cancel, and when not? It depends on the way that you frame the problem. That is the answer. I know tricks to make you cancel, but in generic environments it is not obvious.

When do people code things as gains and losses? It might seem intuitive, but let me give you an example. Suppose you work as a server. Every Friday night you work at a sticky bar, just slinging drinks, getting beer all over the place. And on average, most Fridays, you take home 200 bucks in tips. And I say, look, at the beginning of your shift, I am going to give you a coin flip. If it comes up heads, I will give you $150. If it comes up tails, I will give you $300. But I am going to take your tips. Are those gains or losses?

A natural coding, in my opinion, is that the $150 is coded as a loss of 50. Why? Because on average you get $200 in tips. I am offering you $150. That is $50 short. And so a natural interpretation is that this is a lottery that includes a mix of some gains, a gain of 100, and a loss of 50. But that is not immediately obvious. That depended on this whole wind-up, the fact that you work at the bar, you have got beer on your shoes, and it is all sticky and gross. Absent that, if I just described that lottery, those are obviously gains. And that is why it is difficult to port this editing stage into a variety of different settings.

Finally, they offer, in one of their rare misses, that people eliminate dominated alternatives. This is not true. They said that it was a thing. It is a natural thing you could imagine being true. It does not seem to happen in data, but they offer it as one of the suggestions of what might be happening in the editing stage. Again, this is the type of thing I am not going to spend much time on, because it lacks a systematic nature and depends a lot on the environment, but it was a lot of the process, largely from their background in psychology.

Now the evaluation stage is going to be the meat and potatoes. It is a weird expression. What is a more culturally neutral base? Well, if we were going by global population, it would be rice. It is the rice. The rice and bread. There you go. Everybody eats bread of some kind.

One of the key bits of the evaluation stage I have already previewed, which is the notion that changes relative to a reference point are a basic aspect of human nature. Again, a truncated version of the quote:

Two other features in evaluation that they emphasize are loss aversion and diminishing sensitivity, which is going to be a synonym for that big nasty paragraph in italics. So what I am going to do now, to give you a roadmap, so you and your notes and you and your brain can structure where we are going, is describe features of this $v$ function (you can use whatever letter you like), this function that is kind of like a utility function, it is not exactly like that, but we will call it the value function. And those features are going to capture loss aversion and diminishing sensitivity, which I will, of course, define more precisely in the coming slides. You good? Gucci? Great? Gravy? Other synonyms? Okay.

Loss aversion is really easy. This is one of those ones you do not have to overthink. It is a little bit more than people don’t like losses, because that is certain. A loss is defined as less than what you currently have, or less than, in some sense. So the notion that people do not like it is not sufficient to describe loss aversion. Loss aversion is, in fact, that people dislike losses more than they like an equal-sized gain. People dislike losses more than they like equal-sized gains. We like gains by definition. I like $10. Please give it to me. Damn. Never works. I dislike losing $10. Oh, I don’t have $10. Oof, dodged a bullet there. And I dislike losing $10 more than I like gaining $10. There is not a symmetry there. Sum up how I feel: I have to give you $10, and then a week later you give me $10. I am in the same wealth state. I went up and down. But I am not in the same emotional state. I did not really like that whole process. I would prefer not to do that.

And they are using this notion to capture the idea that the vast majority of people will turn down modest-sized gambles that are better than fair. Lose $600, gain $700 on a coin flip is a positive expected value gamble, and I like $700, but losing $600 would suck, and it would suck more than gaining $700 would feel good. And again, reminder, a similar notion is embedded in expected utility theory. That is, diminishing marginal utility says, well, if I lose money from my current wealth, each of those utils is worth more than the utils gained. But it is just counterfactually large. Because I could lop off two zeros from this example and present you with the coin flip of lose six, gain seven. And if I made you all bring in six dollars, half of you would take it. Something like that. Why? Because losing $6 sucks worse than gaining $6 feels good, or certainly it might be close for $7. And this dislike of losses is most psychologically salient when there is a mixed gamble, that is, a gamble that includes both gains and losses. Why? Because it just stands out.

Now, situations in which loss aversion is important. This is probably the second biggest of the two big ideas in this course. We are going to describe all of these in detail, so I am previewing where we are headed.

One idea is what is often called the endowment effect, or status quo bias. The endowment effect is the notion that I like things that I have more than I like things that I do not have. That seems kind of circular: why did you buy the thing to begin with? But suppose I give you something. I give you a thing, and I give it to you randomly. I say, I am going to give you a mug, and I am going to make you stare at the mug he now has. Because, sweet, he has got a mug. Cool. How much are you willing to pay to buy that mug? You say, I don’t know, four bucks. How much are you willing to accept to sell that mug? Like, I like this. This is my mug now. It is your precious. You start petting it, give it a name. That is the endowment effect. Why? Because we can interpret that as: now, losing the mug is a loss, and losses feel worse than equal-sized gains. Status quo bias in trades comes up in financial transactions, where the managers of portfolios are unlikely to get rid of their losers, and more likely to keep the portfolio that they themselves obtained, even if it sucks. Again, we will describe these in more detail later. Disposition effects are similar. It happens in a lot of different environments, such as houses and investments. I will describe these coming up.

One that I will not describe, but which is definitely true, it is just a little hard to find evidence for: in countries with disinflation, where the value of cash is actually going up with time (this happens, not that often, but it does happen), sometimes people get nominal wage cuts that are real wage flat, or increases. So the idea is, if you were making 100 squiggles last month, and now 100 squiggles are really worth 200 squiggles, because your country is in disrepair, probably because you use squiggles as currency, then the 100 squiggles you were getting as income is now worth 200. They say, well, we are going to cut your income in half. People hate that. Even if they know that this is the same amount of money. And even if you make them more than whole; you say, I am going to take you down to 120.

And likewise, people really, really, really hate consumption declines over their lifetime. This is an important thing for you to realize. The sooner you ramp up your lifetime consumption, the more you get used to that. One of the most useful things I did in my 20s was just be poor. I was not poor in terms of income. I had plenty of income. I just lived like a poor person. Ramen, still doing the grind. And it is not for the boring financial reason that it is good to save money. I mean, it is. But having, over time, a lifetime consumption profile that rises the whole way is way better than one that rises and then falls, even if they end up at the same place. Psychologically, the decline will feel even worse, because you got used to that standard of living, and losses feel worse than equal-sized gains. So just force yourself to poverty living, and then the exponential to the moon. It is my how-to-live-your-life tip number one of 700. That is more like 20 of 700.

Another example we will describe in detail is income targeting. Go back to our hypothetical server at the sticky bar. You might say, I am just going to work until I hit $200 in tips, and then I will quit. Here is a question. We can all have an intuition; we are all econ majors. If you are working and you reach $200 in tips in two hours, the inference you should make is, I should stay. It is a hot night. I am going to make a ton of money. That would be the correct inference by some objective measure. But if falling short of 200 feels bad and gaining above the 200 feels good, then on average you might have this propensity to stop right at the kink. And I will describe evidence that suggests that is true, in the taxi market, or Uber.

Loss aversion in action, a great example: Mahomes, absolute loss. Any time you lose the Super Bowl, it is the worst feeling in the world. They hurt probably more than the wins feel good. It is kind of all you need.

So our first key idea is that what we want here is to somehow embed, in this hypothetical function that transforms outcomes, something that is more sensitive to losses than to equal-sized gains.

The second key notion is that we want to embed something with respect to a reference point. For this, you do not need to write these down. This is just a vibes moment. I am going to introduce this second idea, which is called diminishing sensitivity, by example, and in each of these I ask you which feels like the bigger difference.

101 feet away versus 100 feet away, versus 0 feet away and 1 foot away. They are the same difference. A difference of one foot. They are the same. But the first one feels kind of negligible. To me. I will own my feelings. Versus 0 and 1 feet. Walk 100 feet and it will be the same thing.

Gaining 101 versus gaining 100, versus gaining 0 versus gaining 1. The same intuition; just turn these into dollars. Something about the marginal, air quotes, utility (this is not the same notion of marginal utility), so the marginal value of a dollar, when we are close to our current reference point, feels bigger than when we are farther away. It does not have anything to do with wealth, because this is $100. It is negligible in our lifetime wealth.

Likewise, there is something kind of intriguing about reflecting over the axis that does not come from a concave utility function. When we reflect over the axis, that is, we turn things negative, losing a dollar feels like a big increment relative to not losing anything. That first loss is the biggest bit, and then each incremental bit feels smaller. The difference between losing 100 and losing 101, I don’t know, it is all losses now. Versus 0 and 1 feels like a very discrete jump. And it is not just a 0 and 1 thing. I can go to 2 and 1 versus 101 and 100. I still think, at least to me, the second difference, the difference between losing 2 and 1, feels larger. Do we all agree? We are all hot and sleepy.

Now, the same notion applies to time, by the way, which we are going to leverage in the second section of the class. I am going to give you some monetary gain. We fix the amount, but ask: do you want it now or tomorrow? Versus 100 days and 101 days from now? Those feel different. How about saving $10 on a $1,000 item versus saving $10 on a $20 item? When we start embedding this seemingly simple notion into different examples, we get something that feels like thinking relatively. That is, there is a division that is happening in our heads, in some sense.

And we can keep doing this. My favorite example is carrying a suitcase 21 blocks versus 20 blocks, versus carrying a suitcase two blocks versus one block. If you have ever carried a suitcase, it is actually harder; the marginal effort goes up with time. So the true difference is definitely bigger for 20 versus 21. But if you get out of a train or something and you have to walk to your hotel, you might agonize over the choice where one of them is two blocks away and one of them is only one block away. The other choice, you are like, I don’t know, they are all far away. That is going to be harder. You should either not care at all, or that marginal block after you have walked 20 is going to feel like torture.

It also happens in probability space, a little bit. 18 and 19% chances versus 0 and 1% chance. So you’re saying there’s a chance. And 100 versus 99: so you’re saying there’s not a chance. Also in mixes of outcomes. This is a very widespread phenomenon. We call it diminishing sensitivity.

More formally, now we can get our heads back into some mathematics, which we were all excited about, at the edge of our seats. Diminishing sensitivity is the notion that people pay less attention to incremental differences when those differences are farther away from some reference. Say again: people pay less attention to an incremental difference, a dollar, a foot, a percentage in probability, when they are farther away from some reference point, whatever that reference point may be.

How does this play out? Let us think about something for a second. Imagine the choice of $420 for sure (why 420? because it is funny) versus a 50-50 chance of $900 or zero. A natural reference point that I suspect many of you have in your head is your current wealth, worth zero. In which case $900 is kind of far away. And therefore, according to some function $v$, where now this is zero, 420 may give you most of the increase, versus going another increment out to 900. I did not measure very well on the board the first time. Here it is again with proper axes. Make it so that 900 is the width of a paper, and then 420 is a little less than half. Notice that is more than halfway up.

If we do the same thing on the loss side, we notice the same notion does not mean that you keep continuing. No, no. If the marginal sensation close to the reference point, the zero, is steepest, that means we get something like the cup-up shape on the left. And indeed this is essentially going to be the only picture you need to know to understand prospect theory. The value function has this kind of squiggly S shape. Concave over gains (we can label these as gains, and these as losses) and convex over losses is another way to write down this basic notion. And it reflects a big and general fact about humans. We think in proportions rather than absolutes. And we think in changes rather than absolutes as well.

So I promised you something like a transformation of the way a person evaluates lotteries. I am going to change the notation to match the slide. We are going to use a function. We are going to call it $\pi$. It is still a function, so it is not the number pi. I am just out of letters. Consider a lottery with, say, two outcomes, with probabilities $p$ and $q$. Notice, though, that $p$ and $q$ may not sum to 1 here. Why not? Because, say, there is a third outcome, which is no change, and we are going to say that the function $v$ goes through the origin. I will formalize that in a moment.

So prospect theory says that a person evaluates a prospect according to the following. First, they take probabilities not as their objective number, but as possibly screwed up. I find the easiest way for students to remember this is for me to swear: people f&$# up probabilities. That is what $\pi$ is. I say there is a 10% chance you are going to get an A on the project, and you go, got it, there is a 90% chance I get an A on the project. No. I said 10. Got it. No. That is what is happening with $\pi$. And then $v$ is something akin to a utility function, which we are calling a value function, that is going to have this particular shape, capturing both diminishing sensitivity and loss aversion.

I will flesh this out. But we have gotten all the ideas. Now we just have to put them together in mathematics. Before I do that, let us just compare. Let us zoom out and say, what have we learned? How does this look different from applying the same thing under expected utility theory? Well, expected utility theory says you take the probabilities as objective, you multiply them by the utility function, where you incorporate the wealth, and you do that for each of the possible outcomes, including the outcome of no change. That can be omitted in prospect theory because, much like David Bowie, it is all about the ch-ch-changes. Somebody else knew it. It is a really good song.

So what is new? One, this $\pi$ function has the somewhat geeky name of a probability weighting function. $\pi$, if you like mathematics, is a function that takes the $[0,1]$ interval back into the $[0,1]$ interval. So it is only going to return numbers in $[0,1]$. But it is going to distort them. It is not the identity function, the one that takes the number you put in and gives you the number out. If $\pi(p) = p$, that means no probability weighting. If $\pi(p) \neq p$, it means they f&$# up probabilities. And then the second bit is that we go from this utility function that encompasses wealth to a value function, which is going to have different features, which are visually displayed on the board with this little squiggly S guy.

How are we doing? Making a face. I don’t know what the face means, just that you are making one. A lot is happening? A lot is happening, yes. To make you feel less stable, thereby making you feel more stable, we are also catapulting through space and time at a catastrophic rate that is difficult for us to even fathom. So that is happening to you also.

A question: if $\pi(p)$ does not equal $p$, that means people f&$# up the probability? That is correct, in general. The identity function means you put anything in the box and it gives you the thing back. In probability world, in psychology, it means if I put some objective probability into the box, I tell you the probability is 10%, then you say, got it, it is 10%. Any distortion of that means you screwed it up. Anything that does not equal this, you messed it up. And, as we are going to explore in the later bits of this lecture, we will say not just that they might screw it up. It is not just noise. It is not mean-zero error. Rather, people may screw up probabilities in a systematic way. We are going to specifically describe people who overweight small probabilities and underweight large probabilities. So I treat 99% as if it is 80%, and I treat 1% as if it is 10%. Something akin to that. But in principle, once you say $\pi(p) \neq p$, a person could screw it up in lots of different ways. They could just mess up numbers. I tell you 10%, you say, got it, 40%. Whatever. I will flesh these out in a moment.

Now, if you like big sum notation, for symmetry: the three formulas above are exactly what I wrote on the board, and there is a nice mapping between the various ideas in the class. Again, the expected utility one should have a plus $w$. Every time, I forget to edit this after class, and then these people are talking to me, and I remember the next time I talk about it in class. So put a plus $w$ in there in your head. An expected value is, take the product of probabilities and outcomes and sum them up. In a sense, this is the entire first exam. We are kind of done. Now we are going to have to know how to use them, but if you were a very smart math major, you would be done. And at least one or two of you are. So I have nothing more interesting to say. That is not true. I think. Who knows?

All right. Let us turn a squiggle, an S shape, into some mathematical features. There are three key features in the value function.

First, the carriers of value are changes in wealth, and $v(0) = 0$. This has two little bits going on in the sentence. The first thing is that the argument of the function, the thing that goes in there, is the changes. The David Bowies. And if there is no change, then there is no change in value. Everybody okay with the first one? Yeah, maybe. Implicit in this particular assumption is that the reference point is the current wealth. We will relax this assumption as we go on. I am never going to put you in a situation where you have to do something counterintuitive with the reference point. I promise. Just think like a person. Have your own intuitive reference points. It is always going to be right. If I surprise you with a question of what a person would do when they are surprised with some lottery, then obviously the reference point is just what they had before they were surprised. There are a lot of examples where this is a bad assumption, and we will get into those. I already gave one. The server who expects to earn $200 on her shift, earning $150 at the end of her shift, is not a gain. Of course it is not. But in no real sense did she have the $200. Except the psychological sense, and that is why this class is about psychology and economics. We are going to start showing how notions that appear only in your head, things like changes relative to some reference point, are going to shape behaviors which are observable to us, the economic analysts.

Second, diminishing sensitivity. If I can introduce this intuitively: 0 and 1 versus 100 and 101. This is the example we want to have in our head. Formally it has two bits. One is boring: the value function is increasing, its first derivative positive. I write things in mathematics because it is shorter. You can write this however you find easiest. You can just write that the value function is increasing. You can write that the derivative is positive. Whatever works for your brain. I encourage the version that is on the board, if only to force the left brain and the right brain to start co-mingling. But you do you. The second bit is that diminishing sensitivity, as the figure illustrates, can be captured by a value function that is concave in the positive domain and convex in the negative domain. Remember: concave, cup down; convex, cup up.

And loss aversion, finally. Losses loom larger than equal-sized gains. I am going to caution you: I am going to write the mathematical definition here, and you are never really going to have to use it. The version I have said verbally over and over again, losses loom larger than equal-sized gains, is all you need. There is a loose way to write it, which is that the value of any gain is smaller in absolute value than the value of the corresponding loss, $v(x) < -v(-x)$ for $x > 0$. This is not exactly right. There is a geeky technical reason why it is not exactly right. The exactly right way is: take a gain that is bigger than some other gain. If you have the bigger gain and its associated loss, you end up less than the smaller gain and the smaller loss:

You will not have to use these mathematical definitions. I am just providing them for completeness. What you will absolutely have to use is what is coming on the next couple of slides. I am going to linger, because I see a couple of people writing.

One thing that is a little hard to draw, and that I have exaggerated in the figure: loss aversion means that when we are close to the origin, each slope in the loss domain is steeper than the associated slope in the gain domain. So it is not a true S shape. It is a kinky S.

A functional form that is often used is the following. We take outcomes that are gains and raise them to some exponent, and outcomes that are losses and raise them to a different exponent, and multiply by something:

Now you might be annoyed (again, a couple of math people in the class; I am trying not to look at you) that this is multiply defined at zero. But $v(0) = 0$ either way, so it does not really matter. Pick one. If you are not familiar with this notation, this is a piecewise function. It just means you do the top bit if, the bottom bit if.

A question: would it be negative when the losses go down? Notice the if statement is going to capture that. We only put negative numbers in for $x$ on the bottom, so we get the negative out of that. Now, you may have been catching something here. We have to be a little careful. We should be putting a negative inside, $(-x)$, and then putting a negative outside, to make sure that we are not raising negative numbers to exponents. I find people find that more confusing. I hate the fact that I have a square foot of board to use in this class. The bottom branch technically should be what is written above, just to avoid shenanigans with negative numbers to fractional exponents. Whatever. Does not matter.

Now, in this mathematical form, we are turning a psychological idea (this is a theme that is going to come up over and over and over again), a thing that is a description of humans, into mathematics. How are we doing it? We have this idea that an incremental dollar feels different as it is farther away from some reference point: diminishing sensitivity. And we are capturing that by raising things to powers that are less than 1. Why less than 1? Because it means it can be concave over gains and convex over losses. And the only way to do that in this form is to make $\alpha$ and $\beta$ less than 1.

But raising things to powers is annoying. And so, maybe in some exercises you do in problem sets or exams or things like this, we will just turn that off. We turn it off by thinking about limits; recall last lecture. The limit is not zero, because then the whole thing is just equal to one. Nor is it positive infinity, because that is some very cup-up function that does not work. The limit is as we take $\alpha$ to 1: we make diminishing sensitivity loom less and less large, and we isolate the effect of loss aversion. An even easier version that we will use turns off those exponents:

In this framework, loss aversion is captured by the parameter $\lambda$. And therefore $\lambda$, in order to capture loss aversion, has to be bigger than 1. We can eyeball this. If your intuitions are struggling to get this, the kinked linear formation is just a line and a different line. And if $\lambda$ were less than 1 in this simple linear formulation, that means they are less sensitive to losses than to equal-sized gains. That is sort of weird psychology. So we are going to assume that $\lambda$ is greater than 1, in order to capture loss aversion. That is what we will look at in a lot of environments, because dealing with that latter function is quite simple. It is easy to do math. And heuristically we will often just set $\lambda$ equal to two: losses loom twice as large as equal-sized gains. If you had to guess about humans, that is a good guess. That is a really good guess. The average of human behavior is that losses loom twice as large as equal-sized gains. Depends a little bit.

This is a practice problem. That is a hypothetical problem. That is definitely a problem that is on the problem set. Suppose that a person has a prospect theory function like the following. If it is a gain, we take the gain to the 0.88 power. And if it is a loss, we multiply it by 2.25, we do this little sign shenanigans, and also take it to the 0.88 power:

And they have no probability weighting, i.e., $\pi(p) = p$. That is a typo on the slide. Suppose the person holds an asset that yields payoffs described by the following gamble. There is a 50% chance of gaining 100. There is a 25% chance of gaining 0. There is a 25% chance of minus 50.

Let us just illustrate briefly the plugging and chugging, the every-econ-class way. $v(x)$ is equal to the stuff above; I am not going to rewrite it. The first little bit is knowing what the outcomes are. Well, 100 is an outcome. So I will write the value of this lottery. It is kind of like the expected utility; the similar notion is equal to the sum of the probability times $v$ of $x_m$. Why is it $p$ here? Because $\pi(p) = p$, the typo on the board. So we are going to take 0.5 times $v(100)$, plus 0.25 $v(0)$, plus 0.25 $v(-50)$:

Pro tip. The number one way that people screw up problems in this class is not doing this intermediate step. It took me three seconds. I literally just did it verbally and wrote it at the same time. Just do this step. Because now all I have got to do is assess a function. I do not even have the number in the spot yet. So it is 0.5 times 100 to the 0.88, plus 0.25 times 0 ($v(0) = 0$), plus 0.25 times 2.25 times minus 50, and we have got to do the minus-minus thing, so the minus from there, and the minus goes out here, to the 0.88:

And then you would do whatever the question demanded. Which you will do. I believe this is question three or four or something; I do not remember. If you find this a thing that you could not do without it being on the board, but could with it, come see me. Then something is not going right, and it is going to be tough sledding. That does not mean it is not fixable. I am pretty smart. I am sure I can fix it. With time, maybe. I mean, I will try. That is all I can ask.

These are the key things. Not the mathematics of what the value function is; it is the implications. We care about what people do, not some geeky mathematics garbage. Here is a fact. If a person maximizes her preferences meeting the assumptions, with a value function having the three features I previously described (and you will see reference to the value function with the three features on problem sets and exams, things like this; this is where to look), then:

One of the implications: she shall turn down any 50-50 lose $X$, gain $X$ bet, for any value of $X$. Why? It is just an implication of loss aversion. It is loss aversion. Remember, loss aversion is that a loss feels worse than its equal-sized gain. That is what is happening here. They would rather have zero, that is, stay at their current wealth. This is a fair bet. It does not mean that she is averse to all fair bets. I will describe that in a moment.

Second: is she risk averse amongst bets involving only gains? Do you know what this definition is? Do not need to do hands again. You know if you do not know. Again, the definition exactly as it is written. No added words, no vibes. A person whose value function exhibits the three properties previously described has, as I mentioned, this squiggly thing. And it is concave amongst only gains. Because it is concave amongst only gains, a person will exhibit risk aversion amongst bets involving only gains. This is a consequence of diminishing sensitivity. Everybody with me? Maybe with me. Hot and sleepy.

And analogously, a person is risk loving amongst bets involving only losses. And here we really deviate from expected utility theory. Why? Well, it is cup-up over here, on my crumbly drawn figure. This S-shaped value function is convex amongst only losses. Convexity is synonymous with risk loving. And psychologically, what is happening is that the marginal dollar close to zero is the marginal dollar that feels the biggest. In the gains domain, that means that ensuring some degree of certainty is good. In the loss domain, that means ensuring some degree of certainty is bad. They would rather gamble to get back to zero. It is also a direct implication of diminishing sensitivity.

The final one is geeky, and I am debating how geeky I want to be. I will be geeky. You can choose your adventure. If you feel like being a geek, listen for three minutes. If you feel like just chilling, let it wash over you like a warm something. The last one requires an important assumption. First-order risk aversion means that as you zoom in to zero, you maintain some degree of loss aversion. I have drawn this in all the figures, but some of these functions are not well defined. When you define a piecewise function, it does not have a limit; it is just geeky math garbage. And therefore I have to directly assume, rather than having it from that piecewise function, that the limit when you zoom in to the origin is something that is greater than one:

When you assume that, you get first-order risk aversion. It is in a lot of the functional forms I will use. End of the slide.

So, zooming out. Prospect theory. We are almost done. The header of the slide suggests we have got one more bit. We described the value function. It has three features: $v(0) = 0$, concave over gains and convex over losses, and losses loom larger than equal-sized gains.

The probability weighting function we get to be a little more hand-wavy about. So this is not going to have as much math. Why? The swearing is intentional. It means that people f&$# up the probabilities. Once you are screwing things up, people do weird stuff in this domain. This is not exactly akin to how they feel about these probabilities. People just objectively treat probabilities wrong. Note again, expected utility theory says $\pi(p) = p$. No typo here. That is, people do not screw up probabilities.

Here are some boring assumptions. We have to anchor it. So we are going to say, if I tell you that the probability of something is truly zero, that you believe. If I tell you the probability is truly one, that you believe. It is all the interior bits that you might screw up. And whatever this function is that distorts probabilities, it is at least increasing. The bigger the numbers I tell you, the bigger the numbers; again, you might screw it up, but you are never getting the order of the numbers wrong. You know that if I tell you it is a 40% chance and then I tell you it is a 45% chance, the 45 is bigger than 40. Now, you might treat those as if they are 70% and 72%, respectively, but at least you get the order right.

Now, here are some loose things that people actually do. They are loose because I am not going to define what small and large are. Small $p$ and large $p$: you know it when you see it. For small $p$, $\pi(p)$ is bigger than $p$. That is, people distort small probabilities as if they are larger. I have been giving verbal examples that are not misleading this whole time. I tell you something is 1%, and you say, got it, 10%. There is a very small chance that you will win the Powerball if you buy a Powerball ticket. That probability is not truly zero. It is effectively zero. But people treat it as if it is many orders of magnitude larger than it actually is. For lots of things this is true. If you are a Tesla stan or something: what are the odds that Tesla stock gets to 10,000? It is not zero. It just isn’t. And some finance professor will say, look, that is a one-in-a-million chance. And that person, the Tesla stan, says, got it, one in 10. I said one in a million. No, got it, got it, one in 10. I get it, it is small. You said one in a million; well, like, one in 10. This is the same thing. No, they are not. One in a million is things that don’t happen. It is just statistically never happening.

And, of course, analogously, big probabilities are distorted downwards. So in this instance it is like, look, there is a 96% chance that it is going to rain tomorrow. And you are like, got it. So it is like 50-50. No, it is not. It is 96. Yeah, 50-50. This is capturing the notions of diminishing sensitivity, in a sense, about the anchor points. Changes about these anchors loom very large.

Here is another fun fact. If you take a probability and its complement, so a 40% chance and a 60% chance, there is a 40% chance of this and a 60% chance of that, and you say, all right, got it, so there is still a little chance of something else. Well, there isn’t. That means all the chances. This is often called subcertainty, $\pi(p) + \pi(1-p) < 1$. And subproportionality is a notion we have actually already covered. It is the idea that if I take some probabilities and multiply them through by some common factor, that can distort them. These mathematical notions are not going to be on a thing. They are useful. They can simplify your tasks and present you with intuitions that match mathematics, which I find helpful, but you may not. So you need not memorize them, but they may come in handy.

A way to think about it is with a picture. One way the probability weighting function could look is something like this. Small probability (what constitutes small, I don’t know, ask your friends), something in here gets distorted upwards, and everything else is kind of bowed downwards. This was just them drawing a picture. They had no data, just intuitions, and research started filling this in, and you get this factoid that at about a third, people get probabilities right. So a third, people sort of understand: it is less than a half, it is not that likely to happen, and the heuristics sort of balance out. People really screw up intermediate certainties; people find that very off-putting. And also, by orders of magnitude, distort things close to the ends.

A couple of themes, and then I will wrap. What have we got? We can think of this in a geeky way as a non-linear decision weight. If we multiply not by the objective probability, which is shown on the diagonal line, but by this distorted probability, then we get some sort of screwed-up decision weight. Or, again, in swearing, which is not misleading: people might f&$# up probabilities. We have also introduced this idea of changes relative to some reference point, or reference dependence. And that losses relative to that reference point might loom larger than equal-sized gains. And along the way, a couple of things are going to start jumping out. That the way that you formulate a problem matters, because it is going to shape the way that it is coded as either losses or gains. This is what is known as the framing effect. And then the process that a person goes through to code things, to sort things into bins in their head, which is known as mental accounting, is going to be very important, because it is going to mean what do we put in these functions. We are going to explore these on Thursday.

Next time, we will cover a couple of applications of prospect theory in the real world, which will help us both get our sense of the mathematics and understand the real-world phenomena we are interested in. The problem set is due Thursday the 24th, unless you want to test your self-control, in which case it is due before the exam. Answers will be posted on Thursday, September 24th. And again, don’t do problem seven unless you want to test yourself, in which case you can look at it. Solutions should be up for the last problem set. I am going to check right now and see if my coding works.

Are we going to get these back? Yes. Will it be before the exam? Yes. They will come back probably at the latest next Tuesday, and I recommend you just wait until then.