He is accelerating all the time. That last lap was run in 64 seconds compared to the one before in 62 seconds.
John Coleman, sports commentator
A theme in chapters two and three was uncertainty and risk. For instance, in chapter two I talked about Anna not knowing whether she would prefer the Nutty breakfast cereal or the Honey cereal. In chapter three I talked about Alan deciding whether or not to fully insure his car without knowing whether his car would be stolen. New information should remove some of this uncertainty. For example, if Anna tastes the breakfast cereals, or Alan fi nds out the number of cars stolen in his area in recent years, then they should be able to make more informed decisions. This looks like a good thing.
It should, therefore, be a good thing that we are continually bombarded with new information, from the TV, radio, internet, newspapers, friends, family, colleagues, and our own experiences! New information is only useful, however, if it is used wisely. It is clear that it can infl uence behavior in signifi cant ways. Why else would fi rms spend so much money on advertisements? Why else would someone ask friends, or look on the internet, for advice before making a decision? What we need to think about is how wisely people use new information and the consequences this has for how they behave. That will be the focus of this chapter.
5.1 Bayesian updating and choice with uncertainty
Imagine someone called John, who is thinking of buying a new car and has narrowed it down to two choices, a ‘Sporty’ car or ‘Comfort’ car. Either car is approximately the same price but John does not know which will give him higher utility. In the process of deciding he tries to become better informed by asking his friends, doing a test drive, searching for advice on the internet, and so on. What should he do with this new information?
The benchmark for what he should be doing is Bayesian updating . Let p denote the prior probability John puts on the Sporty giving the higher utility. Any new information he gets we call a signal . For example, a signal might be his
friend saying ‘buy the Comfort’ or him doing a test drive and ‘liking the Sporty’. Let θ denote the signal precision or probability he would got the signal if Sporty does give the higher utility. The value of θ should capture things like the trust that John puts in his friends opinion or expertise, or how relevant he thinks a test drive is. If John uses Bayes rule then the posterior probability he should put on the Sporty giving the higher utility is:
(5.1)
For example, if John started relatively indifferent between the two, say p = 0.4, but his friend said that Sporty is better and he trusts his friend’s opinion, say θ = 0.9, then he should put posterior probability 0.86 on the Sporty being better for him. This new information changes his beliefs and in doing so may change his behavior from buying the Comfort to buying the Sporty.
Bayesian updating is what John should do if he wants to make the most informed decision. Every new piece of information he gets, it makes sense to use Bayes rule. We have, therefore, a prediction for how John should use any new informa- tion. Bayes updating can, however, be a bit more complicated than equation (5.1) might suggest. To explain why, it is useful to relate John’s choice to what we did in chapters two and three .
We can think of John as not knowing what car will give him highest utility and so he searches for more information to become better informed. He is never going to be completely sure what is best, but new information allows him to update his beliefs and eventually make a more informed decision. Throughout this process John is making risky choices, particularly when he fi nally buys a car. For example, we can think of there being a ‘Sporty prospect’ and a ‘Comfort prospect’ where each prospect details the possible outcomes from buying that car. New informa- tion allows John to change the numbers in one or more of the prospects. For example, if he test drives the Comfort he can update the Comfort prospect. By updating the Comfort prospect he can then update his beliefs the Comfort is best. Table 5.1 illustrates how this might work in practice when John test drives the Comfort.
Framing things in this way we can see two potential issues. First, it becomes complicated to update beliefs. When I introduced Bayes updating, the focus was on the probability the Sporty is best for John. Now, we are starting to get down to fi ner details about each car. This seems natural but means that John has some work to do in order to turn new information into updated beliefs. He has to update his beliefs about the merits of each car, and then update his belief about which car is best. This can easily get quite complicated. Can John realistically be expected to do it? We will get back to that question in section 5.2.
A second issue is that of ambiguity and uncertainty. At the start of chapter three I mentioned the distinction between risk and uncertainty and you might recall that if a person has a choice with uncertainty then they may not know the probability of possible outcomes. John can presumably predict that he could like the Comfort,
not like the Comfort, and so on. What would be much harder for him to do is put specifi c numbers on, say, the probability that he will like the Comfort, and the value he would put on it if he did like it. This means that John’s choice of car is a choice with uncertainty. It also means we need to think about where the numbers in Table 5.1 could come from because without them we cannot possibly expect John to use Bayesian updating. We need, therefore to think about choice with uncertainty.
5.1.1 Models of choice with uncertainty
I shall start by talking about a relatively simple example to illustrate the most important issues, and then relate things back to John buying a car. Imagine a box containing ten balls. One of the balls is going to be drawn randomly from the box, and you have to bet on what color the ball will be. If you get it correct you win $2. Now contrast these two different possibilities:
Box 1: Contains fi ve red balls and fi ve black balls.
Box 2: Contains an unknown number of red balls and black balls. It could be zero red and ten black, ten red and zero black, or anything in between.
What box would you rather bet on?
Box 1 provides a simple choice with risk like that we looked at in chapter three . If John bets on black he faces the prospect (0.5, $2; 0.5, $0), so with a 50 percent chance he will be correct and win the $2. His expected utility is then:
U (black) = 0.5u (w + 2) + 0.5u (w)
Table 5.1 John updates his beliefs after a test drive. The Comfort is best if it is worth $12,000. Initially he puts probability 0.7 on this being the case so Sporty is best with prob- ability 0.3. He test drives the Comfort and did not like it. This lowers the probability he puts on the Comfort being worth $12,000 down to 0.6
Event John’s opinion Summary
Initial prior The Sporty is worth $10,000 with probability 0.9 and $9,000 with probability 0.1.
The Comfort is worth $12,000 with probability 0.7 and $4,000 with probability 0.3.
Sporty is best with probability 0.3.
Test drives the Comfort and gets signal ‘did not like the Comfort’
The signal precision, i.e. probability this would happen if the Comfort is worth 12, is 0.39.
He updates his beliefs about the Comfort to being worth $12,000 with probability 0.6 and $4,000 with probability 0.4.
Sporty is best with probability 0.4.
where w is his wealth. What about box 2? This time I cannot write down the pros- pect because I cannot say what probability John has of winning if he bets on black. The probabilities are ambiguous . If the probabilities are ambiguous then we cannot work out John’s expected utility, use prospect theory, or do anything else we did in chapter three . We need something new.
The simplest thing we can do is use subjective expected utility in which we go with what John thinks will happen. If John thinks that box two will contain four red balls and six black balls then his subjective probability of drawing a black ball is 0.6 and his subjective expected utility is:
U (black) = 0.6u(w + 2) + 0.4u(w)
That looks like a simple solution! The problem is that John could have any beliefs he wanted and not be wrong. He could think none of the balls are black, all of them are black, and we have no way to tie down what his beliefs should be, or are likely to be.
Consider now another possible box:
Box 3: A number between zero and ten will be randomly drawn and deter- mine the number of red balls in the box. The rest of the ten balls will be black. Does this box provide a choice with risk or with uncertainty? The correct answer is that it provides a choice with risk. It does, however, look different to box one. That’s because it involves a compound lottery. A compound lottery is when there are two or more consecutive lotteries to determine the outcome. Box three is a compound lottery because there is a lottery to determine how many balls in the box, and then a lottery when John comes to pick a ball. Just to confuse, the fi nal stage, where John picks a ball, is typically called the fi rst order lottery or second stage lottery . The preliminary stage, where the number of balls in the box is determined, is called the second order lottery or fi rst stage lottery . Two stages are enough in this case but there can be three or more stages.
To try and capture the lottery of box three we could write down one long pros- pect (0.09, $2; 0.09, (0.9, $2; 0.1; $0); 0.09, (0.8, $2; 0.2, $2); . . .) where we recognize there is a one in eleven chance all the balls are black and John wins $2, a one in eleven chance nine of the balls are black and John wins $2 with prob- ability 0.9, and so on. This looks a bit messy. We could, therefore, try reducing the compound lottery to a simple lottery. To do this we work out the overall probability that a ball will be black or red. If you do this you should fi nd that there is a 50 percent chance a ball will be black and a 50 percent chance it will be red. This suggests betting on black with box three gives the same prospect (0.5, $2; 0.5, $0) as it did with box one!
It is not clear, however, that we should be reducing compound lotteries to a simple lottery. For instance we could do something called recursive expected utility and work out the expected utility of each lottery and compound them together. To do this, we start with the fi rst order lottery where John picks a color.
If there is one black ball and nine black balls his expected utility of betting on black is:
U ( black | one black) = 0.1u(w + 2) + 0.9u(w).
We need to do this for all possible combinations of black and red balls. We then go to the second order lottery where the number of balls is determined. For reasons we do not need to worry about, Kilbanoff, Marinacci and Mukerji (2005) show that his expected utility can be written:
U (black) = 0.09 φ ( U (black | no black)) + 0.09 φ ( U (black | one black)) + . . . where φ is some function that captures preferences over second order lotteries.
The key to this way of doing things is that we may want to distinguish risk attitudes towards fi rst and second order lotteries. In particular, there are good arguments for why John may be more risk averse in a second order lottery than a fi rst order lottery. If so, that means we should not be reducing compound lotteries but instead using recursive expected utility. It also means that John will be more reluctant to bet on box three than box one because he essentially does not like compound lotteries.
It is now interesting to go back and look again at box two. When the probabili- ties are ambiguous, as they are in box two, it seems more natural to think in terms of a compound lottery. That is, it seems natural to think of John having subjective beliefs about an additional stage in which balls are added to the box. For example, he might put subjective probability 0.5 on there being fi ve red and fi ve black balls, and probability 0.25 that all the balls are black, and so on. It may, therefore, be most apt to use recursive subjective expected utility to think about choice with uncertainty. To see whether that is the case we need to look at some data.
5.1.2 The Ellsberg Paradox
I am going to look at the results of a study by Halevy (2007). In the study, subjects were asked to bet on boxes one to three, and a fourth box:
Box 4: A fair coin will be tossed to determine whether all the balls in the box are black, or all of them are red.
The interesting thing about box four is that all risk is resolved in the fi rst stage. So, we have box one where risk is resolved in the second stage, box four where it is resolved in the fi rst stage, and box three which is a true compound lottery. If people are more risk averse about second order risk, we should be able to pick this up.
In the study, rather than just have subjects bet on a color, subjects were given the chance to sell bets by saying the amount they were willing to accept to forgo
the bet. Figure 5.1 plots the average, minimum amount subjects were willing to accept. Remember they would win $2 if they won the bet, so a risk neutral person would want $1. The most important thing to pick out is the greater willingness to sell a bet on box two than a bet on box one. This difference was fi rst pointed out by Ellsberg (1961), and so goes by the name of the Ellsberg Paradox . (For another Ellsberg Paradox, see review question 5.2.) The Paradox is why people do not like betting on box two. It seems that people are ambiguity averse in prefer- ring a choice with risk to one with uncertainty. The results for boxes three and four suggest, however, we need to qualify this somewhat because subjects also seem averse to a second order lottery.
To delve a little deeper, Halevy looks at the choices of individual subjects and tries to classify those choices. Table 5.2 summarizes his fi ndings. Let’s go through the table row by row.
First, there were subjects willing to sell a bet on any box at the same price. They did not seem to be bothered by ambiguity or a second order lottery and so their choices seem consistent with subjective expected utility maximization.
Second are those who were more willing to sell a bet on box four than on box three than on box one. These subjects seemed to be risk averse about a second order lottery and so their choices are most consistent with recursive expected utility.
Third are those who are willing to sell a bet on boxes one and four at the same price but more willing to sell a bet on box three. It does not look as though these subjects are risk averse about a second order lottery, otherwise they would be more willing to sell a bet on box four. Instead, it seems they disliked the
Figure 5.1 The willingness to sell a bet on each of the boxes. The Ellsberg Paradox is why subjects want less to sell a bet on box two than on box one.
compounding of lotteries necessary with box three. I will not go into the details, but this is consistent with recursive non-expected utility in which a person uses rank dependent utility (as introduced in chapter three ).
Finally, we come to those who simply do not like ambiguity. In his original paper Ellsberg suggested that when there is ambiguity, people are pessimistic and go with the worst thing that can happen. Hence we get a story of maximizing the minimum. Only one subject in this study behaved like that in being more willing to sell box two, but not boxes three or four.
Let’s go back now and think about John deciding what car to buy. First of all, we now know that we should view the numbers in Table 5.1 as John’s subjective beliefs about what he thinks might happen. We can also now see that it is natural to think of his choice as a compound lottery where, say, the quality of each car is determined in a second order lottery and the car he would most prefer is deter- mined in a fi rst order lottery.
The subjective nature of John’s beliefs means it will be very hard for us to know why John would choose one car over another and whether he made the best choice. We do see, however, in Table 5.2 that the majority of people do not seem to like compound lotteries, or are ambiguity averse. This does have clear implications: John may be reluctant to gamble on which car is best, and so put off buying a car; he should also be eager to become better informed about each car in order to reduce the ambiguity or second order risk he faces. It is this last possibility I want to focus on fi rst.
5.2 Two cognitive biases
If John does not like ambiguity or second order risk then he needs to become better informed, and so he should be on the look-out for new information. What we need to ask is whether he will use that information wisely. Will he, for example, use Bayes rule? To do so, would be quite complicated and so it would not be a huge surprise to see some mistakes and biases in how he does interpret new infor- mation. In this section I will look at two of the more common biases we do seem to observe: the confi rmatory bias and law of small numbers.
Table 5.2 The proportion of subjects fi tting each model of preferences and the average willingness to sell
Model of preferences Proportion of
subjects Willingness to accept to forgo bet box 1 box 2 box 3 box 4
Subjective expected utility 19.2% 1.03 1.03 1.03 1.03 Recursive expected utility 40.3% 1.04 0.78 0.96 0.79 Recursive non-expected utility 39.4% 1.11 0.91 0.85 1.07 Maxmin expected utility 0.1% 0.90 0.80 0.90 0.90
5.2.1 Confi rmatory bias
Suppose John asks his friend’s advice and is told: ‘The Comfort is a great car, but