# Expected utility theory and prospect theory

In this chapter; /;

• Extend the heuristics and biases literature presented in the previous chapter:
• Provide a quick overview of the canonical expected utility theory of economics:
• Discuss the Alla is Paradox, that calls into question key assumptions of the expected utility model:
• Explore deviations from the canonical expected utility model:
• Provide an overview of Kahneman and Tversky’s prospect theory, including the concepts of reference points, differential treatment of gains and losses and loss aversion:
• Utilize these concepts to discuss the principles of the endowment effect and. overconfidence and their applications to economics.

## Introduction

You have all seen the signs specifying penalties for littering along highways. Have you ever wondered why the fines are so large? For instance, one sign might say that the fine for littering is US \$2,000! Why is the fine for flicking a candy wrapper out of your car window so high? The answer is that, while the magnitude of the fine is very large, the expected fine is small. What is the expected fine? It is the magnitude of the fine multiplied by the probability of getting caught. You see, if everyone who littered got caught, then we could impose a \$50 fine on everyone and this fine would likely be enough to deter litterbugs. But on a highway, what are the chances of actually getting caught? Is it 1 in 100? If so, then the expected fine for littering is (1/100)*\$2,000 = \$20. This is much smaller than \$2,000 but still a substantial amount. Given the low probability of getting caught, in order to deter littering, the amount of the fine needs to be large.

This example demonstrates the concept of expected value. Lots of things in life are probabilistic. When I buy a big 75-inch OLED television, there is a chance that it could topple over and break, which may imply the need to have the super-cover warranty on it. If I go mountain climbing or downhill skiing I could fall and break a leg. This may require having adequate health insurance. Should I get flood insurance on my house? Well, that depends on the location of the house; how flood prone is the area? What is the probability of a flood? What is the expected loss in the event of a flood? How much will it cost me in ongoing insurance premia?

In New Zealand, as in many other countries around the world, such as Canada, healthcare is provided by a public system. This essentially means that if you have a heart attack or an accident, you can show up at your local hospital emergency room and they will take care of you. New Zealand does provide for private insurance. If you require treatments for non-life-threatening problems such as cataracts, hip replacements, kidney stones or a tonsillectomy, and you go to the public system, you may have to wait for a while until a suitable opening is available. But if you have private insurance, then you can get this addressed faster via the private system. So, should you have private insurance or not? It depends to an extent on your risk tolerance. Are you happy to wait for an opening in the public system or do you want to get this done immediately? As a result, a lot of people I know in New Zealand do not have private insurance. Some of them keep a sum of money aside in case of emergencies since you always have the option of paying out of your pocket in the private system. But it all comes down to expectations. What are your chances of developing kidney stones? What is your pain threshold? Are you willing to wait for the public system? Or do you want to spend the extra money to take out private insurance so that you can be treated immediately? Most of these outcomes pose a degree of uncertainty and what you decide to do is a matter of your perceptions of these risks.1

So, suppose you are asked to place a bet that a random card drawn from a deck of 52 cards is an ace. If the card turns out to be an ace then you win \$5; otherwise you get nothing. What are the chances that a random card will turn out to be an ace? Four out of 52, or 1/13- So, there is only a 1 in 13 chance of you winning this bet. This means that if the prize is worth \$5, then your expected winnings are (1/13)*\$5, or just 38 cents. This means that, in expectation, you should expect to win only \$0.38. So, if you paid \$1 to buy a ticket for this lottery with a winning prize of \$5, then you would expect to lose about \$0.62. Even if the winning prize was \$10 and you paid \$1 for a ticket, you should expect to lose around \$0.23. This is because with a prize of \$10, your expected winnings are (1/13)*\$ 10, which is around \$0.77.

This is true of most gambling games. Even if the prize is very large, the probability of winning that prize is very small; meaning the expected value of the win is relatively small, which is why fines for littering need to be large in magnitude. So, suppose someone offers you the following bet. A ten-sided die is tossed. If the number "10" comes up, then you win \$25. But if any of the other numbers, such as 1 through 9, come up, then you get nothing. Lets say that you can buy a ticket for \$3 in order to accept this bet. Should you accept it? Probably not. Because, the chance of winning the \$25 prize is 1/10. This means that your expected winnings is (1/10)*\$25 = \$2.50. But you are paying \$3 to enter the game. Of course, once in a while someone will earn the grand prize of \$25, but on average, you would expect to lose 50 cents per game: the expected win of \$2.50 minus the ticket price of \$3-

The same is true for most lotteries or casino games at a much larger scale. Even if the jackpot is very large, the probability of winning is vanishingly small. For example, suppose the jackpot for a particular lottery is \$40 million and the ticket costs \$25. Should you buy a ticket? It depends. Suppose, 5 million people buy tickets. Then your expected probability of winning is 1/5,000,000. This means that even if the jackpot is \$40 million, the amount you expect to win is: (l/5,000,000)*(40,000,000) or \$8. Of course, someone, or more than one, will win, but, on average, most people will lose and their expected loss is (\$25-\$8) =

\$17. This implies that the expected value of any win is typically less than the price of the lottery ticket or the price of entering the casino game. This is why the house usually wins. Of course, people do routinely accept these bets and we will see shortly how this decision may be justified.

The expected value of a gamble is the sum of the value of each possible outcome multiplied by the probability of that outcome. For example, if there is a 70% chance of winning \$500 and a 30% chance of losing \$100, then the expected value of the gamble is (0.70)*(500) + (0.30)*(-100) = \$320. The expected value is the amount I would earn per event on average if the event were repeated many times. Let us look at two different lotteries. Which would you prefer? (A) 50% chance of losing \$100 and 50% chance of winning \$500 or (B) 25% chance of losing \$400, 25% chance of winning \$100 and 50% chance of winning \$600. What is the expected value of lottery A? 0.50*(-100) + 0.50*(500) = \$200. The expected value of lottery В is 0.25*(-400) + 0.25*000) + 0.50*(600) = \$225. The expected value of the second lottery (B) is higher than the first (A) and so you should choose the second over the first.

Let us take another example. Which would you prefer? (C) 50% chance of losing \$100 and 50% chance of winning \$600 or (D) 25% chance of losing \$400, 25% chance of winning \$100 and 50% chance of winning \$600. What is the expected value of lottery C? 0.50*(- 100) + 0.50*(600) = \$250. The expected value of lottery D is 0.25*(-400) + 0.25*(100) + 0.50*(600) = \$225. In this case you should prefer the former (C) over the latter (D). So, in all these cases, the answer is clear. You should pick the lottery with the higher expected value.

One final example. Which would you prefer? (E) 50% chance of losing \$100 and 50% chance of winning \$500 or (F) 25% chance of losing \$500, 25% chance of winning \$100 and 50% chance of winning \$600. What is the expected value of lottery E? 0.50*(-100) + 0.50*(500) = \$200. The expected value of lottery F is 0.25*(-500) + 0.25*(100) + 0.50*(600) = \$200. The expected values are equal and so in this final example, you should be indifferent between the two lotteries.