Expected value, expected utility and multi-attribute utility theory

TRADING OFF PROBABILITIES AND PAYOFFS: Expected value and expected utility theory




Definition of expected value: In a gamble in which there is a p% chance of  winning $X, the expected value is equal to pX. If there is a p% chance of X and a q% chance of Y, then EV=pX+qY.


Expected value theory says you should always choose the option with the HIGHEST EXPECTED VALUE.



1.Calculate the expected value of the following gambles:

a. 10% chance of $90 *[If you are given probabilities that add up to less than 100%, you can assume the payoff otherwise equals 0]

b.50% chance of $200

c. 15% chance of losing $100

d. 50% chance of $10, 50% chance of $50

e. 30% chance of $30, 70% chance of $90

f. 25% chance of $50; 75% chance of losing $20


2. According to EV theory, which should you prefer:
a. $100

b. 50% chance of $300, 50% chance of losing $40


3. According to EV theory, which should you prefer:
a. lose $40

b. 40% chance of losing $100


4. According to EV theory, which should you prefer: 1c or 1f?


5. According to EV theory, how much should you pay for a lottery ticket with a 1% chance of winning $50,000?


6. Imagine your car is worth $10,000 and you think there is a .1% chance it will be stolen.  According to EV theory, how much should you be willing to pay for insurance against this loss?


The problem with expected value theory


Expected value theory conflicts with people’s intuitions. Researchers attribute this discrepancy to a flaw in the theory.



A $1 million

 B 50% chance of $3 million


EV says you should prefer B to A. Many people prefer A to B.


Why do people prefer A to B, in contrast to the advice given by EV theory?

Answer 1: A is a SURE THING. B is a GAMBLE.

This explanation won’t hold up. I can easily come up with an example in which you would choose a gamble over a sure thing.

A. 99% chance of $1 million

B. $1 for sure


Most people would choose the gamble A over the sure thing B.


Another way to see that the sure thing vs. gamble idea can't explain the pattern of preferences in Example 1 is to consider the following two options:

A. 95% chance of $1 million

B. 50% chance of $3 million


In this situation, most people would prefer A. But BOTH A and B are gambles.


Answer 2: $3 million is not really three times as desirable a consequence as $1 million. This is the answer given by expected utility theory.


The solution: Expected utility theory


Most decision researchers explain the pattern of choices in Example 1 by saying that the satisfaction we’d get from $3 million isn’t that much greater than the satisfaction we’d get from  $1 million. We can construct a scale, called a utility scale in which we try to quantify the amount of satisfaction (UTILITY) we would derive from each option. Suppose you use the number 0 to correspond to winning nothing and 100 to correspond to winning $3 million. What number would correspond to winning $1 million?

Suppose you said  80.  This means that the difference between what you have now and a million extra dollars is four times as great as the difference between a million and three million extra dollars. We can express the original choice between A and B in terms of these units (UTILES) instead of dollars.


A 80 utilies for sure

B 50% chance of 100 utiles


You calculate expected utility using the same general formula that you use to calculate expected value. Instead of multiplying probabilities and dollar amounts, you multiply probabilities and utility amounts. That is, the expected utility (EU) of a gamble equals probability x amount of utiles.

So EU(A)=80. EU(B)=50. Expected utility theory says if you rate $1 million as 80 utiles and $3 million as 100 utiles, you ought to choose option A.


EU theory captures the very important intuition that there is DIMINISHING MARGINAL UTILITY of MONEY.  Definition of DMU: The value of an additional dollar DECREASES as total wealth INCREASES.  The change in your life when you go from 0 to 1 million is larger than the change in your life when you go from 1 million to 2 million.


Summary of the difference between EV and EU


In expected value theory, the correct choice is the same for all people. In expected utility theory, what is right for one person is not necessarily right for another person.


Consider a decision of the following form:

Option A: p% chance of $X

Option B: q% chance of $Y


Once you specify what p,q,X,Y are, then EV theory will give VERY SPECIFIC ADVICE. It will say one of the following things:

1 You should choose A

2 You should choose B

3 You should be indifferent between A and B


In contrast, EU theory says


1 You might choose A

2 You might choose B

3 You might be indifferent between A and B

It all DEPENDS on the utilities you assign to X and Y.



Option A: 90% chance of $100

Option B: 20% chance of $500


EV theory says that you SHOULD prefer B. EV(A)=$90. EV(B)=$100.


EU theory says that before you can decide which one you prefer, you need to determine the UTILITY of $100 vs. $500.


Person 1 might say:  $500 is better than $100, but not five times better. It's really only three times better. So I'll rate $100 as 10 and $500 as 30. (This person demonstrates the typical diminishing marginal utility for money.) Therefore, EU(A)=9 and EU(B)=6. EU theory would say that Person 1 should choose A.


In contrast to EV theory (which says EVERYONE SHOULD CHOOSE THE SAME THING), EU theory says different people might choose different things.


Person 2 might say: I really want to buy skis that cost $500. Winning $100 would be nice, but wouldn't let me achieve that goal. But if I won $500, I could buy the skis. This person might think that $500 is MORE than five times as good as $100. This person might assign a utility of 10 to $100 and a utility of  70 to $500. For this person, EU(A)=9, EU(B)=14. According to EU theory, Person 2 should choose B.


It is very easy to show that a person's choices conflict with expected value theory. This is because EV theory always specifies the "RIGHT" answer to the question "Which option do you prefer?" In contrast, it is very difficult to show that a person's choices conflict with expected utility theory.  Next time, we will talk about some clever examples demonstrating that people sometimes make choices that conflict with EU theory.