1. Behavioral Finance
Economics 437

2. Choices When Alternatives are Uncertain
Lotteries
Choices Among Lotteries
Maximize Expected Value
Maximize Expected Utility
Allais Paradox

3. What happens with uncertainty
Suppose you know all the relevant probabilities
Which do you prefer?
50 % chance of $ 100 or 50 % chance of $ 200
25 % chance of $ 800 or 75 % chance of zero

4. Expected Value Calculates the Average Value:
50 % chance of $ 100 or 50 % chance of $ 200
Expected Value = ½ times $100 plus ½ times $200, which equals $ 150
25 % chance of $ 800 or 75 % chance of zero
Expected Value = ¼ times $ 800, which equals $ 200
These two have the same “expected value.” Are you indifferent between them?

5. How to decide which to choose?
Would you simply pick the highest “expected value,” regardless of how low the probability of success might be, e.g.
1/10th chance of $ 2,000 or 9/10th chance of zero has an “expected value” of $ 200.
Would you pick this over the two previous choices, both of which have an “expected value” of $ 200?
If you are still indifferent between the three choices, then you probably order uncertain choices by their expected value.

6. Bernoulli Paradox
Suppose you have a chance to play the following game:
You flip a coin. If head results you receive $ 2
Expected value is $ 1 dollar
Suppose you get to continue flipping until your first head flip and that you receive 2N dollars if that first heads occurs on the Nth flip.
Exp Value of the entire game is:
$ 1 plus ¼($4) plus 1/8($ 8) plus ……….
Infinity, in other words
This suggest you would pay an arbitrarily large amount of money to play this flipping game
Would you?

7. So, how do you resolve the Bernouilli Paradox?

8. This lead folks to reconsider using “expected value” to order uncertain prospects
Maybe those high payoffs with low probabilities are not so valuable
This lead to the concept of a lottery and how to order different lotteries

9. Lotteries A lottery has two things:
A set of (dollar) outcomes: X1, X2, X3,…..XN
A set of probabilities: p1, p2, p3,…..pN
X1 with p1
X2 with p2
Etc.
p’s are all positive and sum to one (that’s required for the p’s to be probabilities)

10. For any lottery We can define “expected value”
p1X1 + p2X2 + p3X3 +……..pNXN
But “Bernoulli paradox” is a big, big weakness of using expected value to order lotteries
So, how do we order lotteries?

11. “Reasonableness” Four “reasonable” axioms:
Completeness: for every A and B either A ≥ B or B ≥ A (≥ means “at least as good as”
Transitivity: for every A, B,C with A ≥ B and B ≥ C then A ≥ C
Independence: let t be a number between 0 and 1; if A ≥ B, then for any C,:
t A + (1- t) C ≥ t B + (1- t) C
Continuity: for any A,B,C where A ≥ B ≥ C: there is some p between 0 and 1 such that:
B ≥ p A + (1 – p) C

12. Conclusion
If those four axioms are satisfied, there is a utility function that will order “lotteries”
Known as “Expected Utility”

13. For any two lotteries, calculate Expected Utility II
p U(X) + (1 – p) U(Y)
q U(S) + (1 – q) U(T)
U(X) is the utility of X when X is known for certain; similar with U(Y), U(S), U(T)

14. Expected Utility Simplified
Image that you have a utility function on all certain prospects
If only money is considered, then:
Utility
Money

15. Assume that Utility Function
Has positive marginal utility
Diminishing marginal utility (which means “risk aversion”)

16. So, begin with a utility function that values certain dollars
Then consider a lottery
Calculate Average Utilities
Lotteries involving $ 1 and $ 2
$ 1
$ 2

17. If th

18. Now, try this: Choice of lotteries Lottery C Or, Lottery D:
89 % chance of zero
11 % chance of $ 1 million
Or, Lottery D:
90 % chance of zero
10 % chance of $ 5 million
Which would you prefer? C or D

19. Back to A and B If you prefer B to A, then Choice of lotteries
Lottery A: sure $ 1 million
Or, Lottery B:
89 % chance of $ 1 million
1 % chance of zero
10 % chance of $ 5 million
If you prefer B to A, then
.89 (U ($ 1M)) (U($ 5M)) > U($ 1 M)
Or *U($ 5M) > .11*U($ 1 M)

20. And for C and D If you prefer C to D:
Choice of lotteries
Lottery C
89 % chance of zero
11 % chance of $ 1 million
Or, Lottery D:
90 % chance of zero
10 % chance of $ 5 million
If you prefer C to D:
Then .10*U($ 5 M) < .11*U($ 1M)

21. Allais Paradox Choice of lotteries Lottery A: sure $ 1 million
Or, Lottery B:
89 % chance of $ 1 million
1 % chance of zero
10 % chance of $ 5 million
Which would you prefer? A or B

22. Now, try this: Choice of lotteries Lottery C Or, Lottery D:
89 % chance of zero
11 % chance of $ 1 million
Or, Lottery D:
90 % chance of zero
10 % chance of $ 5 million
Which would you prefer? C or D

23. So, if you prefer B to A and C to D It must be the case that:
.10 *U($ 5M) > .11*U($ 1 M)
And
.10*U($ 5 M) < .11*U($ 1M)

24. The End