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. 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)

5. 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?

6. “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

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

8. 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)

9. 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

10. 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

11. 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)

12. 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)

13. 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)

14. First Mid-Term Exam: Tuesday, Feb 20, 2018
In Wilson Auditorium, 9:30 – 10:45 AM
All readings, lectures and powerpoint slides through and including Thursday, Feb 8th
Readings, specifically:
Shleifer: Chapters 1 and 2 Articles by Black, Shiller, Malkiel, Fama Burton and Shah, pp. 1-51
But not Kahneman: Thinking: Fast and Slow
Not lectures on Feb 13 and Feb 15

15. The End