Behavioral Finance Economics 437

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

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

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)

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?

“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

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

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)

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

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

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)) + .10 (U($ 5M)) > U($ 1 M) Or .10 *U($ 5M) > .11*U($ 1 M)

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)

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)

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

The End