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

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?

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.

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?

So, how do you resolve the Bernouilli Paradox?

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

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)

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

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

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

If th

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)

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

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)

The End