Decisions Under Uncertainty

1. Decisions Under Uncertainty • explicitly consider probability • again contrast normative (economic theory) and descriptive models • examine expected value and expected utility models • represent our own utility functions • try some numerical examples $

2. Decisions Under Uncertainty • Suppose I flip a coin. If it is heads, you win $15. Tails, you lose $10. Play? • What if you believe there is a 10% chance that the coin is “fixed”? • How should we choose? • How would we know whether or not to play the Massachusetts lottery?

3. Expected Value Principle • 18th c. court mathematicians EV = p * V play any gamble if EV > 0 choose so EV is maximized

4. Events Heads Tails 15 -10 Play Acts 0 Don’t Play 0 Matrices and Trees • Payoff Matrix Events = mutually exclusive and exhaustive set of “States of Nature”, e.g., snow/none Outcomes = consequences (money, pleasure?) Acts = choices, decisions taken Independence Assumption: acts affect outcomes but not events

5. Matrices and Trees, continued * Tree

6. Behavior Doesn’t Match EV • People find “gambles” attractive even if the EV < 0, e.g., Mass Lottery • People find gambles unattractive even if the EV > 0, e.g., subsidized insurance • St. Petersburg Paradox: Flip a coin until heads comes up. Payoff outcome is $2n (n= # flips). What would you pay to play once?

7. Expected Utility • People act as if gaining more money has diminishing returns as wealth increases • Bernouilli (1738): EU = p * U • U(W) = b * log(W) • U(gain X) = U(W+X) - U(W) = b*log(W+X) - b*log(W) = b*log[(W+X)/W)] • people reject fair gambles (risk aversion), since U(W) > .5*U(W+X) + .5*U(W-X)

8. U(W+X) U(W) U(W-X) 0 0 W-X W W+X Diminishing Returns

9. Psychophysics of Wealth • Suppose your current level of total monetary and non-monetary wealth (your “life situation score”) is 1000 points. If I give you $10,000, what would your life situation score be? • What if I give you $20,000? • What if I took away $10,000?

10. 1000 0 0 W-X W W+X W+2X Psychophysics of Wealth

11. U(W+2X) U(W+X) 1000 U(W-X) 0 0 W-X W W+X W+2X Psychophysics of Wealth, cont.

12. Von Neumann - Morgenstern Axioms • Completeness either X > Y, Y > X, or X ~ Y • Transitivity if X >~ Y and Y >~ Z, then X >~Z • Probability Mix I if X > Y, then X > (p,X; 1-p,Y) > Y

13. More VNM Axioms • Substitutability if X ~ Y, then (p,X; 1-p,Z) ~ (p,Y; 1-p,Z) • Probability Mix II if X > Y > Z, there must be p such that Y ~ (p,X; 1-p,Z) • Solvability of Complex Gambles [p(q,X; 1-q,Y); 1-p,Z] ~ (pq,X; p-pq,Y; 1-p,Z)

14. Why Axioms? • If the axioms are satisfied, then there exists a utility function U(X) such that the ordering of lotteries by utilities is equivalent to the ordering of preferences, and U(X) is interval. It can have any monotonic shape. • Then we can measure U(X) - Certainty equivalent method - Probability equivalent method

15. Certainty Equivalent Method • What is X: X ~ (.5,$0; .5,$10,000) ? i.e., what is your minimum selling price for a ticket worth a 50% chance at $10,000? • This procedure identifies U(X) = .5 * U(0) + .5 * U(10,000) • We are free to choose a scale for U, usually U(0) = 0 and U(10,000) = 100, thus U(X) must be 50 (we are really looking at a segment of your utility curve above W)

16. Certainty Equivalent, continued • What is Y such that: Y ~ (.5, $0; .5, X) ? this defines U(Y) = 25 • What is Z: Z ~ (.5,X; .5,$10,000) ? by the axioms, U(Z) = 75 • Now, plot a utility function with dollars on the X-axis and utility on the Y-axis • Concave is diminishing marginal utility or risk-aversion; convex is risk-seeking

17. Your Utility Curve 100 75 50 25 0 $5,000 $10,000 $0

18. Probability Equivalent • What is the p at which (1-p,$0; p,$10,000) ~ (.5,$0; .5,$5,000) ? this equates the two utilities, so (1-p)*U(0)+p*U(10,000)=.5*U(0)+.5*U(5,000) or, 0 + 100*p = 0 + .5*U(5,000) therefore, U($5,000) = 200*p • Try $2,000 and $8,000, etc. • This plots another utility curve

19. Numerical Examples • assume U(X) = 30+X • what is the certainty equivalent or minimum selling price X for (.33,$6; .67,$19) ? U(X) = .33*U(6) + .67*U(19) = .33 * 36 + .67* 49 = 2 + 4.67 = 6.67 utiles; but X in $ ? • U(X) = 6.67 = 30+X X = $14.44 (note, EV is $14.67)

20. Numerical Examples, continued • What is the certainty equivalent of playing the same lottery twice? $28.88 ? • Outcomes are: (.11,$12; .44,$25; .44,$38) • U(XX)= .11*U(12) + .44*U(25) + .44*U(38) = .11 42 + .44 55 + .44 68 = 7.68 utiles = 30+X XX = 7.68*7.68 - 30 = $29.02

21. Numerical Examples, continued • What’s the minimum bid for the simple lottery? Is it the certainty equivalent (X)? • If you play, you get $6 or $19, but you have already paid your bid $b; not play = U(0) U(not play) = .33*U(6-b) + .67*U(19-b) 30 = .33 36-b + .67 49-b 9b*b + 1740b - 26,780 = 0 b = $14.33 (if U(X) is exponential, b=X)

22. Summary • Normative economic model has changed over time (from EV to EU) to better represent decision makers’ preferences • As we will see, EU is still not a complete descriptive theory • The EU model does provide a structure and benchmark for analyzing decisions

23. More on Job Choice Exercise • Most (but not all) trust the intuitive model, and try to adjust the linear models to agree • Does using intuition first bias the model? • Weight ranges varied greatly • Weights may be hidden in attribute ratings • What would you do if this really mattered? • Modeling for learning, not for choice!