Decisions Under Uncertainty v explicitly consider probability v
Decisions Under Uncertainty v explicitly consider probability v again contrast normative (economic theory) and descriptive models v examine expected value and expected utility models v represent our own utility functions v try some numerical examples $
Decisions Under Uncertainty v Suppose I flip a coin. If it is heads, you win $15. Tails, you lose $10. Play? v What if you believe there is a 10% chance that the coin is “fixed”? v How should we choose? v How would we know whether or not to play the Massachusetts lottery?
Expected Value Principle v 18 th c. court mathematicians EV = p*V play any gamble if EV > 0 choose so EV is maximized
Matrices and Trees v Payoff Matrix Events Heads Tails Play 15 -10 Don’t Play 0 0 Acts 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
Matrices and Trees, continued * Tree
Behavior Doesn’t Match EV v People find “gambles” attractive even if the EV < 0, e. g. , Mass Lottery v People find gambles unattractive even if the EV > 0, e. g. , subsidized insurance v St. Petersburg Paradox: Flip a coin until heads comes up. Payoff outcome is $2 n (n= # flips). What would you pay to play once?
Expected Utility v People act as if gaining more money has diminishing returns as wealth increases v Bernouilli (1738): EU = p*U U(W) = b * log(W) v U(gain X) = U(W+X) - U(W) = b*log(W+X) - b*log(W) = b*log[(W+X)/W)] v people reject fair gambles (risk aversion), since U(W) >. 5*U(W+X) +. 5*U(W-X) v
Diminishing Returns U(W+X) U(W-X) 0 0 W-X W W+X
Psychophysics of Wealth v 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? v What if I give you $20, 000? v What if I took away $10, 000?
Psychophysics of Wealth 1000 0 0 W-X W W+X W+2 X
Psychophysics of Wealth, cont. U(W+2 X) U(W+X) 1000 U(W-X) 0 0 W-X W W+X W+2 X
Von Neumann - Morgenstern Axioms v Completeness either X > Y, Y > X, or X ~ Y v Transitivity if X >~ Y and Y >~ Z, then X >~Z v Probability Mix I if X > Y, then X > (p, X; 1 -p, Y) > Y
More VNM Axioms v Substitutability if X ~ Y, then (p, X; 1 -p, Z) ~ (p, Y; 1 -p, Z) v Probability Mix II if X > Y > Z, there must be p such that Y ~ (p, X; 1 -p, Z) v Solvability of Complex Gambles [p(q, X; 1 -q, Y); 1 -p, Z] ~ (pq, X; p-pq, Y; 1 -p, Z)
Why Axioms? v 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. v Then we can measure U(X) - Certainty equivalent method - Probability equivalent method
Certainty Equivalent Method v 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? v This procedure identifies U(X) =. 5 * U(0) +. 5 * U(10, 000) v 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)
Certainty Equivalent, continued v What is Y such that: Y ~ (. 5, $0; . 5, X) ? this defines U(Y) = 25 v What is Z: Z ~ (. 5, X; . 5, $10, 000) ? by the axioms, U(Z) = 75 v Now, plot a utility function with dollars on the X-axis and utility on the Y-axis v Concave is diminishing marginal utility or risk-aversion; convex is risk-seeking
Your Utility Curve 100 75 50 25 0 $0 $5, 000 $10, 000
Probability Equivalent v 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 v Try $2, 000 and $8, 000, etc. v This plots another utility curve
Numerical Examples v assume v 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 $ ? v U(X) = 6. 67 = 30+X X = $14. 44 (note, EV is $14. 67)
Numerical Examples, continued v What is the certainty equivalent of playing the same lottery twice? $28. 88 ? v Outcomes are: (. 11, $12; . 44, $25; . 44, $38) v 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
Numerical Examples, continued v What’s the minimum bid for the simple lottery? Is it the certainty equivalent (X)? v 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 9 b*b + 1740 b - 26, 780 = 0 b = $14. 33 (if U(X) is exponential, b=X)
Summary v Normative economic model has changed over time (from EV to EU) to better represent decision makers’ preferences v As we will see, EU is still not a complete descriptive theory v The EU model does provide a structure and benchmark for analyzing decisions
More on Job Choice Exercise v Most (but not all) trust the intuitive model, and try to adjust the linear models to agree v Does using intuition first bias the model? v Weight ranges varied greatly v Weights may be hidden in attribute ratings v What would you do if this really mattered? v Modeling for learning, not for choice!
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