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
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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?
Expected Value Principle
 18th
c. court mathematicians
EV =
p*V
play any gamble if EV > 0
choose so EV is maximized
Matrices and Trees
 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
Choice
Play
Acts
Events
Outcomes
Don't Play
Heads
Tails
Heads
Tails
+15
-10
0
0
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?
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)

Diminishing Returns
U(W+X)
U(W)
U(W-X)
0
0
W-X
W
W+X
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?
Psychophysics of Wealth
1000
0
0
W-X
W
W+X
W+2X
Psychophysics of Wealth, cont.
U(W+2X)
U(W+X)
1000
U(W-X)
0
0
W-X
W
W+X
W+2X
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
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)
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
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)
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
Your Utility Curve
100
75
50
25
0
$0
$5,000
$10,000
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
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)
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
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
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
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