Problems With Expected Utility
or
if the axioms fit, use them
but...
Completeness (Consistency)
 Which
do you prefer?
A: a 1 out of 100 chance of losing $1000
B: buy insurance for $10 to protect you
from this loss
 Which
do you prefer?
C: a 1 out of 100 chance of losing $1000
D: lose $10 for sure
Completeness - Response Modes
Suppose you own a lottery ticket that gives
you a 8/9 chance to receive $4. What is the
lowest price you would take to sell this
ticket?
 Suppose you own a lottery ticket that gives
you a 1/9 chance to receive $40. What is the
lowest price you would take to sell this
ticket?
 Suppose you can choose only one of the
above lotteries. Which do you prefer?

Intransitivity
 Consider
the following five gambles:
A) 7/24 chance of receiving $5.00
B) 8/24 chance of receiving $4.75
C) 9/24 chance of receiving $4.50
D) 10/24 chance of receiving $4.25
E) 11/24 chance of receiving $4.00
 Do you prefer A or B? B or C? C or D?
D or E? A or E?
The Money Pump
 Your
grocery store has three brands of
canned peas, each 12 oz.:
National brand name $2.49
Store brand
$2.09
Generic
$1.69
If you prefer N>S>G, but G>N, then I can
trade you S for G and $.41, N for S and
$.41, G and $.79 for N, etc.
Allais Paradox
(Savage’s Independence Principle)


Which would you prefer:
A. receive $1 Million for sure
B. a gamble in which you have an 89% chance of
getting $1 Million, 10% chance of getting $5
Million, and 1% chance of receiving nothing
Which would you prefer:
C. a gamble in which you have an 11% chance of
getting $1 Million, otherwise nothing
D. a gamble in which you have a 10% chance of
getting $5 Million, otherwise nothing
Lottery Ticket Formulation
Lottery Tickets
A
1
$1 M
2-11
$1 M
12-100
$1 M
B
$0
$5 M
$1 M
C
$1 M
$1 M
$0
D
$0
$5 M
$0
Ellsberg’s Paradox
Two jars contain Red and White balls. You
may choose Red or White and pick one ball
from one jar. If you pick your chosen color
you will be paid $100.
 Jar 1 contains 50% Red and 50% White
 Jar 2 contains a randomly chosen unknown %
of Red and White
Which jar do you choose to pick from?
Solvability
Which would you prefer,
A. win $30 for sure
B. play a gamble in which you have an 80%
chance of winning $45, otherwise $0
 Which would you prefer,
C. play a gamble with a 75% chance of
winning $30, otherwise $0
D. play a gamble with a 60% chance of
winning $45, otherwise $0

Solvability, continued
Suppose you must choose the second part of
a two-stage gamble now. In the first stage,
there is a 25% chance that you will get $0, and
a 75% chance that you will go on to the
second stage. In the second stage, you either
A. win $30 for sure
B. play a gamble in which you have an 80%
chance of winning $45, otherwise $0
 You must choose A or B before you know the
first-stage result. Which do you choose?

Errors and Biases
The above examples illustrate persistent
problems with the EU model
 People behave in ways that are inconsistent
with the axioms, predictably, not randomly
 Some theorists have developed models that
retain the form of the EU rule with additional
modifications to probability and value
functions, while others catalog decision rules
that bear no direct resemblance to EU

Exercise #2: Discussion
Sales (Events)
1000
5000
9000
$4000
-2000
machine
$10000
-6400
machine
p
.3
6000
14000
8000
22400
.5
.2
Outcome = price*sales - fixed cost - var. cost*sales
e.g., $4*5000 - $4000 - $2*5000 = $6000
Choosing by Expected Value
 For
the $4000 machine,
EV = .3(-2000) +.5(6000) +.2(14000) =
$5200
 For the $10000 machine,
EV = .3(-6400)+.5(8000)+.2(22400) = $6560
 Thus,
on EV grounds, go with the more
expensive machine!
Choosing by Expected Utility
Risk averse utility:
 U($0) =
0
 U($2000) = 1000
 U(-6400) = -4000
 U(-2000) = -1500
 U(6000) = 2500
 U(8000) = 3200
 U(14000) = 4800
 U(22400) = 6500
4
0
-4
-6.4
0
$22.4
Choosing by Expected Utility
 For
the $4000 machine
EU = .3(-1500) +.5(2500) +.2(4800) =1760
 For the $10000 machine
EU = .3(-4000) +.5(3200) +.2(6500) =1700
Therefore, choose the $4000 machine!
Additional Complexities
Within the model:
continuous sales estimates
ambiguous probabilities
other costs (your time?)
other benefits (polish your resume?)
 Outside the model:
self-esteem, anxiety
change of self-concept/preferences
reputation/respect/status
