Problems With Expected Utility

orif the axioms fit, use them

but...

Completeness (Consistency) Which do you prefer?A: a 1 out of 100 chance of losing $1000B: buy insurance for $10 to protect you

from this loss

Which do you prefer?C: a 1 out of 100 chance of losing $1000D: 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 $5.00B) 8/24 chance of receiving $4.75C) 9/24 chance of receiving $4.50D) 10/24 chance of receiving $4.25E) 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 $2.49Store brand $2.09Generic $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

Which would you prefer:A. receive $1 Million for sureB. 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 nothingD. a gamble in which you have a 10% chance of getting $5 Million, otherwise nothing

(Savage’s Independence Principle)

Lottery Ticket Formulation

1 2-11 12-100A $1 M $1 M $1 M

B $0 $5 M $1 M

C $1 M $1 M $0

D $0 $5 M $0

Lottery Tickets

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 WhiteWhich jar do you choose to pick from?

Solvability

Which would you prefer,A. win $30 for sureB. 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 $0D. 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 eitherA. win $30 for sureB. 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

1000 5000 9000

$4000machine

-2000 6000 14000

$10000machine

-6400 8000 22400

p .3 .5 .2

Sales (Events)

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 UtilityRisk 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

0

0

$22.4-6.4

-4

4

Choosing by Expected Utility For the $4000 machineEU = .3(-1500) +.5(2500) +.2(4800)

=1760 For the $10000 machineEU = .3(-4000) +.5(3200) +.2(6500)

=1700

Therefore, choose the $4000 machine!

Additional Complexities Within the model:

continuous sales estimatesambiguous probabilitiesother costs (your time?)other benefits (polish your resume?)

Outside the model:self-esteem, anxietychange of self-concept/preferencesreputation/respect/status