The Allais Paradox

Situation X
Option 1	$1,000,	1.00
Option 2	$1,000,	.89
	$2,000,	.10
	$0,	.01
Situation Y
Option 3	$1,000,	.11
	$0,	.89
Option 4	$2,000,	.10
	$0,	.90

Allais Paradox as a lottery

	Ball numbers
	1	2-11	12-100
Situation X
Option 1	$1,000	$1,000	$1,000
Option 2	$0	$2,000	$1,000
Situation Y
Option 3	$1,000	$1,000	$0
Option 4	$0	$2,000	$0

Expected utilities in the Allais Paradox

Situation X	Option 1:	.01 u($1,000)	+ .10 u($1,000)	+ .89 u($1,000)
	Option 2:	.01 u($0)	+ .10 u($2,000)	+ .89 u($1,000)
Situation Y	Option 3:	.01 u($1,000)	+ .10 u($1,000)	+ .89 u($0)
	Option 4:	.01 u($0)	+ .10 u($2,000)	+ .89 u($0)

If we follow EU and prefer option 1 to option 2 (dropping .89 u($1,000)):
.01 u($1,000) + .10 u($1,000) > .01 u($0) + .10 u($2,000)

If we prefer option 4 to option 3 (dropping u($0)):
.01 u($1,000) + .10 u($1,000) < .01 u($0) + .10 u($2,000)

Prospect Theory will explain this inconsistency.

Prospect theory: Certainty effect

($30) vs. ($45, .80)

($30, .25) vs. ($45, .20)

Two stage game: ... If you reach the second stage, you have a choice between ($30) and ($45, .80). However, you must make this choice before either stage of the game is played.

p (pi) function

Overweighing small probabilities

($1000, .001) vs. ($1)

(-$1000, .001) vs. (-$1)

Value function

Asian disease problem

Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimate of the consequences of the programs are as follows:

Program A: (200 saved)
Program B: (600 saved, .33)

Program A: (400 die)
Program B: (600 die, .67)

Can PT explain the Allais Paradox?

Situation X
Option 1	$1,000,	1.00
Option 2	$1,000,	.89
	$2,000,	.10
	$0,	.01
Situation Y
Option 3	$1,000,	.11
	$0,	.89
Option 4	$2,000,	.10
	$0,	.90

Risk aversion for gains, risk seeking for losses

Loss aversion: the status-quo effect

(a) Assume you had been exposed to a disease which if contracted leads to a quick and painless death within a week. The probability you have the disease is .001. What is the maximum you would be willing to pay for a cure?

(b) Suppose volunteers were needed for research on the above disease. All that would be required is that you expose yourself to a .001 chance of contracting the disease. What is the minimum payment you would require to volunteer for this program? (You would not be allowed to purchase the cure.)

Kahneman, Knetch, & Thaler (1990)

Sellers had mug:
"Would you sell it for $0.25? for $0.50? ... for $9.25?"
Median $7.12

Buyers:
"Would you buy a mug for $0.25? for $0.50? ... for $9.25?"
Median $2.87

Choosers:
"For each amount, would you rather have that amount in cash or a mug?"
Median $3.12

Implications of status-quo (endowment) effect

Libertarian paternalism

(Choose good defaults.}

Confounded with omission bias and default bias.