(continue with slides from last lecture)

Expected Value

• Probability x objective value
• Use objective value instead of _______
• E.g., flip a coin, $10 for head, $0 for tails
• EV = $_____
• E.g., Roll one die.  If even, you get the amount on face in dollars. If odd, you pay the amount on face in dollars.
• 1/6 chance of $2
• 1/6 chance of $4
• 1/6 chance of $6
• 1/6 chance of -$1
• 1/6 chance of $-3
• 1/6 chance of $-5
•  EV = $_______

• Option A: 100% chance of $50
• Option B: 50% chance of $100 (50% chance of $0)
• How should Expected Utility Theory be applied to this choice?
• Expected Value (EV) = $____ for both
• What if you prefer A? Is that rational?
• _____! Depending on your utility function for money.

St. Petersburg Paradox

• Toss coin until the first time it comes up heads.
• Heads on 1st toss pays $1
• Heads on 2nd toss pays $2
• Heads on 3rd toss pays $4
• Heads on 4th toss pays $8, etc.
• What’s the EV?
• EV is ___________:
• 1st toss:  $1 x 0.50 = $0.50
• 2nd toss: $2 x 0.25 = $0.50
• 3rd toss: $4 x 0.125 = $0.50, etc.
• EV = .5 + .5 + .5 . . . . = _________________
• So, would you pay your life savings to play this lottery?  Why not?

• Bernoulli argued that the paradox is due to ________________________________ for money.
• Each dollar is worth less than the last.
• This utility function for money results in risk ___________________.
• SEU allows any shape of utility function

Why isn’t this normative?

	Tickets:	1	2-10	11-100
Choice 1	Lo	$1000	$1000	$____________
	Hi	$0	$5000	$____________
Choice 2	Lo	$1000	$1000	$____________
	Hi	$0	$5000	$____________

Allais Paradox

• Violates the _______________ axiom
• The 3rd column is the same for both options
• Thus, the particular value in the 3rd column should not influence the choice.
• But it does.

• An urn contains 90 balls
• 30 are red
• The other 60 are black and yellow, in unknown proportions
• One ball is randomly drawn
• Do you want to bet that it’s red or that it’s black?

Ellsberg Paradox

• _________ = an unknown probability
• _______ order uncertainty
• Decision makers are usually ambiguity averse
• Prefer to bet on the option with _______ likelihood to win
• In the Ellsberg paradox, this tendency violates the __________ axiom

• St. Petersburg Paradox
• Allais Paradox
• Ellsberg Paradox

Intransitivity

• A>B>C>D>E
• But ___ > _____
• Intelligence is more important than emotional stability and social facility
• In ___ vs. ____  the difference in intelligence is minor,
• But in ____ vs. ____, the difference intelligence is large enough to overrule the other dimensions

Gain Frame

• If Program A is adopted, 200 people will be saved. (72%)
• If Program B is adopted, there is a one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved. (28%)

• If Program C is adopted, 400 people will die. (22%)
• If Program D is adopted, there is a one-third probability that nobody will die and a two-thirds probability that 600 people will die. (78%)