Problems With Expected Utility or if the axioms fit, use them but. . .
Completeness (Consistency) v 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 v 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? v 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? v Suppose you can choose only one of the above lotteries. Which do you prefer? v
Intransitivity v 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 v Do you prefer A or B? B or C? C or D? D or E? A or E?
The Money Pump v 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) v v 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
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. v Jar 1 contains 50% Red and 50% White v 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 v 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 v
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 v You must choose A or B before you know the first-stage result. Which do you choose? v
Errors and Biases The above examples illustrate persistent problems with the EU model v People behave in ways that are inconsistent with the axioms, predictably, not randomly v 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 v
Exercise #2: Discussion Sales (Events) 1000 $4000 machine $10000 machine p 5000 9000 -2000 6000 14000 -6400 8000 22400 . 3 . 5 . 2 Outcome = price*sales - fixed cost - var. cost*sales e. g. , $4*5000 - $4000 - $2*5000 = $6000
Choosing by Expected Value v For the $4000 machine, EV =. 3(-2000) +. 5(6000) +. 2(14000) = $5200 v For the $10000 machine, EV =. 3(-6400)+. 5(8000)+. 2(22400) = $6560 v Thus, on EV grounds, go with the more expensive machine!
Choosing by Expected Utility Risk averse utility: v U($0) = 0 v U($2000) = 1000 v U(-6400) = -4000 v U(-2000) = -1500 v U(6000) = 2500 v U(8000) = 3200 v U(14000) = 4800 v U(22400) = 6500 4 0 -4 -6. 4 0 $22. 4
Choosing by Expected Utility v For the $4000 machine EU =. 3(-1500) +. 5(2500) +. 2(4800) =1760 v 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? ) v Outside the model: self-esteem, anxiety change of self-concept/preferences reputation/respect/status v
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