Topic2a - Topic II: Choice Under Uncertainty Topic IIA: The...

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Topic II: Choice Under Uncertainty Topic IIA: The Standard Model: Expected Utility Step 1: De f ning the Environment De f nition :A lottery (or gamble or risky prospect )isaset of possible outcomes and a probability of each outcome occurring. —————————– Example A: If you go to a roulette table and place $10 on BLACK, you have just purchased a lottery. With probability 18 / 38 you will receive $20, and with probability 20 / 38 you will receive $0. Example B: When you buy a new car, you are buying a lottery. For instance, it might be that with probability 1 / 2 it will be a car that you love (high value), with probability 2 / 5 it will be a car that only adequately serves your needs (low value), and with probability 1 / 10 it will be a “lemon” (worthless). Example C: More abstractly, you might get x 1 with probability p 1 , x 2 with probability p 2 ,and x 3 with probability p 3 (where p 1 + p 2 + p 3 =1 ). 1
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Ways to write lotteries: As a probability tree. As a vector of outcome ­ probability pairs: Example A: Payoff =($20 , 18 / 38 ; $0 , 20 / 38 ) . Example B: Car Value = ( high , 1 / 2; low , 2 / 5; worthless , 1 / 10) . Example C: Outcome =( x 1 ,p 1 ; x 2 ,p 2 ; x 3 ,p 3 ) . As a type of equation: Example A: Payoff = ( $20 with prob 18 / 38 $0 with prob 20 / 38 Example B: Car Value = high value with prob 1 / 2 low value with prob 2 / 5 worthless with prob 1 / 10 Example C: Outcome = x 1 with prob p 1 x 2 with prob p 2 x 3 with prob p 3 2
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In the realm of choice under uncertainty, lotteries are the objects of choice. Step 2: Models of Behavior Suppose you face a choice between two lotteries. How do you decide? A possible model: People choose the option with the largest “expected value”. De f nition :The expected value of a lottery x =( x 1 ,p 1 ; ... ; x n ,p n ) is EV ( x ) n X i =1 p i x i . 3
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Consider the following bet: I’m going to ip a coin, and I’m goingtokeepon ipping it until I ip a HEADS. Then you’ll be paid as a function of how many times we ip. Speci f cally: If I immediately ip a HEADS, you get $2. If I ip one TAILS and then a HEADS, you get $4. If I ip two TAILS and then a HEADS, you get $8. If I ip three TAILS and then a HEADS, you get $16. And so forth. ... How much are you willing to pay for this bet? ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The “paradox”: The EV of this bet is , but people are unwilling to pay much for it. Hence, the EV criterion is not a good description of people’s choices. 4
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Topic2a - Topic II: Choice Under Uncertainty Topic IIA: The...

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