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Unformatted text preview: Uncertain Prospects Suppose you have to eat at a restaurant and your choices are: chicken quiche Normally you prefer chicken to quiche, but ... Now youre uncertain as to whether the chicken has salmonella. You think its unlikely, but its possible. Key point : you no longer know the outcome of your choice. This is the common situation! How do you model this, so you can make a sensible choice? 1 States, Acts, and Outcomes The standard formulation of decision problems involves: a set S of states of the world, state : the way that the world could be (the chicken is infected or isnt) a set O of outcomes outcome : what happens (you eat chicken and get sick) a set A of acts act : function from states to outcomes A decision problem with certainty can be viewed as the special case where there is only one state. There is no uncertainty as to the true state. 2 One way of modeling the example: two states: s 1 : chicken is not infected s 2 : chicken is infected three outcomes: o 1 : you eat quiche o 2 : you eat chicken and dont get sick o 3 : you eat chicken and get sick Two acts: a 1 : eat quiche * a 1 ( s 1 ) = a 1 ( s 2 ) = o 1 a 2 : eat chicken * a 2 ( s 1 ) = o 2 * a 2 ( s 2 ) = o 3 This is often easiest to represent using a matrix, where the columns correspond to states, the rows correspond to acts, and the entries correspond to outcomes: s 1 s 2 a 1 eat quiche eat quiche a 2 eat chicken; dont get sick eat chicken; get sick 3 Specifying a Problem Sometimes its pretty obvious what the states, acts, and outcomes should be; sometimes its not. Problem 1: the state might not be detailed enough to make the act a function. Even if the chicken is infected, you might not get sick. Solution 1: Acts can return a probability distribution over outcomes: If you eat the chicken in state s 1 , with probability 60% you might get infected Solution 2: Put more detail into the state. state s 11 : the chicken is infected and you have a weak stomach state s 12 : the chicken is infected and you have a strong stomach 4 Problem 2: Treating the act as a function may force you to identify two acts that should be different. Example: Consider two possible acts: carrying a red umbrella carrying a blue umbrella If the state just mentions what the weather will be (sunny, rainy, ...) and the outcome just involves whether you stay dry, these acts are the same. An act is just a function from states to outcomes Solution: If you think these acts are different, take a richer state space and outcome space. 5 Problem 3: The choice of labels might matter. Example: Suppose youre a doctor and need to decide between two treatments for 1000 people. Consider the following outcomes: Treatment 1 results in 400 people being dead Treatment 2 results in 600 people being saved Are they the same?...
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This homework help was uploaded on 02/19/2008 for the course ECON 4760 taught by Professor Blume/halpern during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 BLUME/HALPERN

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