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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 you’re uncertain as to whether the chicken has salmonella. You think it’s unlikely, but it’s 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 isn’t) • 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 don’t 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; don’t get sick eat chicken; get sick 3 Specifying a Problem Sometimes it’s pretty obvious what the states, acts, and outcomes should be; sometimes it’s 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 you’re 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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- Fall '06
- Game Theory, Analysis of algorithms, Computational complexity theory, fast memory, competitive ratio