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chapter16 - Rational decisions Chapter 16 Chapter 16 1...

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Rational decisions Chapter 16 Chapter 16 1
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Outline Rational preferences Utilities Money Multiattribute utilities Decision networks Value of information Chapter 16 2
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Preferences An agent chooses among prizes ( A , B , etc.) and lotteries , i.e., situations with uncertain prizes Lottery L = [ p, A ; (1 - p ) , B ] L p 1-p A B Notation: A B A preferred to B A B indifference between A and B A B B not preferred to A Chapter 16 3
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Rational preferences Idea: preferences of a rational agent must obey constraints. Rational preferences behavior describable as maximization of expected utility Constraints: Orderability ( A B ) ( B A ) ( A B ) Transitivity ( A B ) ( B C ) ( A C ) Continuity A B C ⇒ ∃ p [ p, A ; 1 - p, C ] B Substitutability A B [ p, A ; 1 - p, C ] [ p, B ; 1 - p, C ] Monotonicity A B ( p q [ p, A ; 1 - p, B ] [ q, A ; 1 - q, B ]) Chapter 16 4
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Rational preferences contd. Violating the constraints leads to self-evident irrationality For example: an agent with intransitive preferences can be induced to give away all its money If B C , then an agent who has C would pay (say) 1 cent to get B If A B , then an agent who has B would pay (say) 1 cent to get A If C A , then an agent who has A would pay (say) 1 cent to get C A B C 1c 1c 1c Chapter 16 5
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Maximizing expected utility Theorem (Ramsey, 1931; von Neumann and Morgenstern, 1944): Given preferences satisfying the constraints there exists a real-valued function U such that U ( A ) U ( B ) A B U ([ p 1 , S 1 ; . . . ; p n , S n ]) = Σ i p i U ( S i ) MEU principle : Choose the action that maximizes expected utility Note: an agent can be entirely rational (consistent with MEU) without ever representing or manipulating utilities and probabilities E.g., a lookup table for perfect tictactoe Chapter 16 6
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Utilities Utilities map states to real numbers. Which numbers? Standard approach to assessment of human utilities: compare a given state A to a standard lottery L p that has “best possible prize” u > with probability p “worst possible catastrophe” u with probability (1 - p ) adjust lottery probability p until A L p L 0.999999 0.000001 continue as before instant death pay $30 ~ Chapter 16 7
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Utility scales Normalized utilities : u > = 1 . 0 , u = 0 . 0 Micromorts : one-millionth chance of death useful for Russian roulette, paying to reduce product risks, etc. QALYs : quality-adjusted life years useful for medical decisions involving substantial risk Note: behavior is invariant w.r.t. +ve linear transformation U 0 ( x ) = k 1 U ( x ) + k 2 where k 1 > 0 With deterministic prizes only (no lottery choices), only ordinal utility can be determined, i.e., total order on prizes Chapter 16 8
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