Rational preferences. Rational decisions. Outline. Rational preferences contd. Maximizing expected utility. Preferences

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1 Rational preferences Rational decisions hapter 16 Idea: preferences of a rational agent must obey constraints. Rational preferences behavior describable as maximization of expected utility onstraints: Orderability ( ) ( ) ( ) Transitivity ( ) ( ) ( ) ontinuity p [p, ; 1 p, ] Substitutability [p, ; 1 p, ] [p, ; 1 p, ] Monotonicity (p q [p, ; 1 p, ] [q, ; 1 q, ]) hapter 16 1 hapter 16 4 Outline Rational preferences contd. Rational preferences Utilities Money Multiattribute utilities Decision networks Value of information 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, then an agent who has would pay (say) 1 cent to get If, then an agent who has would pay (say) 1 cent to get If, then an agent who has would pay (say) 1 cent to get hapter 16 2 hapter 16 5 Preferences n agent chooses among prizes (,, etc.) and lotteries, i.e., situations with uncertain prizes Lottery L = [p, ; (1 p), ] Notation: preferred to indifference between and not preferred to L p 1p 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() U() U([p 1, S 1 ;... ; p n, S n ]) = Σ i p i U(S i ) MEU principle: hoose 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 hapter 16 3 hapter 16 6

2 Utilities Utilities map states to real numbers. Which numbers? Standard approach to assessment of human utilities: compare a given state 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 L p pay $30 ~ L continue as before instant death Student group utility For each x, adjust p until half the class votes for lottery (M=10,000) p $x hapter 16 7 hapter 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. QLYs: quality-adjusted life years useful for medical decisions involving substantial risk Note: behavior is invariant w.r.t. ve linear transformation U (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 Decision networks dd action nodes and utility nodes to belief networks to enable rational decision making ir Traffic Litigation onstruction irport Site Deaths Noise ost lgorithm: For each value of action node compute expected value of utility node given action, evidence Return MEU action U hapter 16 8 hapter Money Money does not behave as a utility function Given a lottery L with expected monetary value EMV (L), usually U(L) < U(EMV (L)), i.e., people are risk-averse Utility curve: for what probability p am I indifferent between a prize x and a lottery [p, $M; (1 p), $0] for large M? Typical empirical data, extrapolated with risk-prone behavior: U o o o o o o o o $ o o o o 150,000 o 800,000 o Multiattribute utility How can we handle utility functions of many variables X 1... X n? E.g., what is U(Deaths, Noise, ost)? How can complex utility functions be assessed from preference behaviour? Idea 1: identify conditions under which decisions can be made without complete identification of U(x 1,..., x n ) Idea 2: identify various types of independence in preferences and derive consequent canonical forms for U(x 1,..., x n ) o hapter 16 9 hapter 16 12

3 Strict dominance Label the arcs or Typically define attributes such that U is monotonic in each Strict dominance: choice strictly dominates choice iff i X i () X i () (and hence U() U()) X 2 This region dominates X 2 D Deterministic attributes X 1 Uncertain attributes X 1 Strict dominance seldom holds in practice hapter hapter Probability Negative cost Stochastic dominance S1 S2 Probability 1 S1 S Negative cost Distribution p 1 stochastically dominates distribution p 2 iff t t p 1(x)dx t p 2(t)dt If U is monotonic in x, then 1 with outcome distribution p 1 stochastically dominates 2 with outcome distribution p 2 : p 1(x)U(x)dx p 2(x)U(x)dx Multiattribute case: stochastic dominance on all attributes optimal Label the arcs or hapter hapter Stochastic dominance contd. Stochastic dominance can often be determined without exact distributions using qualitative reasoning E.g., construction cost increases with distance from city S 1 is closer to the city than S 2 S 1 stochastically dominates S 2 on cost E.g., injury increases with collision speed an annotate belief networks with stochastic dominance information: X Y (X positively influences Y ) means that For every value z of Y s other parents Z x 1, x 2 x 1 x 2 P(Y x 1, z) stochastically dominates P(Y x 2, z) Label the arcs or hapter hapter 16 18

4 Label the arcs or Preference structure: Deterministic X 1 and X 2 preferentially independent of X 3 iff preference between x 1, x 2, x 3 and x 1, x 2, x 3 does not depend on x 3 E.g., Noise, ost, Safety : 20,000 suffer, $4.6 billion, 0.06 deaths/mpm vs. 70,000 suffer, $4.2 billion, 0.06 deaths/mpm Theorem (Leontief, 1947): if every pair of attributes is P.I. of its complement, then every subset of attributes is P.I of its complement: mutual P.I.. Theorem (Debreu, 1960): mutual P.I. additive value function: V (S) = Σ i V i (X i (S)) Hence assess n single-attribute functions; often a good approximation hapter hapter Label the arcs or Preference structure: Stochastic Need to consider preferences over lotteries: X is utility-independent of Y iff preferences over lotteries in X do not depend on y Mutual U.I.: each subset is U.I of its complement multiplicative utility function: U = k 1 U 1 k 2 U 2 k 3 U 3 k 1 k 2 U 1 U 2 k 2 k 3 U 2 U 3 k 3 k 1 U 3 U 1 k 1 k 2 k 3 U 1 U 2 U 3 Routine procedures and software packages for generating preference tests to identify various canonical families of utility functions hapter hapter Label the arcs or Value of information Idea: compute value of acquiring each possible piece of evidence an be done directly from decision network Example: buying oil drilling rights Two blocks and, exactly one has oil, worth k Prior probabilities 0.5 each, mutually exclusive urrent price of each block is k/2 onsultant offers accurate survey of. Fair price? Solution: compute expected value of information = expected value of best action given the information minus expected value of best action without information Survey may say oil in or no oil in, prob. 0.5 each (given!) = [0.5 value of buy given oil in 0.5 value of buy given no oil in ] 0 = (0.5 k/2) (0.5 k/2) 0 = k/2 hapter hapter 16 24

5 General formula urrent evidence E, current best action α Possible action outcomes S i, potential new evidence E j EU(α E) = max a Σ i U(S i ) P (S i E, a) Suppose we knew E j = e jk, then we would choose α ejk s.t. EU(α ejk E, E j = e jk ) = max a Σ i U(S i ) P (S i E, a, E j = e jk ) E j is a random variable whose value is currently unknown must compute expected gain over all possible values: V P I E (E j ) = ( Σ k P (E j = e jk E)EU(α ejk E, E j = e jk ) ) EU(α E) (VPI = value of perfect information) hapter Properties of VPI Nonnegative in expectation, not post hoc j, E V P I E (E j ) 0 Nonadditive consider, e.g., obtaining E j twice V P I E (E j, E k ) V P I E (E j ) V P I E (E k ) Order-independent V P I E (E j, E k ) = V P I E (E j ) V P I E,Ej (E k ) = V P I E (E k ) V P I E,Ek (E j ) Note: when more than one piece of evidence can be gathered, maximizing VPI for each to select one is not always optimal evidence-gathering becomes a sequential decision problem hapter Qualitative behaviors a) hoice is obvious, information worth little b) hoice is nonobvious, information worth a lot c) hoice is nonobvious, information worth little U U U (a) (b) (c) hapter 16 27