Chapter 16: Making Simple Decision March 23, 2004

16.1 Combining Belief and Desires Under Uncertainty Utility: captures the desirability of a state, U(S) A: action E: evidence EU: expected utility EU(A|E) = ∑ P(Result i (A) | Do(A), E) * U(Result i (A))

MEU: maximum expected utility A rational agent should choose an action that maximizes the agent’s expected utility This is a framework where all of the components of an AI system fit

16.2 The Basis of Utility Theory 6 Preference Axioms Orderability (A > B) or (B > A) or (A ~ B) Transitivity (A > B)  (B > C) -> (A > C) Continuity (A > B > C) ->  p [p, A; 1-p, C] ~ B (“lottery”) 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] Decomposability [p, A, 1-p, [q, B; 1-q, C]] ~ [p, A; (1-p)q, B; (1-p)(1-q), C]

Utility Principle If an agent’s preferences obey the axioms of utility, then there exists a real-valued function U that operates on states such that U(A) > U(B) if and only if A is preferred to B and U(A) = U(B) if and only if the agent is indifferent to A and B. U(A) > U(B) A > B U(A) = U(B) A ~ B

Maximum Expected Utility Principle The utility of a lottery is the sum of the probability of each outcome times the utility of that outcome. U([p 1, S 1 ; … ; p n, S n ]) = ∑ p i U(S i )

16.3 Utility Functions Utility of money, Figure 16.2 Monotonic preference Expected Monetary Value, EMV –take $1000 or 50% chance of $3000 –EU(Accept) =.5 *U(S k ) +.5*U(S k+3000 ) –EU(Decline) = U(S k+1000 )

Kahneman and Tversky (1982) A: 80% chance of $4000 B: 100% chance of $3000 Subjects prefer B,.8U(4000) < U(3000) C: 20% chance of $4000 D: 25% chance of $3000 Subjects prefer C,.2U(4000) >.25U(3000)

Utility Scales Utility functions are not unique U’(S) = k 1 + k 2 * U(S), k 2 > 0 u┬, best possible outcome, 1 u┴, worst possible outcome, 0 Micromort (1:1,000,000 chance of death), $20 in 1980 QALY: quality-adjusted life year