is coser to the city dominates 5.on cost with distance from city Stochastic dominance contd. Stochastic dominance Typically define attributes such that[is monotonic in each Strict dominance Label the arcs or the +9 号Strict dominance Typically define attributes such that U is monotonic in each Strict dominance: choice B strictly dominates choice A iff ∀ i Xi(B) ≥ Xi(A) (and hence U(B) ≥ U(A)) 1 X 2 X A B C D 1 X 2 X A B C dominates This region A Deterministic attributes Uncertain attributes Strict dominance seldom holds in practice Chapter 16 13 Stochastic dominance 0 0.2 0.4 0.6 0.8 1 1.2 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 Probability Negative cost S2 S1 0 0.2 0.4 0.6 0.8 1 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 Probability Negative cost S2 S1 Distribution p1 stochastically dominates distribution p2 iff ∀ t Z −∞ t p1(x)dx ≤ Z −∞ t p2(t)dt If U is monotonic in x, then A1 with outcome distribution p1 stochastically dominates A2 with outcome distribution p2: Z −∞ ∞ p1(x)U(x)dx ≥ Z −∞ ∞ p2(x)U(x)dx Multiattribute case: stochastic dominance on all attributes ⇒ optimal Chapter 16 14 Stochastic dominance contd. Stochastic dominance can often be determined without exact distributions using qualitative reasoning E.g., construction cost increases with distance from city S1 is closer to the city than S2 ⇒ S1 stochastically dominates S2 on cost E.g., injury increases with collision speed Can 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 ∀ x1, x2 x1 ≥ x2 ⇒ P(Y |x1, z) stochastically dominates P(Y |x2, z) Chapter 16 15 Label the arcs + or – SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost Chapter 16 16 Label the arcs + or – SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost + Chapter 16 17 Label the arcs + or – SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost + + Chapter 16 18