Non-Additive Measures and their Applications to Decision Theory Jean-Yves Jaffray, LIP6 , UPMC-Paris6 2nd SIPTA Summer School – Madrid, July 24-28, 2006
The decision-theoretic Approach Its specific features • A representation of data/beliefs concerning a family of events (e.g., subjective probabilities ) is of no interest by itself • It only becomes interesting when it gets imbedded into a decision model (e.g., subjective probabilities into subjective expected utility theory) 2 / 1
Choice vs Preference P.A.S AMUELSON , Revealed Preference Theory • People make choices: a choice consists in selecting a decision d chosen from a feasible decision set D . Choices are observable (at least in principle) • Decision theory hypothesizes the existence of preferences: assumed preference relation � determines the preferred element d pref of decision set D • Choice function Ch : D �→ Ch ( D ) = d chosen reveals preference relation � when for every decision set D , d pref = d chosen . 3 / 1
Descriptive vs. Normative Decision Theory • The objective of descriptive decision models is to explain parcimoniously people’s observed choices as well as to predict accurately their future choices • The goal of normative decision models is to help people make better decisions by: (i) providing guidelines for rational behavior; and (ii) being operational in real applications • Normative decision models generally serve as references for building descriptive models 4 / 1
Representation Theorems in Decision Theory • When possible, preferences are expressed by a decision criterion V which reduces the comparison of two decisions a and b to that of two numbers V ( a ) and V ( b ) ; V is a utility function . • A representation theorem states that a system of axioms - a list of properties of preference relation � - is sufficient (resp. necessary and sufficient, in the best cases) for the existence of a decision criterion representing � • The axiom system generally comprises rationality axioms, behavioral axioms and technical axioms • A representation theorem unveils the implicite assumptions on which the criterion rests; they can then be tested separately (descriptive models) or discussed (normative models) 5 / 1
Decision Making under Risk • Risk denotes a decision situation in which events are endowed with extraneous probabilities Π known from the DM • Whether or not the DM accepts these probabilities (makes them his own subjective probabilities) can be revealed by his choices • Each decision d generates a probability distribution P on the consequence set C : P ( C ) = Π ( { d yields a consequence in C } ) , C ⊆ C • A finite range probability with support S = { c i , i ∈ I } , I finite, is called a lottery ; it is completely defined by { P ( { c i } ) , i ∈ I } • Preferences are assumed to depend only on these probability distributions: � is defined on P = { P } 6 / 1
Expected Utility (EU) criterion under Risk • Linear utility theory ( VON N EUMANN and M ORGENSTERN , 1947) derives from a series of axioms (Ordering, Independence, Continuity, Dominance) the validity of the Expected Utility (EU) criterion: the utility U ( P ) of probability P is the mathematical expectation of a function u with respect to P , � U ( P ) = C u ( c ) dP • Case of a lottery P with support S = { c i , i ∈ I } , U ( P ) = � n i = 1 P ( { c i } ) u ( c i ) • u : C −→ R , is the von Neumann- Morgenstern (vNM) utility function . Its shape is interpreted as indicating the DM’s attitude w.r.t. risk; e.g., u concave ⇐⇒ Risk-Aversion. 7 / 1
Decision Making under Uncertainty • decisions are identified to acts ,which are mappings Ω −→ C , where Ω is the set of states of nature and events are subsets of it • Information concerning the events can take various forms, from purely objective to purely subjective ones, and be more or less precise: complete ignorance, objective (resp. subjective) upper \ lower probabilities, etc. • In the purely subjective case, in which beliefs concerning the events are totally implicite, the standard axiomatic model, due to L.J. S AVAGE (1954), justifies a Subjective Expected Utility (SEU) criterion 8 / 1
Subjective Expected Utility (SEU) (I) L.J. S AVAGE (1954) • In S AVAGE ’s model, preferences � are defined on the set of all acts A and required to satisfy a list of axioms • If act f E offering prize ( M , m ) on E , i.e. f E ( ω ) = M , ω ∈ E ; f E ( ω ) = m , ω � E is preferred to similar act f E ′ , then E is declared qualitatively more probable than E ′ . This relation turns out to be a comparative probability which is uniquely representable by a (quantitative) probability Π • Given Π , decisions d generate probability distributions P d on C , on which � induces a relation satisfying the vNM axioms 9 / 1
Subjective Expected Utility (SEU) (II) • Since probability distributions P d on C are ordered by relation � satisfying the vNM axioms • The utility V ( d )) of decision d is thus � � V ( d ) = U ( P d ) = u ( c ) dP d = u ( d ( c )) d Π C Ω • or, case of a finite range n � Π ( d − 1 { c i } ) u ( c i ) V ( d ) = i = 1 10 / 1
The Sure Thing Principle (I) This is S AVAGE ’s key axiom • If acts a , b , a ′ , b ′ and an event E satisfy a | E = a ′ b | E = b ′ a ′ | E c = b ′ | E , | E , a | E c = b | E c , | E c , then a � b ⇐⇒ a ′ � b ′ • In words, Common modifications of common parts of acts should not change their ranking 11 / 1
The Sure Thing Principle (II) A common modification of common part of acts a and b does not change their ranking : a � b ⇐⇒ a ′ � b ′ . b , b ′ a , b a , a ′ a ′ , b ′ E c E 12 / 1
The A LLAIS Paradox M.A LLAIS (1953) • Consider lotteries A, B, C, and D A : 10 k e with certainty ; B : 50 k e with probability 10 11 ; nothing otherwise ; C : 10 k e with probability 11 100 ; nothing otherwise ; D : 50 k e with probability 10 100 ; nothing otherwise • U ( A ) = u ( 10 ) ; U ( B ) = 10 11 u ( 50 ) + 1 11 u ( 0 ) U ( C ) = 11 100 u ( 10 ) + 89 100 u ( 0 ) = 11 100 U ( A ) + 89 100 u ( 0 ) U ( D ) = 10 100 u ( 50 ) + 90 100 u ( 0 ) = 11 100 U ( B ) + 89 100 u ( 0 ) • EU predicts (property of means): A ≻ B ⇐⇒ C ≻ D • Experimental result: more than half of the subjects choose A against B and D against C ; EU is not good descriptive model ! 13 / 1
Rank Dependent Utility (RDU) under Risk • Rank Dependent Utility (RDU) theory (J.Q UIGGIN , 1982) is an axiomatic model which is more flexible than EU and accommodates the A LLAIS paradox. • Beside the vNM utility, u , RDU possesses an additional parameter, which is a function ϕ operating on the decumulative distribution functions G ( c ) = P ( { c ′ : u ( c ′ ) > u ( c ) } ) • ϕ is called the weighting function ; ϕ is strictly increasing from [ 0 , 1 ] onto [ 0 , 1 ] (thus ϕ ( 0 ) = 0 and ϕ ( 1 ) = 1). 14 / 1
A typical weighting function which is consistent with certainty and potential effects 1 potential effect ϕ certainty effect 0 0 1 A 15 / 1
Expression of the RDU criterion: the finite case • In EU, lottery P with support { c i , i = 1 , .., n } , where u ( c 1 ) � u ( c 2 ) � .. � u ( c n ) has utility U ( P ) = � n − 1 i = 1 [ � n j = i P ( { c j } ) − � n j = i + 1 P ( { c j } )] u ( c i ) + P ( { c n } ) u ( c n ) = u ( c 1 ) + � n − 1 i = 1 ( � n j = i + 1 P ( { c j } ))[ u ( c i + 1 ) − u ( c i )] • Operating transformation ϕ on the decumulated sums we get V ( P ) = � n − 1 i = 1 [ ϕ ( � n j = i P ( { c j } )) − ϕ ( � n j = i + 1 P ( { c j } )] u ( c i ) + φ ( P ( { c n } )) u ( c n ) = u ( c 1 ) + � n − 1 i = 1 ϕ ( � n j = i + 1 P ( { c j } ))[ u ( c i + 1 ) − u ( c i )] • The dominant behavioral pattern of the A LLAIS Paradox is now acceptable; it requires a weighting function ϕ satisfying 11 ) < ϕ ( 10 100 ) ϕ ( 10 ϕ ( 11 100 ) 16 / 1
Expression of the RDU criterion: the general case • In EU, the utility U ( P ) of probability distribution P of random variable c can be written under equivalent form: � U ( P ) = C u ( c ) dP = � 0 � ∞ − ∞ [ P (( u ( C ) > t ) − 1 ] dt + 0 P (( u ( c ) > t ) dt • Operating transformation ϕ on the decumulated sums we get � 0 � ∞ V ( P ) = − ∞ [ ϕ ( P (( u ( c ) > t )) − 1 ] dt + 0 ϕ ( P (( u ( c ) > t )) dt • As we shall see later, this is a particular case of a C HOQUET integral, where the capacity involved is a probability transform. 17 / 1
The E LLSBERG Experiment E LLSBERG (1961) • An urn contains 90 balls: 30 are red, and 60 are blue or yellow (in unknown proportions). A ball is to be drawn at random (event R = ”a Red ball is drawn”; etc...), and the DM has to compare: (i) alternatives f R and f B ; (ii) alternatives f R ∪ Y and f B ∪ Y ; with: f R : win M conditionally on event R ; f B : win M conditionally on event B ; f R ∪ Y : win M conditionally on event R ∪ Y ; f B ∪ Y : win M conditionally on event B ∪ Y . • E LLSBERG has observed the predominant preference pattern: f R ≻ f B and f B ∪ Y ≻ f R ∪ Y 18 / 1
Recommend
More recommend
