Decisions under risk and partial knowledge modelling uncertainty and - PowerPoint PPT Presentation

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Decisions under risk and partial knowledge modelling uncertainty and risk aversion Giulianella Coletti a Davide Petturiti a , b Barbara Vantaggi b a University of Perugia b “La Sapienza” University of Rome 9 th Int. Symp. on Imprecise Probability: Theories and Applications July 20-24, 2015 Pescara, Italy

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Classical decision theory under risk • L = { L = ( X L , P L ) } , a set of lotteries on X = { x 1 , . . . , x n } , where P L is a probability distribution with support X L ⊆ X • � , a preference relation on L von Neumann-Morgenstern’s axioms are equivalent to the existence of a linear function U : L → R (unique up to positive linear transformations) representing � , i.e., for L , L ′ ∈ L L � L ′ ⇔ U ( L ) ≤ U ( L ′ ) ⇒ If L contains the degenerate lotteries L 0 = { δ x : x ∈ X } then there exists u : X → R s.t. for L ∈ L � U ( L ) = E P L ( u ) = u ( x ) P L ( x ) x ∈ X

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Problems • The DM cannot consider just a finite set of lotteries L • The DM has to provide comparisons between “certainty equivalents” and “risky prospects” • It is not possible to consider imprecise probabilities Imprecise information In order to deal with imprecise information some axiomatizations, that generalizes von Neumann-Morgenstern theory, have been provided (see e.g. Jaffray (1989), Gaidos et al. (2004)). Actually, these axiomatizations are not structure free. Aim The aim is to provide a rational criterion where the DM expresses just few preferences.

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Partial preferences on an arbitrary set L Strengthened preference relation Given an arbitrary set of lotteries L , we consider a pair of consistent relations ( � , ≺ ) where none of � or ≺ is assumed to be complete and ≺� = ∅ : • L � L ′ stands for “ L is not preferred to L ′ ”; • L ≺ L ′ stands for “ L ′ is preferred to L ”. Representation We search for a function U : L → R representing ( � , ≺ ), i.e., for every L , L ′ ∈ L L � L ′ ⇒ U ( L ) ≤ U ( L ′ ) and L ≺ L ′ ⇒ U ( L ) < U ( L ′ ) .

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension A paradigmatic example U 1 L 2 ≺ L 1 , L 4 ≺ L 3 P 1 = { P θ } with θ ∈ 0 , 2 � � for no θ there exists u : { 0 , 100 } → R 3 { w } { b } { r } s.t. E P θ ( u ( L )) represents ≺ 1 2 P θ θ 3 − θ 3 w b r Problem: Probability is not suit- L 1 100 e 0 e 0 e able to measure uncertainty in L 2 0 e 0 e 100 e situations as those described 0 e 100 e 100 e L 3 above: non-additive uncertainty L 4 100 e 100 e 0 e measures come to the fore

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Generalized lotteries Definition [Jaffray 1987] A generalized lottery , or g-lottery for short, on a finite set X L is a pair L = ( ℘ ( X L ) , Bel L ) where Bel L is a belief function on ℘ ( X L ), i.e.: (i) Bel L ( ∅ ) = 0 and Bel L ( X L ) = 1; �� n ( − 1) | I | +1 Bel L � � �� � (ii) Bel L i =1 A i ≥ i ∈ I A i for every A i ∈ ℘ ( X L ). ∅� = I ⊆{ 1 ,..., n } ⇒ A g-lottery could be equivalently defined as L = ( ℘ ( X L ) , m L ), where m L is the basic assignment associated to Bel L defined for every A ∈ ℘ ( X L ) as � ( − 1) | A \ B | Bel L ( B ) m L ( A ) = B ⊆ A ⇒ m L is a function from ℘ ( X L ) to [0 , 1] s.t. m L ( ∅ ) = 0 and � m L ( A ) = 1 A ∈ ℘ ( X L ) ⇒ Probability measures on ℘ ( X L ) are particular belief functions

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Generalized convex lotteries Definition A gc-lottery on a finite set X L is a pair L = ( ℘ ( X L ) , ϕ L ) where ϕ L is a convex capacity on ℘ ( X L ), i.e. ϕ ( A ∪ B ) ≥ ϕ ( A ) + ϕ ( B ) − ϕ ( A ∩ B ) . (1) ⇒ A gc-lottery could be equivalently defined as L = ( ℘ ( X L ) , m L ), where m : ℘ ( X ) → R is the basic assignment associated to ϕ L . ⇒ For every A ∈ ℘ ( X ) with | A | ≥ 2 and every { x i , x j } ⊆ A , it satisfies � { x i , x j }⊆ B ⊆ A m ( B ) ≥ 0

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Operations on gc-lotteries ⇒ Given L = { L = ( ℘ ( X L ) , ϕ L ) } , if X = ∪{ X L : L ∈ L} is finite, then all gc-lotteries can be rewritten on X : they reduce to convex capacities on ℘ ( X ) Convex combination of gc-lotteries For L 1 , . . . , L t ∈ L and k = ( k 1 , . . . , k t ) with k i ≥ 0 ( i = 1 , . . . , t ) and � t i =1 k i = 1, the convex combination of L 1 , . . . , L t according to k is the gc-lottery on X � � A k ( L 1 , . . . , L t ) = for every A ∈ ℘ ( X ) \ {∅} . (2) � t i =1 k i m L i ( A ) Set of degenerate gc-lotteries L ∗ 0 = { δ B : B ∈ ℘ ( X ) \ {∅}} , where m δ B ( B ) = 1 for B ∈ ℘ ( X ) \ {∅}

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Jaffray’s linear representation for g-lotteries If L is closed under convex combinations of g-lotteries, von Neumann-Morgenstern axioms are equivalent to the existence of a linear function LU : L → R (unique up to p.l.t.) representing � , i.e., for L , G ∈ L L � G ⇔ LU ( L ) ≤ LU ( G ) ⇒ If L contains also the degenerate g-lotteries , then there exists v : ℘ ( X ) → R s.t. for L ∈ L � LU ( L ) = v ( B ) m L ( B ) B ∈ ℘ ( X ) ⇒ The semantic interpretation of “utility” function v on ℘ ( X ) is not clear, moreover, it requires to specify a number of parameter in the order of 2 card X ⇒ A possible alternative is to search for a function u : X → R and to use the Choquet expected utility functional [Schmeidler 1989] defined for L ∈ L as � CEU ( L ) = C u d Bel L

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Ordered set of prizes Consider • L , set of gc-lotteries • X = � { X L : L ∈ L} = { x 1 , . . . , x n } totally ordered as x 1 < . . . < x n Aggregated basic assignment The aggregated basic assignment of L ∈ L is defined for every x i ∈ X as � M L ( x i ) = m L ( B ) , x i ∈ B ⊆ E i where E i = { x i , . . . , x n } for i = 1 , . . . , n . ⇒ M L is a “pessimistic” probability dis- tribution on X induced by ϕ L ⇒ If u : X → R is strictly increasing u d ϕ L = � n � c i =1 u ( x i ) M L ( x i ) Assumpion on L (AO) L 0 = { δ { x } : x ∈ X } ⊆ L and ∀ x , x ′ ∈ X , x ≤ x ′ ⇔ δ { x } � δ { x ′ }

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Generalized Choquet rationality condition (g-CR) Definition A strengthened preference relation ( � , ≺ ) on an arbitrary set L of gc-lotteries is said to be generalized Choquet rational if it satisfies: For all h ∈ N and L i , L ′ i ∈ L with L i � L ′ i ( i = 1 , . . . , h ), if k ( M L 1 , . . . , M L h ) = k ( M L ′ 1 , . . . , M L ′ h ) (g-CR) with k = ( k 1 , . . . , k h ), k i > 0 ( i = 1 , . . . , h ) and � h i =1 k i = 1, then it can be L i ≺ L ′ i for no i = 1 , . . . , h . ⇒ (g-CR) involves aggregated basic assignments: convex combinations are in the usual sense, i.e., among probability distributions ⇒ If � is complete and L is convex, (g-CR) implies von Neumann-Morgenstern axioms and (*) for every L , L ′ ∈ L , M L = M L ′ ⇒ L ∼ L ′

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension CEU representation theorem Theorem Let L be a finite set of gc-lotteries, X = � { X L : L ∈ L} = { x 1 , . . . , x n } and let ≤ ∗ be a total preorder on X . For a strengthened preference relation ( � , ≺ ) on L satisfying (A0) the following statements are equivalent: (i) ( � , ≺ ) is Choquet rational (i.e., it satisfies (gc-CR) ); (ii) there exists a strictly increasing function u : X → R , whose CEU functional defined, for every L ∈ F n � � CEU F ( L ) = C u F d ϕ L = u F ( x i ) M L ( x i ) i =1 represents ( � , ≺ ) .

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Risk aversion in the case of money payoffs Suppose X = { x 1 , . . . , x n } ⊂ R and ≤ ∗ ≡≤ with x 1 < . . . < x n : M L is a probability distribution on X for every L ∈ L . Assumptions (A1) and (A1*) x i +1 − x i Let k i = ( k i , 1 − k i ) be with k i = x i +1 − x i − 1 ( i = 2 , . . . , n − 1), define � � � � L 1 = k i : i = 2 , . . . , n − 1 δ { x i − 1 } , δ { x i +1 } � � (A1) L 1 ⊆ L and k i ≺ δ { x i } or δ { x i − 1 } , δ { x i +1 } � � k i ∼ δ { x i } . δ { x i − 1 } , δ { x i +1 } (A1*) L 1 ⊆ L and k i � � ≺ δ { x i } . δ { x i − 1 } , δ { x i +1 }

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension Risk aversion in the case of money payoffs Proposition [Risk aversion in case of money payoffs] Assume ( � , ≺ ) satisfies (A0) and (gc-CR) and let u be a utility whose CEU represents ( � , ≺ ). The following statements hold: (i) if (A1) holds then u extends to a strictly increasing concave function v ∈ C 0 ([ x 1 , x n ]); (ii) if (A1*) holds then u extends to a strictly increasing strictly concave function w ∈ C 2 ([ x 1 , x n ]).