Savage in the Market Federico Echenique Kota Saito California - PowerPoint PPT Presentation

Epstein (2000) Necessary Condition for prob. sophistication: if ∃ ( x , p ) and ( x ′ , p ′ ) � (i) p 1 ≥ p 2 � and p ′ 1 ≤ p ′ 2 with at least one strict ineq. (ii) x 1 > x 2 and x ′ 1 < x ′ 2 ⇒ Not Probability Sophisticated { ( x 1 , x 2 ) , ( x ′ 2 , x ′ 1 ) } satisfy conditions in SARSEU: so must have p 1 p ′ 2 ≤ 1 , p 2 p ′ 1 hence can’t violate Epstein’s condition.

A probabilistically sophisticated data set violating SARSEU.

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Maxmin � U ( x ) = min µ s u ( x s ) µ ∈ M s ∈ S M is a convex set of priors.

Maxmin ( x k , p k ) K k =1 is maxmin rational if ∃ ◮ convex set M ⊆ ∆ ++ ◮ and u ∈ C s.t. y ∈ B ( p k , p k · x k ) ⇒ min � � µ s u ( x k µ s u ( y s ) ≤ min s ) . µ ∈ M µ ∈ M s ∈ S s ∈ S

Maxmin Proposition Let S = K = 2 . Then a dataset is max-min rational iff it is SEU rational. Example with S = 2 and K = 4 of a dataset that is max-min rational and violates SARSEU.

Objective Probabilities � µ s u ( x s ) max p · x ≤ I ◮ Observables: µ , p , x ◮ Unobservables: u Varian (1983), Green and Srivastava (1986), and Kubler, Selden, and Wei (2013)

Objective Probabilities Varian (1983), Green and Srivastava (1986): FOC µ s u ′ ( x s ) = λ p s , (linear ”Afriat” inequalities). Kubler, Selden, and Wei (2013): axiom on data.

Objective Probabilities s ) = λ k p k u ′ ( x k s = λ k ρ k s , µ s ◮ ρ k s = p k s /µ s is a “risk neutral” price.

Objective Probabilities (Strong Axiom of Revealed Exp. Utility (SAREU)) k ′ For any ( x k i i ) n s i , x i i =1 s.t. s ′ k ′ 1. x k i s i > x i s ′ i 2. each k appears in k i (on the left of the pair) the same number of times it appears in k ′ i (on the right): we have: n ρ k i � s i ≤ 1 . k ′ ρ i i =1 s ′ i Theorem A dataset is EU rational if and only if it satisfies SAREU.

Savage Primitives: infinite S ; � on acts: information on all pairwise comparisons. Define � to be the rev. preference relation defined from a finite dataset ( x k , p k ): ◮ x k � y if y ∈ B ( p k , p k · x k ) ◮ x k ≻ y if . . . ◮ note: � is incomplete.