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eco331-11-ValueOfInformation2Handout

# eco331-11-ValueOfInformation2Handout - Overview Formal...

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Unformatted text preview: Overview Formal Setup Foundations A Diversion: Ambiguity Eco331: Value of Information — Foundations Marciano Siniscalchi November 11 Overview Formal Setup Foundations A Diversion: Ambiguity Overview Value of Information: Formal Definition Emphasize general case (non-linear utility) Information is good Foundations: Static vs. Dynamic Choice A digression: ambiguity and dynamic choice Overview Formal Setup Foundations A Diversion: Ambiguity Value-of-Information Problem Definition A value-of-information problem ( VOIP ) is a tuple ( S , P , I , A ,ξ ): S is the state space and P is a probability on S ; I is the information partition , with P ( E ) > 0 for all E ∈ I ; A is an arbitrary set of actions ; ξ : S × A → R is the prize function . action a ∈ A and state s ∈ S : get (monetary) prize ξ ( s , a ). actions are labels; interpretable in applications (e.g. prices). actions induce random variables, so we can use EU: Remark (1) For every a ∈ A and s ∈ S, let X a ( s ) = ξ ( s , a ) ; then X a : S → X is a r.v. on S. (2) For every i = 1 ,..., n, every a ∈ A, and every s ∈ E i , let X i a ( s ) = ξ ( s , a ) ; then X i a : E i → X is a r.v. on E i . Overview Formal Setup Foundations A Diversion: Ambiguity Defining the Value of Information Optimize ex-ante, no info: max a ∈ A ∑ s ∈ S P ( { s } ) u ( ξ ( s , a )). Optimize interim given E i , info price M : max a i ∈ A ∑ s ∈ E i P ( { s }| E i ) u ( ξ ( s , a i )- M ). Ex-ante EU given interim optimal actions, info price M : ∑ n i =1 P ( E i ) max a i ∈ A ∑ s ∈ E i P ( { s }| E i ) u ( ξ ( s , a i )- M )....
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eco331-11-ValueOfInformation2Handout - Overview Formal...

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