# Another proof which is more algebraic and less

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Another proof, which is more algebraic and less geometric, builds on the state-dependent utility representation of Theorem 10.3 and appears in Kreps, pp. 110–111, . Exercise 10.16. Prove the necessity ( ) part of Theorem 10.14: if there exists a vNM index v : X R and a probability μ ΔΩ such that s μ ( s ) [ x v ( x ) h s ( x )] is a utility representation of % , then % is independent, Archimedean, and state-independent. Exercise 10.17. Prove the uniqueness part of Theorem 10.14: if there exist h g and U ( h ) = s μ ( s ) [ x v ( x ) h s ( x )] represents % , then U 0 ( h ) = s μ 0 ( s ) [ x v 0 ( x ) h s ( x )] represents % if and 61
only if μ 0 = μ and v 0 = av + b for some a > 0 and b R . Prove that if h g for all h, g H , then the uniqueness result fails. Exercise 10.18. Suppose Ω = { s 1 , s 2 , s 3 } and X = { x 1 , x 2 , x 3 } . The preference relation % is independent, Archimedean, and state-independent. The following are known: δ x 1 δ x 2 δ x 3 and ( δ x 1 , δ x 2 , δ x 3 ) ( δ x 1 , (0 , 1 2 , 1 2 ) , (0 , 1 2 , 1 2 )) (( 2 3 , 1 3 , 0) , ( 2 3 , 1 3 , 0) , δ x 3 ). Find the decision maker’s subjective belief μ . Exercise 10.19. Suppose there exists a vNM index v : X R such that % is represented by U ( h ) = min s Ω X x v ( x ) h s ( x ) . Prove or provide counterexamples to the following statements: % is independent; % is Archimedean; % is state-independent. Exercise 10.20. Suppose there exists a strictly positive probability μ Δ S and a vNM index v : X R such that % is represented by U ( h ) = X s μ ( s ) [min { v ( x ) : h s ( x ) > 0 } ] . Prove or provide counterexamples to the following statements: % is independent; % is Archimedean; % is state-independent. Exercise 10.21. Suppose there exists a closed set of strictly positive probabilities C Δ S and a vNM index v : X R such that % is represented by U ( h ) = min μ C X s μ ( s ) " X x v ( x ) h s ( x ) #! . Prove or provide counterexamples to the following statements: % is independent; % is Archimedean; % is state-independent. In the Anscombe–Aumann Expected Utility Theorem, the preference identifies two things: the utility index over consequences v : X R and the probability on states μ ΔΩ. This contrasts with the von Neumann–Morgenstern Expected Utility Theorem, which only identifies the utility index. Different preferences may imply different beliefs on Ω. 62
11 Basics of Savage expected utility This entire section is totally optional. It is only included to introduce interested students to what Kreps calls the “crowning achievement of singe-person decision theory.” In both the von Neumann–Morgenstern and Anscombe–Aumann models, we assumed the existence of some objective randomizing device. Ideally, all uncertainty in the model would be subjective: no probabilities are assumed a priori and all probabilities are identified by preferences over basic objects that are not contaminated with any assumed objective probability.