Subjective Expected Utility Theory with Costly Actions - Economics ...

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(ii) ) (i) : Assume that there exist positive a¢ne or constant transformations Á a 0 such that ª(a 0 ;u(¢)) = Á a 0 ± ª(a; u(¢)) for all a 0 2 A. Fix an event E ½ S such that E and S ¡ E c are nonnull and suppose that fEw
Two actions, a; a 0 2 A are elementarily linked if there are x; x 0 ;y;y 0 2 X such that (a; x) Â (a; y) ; (a; x) » (a 0 ;x 0 ) and (a; y) » (a 0 ;y 0 ) : Two actions, a and a 0 are said to be linked if there exists a …nite sequence a = a0;a1; :::; an = a 0 such that every aj is elementarily linked <strong>with</strong> aj+1: The axiomatic underpinning of the separately additive speci…cation consists of the preceding axioms and, in addition, the following independence condition, (A.9) (Independence) For all E ½ S; a; a 0 2 A and x; x 0 ;y;y 0 2 X; (a; xEy) < (a 0 ;x 0 E y) if and only if (a; xEy 0 ) < (a 0 ;x 0 E y0 ) : The next theorem characterizes the separately additive representation. Theorem 4 Let < be a binary relation on D and suppose that every pair of actions a; a 0 2 A is linked. Then the following conditions are equivalent: (i) < is a preference relation satisfying (A.1)-(A.9). (ii) There exist a nonatomic probability measure ¼ on S, real-valued continuous functions, u on X and ¹ on A, and a continuous function V representing
Page 1 and 2: Subjective Expected Utility Theory
Page 3 and 4: It seems, therefore, that an examin
Page 5 and 6: 2.2 The main result For each a 2 A
Page 7 and 8: where, for each E µ S; ¼ (E;a) =
Page 9: Proof. (i) ) (ii) : Fix an event E
Page 13 and 14: where (¹a; ¹w) denotes the “out
Page 15: [15] Savage, Leonard, J. (1954) The

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