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

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(A.8) (Action-independent intensity of preferences) For all f;g;h;l 2 F , a; a0 2 A, E ½ S; and w; x; y; z; 2 X, iffEw
Proof. (i) ) (ii) : Fix an event E ½ S such that both E and S ¡ E are nonnull and a 2 A; then, by Theorem 1, there exist f;g;2F andsuchthat Z [ª (a; u(f (s))) ¡ ª(a; u(g (s)))] d¼ (s) =³>0: (10) E Given a0 2 A; by Theorem 1, there exist acts h; l 2 F that satisfy Z [ª (a E 0 ;u(h (s))) ¡ ª(a 0 ;u(l (s)))] d¼ (s) =">0. (11) By continuity of the functions ª(a; ¢) and u; for every ^ ³ 2 [¡³;³]; ^" 2 [¡"; "]; there exist f 0 ;g 0 ;h 0 ;l 0 2F and such that and Z Z [ª (a; u(f E 0 (s))) ¡ ª(a; u(g 0 (s)))] d¼ (s) = ^ ³ (12) [ª (a E 0 ;u(h 0 (s))) ¡ ª(a 0 ;u(l 0 (s)))] d¼ (s) =^". (13) De…ne Á a 0 by ª(a 0 ;u(¢)) = Á a 0 ± ª(a; u(¢)) for all a 0 2 A: Then Á a 0 is a continuous function. To show that Á a 0 is positive a¢ne or constant take ®; ¯; °; ± 2 R such that ¡³ · ® ¡ ¯ = ° ¡ ± · ³ and ¡" · Á a 0 (¯) ¡ Á a 0 (®) · ": Let w; x; y; z 2 R satisfy ª(a; u(x)) ¼ (S ¡ E) =®; ª(a; u(w)) ¼ (S ¡ E) =¯; u (z; a) ¼ (S ¡ E) =° and ª(a; u(y)) ¼ (S ¡ E) =±: Take ^ f;^g 2 F such that Z E h ³ ª a; u( ^ ´ i f (s)) ¡ ª(a; u(^g (s))) d¼ (s) =® ¡ ¯: (14) Thus ^ fEw »a ^gEx and ^ fEy »a ^gEz: Take ^ h; ^ l2F such that, Z E h ³ ª a 0 ;u( ^ ´ ³ h (s)) ¡ ª a 0 ;u( ^ ´i l (s)) d¼ (s) =Áa0 (®) ¡ Áa0 (¯) : (15) Since ª(a 0 ;u(¢)) = Á a 0 ± ª(a; u(¢)) this implies ^ hEw »a 0 ^ lEx: Applying (A.8) twice yields ^ hEy »a0 ^lEz: Thus Z Á a 0 (±) ¡ Á a 0 (°) = E h ³ ª a 0 ;u( ^ ´ ³ h (s)) ¡ ª a 0 ;u( ^ ´i l (s)) d¼ (s) =Áa0 (¯) ¡ Áa0 (®) : (16) By Wakker (1987) Lemma 4.4 this implies that Á a 0 is a¢ne. But Á a 0 is nondecreasing. (To see this, let fEw
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: where, for each E µ S; ¼ (E;a) =
Page 11 and 12: Two actions, a; a 0 2 A are element
Page 13 and 14: where (¹a; ¹w) denotes the “out
Page 15: [15] Savage, Leonard, J. (1954) The

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