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

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