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

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Theorem 1 Let < be a binary relation on D; then the following conditions are equivalent: (i) < is a preference relation satisfying (A.1) - (A.7). (ii) There exist a nonatomic probability measure, ¼; on S, and continuous real-valued functions, u on X and ª on A £ R; such that V represents < and, for all (®; f) 2 D; V (®; f) = nX i=1 ®i Z S ª(ai;u(f (s)) d¼ (s) ; (1) where ª(a; ¢) is a monotonic increasing transformation. Moreover, ¼ is unique <strong>with</strong> ¼(E) > 0 if and only if E is null, and u is unique up to positive linear transformation. Remark: In reality decision makers choose among action-act pairs. Theorem 1 implies that the restriction of the preference relation < to A £ F has the following representation: For all (a; f) and (a 0 ;f 0 ) in A £ F; (a; f) < (a 0 ;f 0 ) , Z S Z ª(a; u (f (s)) d¼ (s) ¸ Hence the representation in (2) is applicable. ª(a S 0 ;u(f 0 (s))) d¼ (s) : (2) Proof. (i) ! (ii) : The proof that (ii) ! (i) is straightforward. I shall prove that Axiom (A.5) and Savage’s (1954) theorem imply that, for every given a 2 A; there exist a nonatomic probability measure ¼ (¢; a) on S and a real-valued function, u; on X such that, Z U(f; a) = u (f (s) ;a) d¼ (s; a) : (3) S where ¼ (¢; a) is unique and u (¢;a) is unique up to positive linear transformation. Moreover, U(f; a) represents
where, for each E µ S; ¼ (E;a) = R E u (x0 ;a) > 0 if and only if u (x; a0 ) ¡ u (x0 ;a0 ) > 0: Hence ¼ (s; a) ds: But, by (A.6), u (x; a) ¡ ¼ (E;a) ¡ ¼ (G; a) ¸ 0 , ¼ (E;a 0 ) ¡ ¼ (G; a 0 ) ¸ 0: (7) But equation (7) implies that, for every E µ S and all a; a 0 2 A; ¼ (E;a) = ¼ (E;a 0 ) = ¼ (E) : To prove the last assertion suppose, by way of negation, that for some E ½ S and a; a 0 2 A; ¼ (E;a) >¼(E;a 0 ) : Because the measures ¼ (¢;a) ;a 2 A are nonatomic, by Liapunov’s theorem 3 there exist K ½ S such that ¼ (E;a) >¼(K; a) =¼ (K; a 0 ) >¼(E;a 0 ) : (To see this observe that the point (¼ (E;a) ;¼(E;a 0 )) is below the diagonal of the unit square. But, by Liapunov’s theorem, the set f(¼ (T;a) ;¼(T;a 0 )) 2 [0; 1] 2 j T 2 2 S g is closed, convex and contains the diagonal. Let r satisfy ¼ (E;a) >r>¼(E;a 0 ) and K be an event whose image under (¼ (¢;a) ;¼(¢;a 0 )) is on the diagonal and satis…es ¼ (K; a) =¼ (K; a 0 )=r:) Then Substituting K for G in equation (7) leads to a contradiction. Hence ¼ (¢; a) =¼ (¢) for all a 2 A: Moreover, it is easy to verify that ¼ (E) > 0 if and only if E is null. Equation (3) together <strong>with</strong> the Theorem of Karni and Safra (2000) imply that V (®; f) = nX i=1 ®i Z u(fi (s) ;ai)d¼ (s) ; S where V and u are continuous functions. But, by (A.6), u(¢;a) is ordinally equivalent to u (¢;a 0 ) for all a; a 0 2 A: Hence, for all ai 2 A; u(¢;ai) =ª(ai;u(¢;a1)) where ª(ai; ¢) is monotonic increasing continuous transformation. Let u(¢) = u(¢;a1) to obtain the representation in (ii) : The uniqueness follows from Savage’s theorem. 2.3 Action-a¢ne utility representation Insofar as the dependence of the utility function on the action is concerned, the result in Theorem 1 is quite general. In some applications the e¤ect of the choice of action on the decision maker’s well-being calls for a more speci…c functional form of the utility function. For example, when actions correspond to levels of e¤ort, it is commonplace to assume some form of separability between the utility of money and the utility of e¤ort. I explore next the axiomatic underpinnings of two alternative, more speci…c, representations of the decision maker’s preferences. The …rst, action-a¢ne utility representation, was used in Grossman and Hart (1983). The next axiom requires that, for any given event, E; the “intensity of preferences” between consequences on the complementary event S ¡E be independent of the action. The formal statement of this idea is an adaptation of Wakker’s (1987) cardinal consistency axiom. 4 3 See Hildenbrand (1974). 4 See Karni (2003, 2003a) for di¤erent adaptations of the same idea. 7
Page 1 and 2: Subjective Expected Utility Theory
Page 3 and 4: It seems, therefore, that an examin
Page 5: 2.2 The main result For each a 2 A
Page 9 and 10: Proof. (i) ) (ii) : Fix an event E
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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