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

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The next axiom requires that the evaluation of coordinates is independent in the sense that preferences among alternatives of the form (e i ; (f;f¡i)); where (h; f¡i) :=(f1;:::;fi¡1;h;fi+1; :::; fn) ; depends solely on the i¡th coordinate of f. Formally, (A.3) (Certainty Principle) For all f; g; h; l 2 F n ; (e i ; (f;f¡i)) < (e i ; (g; g¡i)) if and only if (e i ; (f;h¡i)) < (e i ; (g; l¡i)): In view of the certainty principle, to simplify the notations, I shall henceforth write (ai;f) < (aj;f 0 ) instead of (e i ; (f;f¡i)) < (e i ; (f 0 ; g¡j)): To introduce the next axiom de…ne a partial mixture operation on D: Let (®; f) and (¯;f) in D then (¸® +(1¡ ¸)¯;f) ;¸2 [0; 1] is an element of D such that (¸® +(1¡ ¸)¯)i = ¸®i +(1¡ ¸)¯ i: This may be interpreted as a two-stage lottery in which in the …rst stage the alternatives (®; f) and (¯;f) are obtained <strong>with</strong> probabilities ¸ and 1 ¡ ¸; respectively. In the second stage the coordinate of f is selected by the lottery, ® or ¯, that corresponds to the outcome of the …rst stage. With this interpretation in mind assume that the decision maker prefers (®; f) over (® 0 ; g) and (¯;f) over (¯ 0 ; g): Moreover, assume that if a decision maker faces a choice between the alternatives d =(¸® +(1¡ ¸)¯;f) and d 0 = ¡ ¸® 0 +(1¡ ¸)¯ 0 ; g ¢ he reasons as follows: If the event whose probability is ¸ is realized then he participates in the lottery (®; f) if he has chosen d andinthe lottery (® 0 ; g) if he has chosen d 0 : Conditional on the realization of this event, he is better o¤ <strong>with</strong> d: Bythesamelogichewouldalsopreferd over d 0 conditional on the realization of the event whose probability is 1 ¡ ¸: Consequently, he prefers d over d 0 unconditionally. Formally, (A.4) (Constrained Independence) For all (®; f), (¯;f),(® 0 ; g), (¯ 0 ; g) in D and ¸ 2 [0; 1) if (®; f) » (® 0 ; g) then (¯;f) < (¯ 0 ; g) if and only if (¸® +(1¡ ¸)¯;f) < ¡ ¸® 0 +(1¡ ¸)¯ 0 ; g ¢ : Constrained independence is weaker than the independence axiom of expected utility theory (see discussion in Karni and Safra [2000]). A real-valued function V on D is said to represent < if, for all (®; f) and (¯;g) in D; (®; f) < (¯;g) if and only if V (®; f) ¸ V (¯;g) : Theorem (Karni and Safra [2000]) Let < be a binary relation on D. Then the following conditions are equivalent: (a) < is a preference relation satisfying (A.1)-(A.4). (b) There exist continuous non-constant functions V : D ! R and U (¢; ai) : F ! R; ai2 A, such that V represents < and, for all (®; f) 2 D; nX V (®; f) = ®iU(fi; ai): Moreover, if there are functions W : D ! R and W (¢; ai) :F ! R;ai 2 A; such that W represents < and W (®; f) = Pn i=1 ®iW (fi; ai); then W (¢; ai) = BU (¢; ai)+C; B > 0. 4 i=1
2.2 The main result For each a 2 A de…ne a conditional preference relation,
Page 1 and 2: Subjective Expected Utility Theory
Page 3: It seems, therefore, that an examin
Page 7 and 8: where, for each E µ S; ¼ (E;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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