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

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 with 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