STOCHASTIC

SUBJECTIVE PROBABILITIES AND EXPECTED UTILITIES 1425
Expressions (8) and (9) hold also for all P, Q in 3C0. Letting
it follows from Theorem 3 and
E[E(u, aP(s)+ (l-a)Q(«)),P*l
that, for all P, Q s 3C0 and a £ [0, 1],
w(P) = E[E(u, P(s)),P*\ for all J» e 3C„
= E[aE(u, P{s)) + (1 - a)E(u, Q(s)), P*]
= aE[E{u, P(S)), P*\ + (1 - a)E[E(u, (.)), P*]
P < Q if and only if w(P) g w(Q)
U'(aP + (1 - a)Q) = aw(P) + (1 - a)w(Q).
Then, by Theorem 1, w on 3C0 is a positive linear transformation of the restriction
of v on 3Co. By an appropriate transformation we can, with no loss in generality,
specify that
(10) v(P) = E[E(u,P(s)),P*]
for all P £ 3C0. According to (8) the proof of Theorem 4 can be completed by
proving that (10) holds for all bounded horse lotteries.
Our first step in this direction will be to prove that if P (A) = 1 and if c
and d defined in the following expression are finite then
(11) c = inf {£(«, P(s))\seA\ g v(P) S sup {E{u, P{s))\seA\ = d.
Let Q = P on A and tg Q(s) £ doni'. Since A" is null, Q ~ P and hence
v (P) = w (0) by (8). To show that cSt( n ) Si when c and d are finite suppose
to the contrary that d < v(Q). With c g E(u,Q') £ d and ()' = Q' on S let
i? = aQ + (1 — a)Q' with a < 1 near enough to one so that
d < v(R) = av(Q) + (1 - OL)V(Q') < v(Q).
Then R < Q by (8). But since E(u, Q(s)) g d < w(R) it follows from (8) that
Q(s) < R for all s e S so that axiom AG implies Q 4- R, a contradiction. Hence
d < v(Q) is false. By a symmetric proof, v(Q) < c is false. Therefore
c S «(0) S d.
With i> bounded let A with P*(A) = 1 be an event on which E(u, P(s)) is
bounded and let c and d be defined as in,(11). If c = d then (10) is immediate.
Henceforth assume that c < d. For notational convenience we shall take
c = 0, d = 1.
Let Q be as defined following (11) so that v(Q) — v(P) and
To prove that
E[E(u, Q(s)), P*\ = E[E(u, P(s)), P*}.
v(Q) =E[E(u,Q(s)),P*]
1. EXPECTED UTILITY THEORY 17