STOCHASTIC

MINIMAX POLICIES FOB SELLING AN ASSET AND DOLLAR AVERAGING 383
The second stage of the maximization procedure is to maximize (4) with respect to
t*. This gives the maximum of B for all possible Ap(. It is shown by the following
argument that this maximum occurs either at I* = 0 or t — T. The proof is by contradition.
Suppose that a unique maximum with respect to t is attained for some t
greater than zero and less than T. Suppose further that at this value of t ,
a(l — Zf) fe hz,'.
Since z, is nonincreasing, it must then be true that a(l — Zi) ^ bz, for all t such that
(* £ I S T. Inspection of (4) shows that the assumed value of f cannot give a unique
maximum then, since increasing its value cannot lower the value of R under these
conditions. Suppose next that the inequality is reversed at the assumed maximum
value of t* so that a(l — z(.) < bz,-. From the fact that z, is nonincreasing with t
it follows that o(l — zt) < bz, for all t such that 0 S ( S (. Inspection of (4) indicates
that R will increase as t* is decreased under these conditions. Thus, the assumed
value for t could not give a maximum. Since it has been shown that a value of t
other than t* = 0 or t* = T cannot give a unique maximum to R it follows that the
maximum must occur at either t* — 0 or t = T. Thus, substituting these values
for t* in (4) gives that the maximum of R for all possible values of the Ap, , given
any feasible values of the z,, must be equal to Max [a(T — Ei T z,), !>Ei r «,]. To
obtain the minimax nonsequential strategy the last step is to minimize this maximum
over all feasible values of the 2,. This is easily done. Any feasible set of z, which satisfy
the relation in (5) will minimize Max [a(T — Ei T z,), 6£]i r z,] and give the
required strategy.
(5) Ei'«« = aT/(a + b).
The minimax value of R using this strategy is given in (6):
(6) Min. MaxiriJ = Min, Max [a(T - Ei r «i), &Ei r z