strategies are not represented by the same point. It will be evident
that in choosing a strategy we only need consider strategies that
are not dominated by any other strategy. These are called admissible
strategies. The points representing them will not have any other
point (representing a strategy) to the left and below them.
At this stage we consider randomized strategies. Suppose the
decision maker has come to the conclusion that he considers Si
and s5 to be good strategies, but that he cannot choose between
them. He tosses a coin, having decided beforehand that he will
take si if it falls heads and s5 otherwise. This mixture of strategies
is itself a new type of strategy, a randomized one. It is not difficult
to prove that it is represented by a point lying midway between
the points Si and s5. Using this principle with different probabilities
it is clear that we may consider any point of the line segment
si-s5 to represent a random mixture of Si and s5. It is clear that the
set of all strategies (pure and randomized) is represented by the
convex set S of points bounded by the lines connecting si-S2-S3-s7-
-s8-s5-si. As a result of the admission of randomized strategies, s6
has become inadmissible, because it is dominated by mixtures of
s5 and s8. The admissible strategies are now those represented by
points on the admissible part of the boundary s7-s8-s5-si. From
these a choice will have to be made. This choice will depend on
what one regards as "optimal". The minimax expected loss criterion
is based on the following consideration:
Suppose nature is doing her best to let us suffer as much loss as
possible. Then we should select that strategy whereby the maximum
expected loss is as small as possible. If we first restrict ourselves
to the expected losses of the pure strategies in table 5, we see that
the maximum losses in each column are
The strategy Si has the smallest maximum loss, which is 25,
occurring when O2 is true. This is the minimax expected loss strategy
among the pure strategies. If we take the randomized strategies
into account, we find the minimax expected loss strategy by letting
a square, whose left bottom corner lies in the origin, increase in
size until some part of it touches the convex set 5 of the points
representing the strategies. This is illustrated in fig. 2, where the
procedure yields s9 as the minimax expected loss strategy. Clearly,
s9 is a random mixture of Si and s5. From the ratio of the segments
sis9 and s5s9 it follows that s9 chooses Si with probability 0.826 and
s5 with probability 0.174. From table 2 we recall: if fa, fa, fa are
observed, Si takes the actions «1, a\, ai and s5 takes «2, #1, «1- The
strategies differ only if fa is observed, so s9 also takes «1 when fa or
fa is observed. If fa is observed, s9 uses a chance mechanism for
the mentioned probabilities to choose between s 1 and s5.
i3i
Si
S2
S3
*4
S6
58
25
25-24
31-76
26.51
25-24
3°-49
37
3I-76
