The Value of the Two Cable Ambush Game - Search Theory: A Game Theoretic Perspective - page 131

Its value is 3/8 (see [ 8 ]) so it
is easy to check that an optimal strategy for Defender is given in Fig. 9.1 in which
each (horizontal) line of rectangles represents a pure Defender strategy played with
probability 1/8. Notice that lines 3 and 4 represent the same strategy as do lines 5
and 6. Most vertical lines intersect five rectangles although some intersect more; in
particular vertical lines from points between 7/20 and 8/20 meet seven rectangles
so we could move the rectangles of length 5/20 in rows 5 and 6 which start at 7/20
to the right so that they start at 8/20 without affecting optimality. Doing this means
that they overlap the rectangles of length 8/20 on the same line; with regard to the
proof of the theorem, it is undesirable for rectangles on the same line to intersect
except possibly at a point. Thus the rectangles of length 8/20 in rows 5 and 6 are
moved to the right so that they start at 13/20. From a strictly pedantic standpoint this
is not permissible as they end beyond 1 but this is not a problem because we are at
an interim stage and, when the final stage is reached, they can be moved to the left
so that the intervals lie in [0,1]. Having moved the rectangles on lines 5 and 6, we
see that the vertical lines between 12/20 and 13/20 still intersect seven rectangles so
we perform a similar process to the above on lines 7 and 8 to arrive at the position
in Fig. 9.2 . in which every vertical line between 0 and 1 not only intersects precisely
five rectangles but also precisely one rectangle of each shading. Thus rectangles
with the same shading cover the interval [0,1].

Consider the case when a = 5 / 20 and b = 8 / 20 .

0
20

5
20

8
20

10 12

20

13
20

15
20

20
20 20

Fig. 9.2 The modified Defender strategy when a = 5 / 20 and b = 8 / 20

Notice that the rectangles with no shading is a minimal cover of [0,1] by rectan-
gles of length 5/20. The other rectangles with a particular shading all comprise two
rectangles of length 8/20 and one of length 5/20 and so provide minimal covers of
[0,1] when precisely two rectangles of length 8/20 can be used; although these cov-
ers are different, this is not significant in the analysis and they will all be regarded
as the same type, say C where 1 and 2 represent the number of rectangles of

length a and b respectively in the cover.

( 1 , 2 ) ,