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Chapter 9
The Value of the Two Cable Ambush Game
Vic Baston and Ian Woodward
Abstract
This chapter finds the value of Ruckle's much studied two-person zero-
sum ambush game which involves an Infiltrator attempting to traverse a channel
undetected when the channel is protected by two lengths of electronic cable. The
value is obtained by exploiting the structure of a Defender optimal strategy without
explicitly constructing an optimal strategy for either player. It is expressed as the
minimum of a finite number of simple expressions, each of which is easily calcu-
lated from the values of the cable lengths. This result identifies ranges of values for
the lengths of the two cables over which the games not only have the same value but
also have Defender optimal strategies with the same basic structure.
9.1 Introduction
In [
10
,
11
] Ruckle introduced a number of search and ambush games. Some of the
games he described have generated a considerable amount of interest and one game
in particular has been the subject of a number of papers. It can be described as
follows.
Infiltrator wishes to travel down a channel without being detected by Defender
who has two electronic cables of lengths
a
and
b
which can be placed in the channel.
Infiltrator knows the lengths of the cables but has no means of determining where
they have been laid and he will be detected if he crosses them. As it is natural for
Defender to place the cables across the channel where its width is minimal, the prob-
lem can be formulated mathematically by letting Infiltrator choose a point
x
in the
interval
[
0
,
1
]
and Defender choose closed intervals
[
y
,
y
+
a
]
and
[
z
,
z
+
b
]
in
[
0
,
1
]
.
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