The Value of the Two Cable Ambush Game - Search Theory: A Game Theoretic Perspective - page 130

1 − ib 1 − ib

≤ λ < + 1 . (9.2)

i

a a

Thus a covering of [0,1] having precisely i intervals of length b must have at least
λ i intervals of length a .

Since our (mixed) Defender strategy in will involve the same “number”

of intervals of length a as of length b , the associated coverings will be a balance of
two types;

Γ F ( a , b )

(i) Those that have more intervals of length b than intervals of length a
and

(ii) Those that have at least as many intervals of length a as those of length b .

Λ + and Λ − defined by As a consequence the sets

Λ + = Λ + ( Λ − = Λ − ( a , b )= { i : i > λ i } and a , b )= { i : i ≤ λ i } (9.3)

will play a prominent role and feature in our expression for the game value.

9.3 An Illustrative Example

In the next section we prove a theorem that gives a lower bound for the value of our
game. Its proof takes an arbitrary optimal strategy for Defender and shows that it
must have a certain structure. The arguments to do this are somewhat technical so,
in this section, we provide an example which illustrates them.

0
20

5
20

7
20

8
20

10 12 13 15 20

20 20 20 20 20

Fig. 9.1 An optimal Defender strategy when a = 5 / 20 and b = 8 / 20