Tools to Manage Search Games on Lattices - Search Theory: A Game Theoretic Perspective - page 35

if a pure strategy C has positive probability, all the pure strategies obtained from C
by successive applications of transformation T , TC , T 2 C , T 3 C , ..., T m − 1 C ,have
the same probability. A similar result is obtained in [ 1 ].

Fig. 3.4 Effect of transformation T

3.4 Two Games with Invariance Properties

We solve two games satisfying ( 3.4 ). Some seminal results for the first of them,
the one-element intersection game, can be found in [ 8 ] and the weighted inspection
game, which is the second game, has been studied in [ 11 ].

3.4.1 One-Element Intersection Game

The one-element intersection game (OEIG), is a game ( X , Y , M ) on a lattice L satis-

fying the following conditions:

X = F

Y = { B : B ⊂ L , | B | = s }

1

, if | A ∩ B | = 1

M ( A , B )=

.
The intepretation of this game is that Player I wins if he finds just one point of
B. When he finds this first point, however, an alarm system is activated, so that if he

0 , otherwise