The Cardinality of the Sets Involved in Lattice Games - Search Theory: A Game Theoretic Perspective - page 110

then we can write

A ∈ X 1 1 + A ∈ X 2 2 + ... +

kn

1
F

∑

n , m

M ( A , α )=

X n − 1 (

n − 1 )+

A ∈

2

kn 3 n − 2 3 n − 3 3 n − 4

· + 2 · · 2 + 2 · · 3 + ... + 2 · 3 · ( n − 2 )+ 2 ( n − 1 )+

F

n , m

=

n − 1

h = 1 h 3 n − h − 1

2
F

kn
F

m m

= +

n n

, ,

n − 1

h = 1 h 3 − h

3 n

3 n − 1
F

3 n − 1
F

kn
F

− 2 n − 1 kn

F

m m = m m

= 2 + 2 +

3 n − 1 2 2

n n n n

, , , ,

3 n − 2 n ( 1 − k ) − 1

2 F

n , m

=

and this is the value of the game.

7.4 Further Results

Lattice games were introduced by Ruckle in [ 5 ]. These are games on the lattice
where at least one of the players can move only from one point to an adjacent lat-
tice point. This restriction on the movements of the player is realistic because it
expresses that his movements are difficult. Although different results have been ob-
tained for games on the lattice since the topic of Ruckle, the work on lattice games
is very scarce, and none of the problems set up there has been totally solved. In this
chapter we obtain the cardinality of the sets of strategies for the players of lattice
games, this is the first of the problems proposed by Ruckle, and we hope, as him,
that it will be of value in attacking such games.

References

1. Alpern, S., Morton, A. and Papadaki, K.: Patrolling games. Oper. Res. 59, 1246-1257 (2011)
2. Garnaev, Y.: A remark on a helicopter and submarine game. Naval Res. Logistics 40, 745-753
(1993)
3. Krattenthaler, C. and Mohanty, S. G.: Lattice path combinatorics - applications to probability
and statistics. http://www.mat.unuvie.ac.at/~kratt/artikel/encystat.
html
4. Meyer, K.: Matrix Analysis and Applied Linear Algebra. Published electronically at http://
matrixanalysis.com/DownloadChapters.html