The Cardinality of the Sets Involved in Lattice Games - Search Theory: A Game Theoretic Perspective - page 98

Chapter 7
The Cardinality of the Sets Involved in Lattice
Games

Noemí Zoroa, María-José Fernández-Sáez, and Procopio Zoroa

Abstract Lattice games were introduced by Ruckle in (Geometric games and their
applications. Pitman Advanced Publishing Program, 1983). These are games on the
Lattice games where at least one of the players can move only from one point to
an adjacent lattice point. This restriction on the movements of the player is realistic
because it expresses that his movements are difficult. Although different results have
been obtained 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 cardinalities 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
does he that it will be of value in attacking such games.

7.1 Introduction

In this chapter we deal with two-person zero-sum games on the lattice

= { , ,..., }×{ , ,..., }

L 1 2 n 1 2 m

in which, one of the sets of strategies for the players is the set of all the functions
from

{ , ,..., } { , ,..., } ( + )

1 2 n into 1 2 m such that f i 1 equals one of the three values

or a subset of it. Ruckle, in his interesting topic [ 5 ]on
geometric games, includes some games of these kind and calls them lattice games.
In Chap. 3 we considered algorithms for solving search games on a lattice. In this
chapter, we consider the cardinality of the strategy space.

f ( i ) , f ( i + 1 ) ,or f ( i − 1 )