Some Cinderella Ruckle Type Games - Search Theory: A Game Theoretic Perspective - page 94

structure, illustrating that it may not be easy to home in on particular BLUE optimal
structures for other cases. It is probably over-optimistic to hope for a complete so-
lution of the game without further inroads into special subcases being made first.
Zoroa, Fernández-Sáez and Zoroa have solved the game when q is relatively small
so it would seem that the two subcases that present the best chance of progress on
an analytical front are:

Problem 3. Solve The Integer Number Hides Game with Capacities for compara-
tively large q .

Problem 4. Solve The Integer Number Hides Game with Capacities for c = 2 .

Like the Several Intervals Game, one feels that a comprehensive solution of this
game may need the insight given by computer generated solutions where the com-
puter has been programmed to target certain types of solution.
A natural variation of the above game in which BLUE has an amount q not nec-

essarily an integer, of divisible material to hide was introduced by Zoroa, Fernández-
Sáez and Zoroa in [ 8 ]. It can be formulated as follows.

,

BLUE has an amount q not necessarily an integer, of divisible material to hide in the integer

interval I and must choose a subinterval B of I with length at most b in which to do so under
the restriction that an amount of at most c is allocated to each point of B

,

Simultaneously
RED picks a subinterval R of length r and gets a payoff equal to the amount of material that
BLUE allocated to the points of R

.

.

It is not easy to give a summary of the theorems obtained in [ 8 ] which does justice
to them without involving detailed notation so the reader is encouraged to read the
paper itself. Although a complete solution appears to be extremely difficult, many
open questions regarding partial results suggest themselves. Note that, in this game,
BLUE is allowed to put an amount zero at some of the chosen points so it is not
totally obvious that there is a close connection between this game and the previous
one. Thus it is of interest that [ 8 ] points out that there are similarities between the
two in some cases.

6.7 Hiding in a Disc Game

The Hiding in a Disc game is very simple to state and easy to understand but seems
difficult to solve. It can be described as follows.

Without knowing each other's choices, RED and BLUE choose points r and b in a disc D
with centre O and radius one. The payoff to RED in this zero-sum game is one if | r − b |≤ c

and zero otherwise.

/ √ 2

When 1 the value is the ratio of the length of the arc whose chord has

length c to the circumference of D ; optimal strategies for BLUE and RED are to
choose a point according to a uniform distribution on the circ umfere nce of D and

≤ c < 1 ,

on the circumference of a circle with centre O and radius √ 1

− c 2 respectively.