Some Cinderella Ruckle Type Games - Search Theory: A Game Theoretic Perspective - page 88

latter games, the crucial factor that determines whether BLUE's path intersects the
convex set C chosen by RED is the projection of C onto the y -axis. In particular,
in this restricted form of Ruckle's area greedy game, in order to play optimally
RED will choose a rectangle with base of length one so his choice is effectively to
choose the height of the rectangle (which is a line segment) while BLUE is effec-
tively choosing a point (the value of the constant). Perhaps the next natural step is
to tackle the following problem.

Problem 1. Solve the Area Greedy Game on the unit square when BLUE can take
any path of length one and RED can choose any compact convex set in the unit
square.

6.5 The Numbers Hides Game

Like the Several Intervals Game the Number Hides Game is a two person zero-sum
game which was posed as a problem in [ 4 ] with some special cases being solved. Its
formulation is particularly simple.

RED and BLUE choose sequences of p and q consecutive integers respectively between
1and n . The payoff to RED is the number of integers in the intersection of the chosen

intervals.

The game has proved more tractable than the Several Intervals Game. When Baston
and Bostock submitted their solution to the proceedings of the American Mathe-
matical Society, they were told that Ferguson had also solved it so a three author
paper [ 3 ] was written in the style of Ferguson which the editors preferred. They
subsequently learned that Zoroa, Zoroa and Ruiz had also found a solution [ 6 ].
The game is the discrete version of the Interval Overlap Game in which RED
and BLUE choose intervals of lengths at most

α and at least β respectively in the

unit interval I and RED has a payoff of the measure (length) of the intersection

of the chosen intervals; this game was solved in [ 4 ]. We have seen in Sect. 6.2 that
the Several Intervals Game is essentially equivalent to a corresponding finite game
so it is natural to ask whether a similar situation pertains here. Although no formal
justification has been given, the answer is probably yes as it was remarked in [ 3 ]that
the ideas used to solve the Numbers Hide game carry over to the Interval Overlap
Game; in fact these ideas enabled a fault in the analysis of BLUE's optimal strategies
in [ 4 ] to be corrected. The games where the Number Hides Game is modified so that
one or both players are permitted to choose an arbitrary set of integers rather than a
set of consecutive integers have been solved in [ 3 ]and[ 7 ]. The game in which both
players can choose an arbitrary set of integers is called the Simple Point Catcher
Game. As pointed out in [ 3 ], its value is pq

,

when RED chooses at most p integers
and BLUE chooses at least q but Ruckle [ 4 ] gave a more complicated expression,
namely

/ n ,