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What intrigued me when I first read Ruckle's topic was that the solution for
1
/ 2 was still open, particularly so because the value when c
2
is known. Ruckle showed that, in the range, RED can guarantee a payoff of at least
(
/
2
<
c
<
1
=
1
/
demonstrating that a solution communicated to the American
Mathematical Monthly was incorrect. The RED strategy which ensured this payoff
seemed an intuitively natural one so all that was needed was to produce a BLUE
strategy showing RED could do no better. However the “all” proved elusive and,
after prolonged efforts, I gave the problem best. On re-reading Ruckle's topic re-
cently I was curious whether progress had been made on the problem but I have
been unable to find references to it.
It is natural to wonder whether a symmetry argument which is standard in the
literature might be of use for this game; for a formal group theoretic justification of
the following process see [ 1 ]. Let
1
/ π )
arccos
(
1
/
2 c
)
A
denote the set of all rotations about the cen-
tre and let
denote the symmetized version of the game in which, after RED and
BLUE choose strategies r and b respectively, a random (equiprobable) member
Γ
γ
is
, γ 1 b
selected from
A
and the payoff P
( γ
r
,
b
)=
P
(
r
)
is assigned to RED. Observe
that either player can ensure that
is played by applying a random automorphism to
his own strategy so its value must be the same as that of the original game. Hence the
Hiding in a Disc game is solved once
Γ
Γ
is solved. We may therefore regard mixed
strategies of the players in
so
that the strategy spaces are represented by the unit interval. The optimal strategies
of the game given in [ 4 ] can all be expressed in
Γ
as distributions over the equivalence classes of
A
as probability distributions over
a finite number of points in the unit interval. Unfortunately the payoff of the sym-
metrized game is much more complicated than that for the original game so there
may be few practical benefits of symmetrising this particular game. However it does
highlight a question that is of interest.
Γ
Question 2. In the symmetrized Hiding in a Disc game, do there always exist opti-
mal strategies for the players which are probability distributions over a finite number
of points in the unit interval?
6.8 A General Ruckle-Type Game
Ruckle proposed the problem of solving the Hiding in a Disc Game played on a set
S more general than the circular disc. As that game appears to be still unsolved, it
might seem somewhat bizarre to give a game formulation of which it is a special
case. However the game we now introduce does show that a number of win-lose
Ruckle-type games (payoff 0 or 1) do have a common structure and that there may
be interesting research to be done on them in topologies other than the Euclidean
one. The reader is reminded that a closed ball with centre c and radius r is the set of
points which are at a distance less than or equal to r from c
.
Let
denote the following two-person zero-sum game played on a convex
compact subset S of R n
Γ S (
b ; r 1 ,...,
r k )
endowed with a topology from a metric. BLUE chooses a closed
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