A Location Game On Disjoint Circles

3 Results

We first show that for any game hN, {Cj}_j^K₌₁i with K ≥ 2 and N ≥ K,
R, i.e. the first mover, has a tying strategy. We will consider two cases
separately: K-N and K | N . We start by demonstrating a tying strategy for
R for the first case. The general idea of this strategy is for R to capture key
positions on the circles. Key positions on each circle will be determined by
the first point placed on the circle and either dN/Ke or bN/Kc, depending
on the situation (and G’s play, in particular). Let r be a round and let L(r)
be the number of circles occupied after R places the first of the points he is to
place in round r. Key positions on the occupied circles are determined with
respect to dN/Ke and the first point placed on the circle. The number of
total key positions on these circles is L(r)dN/Ke and the number of vacant
key positions is

V(r) = L(r)dN/Ke -Y(r),

where Y (r) is the number of key position already occupied after R places
the first point in round r. Let φ(r) stand for the number of points R is
left with if after placing his first point in round r he would have covered all
vacant key positions in the occupied circles, that is

φ(r) = N — r — V (r).

We first prove the following lemmas and a corollary, which are gener-
alizations of the lemmas presented in Ahn et al. [2004] for more than one
circle.

Lemma 1 Let {Ck}_k^K₌₁ be a family of circles. Then

K   K   KK

^XI^R(Ck)—^XI^G(Ck))=^Xr^Ck —^Xg^Ck,
k=1           k=1            k=1       k=1

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