A Location Game On Disjoint Circles

the other so that a player wins if and only if the area it secures is strictly
the largest one. Otherwise there is a tie. It is shown in Ahn et al. [2004]
that in this game G always has a winning strategy, though R can bring its
length of influence as close as that of G’s. Our objective is to check if such
a second mover advantage prevails when there are more that one disjoint
identical circles. We now present these ideas and the finding in Ahn et al.
[2004] formally.

Let {R, G} be a set of players, where R stands for Red and G stands
for Green. The game on the family of disjoint circles is defined by a pair
hN, {Cj}_j^K₌₁ i, such that N > K ≥ 1 and {Cj}_j^K₌₁ is a family of K disjoint
circles. Notice that the game studied in Ahn et al. [2004] is the special case
where K = 1. Throughout the game each player p ∈ {R, G} will select a total
of N points on K circles. The set of points selected by R is Γ ⊆ ^S_j^K₌₁ Cj
and the set of points selected by G is Ω ⊆ SjK=₁ Cj. Players re-arrive in
alternating sequence with R moving first, and are in principle allowed to
place points in batches. Let Γr be the set of points that R places in round
r ≥ 1 while Ω_r be the same for G. The game ends when all 2N points are
placed on the circles.⁴ We will use w ∈ Γ (b ∈ Ω) to denote a point placed
by R (G) during the game. We will call points placed by the player R red
points and those placed by the player G green points.

As discussed above, the game has the following conditions:

1. ∣Γ_r∣, ∣Ω_r∣ ≥ 1 for every r ≥ 1.

2. ∣Γι ∣ < N.

3. P^r=ι ∣Γi∣ ≥ ^pr=ι ∣Ωi∣ for every r ≥ 1.

⁴ Please note that we put no restriction on how players distribute these points across
the circles (some circles are allowed to remain empty in which case it is ignored while
computing payoffs).

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