A Location Game On Disjoint Circles

4. D≥ι ∣Γi∣ = ∑i≥ι ∣Ωi∣ = N.

The endogenously determined number of rounds in a given play of the
game will be denoted by Z. Obviously Γ = SZ=₁ Γ_r and Ω = SZ=₁ Ω_r.
Notice that the restrictions of the game imply that Z ≥ 2.

Let C be any circle and let (x, y) be an ordered pair of elements of C.
We will use (x, y) to denote the arc of the circle between x and y in clockwise
direction. Let a(x, y) ∈ [0, 1] denote an angle in clockwise direction between
halflines starting from the center of the circle and going through x and y
(we normalize an angle, so that a full circle has angle equal to 1). Then
d(x, y) = min{a(x, y), a(y, x)} is the angular distance between x and y.
Notice that d(x, y) = d(y, x) ∈ [0, 1/2]. Given an arc (x, y), the length (or a
volume) of (x, y) is a(x, y).

Given a circle Ck , let

AR(Ck) = x ∈ Ck : min d (x, w) < min d (x, b)

be a set of points of Ck that are closer to points placed by R on Ck than
to points placed there by G. Let AG(Ck) be the analogical set defined for
G. Notice that each of these sets is a finite set of arcs of a circle Ck . Let A
be a finite set of arcs and let V (A) denote the volume (sum of lengths, in
angular terms) of arcs in A. When the game ends, each player p receives a
score Sp equal to the volume of the set of arcs constituting the set of points
closest to a position chosen by that player over all circles, that is

K

Sp = ^X V (Ap(Ck))
k=1

for p ∈ {R, G}. Given these scores, the payoff of the players is up(Sp, Sq) =
Sp - Sq, where {p, q} = {R, G}. We say that the game is a tie if Sp = Sq,
while player p wins if Sp > Sq . A strategy in general will be a contingent
plan for every possible history of the game. We do not need to define this

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