A Location Game On Disjoint Circles

responds by playing within the newly created red interval, R takes another
key position in the circle and G responds in the same manner. Since in round
6 player G placed his point in a key position, so player R plays according to
option (a). After he places his point, the number of occupied key positions
at round 7 is Y (7) = 8, the number of vacant key positions on non empty
circles V (7) = 0 and the number of points R would have left if he had covered
all vacant key positions in the occupied circles is φ(7) = 11 — 7 — 0 = 4. Since
this is equal to (11 - L(7))d11/3e = 4, player R can cover key positions with
respect to 4 in the remaining one circle, which he does. Player G answers
placing exactly one point within each newly created red interval. The game
is hence tied.

For the ε-restricted one-by-one version of the game, we take the same
parameters and present an example where players apply their respective Nah
equilibrium strategies Y⁰ (used by the first mover Red) and Y * (used by the
second mover Green) (see Fig. 2). Player R starts by placing a point in an
empty circle (which defines key positions for this circle with respect to the
red point and d11/3e = 4) and G answers by placing a point in a clockwise
neighbouring key position (also assigning key positions for this circle with
respect to the position of the red point and 4). Then both players continue
with taking key positions. When key positions are taken (4 of them on
each circle) player R applies (c’) of his strategy Y⁰ and G responds applying
option (b) of his strategy Y^* by breaking the red interval created by R.
The game goes on in this manner until the last round is reached. In the
last round player R applies option (c’) of his strategy Y⁰ again and player
G responds by applying option (c) of his strategy Y^* and creating a green
interval slightly bigger than the one created by R in this turn. This ends the
game and G wins by the margin ε, the difference between intervals created
in the last round.

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