A Location Game On Disjoint Circles

in each of K circles.

Secondly, observe that if G and R covered all key positions in L circles
(k key positions on each of the circles) using the same number of points,
and so that R’s points are placed on key positions only then G cannot be
winning on these circles. This is because if G is to be winning then there
must be some green intervals on the circles (for if there are not then there
is a tie, as each bichromatic interval is divided equally between G and R).
According to the fact 1, there must be the same amount of red intervals.
Since each red interval is of size 1/k and each green interval is of size ≤ 1/k,
so G cannot be winning.

Assume that in the last round R played according to (d). Then before
R’s move some 1 ≤ L < K circles were covered by G and R so that the
same amount of red and green points where placed there, all key positions
are taken and all red points are placed on key positions (and there are
dN/K e key positions on each circle). Thus there is a tie on these circles.
There are two possible numbers of key positions on each of the K - L circles:
(i) dN/Ke and (ii) bN/Kc. In case (i) the situation analogous to the one
on L circles will be created, resulting in a tie. For case (ii) observe that
answering R’s move G has to place (K - L)bN/K c points. Observe also
that after R’s move (K - L)bN/K c red intervals where created, each of the
size 1/bN/Kc. Since R cannot gain⁷ by playing in any of the L circles that
where occupied before R’s move (as he can gain < 1/dN/Ke, and by playing
within newly created red interval he can gain 1/dN/Ke), he has to place his
points within newly created red intervals. Moreover by placing more than
one point within such interval he can gain < 1/bN/K c while placing one
red point in each of the newly created red intervals he gains 1/bN/Kc and
in this case there is a tie.

⁷By “gain” we mean the area acquired by the player plus the area lost by the opponent.

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