of a tying strategy for the second mover as well, as stated in the theorem
below. This strategy, which we call T0, is simple and is as follows:
Strategy T0
place exactly one point in each red interval created by R in any given
round
Theorem 4 Let hN, {Cj}jK=1i define a game on a family of K disjoint cir-
cles. Then T0 is a tying strategy for G.
Proof. Strategy T0 requires G to place exactly one point in each red
interval created in a given round. It is easy to see that this will ensure that
at the end of the game there is no monochromatic intervals, which is then
a tie. So what requires to be proved is that this strategy is implementable
which we do by induction on the number of the current round. We will
show two things: after R plays in round r , G can place exactly one point
in a red interval and there is no monochromatic interval after G’s move.
Consider the first round. There are no intervals before players move. Assume
that R placed m points. Then, by Lemma 1 there are m red intervals
created and G can place m points, one within each interval. Thus there
is no monochromatic interval after the first round. Now consider a round
r > 1. By induction, there is no monochromatic interval before R’s move
and, by similar argument as in the case of the first round, there is exactly
the same number of newly created red intervals as the number of red points
placed. Thus G can place exactly one point in each of the newly created red
intervals and there are no monochromatic intervals after the round r. This
shows that G’s tying strategy is implementable. ■
The following theorem is an immediate consequence of the results we
have proved so far.
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