A Location Game On Disjoint Circles

We will show that if R plays according to Y ⁰ , he achieves the required
outcome. Similarly to the case where Y * is used by player G, the use of
strategy Y⁰ leads to three stages of the game for player R, though the stages
are slightly different. At first, option (a) is exercised. After all key positions
become occupied, option (b) is used as long as there is a green interval. This
is the second stage. When there is no green interval, options (c’) and (b) are
selected depending on what player G does. If in his move G breaks the red
interval created by the use of option (c’) by R, in the next round R applies
option (c’) again and creates another red interval. Otherwise (which means
that G created a green interval) R applies option (b) and breaks the newly
created green interval.

After the first stage, where option (a) is exercised, R is not loosing. This
is because all his points lie in key positions. Moreover the number of red
key intervals of the size 1/bN/K c cannot be greater than the number of key
intervals of this size (as red key intervals of the size 1/bN/Kc can be created
only on the circles where R placed the first point, cf proof of Claim 2).
Hence if there are monochromatic interval after the first stage, then all red
intervals are at least as big as the existing green intervals. Observe that
since N > K, so option (a) will be applied at least once and there will be
at least one bichromatic interval after the first stage.

In the second stage, where option (b) is exercised, R breaks maximal
green intervals. Observe that throughout this stage after each move of player
R he has an advantage of size of a key interval (either of the size of 1/dN/Ke
or of the size 1/bN/Kc). Moreover G cannot create green intervals of size
≥ 1/dN/Ke, as all key positions are occupied after the firts stage. Thus after
each round of the second stage player R cannot be loosing. This means in
particular that if G is a payoff maximizer, the game will always enter the
third stage.

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