egy profile (Y0,Y*), where the first mover uses strategy Y0 while the second
player uses strategy Y * is a Nash equilibrium. Moreover, in every final con-
figuration resulting from this Nash equilibrium there exists a monochromatic
interval.
We are obviously interested in the case where ε is arbitrarily close to
zero. This theorem will be used to produce equilibrium final configurations
in the one-by-one variant.
4 Examples
In this section we present examples illustrating the game when the players
use the strategies presented above. We start with general game where R
plays according to T * while G plays according to T0. There are two cases
here: K | N and K - N . We illustrate only the second case, which is more
involved. Final configurations in the first case are similar to those of the
second case if both players play these tying strategies.
Let N = 11 and K = 3. 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 an empty circle
(taking a key position and defining remaining key positions for this circle
with respect to 4). The configurations created during the game are presented
in Fig. 1. We use empty discs to depict red points and filled discs to depict
green points. Key positions are depicted with short dashes intersecting the
circles. In the rounds 2-4 player R plays according to option (b) taking free
key positions in the first circle while G places his points within red intervals.
In the round 5 player R plays according to option (d) taking a key position in
a new circle. This determines key positions in the new circle which are taken
with respect to the red point and the number 4 obtained as above. Player G
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