A Location Game On Disjoint Circles

Theorem 5 Let hN, {Cj}_j^K₌₁i define a game on the family of K disjoint
circles. Then strategy profile (T*,T0), where player R plays T* while player
G plays T⁰ , is a pure strategy Nash equilibrium. Moreover, in every final
configuration resulting from this Nash equilibrium profile of strategies, (i)
there is no monochromatic interval on any circle and (ii) all red points lie
on key positions.

There may be other equilibrium points in this game. We shall use the
above equilibrium in the examples we set in section 4 to study equilibrium
configurations. We now deal with a variant of this game where the resource
mobility constraint becomes most binding.

3.1 “One-by-one” variant of the game

A natural variant of the game studied above is the one where players face
very strict resource mobilization constraints so that each places a single
point in each round. Recall that in the tying strategy T^* used by R, it was
crucial for R to place more than one point at some rounds. It turns out that
if players face such a strict resource mobility constraint as the one we are
dealing with now, then G, the second mover has a winning strategy.

The strategy, which we shall call Y ^* , is a generalization of the winning
strategy S^* of the second mover in the one circle case presented in The-
orem 1. Player G first tries to take key positions with respect to dN/Ke
or dN/Ke (depending on the situation described precisely below) and the
first point placed on that circle. Then he breaks biggest red intervals, by
placing a point inside them. In his last move he either breaks the biggest
red interval or plays in a bichromatic interval that is bigger than the biggest
red interval.

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