A Location Game On Disjoint Circles

5 Concluding remarks

We have studied an extension of the two-player Voronoi game of Ahn et al.
[2004] to a playing arena involving multiple disjoint closed curves. Such
games can be used to model important real life situations as highlighted in
the introduction. We have shown that the second mover advantage, albeit
arbitrarily small as shown in Ahn et al. [2004], disappears as we find tying
strategies for both the first and the second mover, thereby enabling us to
demonstrate Nash equilibrium configurations of locations. A general prop-
erty of all such equilibrium configurations is that locations on each circle
alternate in colour. We then study a natural variant of this game where
players face very strict resource mobility constraints to show that the sec-
ond mover advantage, again though arbitrarily small, re-appears. In the
resulting equilibrium configurations of this version of the game, we show
that there exists monochromatic intervals, an interesting difference vis-a-vis
equilibrium configurations in the original game. One may think of the rules
of the game as a mechanism by which distributions of influence between the
two acting players can be affected and in that sense we have shown that
a “literally fair” division is always Nash implementable. Ahn et al. [2004]
has also studied such location games on line segments and it would be in-
teresting to study our game on a family of disjoint line segments. Also, it
would be important to generalize our games to those involving more than
two players. Note also that the tying strategy of the first mover that we
demonstrate depends crucially on the fact the the total number of points N
is known. It would be interesting to extend these environments to incom-
plete information.

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