we show that the second mover advantage as in Ahn et al. [2004] reappears.
We also provide some characterizations of final equilibrium configurations.
The rest of the paper is structured as follows. In section 2 we define the
game. Section 3 states and proves our results. Examples of final equilibrium
configurations are depicted in section 4 and the paper concludes in section 5.
2 The Multiple Circle game
The circle game studied in Ahn et al. [2004] has two players, called1 Red
(R) and Green (G) each having N points to place (or locations to choose)
alternately on a circle with R making the first move. Moreover, (i) each
player must place at least one point in each round, (ii) in the first round
when play begins, R cannot place all N points (perhaps because not all
resources are available at the beginning of the game), 2 (iii) the game ends
only after all players have placed all 2N points, (iv) at any round, the total
points placed so far by G cannot exceed that of R3 , and (v) a location on the
circle cannot serve more than one points. This results in a sequential game
where roles (that is first and second mover identities) cannot be reversed
and the number of rounds is endogenous and can be controlled by R subject
to the restriction that there must be at least 2 rounds. The objective of
each player, as in Voronoi games (a term coined by Ahn et al. [2004]), is to
maximize the total length of the curve that is closer to that player than to
1originally called White and Black in Ahn et al. [2004]
2requirements (i) and (ii) imply that N ≥ 2.
3 This is basically a condition required to preserve the first and second mover identities
over any play. These identites could be preserved even with the assumption that players
place equal number of points in each period. In this sense, the condition given in Ahn
et al. [2004] and used here is general and hence weaker. In Subsection 3.1 we shall study
a natural variant of this game where each player must place exactly one point in each
round.