Location game on disjoint line segments

Abstract

Two players are endowed with resources for setting up N locations on K line segments of identical length, with N > K ≥ 1. The players alternately choose these locations (possibly in batches of more than one in each round) in order to secure the area closer to their locations than that of their rival’s. The player with the highest secured area wins the game and otherwise the game ends in a tie. Earlier research has shown that, if an analogous game is played on disjoint circles, the second mover advantage is in place only if K = 1, while for K > 1 both players have a tying strategy. It was also shown that these results hold for line segments of identical length when rules of the game additionally require players to take exactly one location in the first round. In this paper we show that the second mover advantage is still in place for K ≥ 1 and 2K − 1 ≤ N, even if the additional restriction is dropped, while K ≤ N < 2K − 1 results in the first mover advantage. Our results allow us to draw conclusions about a natural variant of the game, where the resource mobility constraint is more stringent so that in each round each player chooses a single location and we show that the second mover advantage re-appears for K ≤ N < 2K − 1 if K is an even number. In all the cases the losing player has a strategy guaranteeing him arbitrarily small loss.