# Stackelberg solution to $n$-player Hotelling's game on a segment

Suppose that several agents need to place points (one per agent) on the interval $[0,1]$. An agent's goal is to maximize the volume of the Voronoi cell that contains his point. When $n$ agents must place their points sequentially, is an optimal strategy known?

From what I was able to glean from the literature so far, it appears I am asking for the Stackelberg solution to an $n$-player Hotelling's game on a segment.

• I am not asking people to solve this question for me, just wondering what is known about it -- and the obvious extension to 2 dimensions. Mar 20, 2015 at 11:17
• I think these are usually called Voronoi games. I don't know much about them, but I think by searching that keyword you can find out what is known.
– usul
Mar 20, 2015 at 13:15
• Thanks! I was only able to find references for 2-player Voronoi games. Mar 20, 2015 at 13:32
• There may only be research for the 2-player case (don't know)....
– usul
Mar 20, 2015 at 13:36

This game has been studied in

Hee-Kap Ahn, Siu-Wing Cheng, Otfried Cheong, Mordecai Golin, René van Oostrum:
"Competitive facility location: the Voronoi game"
Theoretical Computer Science 310, 2004, pp 457-467

There also are lots of follow-up works, as for instance

Sayan Bandyapadhyay, Aritra Banik, Sandip Das, Hirak Sarkar:
"Voronoi game on graphs"
Theoretical Computer Science 562, 2015, pp 270-282

• Still wondering -- has it been studied for $> 2$ players?
– usul
Mar 22, 2015 at 1:19