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