Game Values and Computational Complexity: An Analysis via Black-White Combinatorial Games

  • Stephen A. Fenner
  • Daniel Grier
  • Jochen Messner
  • Luke Schaeffer
  • Thomas Thierauf
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)


A black-white combinatorial game is a two-person game in which the pieces are colored either black or white. The players alternate moving or taking elements of a specific color designated to them before the game begins. A player loses the game if there is no legal move available for his color on his turn.

We first show that some black-white versions of combinatorial games can only assume combinatorial game values that are numbers, which indicates that the game has many nice properties making it easier to solve. Indeed, numeric games have only previously been shown to be hard for \(\mathsf{NP}\). We exhibit a language of natural numeric games (specifically, black-white poset games) that is \(\mathsf{PSPACE}\)-complete, closing the gap in complexity for the first time between these numeric games and the large collection of combinatorial games that are known to be \(\mathsf{PSPACE}\)-complete.

In this vein, we also show that the game of Col played on general graphs is also \(\mathsf{PSPACE}\)-complete despite the fact that it can only assume two very simple game values. This is interesting because its natural black-white variant is numeric but only complete for \(\mathsf{P}^{\mathsf{NP}[\log ]}\). Finally, we show that the problem of determining the winner of black-white Graph Nim is in \(\mathsf{P}\) using a flow-based technique.


Combinatorial games Computational complexity Graph Nim Poset games Black-white games Numeric games Col 


We would like to thank an anonymous referee for pointing out that one-sided games, including directed bipartite Geography, are vacuously numeric.


  1. 1.
    Berlekamp, E.R., Conway, J.H., Guy, R.: Winning Ways for your Mathematical Plays. Academic Press, New York (1982)zbMATHGoogle Scholar
  2. 2.
    Bouton, C.L.: Nim, a game with a complete mathematical theory. Ann. Math. 3(1/4), 35–39 (1901)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burke, K., George, O.: A PSPACE-complete graph Nim (2011).
  4. 4.
    Cincotti, A.: Three-player Col played on trees is NP-complete. In: International MultiConference of Engineers and Computer Scientists 2009, pp. 445–447 (2009). Newswood LimitedGoogle Scholar
  5. 5.
    Conway, J.H.: On Numbers and Games. Academic Press, New York (1976)zbMATHGoogle Scholar
  6. 6.
    Demaine, E.D., Hearn, R.A.: Constraint logic: a uniform framework for modeling computation as games. In: 23rd Annual IEEE Conference on Computational Complexity, pp. 149–162. IEEE (2008)Google Scholar
  7. 7.
    Faenkel, A.S., Scheinerman, E.R., Ullman, D.: Undirected edge geography. Theoret. Comput. Sci. 112, 371–381 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fenner, S.A., Grier, D., Meßner, J., Schaeffer, L., Thierauf, T.: Game values and computational complexity: an analysis via black-white combinatorial games. Technical Report TR15-021, Electronic Colloquium on Computational Complexity, February 2015Google Scholar
  9. 9.
    Fukuyama, M.: A Nim game played on graphs. Theoret. Comput. Sci. 304, 387–399 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grier, D.: Deciding the winner of an arbitrary finite poset game is PSPACE-complete. In: Fomin, F.V., Freivalds, R.U., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 497–503. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  11. 11.
    Grundy, P.M.: Mathematics and games. Eureka 2, 6–8 (1939)Google Scholar
  12. 12.
    Lichtenstein, D., Sipser, M.: GO is polynomial-space hard. J. ACM 27(2), 393–401 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Schaefer, T.J.: On the complexity of some two-person perfect-information games. J. Comput. Syst. Sci. 16(2), 185–225 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sprague, R.P.: Über mathematische Kampfspiele. Tohoku Math. J. 41, 438–444 (1935–1936)Google Scholar
  15. 15.
    Stockman, G., Frieze, A., Vera, J.: The game of Nim on graphs: NimG (2004).

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Stephen A. Fenner
    • 1
  • Daniel Grier
    • 2
  • Jochen Messner
    • 3
  • Luke Schaeffer
    • 2
  • Thomas Thierauf
    • 4
  1. 1.University of South CarolinaColumbiaUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.UlmGermany
  4. 4.Aalen UniversityAalenGermany

Personalised recommendations