International Journal of Game Theory

, Volume 41, Issue 2, pp 345–367 | Cite as

Analyzing n-player impartial games

  • Walter O. Krawec


Combinatorial game theory is the study of two player perfect information games. While work has been done in the past on expanding this field to include n-player games we present a unique method which guarantees a single winner. Specifically our goal is to derive a function which, given an n-player game, is able to determine the winning player (assuming all n players play optimally). Once this is accomplished we use this function in analyzing a certain family of three player subtraction games along with a complete analysis of three player, three row Chomp. Furthermore we make use of our new function in producing alternative proofs to various well known two player Chomp games. Finally the paper presents a possible method of analyzing a two player game where one of the players plays a completely random game. As it turns out this slight twist to the rules of combinatorial game theory produces rather interesting results and is certainly worth the time to study further.


Combinatorial game theory Impartial games n-person games 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Berlekamp ER, Conway JH, Guy RK (2004) Winning ways for your mathematical plays. A. K. Peters, WellesleyGoogle Scholar
  2. Berlekamp ER, et al (1991) In: Guy RK (ed) Combinatorial games. American Mathematical Society, ProvidenceGoogle Scholar
  3. Cincotti A (2010) N-player partizan games. Theor Comput Sci 411: 3224–3234CrossRefGoogle Scholar
  4. Li S-YR (1978) N-person Nim and n-person Moore’s Games. J Int J Game Theory 7(1): 6–31Google Scholar
  5. Loeb DE (1996) Stable winning coalitions, games of no chance. In: Nowakowski RJ (ed) Proc. MSRI workshop on combinatorial games, CA MSRI Publ. vol 29. Cambridge University Press, Cambridge, pp 451–471Google Scholar
  6. Propp J (2000) Three-player impartial games. Theor Comput Sci 233: 263–278CrossRefGoogle Scholar
  7. Straffin PD (1985) Three-person winner-take-all games with Mc-Carthy’s revenge rule. Coll J Math 16: 386–394CrossRefGoogle Scholar
  8. Sun X (2002) Improvements on Chomp. Integers Electron J Comb Number Theory 2:1–8.
  9. Zeilberger D (2001) Three rowed Chomp. Adv Appl Math 26: 168–179CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity at AlbanyAlbanyUSA

Personalised recommendations