International Journal of Game Theory

, Volume 42, Issue 4, pp 777–788 | Cite as

Algorithms for lattice games

  • Alan GuoEmail author
  • Ezra Miller


This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games (Guo et al. Oberwolfach Rep 22: 23–26, 2009; Guo and Miller, Adv Appl Math 46:363–378, 2010). Given a rational strategy for a lattice game, a polynomial time algorithm is presented to decide (i) whether a given position is a winning position, and to find a move to a winning position, if not; and (ii) to decide whether two given positions are congruent, in the sense of misère quotient theory (Plambeck, Integers, 5:36, 2005; Plambeck and Siegel, J Combin Theory Ser A, 115: 593–622, 2008). The methods are based on the theory of short rational generating functions (Barvinok and Woods, J Am Math Soc, 16: 957–979, 2003).


Combinatorial game Lattice game Convex polyhedron Generating function Affine semigroup Misère quotient 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA

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