Journal of Mathematical Sciences

, Volume 145, Issue 3, pp 4967–4974

Potential theory for mean payoff games

  • Y. M. Lifshits
  • D. S. Pavlov

DOI: 10.1007/s10958-007-0331-y

Cite this article as:
Lifshits, Y.M. & Pavlov, D.S. J Math Sci (2007) 145: 4967. doi:10.1007/s10958-007-0331-y


The paper presents an O(mn2nlog Z) deterministic algorithm for solving the mean payoff game problem, m and n being the numbers of arcs and vertices, respectively, in the game graph, and Z being the maximum weight (the weights are assumed to be integers). The theoretical basis for the algorithm is the potential theory for mean payoff games. This theory allows one to restate the problem in terms of solving systems of algebraic equations with minima and maxima. Also, in order to solve the mean payoff game problem, the arc reweighting technique is used. To this end, simple modifications, which do not change the set of winning strategies, are applied to the game graph; in the end, a trivial instance of the problem is obtained. It is shown that any game graph can be simplified by n reweightings. Bibliography: 16 titles.

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Y. M. Lifshits
    • 1
  • D. S. Pavlov
    • 2
  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia
  2. 2.Department of MathematicsInstitute of Fine Mechanics and OpticsSt. PetersburgRussia

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