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Impartial games with decreasing Sprague–Grundy function and their hypergraph compound

Author

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  • Endre Boros

    (Rutgers University)

  • Vladimir Gurvich

    (National Research University Higher School of Economics)

  • Nhan Bao Ho

    (La Trobe University)

  • Kazuhisa Makino

    (Kyoto University)

  • Peter Mursic

    (Rutgers University)

Abstract

The Sprague–Grundy (SG) theory reduces the disjunctive compound of impartial games to the classical game of NIM. We generalize this concept by introducing hypergraph compounds of impartial games. An impartial game is called SG-decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce hypergraph compounds of SG-decreasing games to hypergraph compounds of single-pile NIM games. We show that this reduction works only if all games involved in the compound are SG-decreasing. A hypergraph is called SG-decreasing if the corresponding hypergraph compound of single-pile NIM games is an SG-decreasing game. We provide some necessary and some sufficient conditions for a hypergraph to be SG-decreasing. In particular, for hypergraphs with hyperedges of size at most 3 we obtain a necessary and sufficient condition verifiable in polynomial time.

Suggested Citation

  • Endre Boros & Vladimir Gurvich & Nhan Bao Ho & Kazuhisa Makino & Peter Mursic, 2024. "Impartial games with decreasing Sprague–Grundy function and their hypergraph compound," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(4), pages 1119-1144, December.
    • Handle: RePEc:spr:jogath:v:53:y:2024:i:4:d:10.1007_s00182-023-00850-7
    DOI: 10.1007/s00182-023-00850-7
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    References listed on IDEAS

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    1. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    2. Endre Boros & Vladimir Gurvich & Nhan Bao Ho & Kazuhisa Makino, 2021. "On the Sprague–Grundy function of extensions of proper Nim," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(3), pages 635-654, September.
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