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FairKalah: Towards Fair Mancala Play

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Computers and Games (CG 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13865))

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Abstract

Kalah (a.k.a. Mancala) is a two-player game of perfect information that has been a popular game for over half a century despite a strong first player advantage. In this paper, we present initial game states that are fair, as well as optimal play insights from analysis of optimal and suboptimal states.

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Notes

  1. 1.

    In this paper, we use “fair” in the sense that two perfect players would draw the game. There is no game-theoretic advantage for either player.

  2. 2.

    Thus, a potential free move is easily seen where the number of pieces matches our numeric label for that pit.

  3. 3.

    There is some controversy over whether “empty captures” were intended, as they were disallowed in Dakon. However, a strict reading of the patent makes no requirement for the number of opponent pieces opposite. Our interpretation is consistent with [5] and most printed rules.

  4. 4.

    When we have a fair game with piece(s) in the score pit, we have essentially created a fair game with fewer piece(s) and a perfect komi compensation for fairness.

References

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  5. Irving, G., Donkers, J., Uiterwijk, J.: Solving Kalah. ICGA J. 23(3), 139–147 (2000). http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.4.7870

  6. Lundberg, S.: beeswarm plot - SHAP latest documentation (2022). https://shap.readthedocs.io/en/latest/example_notebooks/api_examples/plots/beeswarm.html. Accessed: 2022-09-03

  7. Lundberg, S.M., Lee, S.I.: A unified approach to interpreting model predictions. In: Guyon, I., et al. (eds.) Advances in Neural Information Processing Systems, vol. 30. Curran Associates, Inc. (2017). https://proceedings.neurips.cc/paper/2017/file/8a20a8621978632d76c43dfd28b67767-Paper.pdf

  8. Neller, T.W., Neller, T.C.: FairKalah: Fair Mancala. http://cs.gettysburg.edu/~tneller/games/fairkalah/. Accessed: 2022-08-29

  9. Plaat, A., Schaeffer, J., Pijls, W., de Bruin, A.: Best-first fixed-depth minimax algorithms. Artif. Intell. 87(1), 255–293 (1996). https://doi.org/10.1016/0004-3702(95)00126-3

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Author information

Authors and Affiliations

  1. Gettysburg College, Gettysburg, PA, 17325, USA

    Todd W. Neller & Taylor C. Neller

Authors
  1. Todd W. Neller
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  2. Taylor C. Neller
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Corresponding author

Correspondence to Todd W. Neller .

Editor information

Editors and Affiliations

  1. Maastricht University, Maastricht, The Netherlands

    Cameron Browne

  2. IBM Research - Tokyo, Tokyo, Japan

    Akihiro Kishimoto

  3. University of Alberta, Edmonton, AB, Canada

    Jonathan Schaeffer

7Appendix: FairKalah Boards

7Appendix: FairKalah Boards

1.1 7.1Moving 1 Piece

These are all existing FairKalah boards where one moves exactly one piece from 4-pieces-per-pit initial conditions.

figure a

1.2 Moving 2 Pieces

All 251 existing FairKalah boards where one moves exactly two pieces from 4-pieces-per-pit initial conditions are available in SVG image form and as CSV data from the FairKalah website [8]. In the CSV data, comma-separated integers on each line indicate the number of pieces in each pit (including scoring pits) starting from the first (i.e. “south”) player’s leftmost play pit and proceeding counterclockwise.

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Neller, T.W., Neller, T.C. (2023). FairKalah: Towards Fair Mancala Play. In: Browne, C., Kishimoto, A., Schaeffer, J. (eds) Computers and Games. CG 2022. Lecture Notes in Computer Science, vol 13865. Springer, Cham. https://doi.org/10.1007/978-3-031-34017-8_1

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  • DOI: https://doi.org/10.1007/978-3-031-34017-8_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-34016-1

  • Online ISBN: 978-3-031-34017-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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