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Impartial achievement and avoidance games for generating finite groups

Author

Listed:
  • Dana C. Ernst

    (Northern Arizona University)

  • Nándor Sieben

    (Northern Arizona University)

Abstract

We study two impartial games introduced by Anderson and Harary and further developed by Barnes. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions.

Suggested Citation

  • Dana C. Ernst & Nándor Sieben, 2018. "Impartial achievement and avoidance games for generating finite groups," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(2), pages 509-542, May.
    • Handle: RePEc:spr:jogath:v:47:y:2018:i:2:d:10.1007_s00182-017-0602-x
    DOI: 10.1007/s00182-017-0602-x
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    References listed on IDEAS

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    1. Max Albert, 2008. "Introduction," Conferences on New Political Economy,in: Max Albert & Stefan Voigt & Dieter Schmidtchen (ed.), Conferences on New Political Economy, edition 1, volume 25, pages 1-9 Mohr Siebeck, Tübingen.
    2. Anderson, M & Harary, F, 1987. "Achievement and Avoidance Games for Generating Abelian Groups," International Journal of Game Theory, Springer;Game Theory Society, vol. 16(4), pages 321-325.
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