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Intransitivity and meta-intransitivity: meta-dice, levers and other opportunities

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Abstract

Various objects and systems in intransitive relations of superiority (“rock-paper-scissors” relations) are described (intransitive sets of sticks, dice, chess positions, machines, biological species etc.). The concept of meta-intransitivity of superiority is introduced. Meta-intransitivity is a property of systems, which are in intransitive relationships between one another, and, at the same time, each of them contains its own internal, nested intransitive cycles of superiority. The nested cycles�� number is its meta-intransitivity level. We distinguish between two types of meta-intransitivity: qualitative (ordinal, lexicographic), and quantitative. Respectively, we describe two examples of meta-intransitive systems of both types. The first example deals with combinatorics of geometrical shapes in meta-intransitive machines “beating” one another like in a rock-paper-scissors game, and the second one—with combinatorics of numbers in meta-intransitive dice (meta-dice). We show that the formerly known intransitive sets (e.g. intransitive dice) can be considered as intransitive sets with zero-level of meta-intransitivity. A way to build multi-level meta-intransitive sets based on nested Condorcet-like compositions is introduced. Meta-intransitivity of various objects can become a new interesting sub-area of logic and mathematics of intransitive relations. Interactive multiple, intertwining, “rhizomatic” (from the “rhizome” metaphor) intransitivities and meta-intransitivities of dominance in real complex systems seems worthy of special interest in future inter(trans)-disciplinary studies.

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Acknowledgements

The authors are very thankful to anonymous reviewers for their commentaries and recommendations.

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  1. Department of Psychology, HSE University, Moscow, Russia

    Alexander Poddiakov

  2. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    Alexey V. Lebedev

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  1. Alexander Poddiakov
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Correspondence to Alexander Poddiakov.

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Poddiakov, A., Lebedev, A.V. Intransitivity and meta-intransitivity: meta-dice, levers and other opportunities. European Journal of Mathematics 9, 27 (2023). https://doi.org/10.1007/s40879-023-00618-z

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