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On sequences of iterations of increasing and continuous mappings on complete lattices

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  • Olszewski, Wojciech

Abstract

We generalize the famous Tarski result by showing that: if X is a complete lattice, and f:X→X is an increasing and continuous mapping, then for all points x0∈X, the limits of sequences (fn(lim⁡supk⁡fk(x0)))n=1∞ and (fn(lim⁡infk⁡fk(x0)))n=1∞ are fixed points of f. These limits are the tight fixed-point bounds between which sufficiently large iterations fk(x0) are located. We provide an application of this result to studying best-response dynamics.

Suggested Citation

  • Olszewski, Wojciech, 2021. "On sequences of iterations of increasing and continuous mappings on complete lattices," Games and Economic Behavior, Elsevier, vol. 126(C), pages 453-459.
    • Handle: RePEc:eee:gamebe:v:126:y:2021:i:c:p:453-459
    DOI: 10.1016/j.geb.2021.01.011
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    References listed on IDEAS

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    1. Hiroki Nishimura & Efe A. Ok, 2012. "Solvability of Variational Inequalities on Hilbert Lattices," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 608-625, November.
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    4. Echenique, Federico, 2007. "Finding all equilibria in games of strategic complements," Journal of Economic Theory, Elsevier, vol. 135(1), pages 514-532, July.
    5. Vives, Xavier, 1990. "Nash equilibrium with strategic complementarities," Journal of Mathematical Economics, Elsevier, vol. 19(3), pages 305-321.
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    Cited by:

    1. Barthel, Anne-Christine & Hoffmann, Eric, 2023. "On the existence of stable equilibria in monotone games," Journal of Mathematical Economics, Elsevier, vol. 105(C).
    2. Jaeok Park & Doo Hyung Yun, 2023. "Possibilistic beliefs in strategic games," Theory and Decision, Springer, vol. 95(2), pages 205-228, August.

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    More about this item

    Keywords

    Tarski's theorem; Sequences of iterations; Nash equilibria; Adaptive dynamics;
    All these keywords.

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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