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

Wojciech Olszewski

doi:10.1016/j.geb.2021.01.011

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

Wojciech Olszewski

Economics

Research output: Contribution to journal › Article › peer-review

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⁡sup_k⁡f^k(x⁰)))_n=1^∞ and (fⁿ(lim⁡inf_k⁡f^k(x⁰)))_n=1^∞ are fixed points of f. These limits are the tight fixed-point bounds between which sufficiently large iterations f^k(x⁰) are located. We provide an application of this result to studying best-response dynamics.

Original language	English (US)
Pages (from-to)	453-459
Number of pages	7
Journal	Games and Economic Behavior
Volume	126
DOIs	https://doi.org/10.1016/j.geb.2021.01.011
State	Published - Mar 2021

Keywords

Adaptive dynamics
Nash equilibria
Sequences of iterations
Tarski's theorem

ASJC Scopus subject areas

Finance
Economics and Econometrics

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10.1016/j.geb.2021.01.011

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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.",

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author = "Wojciech Olszewski",

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N2 - 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.

AB - 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.

KW - Adaptive dynamics

KW - Nash equilibria

KW - Sequences of iterations

KW - Tarski's theorem

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EP - 459

JO - Games and Economic Behavior

JF - Games and Economic Behavior

SN - 0899-8256

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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