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The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications

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Abstract

We root this tribute to Nicholas Yannelis in Chapter II of his 1983 Rochester Ph.D. dissertation, and in his 1983 paper with Prabhakar: this work strengthens the lower semicontinuity assumption of Michael’s continuous selection theorem to open lower sections, and leads to correspondences defined on a paracompact space with values on a Hausdorff linear topological space. We move beyond the literature to provide a necessary and sufficient condition for upper semi-continuous local and global selections of correspondences, and apply our result to four domains of Yannelis’ contributions: Berge’s maximum theorem, the Gale–Nikaido–Debreu lemma, the Sonnenschein–Shafer non-transitive setting, and the Anderson–Khan–Rashid approximate existence theorem. The last also resonates with Chapter VI of Yannelis’ dissertation, and allows a more general framing of the pioneering application of the paracompactness condition to his current and ongoing work in mathematical economics.

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Authors and Affiliations

  1. Department of Economics, Johns Hopkins University, Baltimore, MD, 21218, USA

    M. Ali Khan

  2. School of Economics, University of Queensland, Brisbane, QLD, 4072, Australia

    Metin Uyanik

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  1. M. Ali Khan
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Correspondence to Metin Uyanik.

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Some of the results reported here were first presented by Khan at the Summer Workshop in Economic Theory (SWET) on October 25, 2018 under the title “Nicholas Yannelis and Equilibrium Theory: Salient Contributions to Economics and Mathematics.” He thanks Bernard Cornet, Ed Prescott and Anne Villamil for discussion and encouragement at his talk; he should also like to acknowledge Greg Duffie and his JHU colleagues for a departmental discussion on proper names in connection with the renaming of the departmental Ely Lectures. This final submission has benefited substantially from stimulating conversation and correspondence with Youcef Askoura, Marcus Berliant, Kalyan Chatterjee, Ying Chen, Marcelo Ariel Fernandez, Aniruddha Ghosh, Rich McLean and Nicholas Yannelis: the authors thank them all, and also record their gratitude to Rich McLean for his 2020 Ischia slides and for giving the first draft a careful reading. Final thanks to Nicholas Yannelis for sharing the final version of his 1983 Rochester Ph.D. dissertation with the authors, and to him and to four anonymous referees for their determined and sympathetic reading of the paper.

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Khan, M.A., Uyanik, M. The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications. Econ Theory 71, 799–840 (2021). https://doi.org/10.1007/s00199-021-01359-4

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