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To Infinity and Beyond: Scaling Economic Theories via Logical Compactness

Author

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  • Yannai A. Gonczarowski
  • Scott Duke Kominers
  • Ran I. Shorrer

Abstract

Many economic-theoretic models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. Such assumptions introduce a conceptual problem, as results that rely on finiteness are often implicitly nonrobust; for example, they may rely on edge effects or artificial boundary conditions. Here, we present a unified method that enables us to remove finiteness assumptions, such as those on datasets, market sizes, and time horizons. We then apply our approach to a variety of revealed preference, matching, and exchange economy settings. The key to our approach is Logical Compactness, a core result from Propositional Logic. Building on Logical Compactness, in a revealed-preference setting, we reprove Reny's infinite-data version of Afriat's theorem and (newly) prove an infinite-data version of McFadden and Richter's characterization of rationalizable stochastic datasets. In a matching setting, we reprove large-market existence results implied by Fleiner's analysis, and prove both the strategy-proofness of the man-optimal stable mechanism in infinite markets, and an infinite-market version of Nguyen and Vohra's existence result for near-feasible stable matchings with couples. In a trading-network setting, we prove that the Hatfield et al. result on existence of Walrasian equilibria extends to infinite markets. Finally, we prove that Pereyra's existence result for dynamic two-sided matching markets extends to a doubly-infinite time horizon.

Suggested Citation

  • Yannai A. Gonczarowski & Scott Duke Kominers & Ran I. Shorrer, 2019. "To Infinity and Beyond: Scaling Economic Theories via Logical Compactness," Papers 1906.10333, arXiv.org, revised Mar 2020.
  • Handle: RePEc:arx:papers:1906.10333
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    References listed on IDEAS

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    1. John William Hatfield & Scott Duke Kominers & Alexandru Nichifor & Michael Ostrovsky & Alexander Westkamp, 2013. "Stability and Competitive Equilibrium in Trading Networks," Journal of Political Economy, University of Chicago Press, vol. 121(5), pages 966-1005.
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    4. Kadam, Sangram V. & Kotowski, Maciej H., 2015. "Multi-period Matching," Working Paper Series rwp15-030, Harvard University, John F. Kennedy School of Government.
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