To Infinity and Beyond: Scaling Economic Theories via Logical Compactness

47 Pages Posted: 27 Jun 2019 Last revised: 9 Nov 2020

See all articles by Yannai A. Gonczarowski

Yannai A. Gonczarowski

Microsoft Research

Scott Duke Kominers

Harvard University

Ran I. Shorrer

Pennsylvania State University

Date Written: November 3, 2020

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 depend upon edge effects or artificial boundary conditions. Here, we present a unified method that enables us to remove finiteness assumptions, such as those on market sizes, time horizons, and datasets. We then apply our approach to a variety of matching, exchange economy, and revealed preference settings.

The key to our approach is Logical Compactness, a core result from Propositional Logic. Building on Logical Compactness, in a matching setting, we reprove large-market existence results implied by Fleiner's analysis, and (newly) prove both the strategy-proofness of the manoptimal 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 in_nite markets. In a dynamic matching setting, we prove that Pereyra's existence result for dynamic two-sided matching markets extends to a doubly infinite time horizon. Finally, beyond existence and characterization of solutions, 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.

Suggested Citation

Gonczarowski, Yannai A. and Kominers, Scott Duke and Shorrer, Ran I., To Infinity and Beyond: Scaling Economic Theories via Logical Compactness (November 3, 2020). Harvard Business School Entrepreneurial Management Working Paper No. 19-127, June 2019, Available at SSRN: https://ssrn.com/abstract=3409828 or http://dx.doi.org/10.2139/ssrn.3409828

Yannai A. Gonczarowski

Microsoft Research ( email )

One Memorial Drive, 14th Floor
Cambridge, MA 02142
United States

HOME PAGE: http://yannai.gonch.name/

Scott Duke Kominers (Contact Author)

Harvard University ( email )

Rock Center
Harvard Business School
Boston, MA 02163
United States

HOME PAGE: http://www.scottkom.com/

Ran I. Shorrer

Pennsylvania State University ( email )

University Park
State College, PA 16802
United States

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