To Infinity and Beyond: Scaling Economic Theories via Logical Compactness

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

Working Paper | HBS Working Paper Series | 2020

To Infinity and Beyond: Scaling Economic Theories via Logical Compactness

by Yannai A. Gonczarowski, Scott Duke Kominers and 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 the 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.

Keywords: Markets; Analysis; Game Theory;

Language: English

Format: Print

43 pages

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Citation:

Gonczarowski, Yannai A., Scott Duke Kominers, and Ran I. Shorrer. "To Infinity and Beyond: Scaling Economic Theories via Logical Compactness." Harvard Business School Working Paper, No. 19-127, June 2019. (Revised March 2020.)

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We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for n ∈{8,24,32,48,56,72,96} every extremal Type II code of length n is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along the way we introduce "t12-designs" as a discrete analog of Venkov's spherical designs of the same name.

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To Infinity and Beyond: Scaling Economic Theories via Logical Compactness

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