To Infinity and Beyond: Scaling Economic Theories via Logical Compactness

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 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 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 [2015] infinite-data version of Afriat's [1967] theorem and (newly) prove an infinite-data version of McFadden and Richter's [1971, 1990] characterization of rationalizable stochastic datasets. In a matching setting, we reprove large-market existence results implied by Fleiner's [2003] 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 [2018] existence result for near-feasible stable matchings with couples. In a trading-network setting, we prove that the Hatfield et al. [2013] result on existence of Walrasian equilibria extends to infinite markets. Finally, we prove that Pereyra's [2013] existence result for dynamic two-sided matching markets extends to a doubly-infinite time horizon.