[Paper Review] To Infinity and Beyond: A General Framework for Scaling Economic Theories

Yannai A. Gonczarowski, Scott Duke Kominers+1 more|arXiv (Cornell University)|Jun 25, 2019

Economic theories and models4 citations

TL;DR

This paper introduces a general logical framework based on model-theoretic compactness to scale economic theorems from finite to infinite settings without altering their proofs. It enables the derivation of infinite-horizon or infinite-agent results from finite-model theorems by verifying only the theorem statement’s logical form, yielding concise, reusable proofs across revealed preference and matching theory applications.

ABSTRACT

Many economic theory models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. We provide a principled framework for scaling results from such models by removing these finiteness assumptions. Our sufficient conditions are on the theorem statement only, and not on its proof. This results in short proofs, and even allows to use the same argument to scale similar theorems that were proven using distinctly different tools. We demonstrate the versatility of our approach via examples from both revealed-preference theory and matching theory.

Motivation & Objective

• To address the conceptual and analytical limitations introduced by finiteness assumptions in economic models, which can obscure intrinsic theoretical properties.
• To develop a general, proof-agnostic method that allows the extension of finite-model results to infinite settings without modifying existing proofs.
• To demonstrate the method’s versatility by applying it to foundational results in revealed preference theory and matching theory.
• To establish that the scalability of a theorem depends solely on its logical structure, not on the complexity of its original proof.
• To provide a unified approach that avoids ad hoc constructions and enables reuse of existing arguments across different economic contexts.

Proposed method

• Leverages the compactness theorem from first-order logic to infer the existence of models in infinite settings from finite submodels.
• Constructs a first-order logical theory encoding the conditions of a given economic model (e.g., preferences, capacities, matchings) using propositional and predicate variables.
• Defines a set of logical formulas that capture all relevant constraints—such as stability, feasibility, and blocking conditions—for a given model instance.
• Applies the finite-subset property: if every finite subset of the theory has a model, then the entire theory has a model, ensuring the existence of an infinite solution.
• Uses logical compactness to derive infinite-horizon or infinite-agent analogues of finite-theorems, such as stable matchings with infinite hospitals or demand functions over infinite datasets.
• Demonstrates that the same logical framework can be reused across different domains (e.g., revealed preference, matching) by encoding domain-specific constraints within the same logical structure.

Experimental results

Research questions

• RQ1 Can finite economic theorems be systematically extended to infinite settings without reworking their proofs?
• RQ2 What logical conditions on a theorem statement ensure its scalability from finite to infinite models?
• RQ3 To what extent can the same proof strategy be reused across different economic theories after scaling?
• RQ4 How can infinite models (e.g., infinite-horizon matching, infinite-demand functions) be constructed from finite ones using logical consistency?
• RQ5 What are the conceptual and technical advantages of using model-theoretic compactness over ad hoc infinite extensions?

Key findings

• The framework enables the derivation of infinite-horizon stable matchings from finite ones using only the logical form of the stability condition.
• The existence of a man-optimal stable matching in infinite settings is established via compactness, without requiring new proof techniques.
• Revealed preference theorems that assume finite data can be extended to infinite demand functions by verifying only the logical structure of the theorem.
• The method produces short, uniform proofs across diverse domains, such as matching and revealed preference, by abstracting away from proof-specific details.
• The framework identifies that scalability depends solely on the logical form of the theorem, not on the complexity of its original proof.
• The approach successfully generalizes results from finite to infinite settings in matching theory, including cases with doubly infinite horizons and infinite capacity constraints.