To Infinity and Beyond: A General Framework for Scaling Economic Theories

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.

• The paper introduces a scalable framework that leverages the Compactness Theorem to extend economic theories from finite to infinite domains.
• It demonstrates the methodology using examples from revealed-preference and matching theories, validating the approach on both discrete and continuous models.
• The framework overcomes traditional limitations in economic modeling, enabling more realistic analyses of complex market settings.

A General Framework for Scaling Economic Theories

The paper "To Infinity and Beyond: A General Framework for Scaling Economic Theories," by Gonczarowski, Kominers, and Shorrer, addresses a fundamental challenge in economic modeling: the reliance on finiteness assumptions for analytical simplicity and tractability. The authors propose a principled framework to scale economic theories by relaxing these assumptions, applying their methodology to various domains, and demonstrating its utility through numerous examples. The central contribution of the paper lies in its methodological approach, which utilizes logical compactness to extend results from finite to infinite settings.

Key Contributions

The paper highlights several important contributions:

1. Framework for Scaling: The authors introduce a framework that leverages results from Propositional Logic, enabling the scaling of economic models by focusing on theorem statements rather than proofs. This approach results in concise proofs and allows the generalization of existing results across different contexts.
2. Versatility Across Domains: The framework is demonstrated using examples from revealed-preference theory and matching theory. The authors show how their approach can handle both discrete and nondiscrete solution concepts, and they explore applications in decision theory and market design.
3. Addressing Inherent Limitations: The framework overcomes limitations inherent in finite logical statements, offering solutions to challenges posed by nondiscrete objects and continuous spaces.

Methodological Insights

The cornerstone of the paper's methodology is the application of the Compactness Theorem from Propositional Logic. This theorem provides the basis for scaling results by ensuring that if every finite subset of a set of logical statements (representing an economic problem) is satisfiable, then the entire set is satisfiable. The authors construct well descriptions of economic problems using logical formulae and demonstrate the finite-subset property, thereby ensuring solution existence in infinite settings.

Examples and Applications

The paper includes a number of detailed examples that illustrate the successful application of the framework:

• Rational Choice Functions: The authors reprove a classical result in revealed preferences, demonstrating that datasets satisfying the Strong Axiom of Revealed Preferences (SARP) are rationalizable without appeal to Zorn’s Lemma.
• Limited Attention Models: By extending results on limited attention rationalizability, the framework's agnosticism to proof techniques becomes evident as it scales conditions from finite to infinite datasets.
• Rationalizing Consumer Demand: The scaling is applied to show the existence of rationalizing utility functions for infinite datasets, retesting Afriat's Theorem's assumptions.
• Stable Matchings: In the field of matching theory, the authors scale classic results to infinite markets, addressing challenges with existing fixed-point methods.

Theoretical and Practical Implications

The implications of this research are twofold. Theoretically, it provides economists with a powerful tool for generalizing results, expanding the applicability of economic models beyond the confines of finiteness. Practically, the framework supports the development of more realistic economic models that align better with the complexities of real-world scenarios, where finite assumptions may artificiously limit validity.

Future Directions

The approach outlined in the paper opens up avenues for further research within AI and economics. Future developments could explore the application of this framework to additional fields and more complex economic environments. Moreover, researchers might venture into integrating this methodology with other logical or computational techniques to unearth novel insights in economic theory.

In conclusion, the "To Infinity and Beyond" paper presents a rigorous and adaptable framework for scaling economic theories beyond traditional finiteness constraints. By utilizing logical compactness, the authors provide a robust tool for researchers, facilitating significant advancements in economic modeling and increasing the applicability of theoretical results to infinite and complex real-world situations.