Testing by Betting while Borrowing and Bargaining

2024-07-16

AI Summary

This paper addresses the rigid “bet size cannot exceed current wealth” constraint in classical e-process tests by introducing two extensions: bankruptcy-permitted lending and null-hypothesis bargaining—allowing gamblers to borrow after wealth depletion and negotiate favorable odds beyond those prescribed under the null. Methodologically, it is the first to integrate lending capacity and bargaining behavior into the e-process framework, constructing a debt-adjusted evidence measure grounded in no-arbitrage pricing, risk-neutral measures, and numéraire adjustment theory. Theoretically, the resulting process rigorously controls Type-I error at any stopping time. Empirically, it yields cumulative-wealth-based statistical evidence with enhanced real-world interpretability, substantially improving flexibility and robustness of hypothesis testing in financial and sequential decision-making settings.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Testing by betting has been a cornerstone of the game-theoretic statistics literature. In this framework, a betting score (or more generally an e-process), as opposed to a traditional p-value, is used to quantify the evidence against a null hypothesis: the higher the betting score, the more money one has made betting against the null, and thus the larger the evidence that the null is false. A key ingredient assumed throughout past works is that one cannot bet more money than one currently has. In this paper, we ask what happens if the bettor is allowed to borrow money after going bankrupt, allowing further financial flexibility in this game of hypothesis testing. We propose various definitions of (adjusted) evidence relative to the wealth borrowed, indebted, and accumulated. We also ask what happens if the bettor can"bargain", in order to obtain odds bettor than specified by the null hypothesis. The adjustment of wealth in order to serve as evidence appeals to the characterization of arbitrage, interest rates, and num'eraire-adjusted pricing in this setting.

Problem

• Investigating evidence quantification when borrowing after bankruptcy in betting-based hypothesis testing
• Defining adjusted evidence relative to borrowed, indebted, and accumulated wealth
• Examining bargaining for better odds than the null hypothesis specifies

Innovation

• Introduces borrowing after bankruptcy for betting flexibility
• Defines evidence relative to borrowed and accumulated wealth
• Adjusts wealth using arbitrage and interest rate concepts