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Testing by Betting while Borrowing and Bargaining

Hongjian Wang, Aaditya Ramdas

TL;DR

The paper addresses how borrowing and bargaining affect evidence in testing by betting, extending the classical coin-betting framework to allow post-bankruptcy borrowing and odds bargaining. It develops a formal structure of evidence using wealth, liabilities, and net wealth, introducing tail evidence, sequential tail evidence, and current e-values that can be computed from observed data without counterfactual assumptions. Key contributions include conditions under which gross wealth and net wealth yield valid evidence, a leverage-invariance result showing borrowing cannot improve standardized evidence in a one-round setting, and an almost supermartingale framework for Bargained NSM Betting. The findings clarify fundamental limits on evidence gains from borrowing and bargaining, while highlighting rich avenues for future work on borrowing-dependent evidences and alternative utility-based criteria, with implications for robust hypothesis testing under financial constraints.

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 numéraire-adjusted pricing in this setting.

Testing by Betting while Borrowing and Bargaining

TL;DR

The paper addresses how borrowing and bargaining affect evidence in testing by betting, extending the classical coin-betting framework to allow post-bankruptcy borrowing and odds bargaining. It develops a formal structure of evidence using wealth, liabilities, and net wealth, introducing tail evidence, sequential tail evidence, and current e-values that can be computed from observed data without counterfactual assumptions. Key contributions include conditions under which gross wealth and net wealth yield valid evidence, a leverage-invariance result showing borrowing cannot improve standardized evidence in a one-round setting, and an almost supermartingale framework for Bargained NSM Betting. The findings clarify fundamental limits on evidence gains from borrowing and bargaining, while highlighting rich avenues for future work on borrowing-dependent evidences and alternative utility-based criteria, with implications for robust hypothesis testing under financial constraints.

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 numéraire-adjusted pricing in this setting.
Paper Structure (12 sections, 11 theorems, 43 equations, 1 table)

This paper contains 12 sections, 11 theorems, 43 equations, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. The Classical Betting Game without Borrowing
  4. Betting Game with Borrowing
  5. Evidence in Borrowed Betting
  6. Gross Wealth as Evidence
  7. Net Wealth as Evidence
  8. Current Evidence
  9. Borrowing as Averaging
  10. The Futile Borrower
  11. Almost Supermartingales and Bargained Bets
  12. Summary

Key Result

Proposition 2.4

The net wealth process $\{ N_t \}$ is a martingale on $\{ \mathcal{F}_t \}$ under $\mathbb P _{\mathsf M}$ and a supermartingale under $\mathbb P _{\mathsf S}$. Further, under $\mathbb P _{\mathsf M}$, $W_t = N_t + L_t$ is the Doob decomposition of the process $\{W_t\}$ into the martingale $\{N_t\}$

Theorems & Definitions (30)

  • Definition 2.1: Coin Betting
  • Definition 2.2: NSM Betting
  • Definition 2.3: Borrowed NSM Betting
  • Proposition 2.4: Doob decomposition of wealth
  • proof
  • Definition 3.1: Tail evidence
  • Definition 3.2: Sequential tail evidence
  • Proposition 3.3
  • proof
  • Proposition 4.3
  • ...and 20 more