Testing by Betting while Borrowing and Bargaining

Paper: arXiv 2407.11465

Authors: Hongjian Wang, Muriel F. Pérez-Ortiz, Wouter M. Koolen, Aaditya Ramdas

Abstract

Testing by betting has been a cornerstone of the game-theoretic statistics literature. One bets against the null hypothesis, and the accumulated wealth $W_t$ quantifies the evidence against the null hypothesis after $t$ rounds, and the null can be rejected at level $α$ whenever $W_t \geq 1/α$. A key assumption permeating the literature is that one cannot bet more money than they currently have (the wealth must stay nonnegative). In this work, we examine the consequences of allowing the bettor to borrow money in each round (for example after going bankrupt). Specifically, we ask how the threshold of $1/α$ must be accordingly adjusted to retain the desired level $α$. Our findings are twofold. First, if the new rejection rule is $W_t \geq g(α,L_t)$ where $L_t$ is the total liability at time $t$, then we show that $g(α,0)>1/α$ if $g(α,L_t)<\infty$ for any $L_t > 0$; in words, we must pay for the possibility of borrowing, even if in fact we do not borrow. Second, and in contrast to the first, if one employs a path dependent threshold $h(α,W_0,L_1,\dots,W_{t-1},L_t)$, that is a function of past leverage ratios, then there is in fact no extra price to pay for the possibility of borrowing.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical expansion of the game-theoretic statistics literature by incorporating leverage/borrowing into testing-by-betting protocols, featuring formal proofs and boundary theorems. However, it is purely mathematical with no empirical backtesting or real-world financial/statistical data evaluation included in the provided text.

Research Flowchart

  flowchart TD
    A[Research Goal: Adjust rejection threshold 1/α when borrowing/leverage is allowed] --> B[Model Setup: Game-theoretic testing by betting with liabilities L_t and wealth W_t]
    B --> C[Analyze Threshold Strategies]
    
    C --> D[Static/Liability-Only Thresholds: g α, L_t]
    C --> E[Path-Dependent Thresholds: h α, W_0, L_1, ..., L_t]
    
    D --> F[Outcome 1: Extra cost required g α,0 > 1/α even if no borrowing occurs]
    E --> G[Outcome 2: No extra cost needed by leveraging past leverage ratios]

Cited References (13)

1. A convergence theorem for non negative almost supermartingales and some applications (1971)
2. Testing by betting: A strategy for statistical and scientific communication (2021)
3. Admissible anytime-valid sequential inference must rely on nonnegative martingales (2020) — link
4. Estimating means of bounded random variables by betting (2024)
5. Probability and finance: it’s only a game! (2005)
6. Etude critique de la notion de collectif (1939)
7. Time-uniform, nonparametric, nonasymptotic confidence sequences (2021)
8. A generalisation of Ville’s inequality to monotonic lower bounds and thresholds (2026) — link

0 of 13 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.