Testing by Betting while Borrowing and Bargaining
Pith reviewed 2026-05-23 23:01 UTC · model grok-4.3
The pith
Allowing borrowing in betting-based hypothesis testing requires raising the rejection threshold above 1/α for liability-dependent rules, but path-dependent rules avoid this cost.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If the rejection rule takes the form W_t ≥ g(α, L_t) for total liability L_t, then g(α, 0) must exceed 1/α whenever g(α, L_t) is finite for L_t > 0. By contrast, a path-dependent threshold of the form h(α, W_0, L_1, …, W_{t-1}, L_t) allows rejection at the original level 1/α with no additional penalty for the possibility of borrowing.
What carries the argument
The threshold function for rejection, either g(α, L_t) which depends on liability alone or the path-dependent h which incorporates historical leverage ratios; the former necessitates a higher bar at zero liability to account for borrowing risk.
Load-bearing premise
The wealth process remains a supermartingale under the null even when borrowing allows negative wealth and betting strategies stay predictable.
What would settle it
Finding a specific g with g(α,0) = 1/α and g finite elsewhere such that the probability of rejecting under the null exceeds α when borrowing is used in the betting strategy.
read the original 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 $\alpha$ whenever $W_t \geq 1/\alpha$. 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 $1/\alpha$ must be accordingly adjusted to retain the desired level $\alpha$. Our findings are twofold. First, if the new rejection rule is $W_t \geq g(\alpha,L_t)$ where $L_t$ is the total liability at time $t$, then we show that $g(\alpha,0)>1/\alpha$ if $g(\alpha,L_t)<\infty$ 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 $h(\alpha,W_0,L_1,\dots,W_{t-1},L_t)$, that is a function of past leverage ratios, then there is in fact no extra price to pay for the possibility of borrowing.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript examines testing by betting when the nonnegativity constraint on wealth is relaxed to permit borrowing. It establishes two main results: (i) any rejection threshold of the form g(α, L_t) with L_t the cumulative liability must satisfy g(α,0) > 1/α whenever g(α, L_t) remains finite for L_t > 0, and (ii) a path-dependent threshold h(α, W_0, L_1, …, L_t) that incorporates the realized history of leverage ratios restores exact α-level control with no additional conservatism.
Significance. If the derivations hold, the work supplies a precise characterization of the price of borrowing in game-theoretic hypothesis testing. The contrast between liability-only and history-dependent thresholds is a clean, useful distinction that clarifies when the classical 1/α rule remains valid. The paper correctly invokes the supermartingale property under predictable strategies and an unchanged null measure; this is a strength.
minor comments (1)
- [Abstract] The abstract states the two findings clearly but does not indicate the precise measurability or predictability conditions imposed on the betting strategies; a one-sentence clarification would help readers locate the relevant definitions in the main text.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending acceptance. Their summary correctly identifies the two main results on liability-dependent versus history-dependent thresholds.
Circularity Check
No significant circularity; derivation self-contained from supermartingale properties
full rationale
The paper's central results follow directly from the definition of supermartingales (conditional expectation property under the null, preserved under predictable strategies regardless of wealth sign) and the standard Markov inequality bound P(W_t >= 1/alpha) <= alpha for nonnegative supermartingales. The adjustment g(alpha, L_t) > 1/alpha when L_t > 0 is shown by contradiction: any finite g for positive liability forces inflation at zero liability to restore type-I control, while path-dependent h restores the bound without inflation. No equations reduce by construction to fitted parameters, self-definitions, or unverified self-citations; the supermartingale property is an external mathematical fact independent of the present claims. This matches the default expectation of non-circularity for a self-contained mathematical argument.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Wealth process is a supermartingale under the null when betting strategies are predictable and wealth is constrained to be nonnegative.
discussion (0)
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