Testing by Betting while Borrowing and Bargaining

TL;DR

This paper investigates how allowing borrowing in betting-based hypothesis testing affects the thresholds for significance, revealing that borrowing influences the threshold unless a path-dependent approach is used.

Contribution

It introduces a novel analysis of borrowing in betting tests, showing how thresholds must be adjusted or can remain unchanged with path-dependent strategies.

Findings

Thresholds increase if borrowing is allowed and fixed.

Path-dependent thresholds eliminate the extra cost of borrowing.

Borrowing impacts the significance threshold unless adaptive strategies are used.

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.