Optimal Priors for Betting Tests with Bounded Payoffs: An Exact Efficiency Characterization

Testing by betting is a powerful framework that recasts hypothesis testing as a betting game. A popular approach uses an $F$-weighted portfolio with a predefined prior $F$. For suitable priors, this approach has recently been shown to achieve the optimal expected rejection time as the error probability goes to zero at a fixed alternative. Yet, the prior $F$ changes how the portfolio test behaves near the null. This motivates the question: Which prior should we use in testing-by-betting algorithms? We study bounded scalar payoffs and introduce an efficiency measure that allows the error probability and alternative to vary jointly. Under mild assumptions, we prove that efficiency equals a logarithmic decay index of the prior near zero. This gives a necessary and sufficient condition for optimality: Beta priors have infinite efficiency, whereas suitable Robbins-type priors attain efficiency 1. As an application, we consider testing a specified mean of a bounded random variable and describe an implementation of the portfolio algorithm with almost-linear computational complexity.