Gaussian-efficient testing by betting on the mean of bounded data

Given adapted $[0,1]$-valued random variables $X_1,\dots,X_n$ with $\mathbb{E}(X_i\mid\mathcal F_{i-1})=\mu$, we propose a new nonasymptotic confidence interval for $\mu$ obtained by inverting terminal e-values generated by a novel betting strategy. Our main conceptual advance is to design betting fractions that track the conditional rejection probability of the most powerful terminal test in an idealized hypothetical Gaussian experiment. The resulting interval combines finite-sample validity under martingale dependence with interval-valued inversion on every sample path. Across our iid experiments, its width rapidly approaches the classical Gaussian benchmark and improves on previous betting intervals, yielding state-of-the-art empirical performance. We further extend the construction to sampling without replacement, where it exhibits the same behavior. These results provide empirical evidence that testing by betting can achieve near-Gaussian efficiency without giving up finite-sample martingale validity.