The multiply iterated law of the iterated logarithm: game-theoretic foundations of sequential detection boundaries

Abstract

Anytime-valid confidence sequences and e-processes are built almost universally from one recipe: average exponential test statistics over a prior on the tilting scale, then invoke Ville's inequality on the resulting nonnegative supermartingale. The mixing prior sets the width of the detection boundary and is usually chosen by hand. We recast the recipe as a two-player game with information as currency. A Learner commits to the prior; Nature adaptively produces a mean-zero score process whose difficulty is priced by a cumulant-generating-function charge. The Learner's mixture wealth obeys a single pathwise Gibbs-variational identity that holds along every realized path with no expectation operator; Ville's inequality, the equalizer condition, the GROW characterization, and the saddlepoint formula are all specializations of it. Three messages organize the rest. First, the law of the iterated logarithm (LIL) is the minimax boundary of this sequential-detection game, not arbitrary combinatorial slack. Second, the optimal prior is not a design choice but the forced equalizer strategy -- the unique law that makes every boundary-crossing time equally costly for Nature -- and it yields the sharp first iterated-log correction in closed form, with coefficient 3/2 = 1 + 1/2 (one for the Erd\H{o}s baseline, one half for the Laplace envelope around the saddle). Third, in the log-log scale chart the equalizer is exactly the Jeffreys prior on the scale-of-scales. The Erd\H{o}s-Kolmogorov integral test is the criterion that selects it. The two-stage finite-time LIL proof, the Howard-Ramdas mixture and stitching constructions, and betting confidence sequences all read as instances of this equalizer principle. A companion empirical evaluation confirms the central identities and locates the Erd\H{o}s threshold at the predicted value.