Competitive optimality in testing by betting via Bell-Cover randomization
Abstract
Bell and Cover showed that an investor who multiplies the initial unit of capital by an independent uniform random variable on (0,2), and then uses the log-optimal portfolio, wins a head-to-head wealth comparison with probability at least one half against every independently randomized competitor. We explain very simply how this result transfers to testing by betting: for any composite null 𝒫 and simple alternative Q, denoting E* as the corresponding numeraire e-variable, we show that UE* exceeds any other e-variable E with probability at least half. Interestingly, we show that this competitive optimality result is actually equivalent to the numeraire inequality 𝔼Q[E/E*]≤1, and in general randomization only helps the numeraire and fails to improve the competitive advantage of an arbitrary e-variable. Under optional stopping with or without knowledge of U, we emphasize a key distinction between e-process validity and competitive optimality. We also show that competitive optimality comes at the price of expected log wealth and power: thresholding UE* at 1/α has sharp size at most α/2, but the factor of two actually disappears under optional stopping. Even after correcting for this factor of two, the test is dominated in conditional rejection probability by randomizing the testing threshold (randomized Markov's inequality). Thus, Bell-Cover randomization is optimal for a specific competitive objective, at the cost of others.
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BibTeX
@misc{ramdas2026competitive,
title = {{Competitive optimality in testing by betting via Bell-Cover randomization}},
author = {Aaditya Ramdas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35714},
eprint = {2609.35714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35714},
}
RIS
TY - UNPB
TI - Competitive optimality in testing by betting via Bell-Cover randomization
AU - Aaditya Ramdas
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35714
UR - https://arxiv.org/abs/2609.35714
AB - Bell and Cover showed that an investor who multiplies the initial unit of capital by an independent uniform random variable on (0,2), and then uses the log-optimal portfolio, wins a head-to-head wealth comparison with probability at least one half against every independently randomized competitor. We explain very simply how this result transfers to testing by betting: for any composite null 𝒫 and simple alternative Q, denoting E^* as the corresponding numeraire e-variable, we show that UE^* exceeds any other e-variable E with probability at least half. Interestingly, we show that this competitive optimality result is actually equivalent to the numeraire inequality 𝔼_Q[E/E^*]≤1, and in general randomization only helps the numeraire and fails to improve the competitive advantage of an arbitrary e-variable. Under optional stopping with or without knowledge of U, we emphasize a key distinction between e-process validity and competitive optimality. We also show that competitive optimality comes at the price of expected log wealth and power: thresholding UE^* at 1/α has sharp size at most α/2, but the factor of two actually disappears under optional stopping. Even after correcting for this factor of two, the test is dominated in conditional rejection probability by randomizing the testing threshold (randomized Markov's inequality). Thus, Bell-Cover randomization is optimal for a specific competitive objective, at the cost of others.
ER -