Competitive optimality in testing by betting via Bell-Cover randomization

2026-09-28

AI Summary

This study addresses the trade-off between competitive advantage and statistical power inherent in randomization strategies within betting tests. By introducing Bell-Cover randomization into the betting test framework, this work integrates log-optimal portfolios, uniform randomization, and e-process theory for inferential analysis. It establishes the competitive optimality of multiplicative e-variables and demonstrates their equivalence to numerical inequalities, while clarifying that randomization benefits only baseline rather than arbitrary e-variables. The findings reveal that although this strategy guarantees a fifty-percent win rate, it sacrifices expected log-wealth and testing power, and is dominated by threshold randomization. Furthermore, its limitations under optional stopping are elucidated.

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 $\mathcal P$ 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 $\mathbb E_Q[E/E^*]\leq1$, 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 $1/\alpha$ has sharp size at most $\alpha/2$, 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.

Problem

• competitive optimality
• testing by betting
• e-variable
• Bell-Cover randomization
• optional stopping

Innovation

• Testing by betting
• E-variables
• Competitive optimality
• Bell-Cover randomization
• Optional stopping