Source-linked AI summary
Competitive optimality in testing by betting via Bell-Cover randomization
Aaditya Ramdas
TL;DR
The paper asks how randomizing initial wealth affects validity, competitive outperformance, and rejection probability, which are distinct objectives. It transfers Bell and Cover’s uniform-lottery result to testing by betting and shows that the numeraire’s competitive guarantee has limits across objectives and settings.
Problem
Whether randomized initial wealth preserves valid evidence, beats competing strategies, or improves rejection probability are distinct questions with different answers.
Source: §1 · p. 1Method
The paper transfers Bell and Cover’s uniform-capital lottery to the numeraire e-variable and studies its competitive guarantee under sequential testing.
Source: §8 · p. 13Results
Against a specified Q, an independently randomized numeraire wins pairwise comparisons with every independently randomized e-variable at least half the time.
Source: §8 · p. 13Takeaways & Limitations
The uniform-capital lottery suits head-to-head competition, while the unrandomized numeraire benchmarks logarithmic evidence and threshold randomization is more favorable for terminal-e-variable power.
Source: §8 · p. 14Randomization is not guaranteed to improve an arbitrary e-variable’s win probability, and competitive optimality for a specified alternative is not a universal finite-sample guarantee.
Source: §5.2 · p. 9
Abstract
from arXiv · showhide
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.
1 Introduction
The paper translates Bell and Cover’s competitive betting result to testing, while distinguishing competitive outperformance from e-validity and increased rejection probability. It studies when randomizing initial wealth helps and how that objective differs from practical statistical benefit.
1 Introduction
- The betting setup starts with one unit of capital, uses it to bet against a null, and interprets resulting wealth as evidence against that null.Repeated bets produce wealth processes, including e-processes for composite nulls.Source: §1 · p. 1
- Randomizing initial wealth raises three distinct questions: whether evidence remains valid, whether a strategy beats another, and whether rejection probability improves.The paper emphasizes that these objectives can have different answers.Source: §1 · p. 1
- The paper shows that the Bell–Cover half-win guarantee transfers to the log-optimal numeraire e-value and is equivalent to the numeraire property.Larsson et al. (2025) established existence of a numeraire for arbitrary composite nulls and point alternatives; the paper presents the equivalence to competitive optimality.Source: §1 · p. 2
- Bell and Cover distinguished competitive protection from increasing investment capital and did not advocate adding the lottery in practice.The paper's distinction between competitive optimality and practical statistical benefit is consistent with that interpretation.Source: §1 · p. 2
2 Statistical setting and the numeraire
The paper sets up e-variables for a composite null and a specified alternative, then defines the numeraire used in competitive comparisons. It also states the assumptions that exclude trivial infinite-wealth rejection and establish the numeraire’s existence.
2 Statistical setting and the numeraire
- An e-variable is a measurable nonnegative statistic whose expectation is at most one under every distribution in the null family.The class of such variables is convex and includes the constant one.Source: §2 · p. 2
- The condition Q≪P excludes events impossible under every null distribution but possible under Q, where an e-variable could otherwise be infinite at no null cost.This is weaker than requiring Q to be absolutely continuous with respect to one particular member of P.Source: §2 · p. 2
- A numeraire exists for every nonempty null family and alternative, is unique up to Q-null sets, and is finite Q-almost surely exactly when Q≪P.This existence and uniqueness result is attributed to Larsson et al. (2025).Source: §2 · p. 3
- Competitive comparisons use independent private random seeds, with both players evaluating their statistics on the same data.The half-probability guarantee averages over the alternative and both lotteries; it is not conditional on every realized dataset.Source: §2 · p. 3
3 The Bell–Cover mechanism on a convex class
The paper uses a uniform lottery to connect competitive outperformance to the numeraire’s expected-ratio and relative log-optimality properties. It transfers this connection to e-variables and clarifies how infinite numeraires require a tie-adjusted game.
3 The Bell–Cover mechanism on a convex class
- Conditioning on the compared wealths gives a uniform-lottery win probability of min{R/2, 1}, yielding a sharp expected-ratio bound.The bound is sharp when the two wealths are equal; the corresponding nonrandomized comparison bound is min{1, 1/c}.Source: §3 · p. 3
3.1 Log-optimality, expected ratios, and competitive optimality
- For every S in C, E_Q[log(S/S★)] ≤ 0, with expectation in [−∞, 0]; independently drawing U∼Unif(0, 2) gives Pr_Q(US★ > S) ≥ 1/2.The passage identifies equivalence (i)–(ii) as covered by Bell and Cover (1988, Theorem 2), and says equivalence to (iii) is new.Source: §3.1 · p. 4
- Relative log-optimality remains meaningful when separate expected logarithms are infinite, provided the relative expectation is well defined.This does not permit the wealth itself to be infinite.Source: §3.1 · p. 4
3.2 The e-variable competitive theorem
- The e-variable competitive theorem makes U E★ a randomized e-variable that wins against every privately randomized competitor at least half the time.Independent randomized numeraires form a saddle point with game value one half.Source: §3.2 · pp. 4–5
3.3 What happens when the numeraire is infinite?
- With an infinite numeraire, the strict-win guarantee can fail, but the tie-adjusted game still has value one half.The extended-value result gives the half-credit guarantee against randomized e-variables and a saddle point for independent randomized numeraires.Source: §3.3 · p. 5
4 The subtleties of optional stopping: validity vs. competitive optimality
This section separates optional-stopping validity from competitive optimality: independent initial randomization preserves e-process validity even when stopping sees the draw, whereas competitive optimality requires independence. It also identifies settings where stopped comparisons hold and examples that rule out stronger claims.
4 The subtleties of optional stopping: validity vs. competitive optimality
- Independent initial capital A with E[A] ≤ 1 preserves e-process validity at bounded stopping times, even when stopping can inspect A.Validity is ex ante; conditional on U = u, the crossing bound is min{αu, 1}, so choosing a favorable draw or redrawing changes the experiment.Source: §4 · p. 6
4.1 With stopping, competitive optimality and validity are different issues
- Competitive optimality at a stopping time requires the comparison time to be independent of U, unlike e-process validity.The criterion permits data-dependent stopping and use of an opponent’s independent seed; drawing a fresh U after stopping ensures the required independence.Source: §4.1 · p. 6
4.2 A simple extension: conditional log-optimal betting
- Conditional relative log-optimality makes the wealth-ratio process a nonnegative Q-supermartingale, yielding competitive optimality at U-independent stopping times within the specified betting class.The allowed wealth multipliers form a conditionally convex class.Source: §4.2 · p. 7
4.3 A simple example: likelihood-ratio processes against a simple null
- For a simple null, stopped likelihood-ratio comparisons give at least a half-win against every P-e-process at comparison times independent of U.This applies to Q-almost surely finite stopping times without requiring global absolute continuity on the infinite-horizon sigma-field; the stopped expectation bound does not imply that every ratio process is a Q-supermartingale.Source: §4.3 · p. 7
4.4 Three ways a stronger claim can fail
- When the comparison time sees U, the randomized benchmark can lose immediately for U < 1 and wins only on tails for U ≥ 1.Null error control still holds, but this example separates competitive optimality from e-process validity and from a guarantee of staying ahead at all times.Source: §4.4 · p. 8
- An opponent who observes U can use a mean-3/4 randomized initial wealth that beats U with probability 3/4.Thus the independent-private-randomization protocol is part of the competitive guarantee.Source: §4.4 · p. 8
- A stopping rule can make the stopped expectation 5/4 > 1, so static optimality at each observation time does not ensure an e-process.The example stops at time one on outcome a and at time two otherwise.Source: §4.4 · p. 8
5 Important caveats
Initial uniform randomization protects the numeraire against lottery-driven wins but has costs in log evidence and does not confer competitive optimality on arbitrary e-variables. Its reported wealth is not itself the numeraire, and repeated randomization can impose a persistent negative growth rate.
5.1 Risks of the competitive objective
- A fair lottery can make V E★ beat E★ with probability 1−ε, arbitrarily close to one, while risking bankruptcy and giving expected log-wealth minus infinity.The uniform strategy protects against this fair-lottery vulnerability in a probability-of-outperformance game.Source: §5.1 · pp. 8–9
5.2 Randomization is only guaranteed to help the numeraire
- Randomization does not improve the competitive win probability of an arbitrary e-variable, and the optimal numeraire generally depends on the specified alternative Q.The competitive guarantee is therefore not a universal finite-sample guarantee across alternatives.Source: §5.2 · p. 9
- The authors omit attempted extensions to unknown Q using asymptotic log-optimality because they added algebra but little new insight.Those attempts combined numeraire competitive optimality with bounds on strategy wealth derived from numeraire wealth.Source: §5.2 · p. 9
5.3 The randomized numeraire is not the numeraire
- On the enlarged experiment, the original E★ remains admissible while randomized U E★ is not a numeraire, so the numeraire optimum and randomized competitive optimum differ.The averaging argument preserves E★'s numeraire role for every randomized e-variable T.Source: §5.3 · p. 9
5.4 An exact log-evidence penalty
- When separate expected logarithms are finite, initial randomization reduces expected log evidence by about 0.307 nats, a geometric multiplier of 2/e ≈ 0.736.The relative calculation remains meaningful even if the separate expected logarithms are infinite.Source: §5.4 · p. 9
- The lottery can at most double wealth but can have an unbounded negative log effect; against the underlying wealth, it raises evidence half the time and lowers it half the time.It therefore provides neither stochastic dominance nor pointwise improvement.Source: §5.4 · pp. 9–10
- A single initial draw does not change asymptotic log growth along positive-wealth paths, whereas repeated independent mean-one draws impose a persistent negative growth rate.The paper advises against inserting the extra randomization at every bet.Source: §5.4 · p. 10
6 Terminal testing: power and sharp calibration
At a fixed terminal evaluation, thresholding randomized capital at 1/α is sharply conservative by a factor of two, while under optional stopping the factor disappears. Randomization does not uniformly improve rejection probability, and the terminal recalibration cannot simply be monitored indefinitely.
6 Terminal testing: power and sharp calibration
- Randomization increases rejection probability when 1/2 < z < 1 but removes some rejections for z ≥ 1, so it does not uniformly improve power.At z = 1, rejection probability falls from one to one half.Source: §6 · p. 10
6.1 Factor of two level conservativeness at fixed times
- The usual threshold 1/α has sharp terminal rejection probability at most α/2 for any e-variable independent of U, including a null-valid stopped e-process evaluated at a stopping time independent of U.The factor is sharp over null distributions and e-variables; this is a testing calibration, not a claim that 2 U E is an e-variable.Source: §6.1 · pp. 10–11
- A terminal recalibration is level α sharply for 0 < α ≤ 1/2.The supplied proposition gives this terminal guarantee; it is not an anytime guarantee.Source: §6.1 · p. 11
6.2 Factor of two conservativeness disappears under optional stopping
- Under optional stopping, the bound α with A = U can be approached arbitrarily closely by a nonnegative null martingale starting at one.The first-crossing time depends on U, so the fixed-time result does not apply; the lowered-threshold terminal recalibration must not be monitored indefinitely at level α.Source: §6.2 · p. 11
7 Randomizing a test threshold instead of initial capital
Randomizing the testing threshold can retain ordinary-test rejections and add rejections below its threshold, while dominating both Bell-Cover calibrations in power. At a stopped endpoint, fresh terminal threshold randomization remains valid, unlike monitoring a lowered threshold indefinitely.
7.1 Comparing to the uniformly randomized Markov’s inequality
- Its null rejection probability is at most αE_P[E] ≤ α, retaining ordinary-test rejections while adding some below the ordinary threshold.The randomized decision threshold is calibrated by the rejection event, not by treating E/V as an e-variable, whose expectation is infinite when E_P[E] > 0.Source: §7.1 · p. 12
- The uniformly randomized Markov test has at least as much power under every data distribution as either Bell-Cover test using the same e-variable.This dominance persists after sharp terminal recalibration, so it is not merely a comparison of level-α/2 and level-α tests.Source: §7.1 · p. 12
7.2 A valid sequential use of terminal threshold randomization
- At a bounded study endpoint, drawing a fresh independent V for the final decision gives a level-α rule; the argument also extends to almost surely finite endpoints.This is the e-process form of the randomized Ville inequality.Source: §7.2 · p. 13
- Monitoring a single randomized lower threshold can inflate size to 3α/2 when α ≤ 1/2, so terminal randomization does not justify repeated lower-threshold monitoring.Fresh uniforms at repeated looks can be still worse.Source: §7.2 · p. 13
8 Conclusion
The uniform initial-capital lottery is suited to head-to-head competitive optimality, but not required for e-validity or optimal for every testing objective. Its costs include reduced expected log evidence, and threshold randomization is more favorable for power from a terminal e-variable.
8 Conclusion
- An independently randomized numeraire wins against every independently randomized e-variable at least half the time; for finite positive numeraires, the statement is a strict-win result.For unrestricted extended-valued numeraires, the corresponding guarantee is a half-tie minimax statement.Source: §8 · p. 13
- Under sequential testing, an independent mean-one initial lottery preserves null validity under stopping rules that observe it, but competitive optimality requires additional conditions.The paper identifies a stopped-ratio bound and a comparison protocol that does not exploit the lottery; conditional log-optimal betting and simple-null likelihood ratios are positive cases.Source: §8 · p. 13
- For finite positive numeraires, initial randomization sacrifices 1 − log 2 nats of expected relative log evidence and abandons the reported wealth’s numeraire property.At the usual terminal threshold it has sharp size at most α/2, while anytime monitoring can in some cases use the full α.Source: §8 · p. 13
- For power from a terminal e-variable, randomizing the testing threshold is more favorable than randomizing initial wealth, even after sharp terminal recalibration.The uniformly randomized Markov test dominates in conditional rejection probability and requires its own sequential safeguards.Source: §8 · p. 14