Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials
Abstract
Blackwell and Cover both proposed the same method for guessing the larger of a pair of two numbers when one can only observe one and the other of the two numbers remains unobserved. As originally stated, the procedure ended with a computation showing that the probability of correctly guessing the larger number was greater than one-half. In this paper, we investigate what we can say when about the number in the pair that was not observed we employ the Blackwell/Cover procedure. Let α1 and α2 be two different numbers in (0,1]. Let G be the uniform distribution on [0, α1] and H be the uniform distribution on [0, α2]. We think of G and H as being represented by the heads probabilities of coins. We use a variant of the Blackwell-Cover strategy ro take pairs of coins chosen from distributions created from G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. We show that there are situations in which recursive application of this process results in a set consisting almost completely of coins from one of the two distributions G and H, and another set consisting almost completely of coins from the other distribution.
Related
- Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials - II
- Adaptive Conformal Inference through the Lens of Blackwell Approachability
- Redundancy and synergy in multivariate Gaussians via the Blackwell order
- Robust Design and Evaluation of Predictive Algorithms under Unobserved Confounding
- Strong failures of club guessing at the successor of a regular cardinal
BibTeX
@misc{stein2026using,
title = {{Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials}},
author = {James Stein},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23820},
eprint = {2609.23820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23820},
note = {CC0},
}
RIS
TY - UNPB
TI - Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials
AU - James Stein
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23820
UR - https://arxiv.org/abs/2609.23820
AB - Blackwell and Cover both proposed the same method for guessing the larger of a pair of two numbers when one can only observe one and the other of the two numbers remains unobserved. As originally stated, the procedure ended with a computation showing that the probability of correctly guessing the larger number was greater than one-half. In this paper, we investigate what we can say when about the number in the pair that was not observed we employ the Blackwell/Cover procedure. Let α1 and α2 be two different numbers in (0,1]. Let G be the uniform distribution on [0, α1] and H be the uniform distribution on [0, α2]. We think of G and H as being represented by the heads probabilities of coins. We use a variant of the Blackwell-Cover strategy ro take pairs of coins chosen from distributions created from G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. We show that there are situations in which recursive application of this process results in a set consisting almost completely of coins from one of the two distributions G and H, and another set consisting almost completely of coins from the other distribution.
ER -