The Test and Find Model
Abstract
We introduce the Test and Find (TF) problem, where a decision maker (DM) faces the following situation: k identical objects are randomly allocated to n distinct boxes (sites) according to some distribution π, with no more than one object to a box. The DM tests all of the boxes. However, the tests are imperfect: they can give false positive or false negative results. DM has m tags, 1≤ m≤ n, and, after testing all boxes, she can place a tag on any box that she thinks has a hidden object. She is rewarded ci for a correct guess and penalized di for a wrong guess in box i. DM knows all of the parameters of the model and her goal is to maximize the expected reward. We give an explicit solution to this problem. We then turn to the symmetric case, for which we derive more computationally efficient results. We also consider several extensions of the TF model and give detailed solutions. One of these extensions is to the realistic case where k, the number of objects, is unknown and random.
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BibTeX
@misc{grabchak2026test,
title = {{The Test and Find Model}},
author = {Michael Grabchak and Xingjie Li and Isaac M. Sonin},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29390},
eprint = {2608.29390},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29390},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Test and Find Model
AU - Michael Grabchak
AU - Xingjie Li
AU - Isaac M. Sonin
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29390
UR - https://arxiv.org/abs/2608.29390
AB - We introduce the Test and Find (TF) problem, where a decision maker (DM) faces the following situation: k identical objects are randomly allocated to n distinct boxes (sites) according to some distribution π, with no more than one object to a box. The DM tests all of the boxes. However, the tests are imperfect: they can give false positive or false negative results. DM has m tags, 1≤ m≤ n, and, after testing all boxes, she can place a tag on any box that she thinks has a hidden object. She is rewarded c_i for a correct guess and penalized d_i for a wrong guess in box i. DM knows all of the parameters of the model and her goal is to maximize the expected reward. We give an explicit solution to this problem. We then turn to the symmetric case, for which we derive more computationally efficient results. We also consider several extensions of the TF model and give detailed solutions. One of these extensions is to the realistic case where k, the number of objects, is unknown and random.
ER -