Isaac Sonin, UNC Charlotte

*Title: *The test and find problem

*Abstract*: We consider the following problem: k objects (balls) are allocated at random among n boxes (sites) according to some initial distribution π, no more than one object to a box. A Decision Maker (DM) can and will test all boxes. The test is not perfect, i.e. it can give false positive and false negative results. DM has m “tags”, 1 ≤ m ≤ n, and after testing all boxes she can place l tags, 0 ≤ l ≤ m on boxes she thinks have hidden objects. She is rewarded (paid) c_i for the correct guess and penalized by d_i for a wrong guess in box i. DM knows all the testing and cost parameters of the model and her goal is to maximize the expected reward. This problem can be easily generalized into a more general problem with three or more kinds of tags: “ball is here, ball is not here, not sure, etc”, and correspondingly with a more general cost structure. This problem can be classified as a problem from Statistical Decision Theory or a problem from Search Theory or as a problem from Mathematical Statistics, or even as a discrete version of an inverse problem if parameters are not completely known to DM. The origin of this model is related to the Locks, Bombs and Testing model that we discussed at the Probability seminar last semester, but the talk will be self contained. The model is quite elementary and can be understood by every graduate and even undergraduate student interested in Applied Probability and Statistics.