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We study threshold testing, an elementary probing model with the goal tochoose a large value out of $n$ i.i.d. random variables. An algorithm can testeach variable $X_i$ once for some threshold $t_i$, and the test returns binaryfeedback whether $X_i \ge t_i$ or not. Thresholds can be chosen adaptively ornon-adaptively by the algorithm. Given the results for the tests of eachvariable, we then select the variable with highest conditional expectation. Wecompare the expected value obtained by the testing algorithm with expectedmaximum of the variables. Threshold testing is a semi-online variant of thegambler's problem and prophet inequalities. Indeed, the optimal performance ofnon-adaptive algorithms for threshold testing is governed by the standardi.i.d. prophet inequality of approximately $0.745+o(1)$ as $n \to \infty$. Weshow how adaptive algorithms can significantly improve upon this ratio. Ouradaptive testing strategy guarantees a competitive ratio of at least$0.869-o(1)$. Moreover, we show that there are distributions that admit only aconstant ratio $c < 1$, even when $n \to \infty$. Finally, when each box can betested multiple times (with $n$ tests in total), we design an algorithm thatachieves a ratio of $1-o(1)$.

Threshold Testing and Semi-Online Prophet Inequalities