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This paper considers elections in which voters choose one candidate each, independently according to known probability distributions. A candidate receiving a strict majority (absolute or relative, depending on the version) wins. After the voters have made their choices, each vote can be inspected to determine which candidate received that vote. The time (or cost) to inspect each of the votes is known in advance. The task is to (possibly adaptively) determine the order in which to inspect the votes, so as to minimize the expected time to determine which candidate has won the election. We design polynomial-time constant-factor approximation algorithms for both the absolute-majority and the relative-majority version. Both algorithms are based on a two-phase approach. In the first phase, the algorithms reduce the number of relevant candidates to O(1), and in the second phase they utilize techniques from the literature on stochastic function evaluation to handle the remaining candidates. In the case of absolute majority, we show that the same can be achieved with only two rounds of adaptivity.

We study threshold testing, an elementary probing model with the goal to choose a large value out of n i.i.d. random variables. An algorithm can test each variable X_i once for some threshold t_i, and the test returns binary feedback whether X_i ≥ t_i or not. Thresholds can be chosen adaptively or non-adaptively by the algorithm. Given the results for the tests of each variable, we then select the variable with highest conditional expectation. We compare the expected value obtained by the testing algorithm with expected maximum of the variables. Threshold testing is a semi-online variant of the gambler’s problem and prophet inequalities. Indeed, the optimal performance of non-adaptive algorithms for threshold testing is governed by the standard i.i.d. prophet inequality of approximately 0.745 + o(1) as n → ∞. We show how adaptive algorithms can significantly improve upon this ratio. Our adaptive testing strategy guarantees a competitive ratio of at least 0.869 - o(1). Moreover, we show that there are distributions that admit only a constant ratio c < 1, even when n → ∞. Finally, when each box can be tested multiple times (with n tests in total), we design an algorithm that achieves a ratio of 1 - o(1).