Quickly Determining Who Won an Election
Lisa Hellerstein, Naifeng Liu, Kevin Schewior
TL;DR
This paper tackles the problem of quickly determining the winner of an election when each vote can be inspected at a known cost, with voters independently assigning probabilities to candidates. By framing the task as a stochastic function-evaluation problem, the authors propose a two-phase framework that first reduces the number of plausible winners to a constant and then applies specialized stochastic-optimization techniques. They achieve constant-factor approximations: a 4-approximation for the absolute-majority case (with two adaptive rounds) and an 8-approximation for the relative-majority case, using Adaptive Dual Greedy and the SBB strategy to handle the remaining uncertainties. The work situates these results within the broader stochastic optimization literature, discusses adaptivity versus non-adaptivity, and outlines several promising directions for future research, including potential NP-hardness results and extensions to other voting rules.
Abstract
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.