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Historically, much of machine learning research has focused on theperformance of the algorithm alone, but recently more attention has beenfocused on optimizing joint human-algorithm performance. Here, we analyze aspecific type of human-algorithm collaboration where the algorithm has accessto a set of $n$ items, and presents a subset of size $k$ to the human, whoselects a final item from among those $k$. This scenario could model contentrecommendation, route planning, or any type of labeling task. Because both thehuman and algorithm have imperfect, noisy information about the true orderingof items, the key question is: which value of $k$ maximizes the probabilitythat the best item will be ultimately selected? For $k=1$, performance isoptimized by the algorithm acting alone, and for $k=n$ it is optimized by thehuman acting alone. Surprisingly, we show that for multiple of noise models, itis optimal to set $k \in [2, n-1]$ - that is, there are strict benefits tocollaborating, even when the human and algorithm have equal accuracyseparately. We demonstrate this theoretically for the Mallows model andexperimentally for the Random Utilities models of noisy permutations. However,we show this pattern is reversed when the human is anchored on the algorithm'spresented ordering - the joint system always has strictly worse performance. Weextend these results to the case where the human and algorithm differ in theiraccuracy levels, showing that there always exist regimes where a more accurateagent would strictly benefit from collaborating with a less accurate one, butthese regimes are asymmetric between the human and the algorithm's accuracy.

When Are Two Lists Better than One?: Benefits and Harms in Joint Decision-making