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One of the most popular existing models for task allocation in ant colonies is the so-called threshold-based task allocation model. Here, each ant has a fixed, and possibly distinct, threshold. Each task has a fixed demand which represents the number of ants required to perform the task.1Thestimulusanant receives for a task is defined as the demand of the task minus the number of ants currently working at the task. An ant joins a task if the stimulus of the task exceeds the ant’s threshold.A large body of results has studied this model for over four decades; however, most of the theoretical works focuses on the study of two tasks. Interestingly, no work in this line of research shows that the number of ants working at a task eventually converges towards the demand nor does any work bound the distance to the demands over time.In this work, we study precisely this convergence. Our results show that while the threshold-based model works fine in the case of two tasks (for certain distributions of thresholds); the threshold model no longer works for the case of more than two tasks. In fact, we show that there is no possible setting of thresholds that yieldsmore »a satisfactory deficit (demand minus number of ants working on the task) for each task.This is in stark contrast to other theoretical results in the same setting [CDLN14] that rely on state-machines, i.e., some form of small memory together with probabilistic decisions. Note that, the classical threshold model assumes no states or memory (apart from the bare minimum number of states required to encode which task an ant is working on). The resulting task allocation is near-optimal and much better than what is possible using joining thresholds. This remains true even in a noisy environment [DLM+18]. While the deficit is not the only important metric, it is conceivably one of the most important metrics to guarantee the survival of a colony: for example if the number of workers assigned for foraging stays significantly below the demand, then starvation may occur. Moreover, our results do not imply that ants do not use thresholds; we merely argue that relying on thresholds yields a considerable worse performance.« less