Generalizing the response and satisfaction threshold models for multiple tasks: A maximal-entropy approach

Response threshold models are often used to test hypotheses about division of labor in social-insect colonies. Each workers probability to engage in a task rapidly increases when a cue associated with task demand crosses some "response threshold." Threshold variability across workers generates an emergent division of labor that is consistent over time and flexibly adaptive to increasing demands, which allows for testable predictions about the shape of hypothetical response-threshold distributions. Although there are myriad different task types in a social-insect colony, the classical response-threshold model is built to understand variability in response to a single type of task. As such, it does not immediately allow for testing predictions about how different workers prioritize different task types or how demand for some tasks interferes with responding to demand for others. To rectify this, we propose a multi-task generalization that degenerates into the standard model for a single task. We replace the classical Hill response probability with a model that draws worker choices from a Boltzmann distribution, which is an approach inspired by multi-class machine learning.