Optimal Scalability-Aware Allocation of Swarm Robots: From Linear to Retrograde Performance via Marginal Gains

In collective systems, the available agents are a limited resource that must be allocated among tasks to maximize collective performance. Computing the optimal allocation of several agents to numerous tasks through a brute-force approach can be infeasible, especially when each task's performance scales differently with the increase of agents. For example, difficult tasks may require more agents to achieve similar performances compared to simpler tasks, but performance may saturate nonlinearly as the number of allocated agents increases. We propose a computationally efficient algorithm, based on marginal performance gains, for optimally allocating agents to tasks with concave scalability functions, including linear, saturating, and retrograde scaling, to achieve maximum collective performance. We test the algorithm by allocating a simulated robot swarm among collective decision-making tasks, where embodied agents sample their environment and exchange information to reach a consensus on spatially distributed environmental features. We vary task difficulties by different geometrical arrangements of environmental features in space (patchiness). In this scenario, decision performance in each task scales either as a saturating curve (following the Condorcet's Jury Theorem in an interference-free setup) or as a retrograde curve (when physical interference among robots restricts their movement). Using simple robot simulations, we show that our algorithm can be useful in allocating robots among tasks. Our approach aims to advance the deployment of future real-world multi-robot systems.