Consensus in the weighted voter model with noise-free and noisy observations

Collective decision-making is an important problem in swarm robotics arising in many different contexts and applications. The Weighted Voter Model has been proposed to collectively solve the best-of-<i>n</i> problem, and analysed in the thermodynamic limit. We present an exact finite-population analysis of the best-of-two model on complete as well as regular network topologies. We also present a novel analysis of this model when agent evaluations of options suffer from measurement error. Our analytical results allow us to predict the expected outcome of best-of-two decision-making on a swarm system without having to do extensive simulations or numerical computations. We show that the error probability of reaching consensus on a suboptimal solution is bounded away from 1 even if only a single agent is initialised with the better option, irrespective of the total number of agents. Moreover, the error probability tends to zero if the number of agents initialised with the best solution tends to infinity, however slowly compared to the total number of agents. Finally, we present bounds and approximations for the best-of-<i>n</i> problem.