Heterogeneous Mean-Field Analysis of Best-of-n Decision Making in Networks with Zealots

Humans and animals often choose between options with different qualities. When the decisions are not determined by one or a few individuals leading a group, a collective can achieve a consensus through repeated interactions among the individuals. Collective decision-making is widely studied in the context of opinion dynamics, showing that individual mechanisms of option selection and the underlying social network affect the outcome. Mathematical techniques, such as the heterogeneous mean-field (HMF) theory, have been developed to systematically analyse the collective behaviour of interconnected agents. Based on the HMF theory, we propose a mathematical model that looks at the combined effects of multiple elements bearing upon the collective decision dynamics, such as the individuals’ cognitive load, the difference in the quality of the options, the network topology, and the location of the zealots in the network. The results of this study show that, in scale-free networks, when individuals employ specific opinion selection mechanisms, characterised by a low cognitive load, the zealots have the ability to steer the consensus towards the option with the lowest quality or to group indecision. This result is reversed when the interaction network is sparsely connected and quite homogeneous – that is, most nodes have few neighbours – and cognitively simple individuals make accurate collective decisions, mostly unaffected by zealots voting for the option with the lowest quality.