We consider an extension of a binary decision model in which nodes make decisions based on influence-biased averages of their neighbors' states, similar to Ising spin glasses with on-site random fields. In the limit where these influences become very heavy-tailed, the behavior of the model dramatically changes. On complete graphs, or graphs where nodes with large influence have large degree, this model is characterized by a new "phase" with an unpredictable number of macroscopic shocks, with no associated critical phenomena. On random graphs where the degree of the most influential nodes is small compared to population size, a predictable glassy phase without phase transitions emerges. Analytic results about both of these new phases are obtainable in limiting cases. We use numerical simulations to explore the model for more general scenarios. The phases associated with very influential decision makers are easily distinguishable experimentally from a homogeneous influence phase in many circumstances, in the context of our simple model.