A simple model for multiple-choice collective decision making

Abstract: We describe a simple model of heterogeneous, interacting agents making decisions between $n\ge 2$ discrete choices. For a special class of interactions, our model is the mean field description of random-field Potts-like models, and is effectively solved by finding the extrema of the average energy $E$ per agent. In these cases, by studying the propagation of decision changes via avalanches, we argue that macroscopic dynamics is well-captured by a gradient flow along $E$. We focus on the permutation-symmetric case, where all $n$ choices are (on average) the same, and spontaneous symmetry breaking (SSB) arises purely from cooperative social interactions. As examples, we show that bimodal heterogeneity naturally provides a mechanism for the spontaneous formation of hierarchies between decisions, and that SSB is a preferred instability to discontinuous phase transitions between two symmetric points. Beyond the mean field limit, exponentially many stable equilibria emerge when we place this model on a graph of finite mean degree. We conclude with speculation on decision making with persistent collective oscillations. Throughout the paper, we emphasize analogies between methods of solution to our model and common intuition from diverse areas of physics, including statistical physics and electromagnetism.