  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.
