  Decision making is a fundamental capability of autonomous systems. As
decision making is a process which happens over time, it can be well modeled by
dynamical systems. Often, decisions are made on the basis of perceived values
of the underlying options and the desired outcome is to select the option with
the highest value. This can be encoded as a bifurcation which produces a stable
equilibrium corresponding to the high-value option. When some options have
identical values, it is natural to design the decision-making model to be
indifferent among the equally-valued options, leading to symmetries in the
underlying dynamical system. For example, when all $N$ options have identical
values, the dynamical system should have $S_N$ symmetry. Unfortunately,
constructing a dynamical system that unfolds the $S_N$-symmetric pitchfork
bifurcation is non-trivial. In this paper, we develop a method to construct an
unfolding of the pitchfork bifurcation with a symmetry group that is a
significant subgroup of $S_N$. The construction begins by parsing the decision
among $N$ options into a hierarchical set of $N-1$ binary decisions encoded in
a binary tree. By associating the unfolding of a standard $S_2$-symmetric
pitchfork bifurcation with each of these binary decisions, we develop an
unfolding of the pitchfork bifurcation with symmetries corresponding to
isomorphisms of the underlying binary tree.
