Dynamical, value-based decision making among $N$ options

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 $SN$ symmetry. Unfortunately, constructing a dynamical system that unfolds the $SN$-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 $SN$. 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 $S2$-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.