Dynamical, value-based decision making among $N$ options:
a constructive approach to unfolding
the symmetric pitchfork bifurcation

The desire for a switching commitment process strongly suggests the use of a bifurcation in the dynamical system, and the symmetry requirement encourages the use of equivariant bifurcation theory [16, 20]. In the case of $N=2$ options, our requirements are satisfied by selecting the vector field $f$ to embed a pitchfork bifurcation as shown in Figure 2. The mechanism can be viewed as follows. Let $x=m_{1}-m_{2}$ represent the degree of commitment for option 1 over option 2. Then, in the symmetric case, selecting $f$ to be a pitchfork bifurcation, i.e.,

will yield the desired behavior. Note that $f$ is an odd function of $x$, so the system (1) obeys the $S_{2}$ symmetry corresponding to switching the option labels. When $\mu$ (which represents the value $v$) is less than zero, the unique equilibrium of (1) is $x=0$, corresponding to equal commitment to both options, and this equilibrium is stable. As $\mu$ increases through zero, the equilibrium at $x=0$ becomes unstable and two new stable equilibria appear at $\pm\sqrt{\mu}$. These facts are summarized in the bifurcation diagram shown in Figure 2(a). Equivariant bifurcation theory predicts that these two branches must appear symmetrically in the post-bifurcation regime; each branch represents a commitment to one option or the other. Due to symmetry, commitment to either option is possible; the option to which the system commits is determined by initial conditions.

The case of asymmetric option values is naturally modeled by breaking the symmetry of the vector field $f$. The study of such symmetry breaking is a core part of equivariant bifurcation theory, and the fundamental concept is the so-called unfolding of the bifurcation. One begins with the concept of a perturbation of a bifurcation. Specifically, a perturbation $F$ of a bifurcation $f$ is a function $F(x,\mu,\alpha)$ such that $F(x,\mu,0)=f(x,\mu)$. A universal unfolding is a $k$-parameter (i.e., $\alpha\in\mathbb{R}^{k}$) perturbation $F$ such that any small perturbation of $f$ can be expressed in terms of the $k$ parameters that define $F$. The two-parameter family

is a universal unfolding of the pitchfork bifurcation (1) [16]. As the parameters $\alpha_{1},\alpha_{2}$ are varied, the bifurcation diagram for $\dot{x}=F(x,\mu,\alpha)$ varies as well. In particular, for appropriate parameter values, the system has a single equilibrium in the post-bifurcation regime, and this equilibrium is both nonzero and stable (see Figure 2(b)). Thus, by choosing appropriate values for the unfolding parameters, one can encode a globally-attracting preference for one option or the other. To complete the model for $N=2$ options, it remains to relate the option values $v_{i}$ to the bifurcation parameter $\mu$ and unfolding parameters $\alpha_{i}$ in (2).

In their paper [1], Seeley et al. study the value-sensitive decision-making problem for $N=2$ options and develop a dynamical systems model. Their model can be shown to embed an unfolded pitchfork [18], thus completing the model whose mechanism we laid out in the preceding section. Concretely, Seeley et al. let the state of their model be $m=(m_{1},m_{2},m_{U})\in\Delta^{2}$ and set $\dot{m}=f(m,v)$, with

for each $i=1,2$, and the dynamics of $m_{U}$ are determined by the normalization constraint. As above, $v_{i}$ denotes the value of option $i$, and $\sigma>0$ is a constant parameter.

In the symmetric case of $v_{1}=v_{2}=v$ the dynamics (3) obey an $S_{2}$ symmetry. Seeley et al. [1] showed that, in the symmetric case, the dynamics (3) exhibits a pitchfork bifurcation as $v$ and $\sigma$ increase above a threshold. In the pre-bifurcation regime, the system has a single stable equilibrium with the symmetry $m_{1}=m_{2}$. Because of the symmetry, the system does not commit to either option, and the equilibrium is said to be a deadlock. In the post-bifurcation regime, the deadlock equilibrium is unstable and two additional equilibria emerge, each representing a decision to commit to one of the two options. In previous work [1], the parameter values at the bifurcation point were found to satisfy

Note that either $\sigma$ or $v$ can be interpreted as the bifurcation parameter, with (4) defining the bifurcation value. In other words, fixing $\sigma$, the bifurcation occurs as $v$ increases through a threshold, while fixing $v$, the bifurcation occurs as $\sigma$ increases.

In the asymmetric case of $v_{1}\neq v_{2}$, the $S_{2}$ symmetry of the dynamics is broken and the pitchfork bifurcation unfolds as studied by Pais et al. [3]. For a fixed value of $\sigma$, the number of equilibria of (3) depends on the parameters $v_{1}$ and $v_{2}$. Certain parameter values result in a single stable equilibrium whose location is biased towards the high-value option, while others result in two stable equilibria representing each option and a saddle point in between. The complete phase diagram of the system is complex, but the two primary findings of [3] can be summarized as follows. First, the dynamics (3) remain deadlocked (i.e., have a single attractor with $m_{1}\approx m_{2}$) when the average option value $\bar{v}=(v_{1}+v_{2})/2$ is small. Second, the dynamics decide for the high value option (i.e., for $v_{1}>v_{2}$, have a single attractor with $m_{1}\gg m_{2}$) when the difference in option values $\Delta v=v_{1}-v_{2}$ is sufficiently large relative to $\bar{v}=(v_{1}+v_{2})/2$. In symbols, we have that the system makes a decision when

where $\kappa(\sigma)$ is a coefficient that depends on the parameter $\sigma$. Pais et al. [3] note that this behavior is analogous to Weber’s law of just-noticeable differences from psychology, which states that the minimum difference in stimulus intensity required to discriminate between two different stimuli varies linearly with their mean intensity.

The implication of these two findings is that the decision-making dynamics (3) has several desirable properties. First, when both options are poor (corresponding to a low value of $\bar{v}$), the system remains deadlocked and avoids making a decision, e.g., to wait for more information. When at least one option is sufficiently satisfactory (corresponding to a high value of $\bar{v}$), the system will quickly commit to an option, and preferentially select the one with a higher value. These are properties that we seek to generalize to the case of $N>2$ options.

As shown in the preceding sections, the Seeley et al. model (3) has desirable characteristics that we seek to generalize. However, the functional form of (3) obscures the unfolding of the pitchfork bifurcation which serves as the fundamental decision-making mechanism. In recent work [18], we studied the dynamics (3) using model reduction techniques to elucidate the unfolding.

The model reductions studied in [18] use singular perturbation theory. Specifically, the reduction approach maps $v_{i}\mapsto Kv_{i}$ for a constant gain $K>0$ and takes the singular limit $K\to+\infty$. This approach is similar to an analysis performed in [3], where the authors studied the limit $\bar{v}\to+\infty$; however, the approach using the gain $K$ preserves the relative difference in values $\Delta v/\bar{v}$. This ratio is key in defining the unfolding of the pitchfork bifurcation embedded in (3).

The bifurcation is more readily analyzed by expressing $m\in\Delta^{2}$ in terms of mean-difference coordinates defined by

which are analogous to the definitions of $\Delta v$ and $\bar{v}$ made above. Note that the definitions of these new coordinates and the definitions of $\Delta^{2}$ and $(v_{1},v_{2})\in\mathbb{R}_{+}^{2}$ imply that $\bar{m},\bar{v}>0$ and that $-2\bar{m}\leq\Delta m\leq 2\bar{m}$ and $-2\bar{v}<\Delta v<2\bar{v}$.

In the mean-difference coordinates, the dynamics (3) of $m=({\Delta m},{\bar{m}})$ take the form

A straightforward application of singular perturbation theory with small parameter $\epsilon=1/K$ and coordinates $x=\Delta m$, and $y=(1-2\bar{m})/\epsilon$ yields the following result.

A standard nonlinear time scaling argument then allows one to eliminate the denominator $6+\alpha x$ from (7) and makes the connection to the unfolding of the pitchfork bifurcation (2) explicit.

The equilibria of the singularly-perturbed system (7) are shown in Figure 3. Note that the system has three equilibria for all possible values of $\alpha\in[-2,2]$. When $\alpha=0$ the options are equally valued, the pitchfork is unperturbed, and the equilibria correspond to those of the standard pitchfork (1) in the post-bifurcation regime. The equilibrium $x=0$ is unstable, while those at $x=\pm 1$ are stable. For nonzero $\alpha$ the pitchfork unfolds. In the singularly-perturbed regime, the non-zero equilibria remain at $x=\pm 1$, while the intermediate equilibrium shifts to $x=-3\alpha/2$. The intermediate equilibrium is unstable for the values of $\alpha\in[-2/3,2/3]$ where it exists. The equilibria at $x=\pm 1$ are stable when the value difference $\Delta v$ is not too biased against the corresponding option. For example, $x=+1$ is a stable equilibrium of (7) for $\alpha\geq-2/3$.

The structure of equilibria shown in Figure 3 determines the decision-making behavior of the model (3) in the singular limit. The state $x=\Delta m=+1$ corresponds to the system committing fully to option 1, i.e., to $m=(1,0,0)$. This state is stable for $\alpha=\Delta v/\bar{v}\geq-2/3$, and globally attracting for $\alpha>2/3$. In other words, the system can commit to option 1 when when $\Delta v\geq-2\bar{v}/3$, and will be globally attracted to committing to option 1 when $\Delta v>2\bar{v}/3$. Switching behavior can occur as $\alpha$ shifts. For example, suppose that the system (7) is initialized with $\alpha<0$ and $\Delta m=-0.9$, representing a commitment to option 2. If $\alpha$ is then raised above the value $2/3$, the state $\Delta m$ will be attracted to the value $+1$, representing a decision to switch and commit to option 1. The rate at which $\Delta m$ is attracted to $+1$ is governed by the parameter $\sigma$, as can be seen from the form of the dynamics (7).

We note that the coefficient $3/2$ arises from the last term $K\Delta v(1-2\bar{m})(1+\bar{m})$ in (5) and can be adjusted by changing the coefficient $1+\bar{m}$ to $\beta+\bar{m}$, which changes the coefficient $3/2=1+1/2$ to $\beta+1/2$. This observation can be used to control the bifurcation properties of the system, e.g., by making it more or less sensitive to the relative value difference $\alpha$. We do not pursue this line of investigation further in the present paper, but recognize that it is a natural point of departure for further work.

In this section, we introduced our requirements for a dynamical system model of value-sensitive decision making We showed how an unfolding of the pitchfork bifurcation can provide the fundamental mechanism for such a model in the case of $N=2$ options, and reviewed a model due to Seeley et al. that embeds such a pitchfork mechanism along with some recent results reducing that model. In the following section we begin to construct a generalization of the Seeley et al. model to the case $N>2$ by parsing the decision among $N$ options into a series of binary decisions represented by a tree structure.

Tree traversal is a fundamental process acting on a tree, wherein the process visits (and carries out an action on) each node in the tree exactly once. Trees may be traversed in either depth-first or breadth-first orders, as depicted in Figure 6. As their names imply, depth-first order operates by going as deep down the tree as possible before going to the next sibling, while breadth-first order operates by going through each sibling before going to a descendant. The nodes in Figure 6 are labeled according to the order in which they will be visited in depth-first or breadth-first traversal.

A path in a finite tree is a finite sequence of edges which joins a sequence of nodes. For a rooted tree, there is always a unique shortest path from the root to any other node. We denote the sequence of nodes along the shortest path from the root to node $i$ as $p_{i}$ and denote its $j^{th}$ element as $p_{ij}$. The sequence $p_{i}$ begins with the root node and ends with the node $i$. The number of nodes in $p_{i}$ is denoted $|p_{i}|$.

Trees may have important symmetries. For example, the nodes of a tree may be rearranged without changing the structure it represents. Two trees which share the same structure are said to be isomorphic. The concept of tree isomorphism is inherited from the concept of graph isomorphism [21], for which tree isomorphisms are a special case.

In words, a rooted tree isomorphism is a mapping between the node sets such that each edge is preserved, along with the root node. An example, Figure 7 shows two isomorphisms of the tree presented in Figure 6(a). Note that isomorphisms of binary trees are generated by flips at nodes, wherein the left and right descendants of a given node are exchanged.

Problems associated with tree isomorphisms arise in many applications. In particular, a standard problem in computer science is to determine whether two rooted trees $T_{1}$ and $T_{2}$ are isomorphic. A classic algorithm due to Aho, Hopcroft, and Ullman [22] solves the problem in $O(n)$ time for trees with $n$ vertices.

The set of isomorphisms of a given tree exhibit a group structure. The elements of the group are generated by the flips at internal nodes described above and the group operation is given by composition. It is clear that each flip is its own inverse, as exchanging left and right descendants of a node twice leaves the tree unchanged. Flips may be carried out in any order, so the operation is associative, and the identity is the element consisting of no flips.

When a tree $T$ is a parsing of a set of $n_{o}$ options, isomorphisms of the tree generate isomorphisms of the option set $\{1,\ldots,n_{o}\}$. Recall that an isomorphism of $T$ is a bijection $\varphi$ from the node set of $T$ to itself. Thus, a node $i$ is mapped to $j=\varphi(i)$ and the set $\{o(i):i\in\{1,\ldots,n_{o}\}\}$ is mapped to $\{o(\varphi(i)):i\in\{1,\ldots,n_{o}\}\}$. The group of all possible isomorphisms of $n_{o}$ objects is $S_{n_{o}}$. Note, however, that not all such isomorphisms can be generated by the set of tree isomorphisms. For example, nodes that are siblings must remain siblings even under isomorphism operations.

Let the tree $T$ be a parsing of a set of $n_{o}$ options. The set of isomorphisms the options that can be generated by isomorphisms of $T$ forms a group $\Gamma_{T}\leq S_{n_{o}}$ whose structure is given by a wreath product of copies of $S_{2}$. This can be seen as follows. Let $i$ be an internal node in the tree $T$ and let $ro(i)$ and $lo(i)$ denote the set of options associated with its right and left descendants, respectively. For example, for the tree in Figure 7(a), $ro(0)=\{o(3),o(4),o(5)\}$ and $lo(0)=\{o(7),o(8)\}$. Let $\gamma_{i}$ represent the action of performing a flip at node $i$. Then $\gamma_{i}$ exchanges the sets $ro(i)$ and $lo(i)$. Explicitly, we have

Note that applying $\gamma_{i}$ twice results in the identity mapping, so $\gamma_{i}$ generates the permutation group $S_{2}$ acting on the set $\{ro(i),lo(i)\}$. Furthermore, any options which are not associated with the descendants of node $i$ are unaffected. Thus, one can think of $\gamma_{i}$ as generating a representation of $S_{2}$ acting on the set $\{1,\ldots,n_{o}\}$; this is trivially a subgroup of $S_{n_{o}}$. Applying $\gamma_{j}$ for another internal node $j\neq i$ generates another representation of $S_{2}$. The group generated by application of both actions $\gamma_{i}$ and $\gamma_{j}$ is the wreath product $S_{2}\wr S_{2}$. This process can be extended for each internal node $i$, yielding a symmetry group which is the repeated wreath product of copies of $S_{2}$. Formally we have the following Proposition.