 \ref{eq:commit-oc} when 
player $\lnot i$ credibly commits to playing $v(t) = t^2$ for the duration of the game and 
player $i$'s final cost function takes the form $\Phi(x) = \Theta(|x| - 1) - \Theta( 1 - |x|)$.
\begin{figure}
\centering
	\includegraphics[width=\columnwidth]{path-integral-control-example.png}
	\caption{
	Result of the path integral Monte Carlo solution method applied to Eq.\ \ref{eq:commit-oc}
		with the final condition $\Phi(x) = \Theta(|x| > 1) - \Theta(|x| \leq 1)$ 
		and $v(t) = t^2$. 
	Approximate value functions are computed using $N = 10 000$ trajectories sampled from
	Eq.\ \ref{eq:abm-const-control} for each point $(x, t)$.
	We display approximate value functions in Panel A
	for $t \in \{0, 0.75, 1 - \dee t\}$ and 
	the corresponding approximate control policies $u(x,t)$ in Panel B, along with smoothed 
	versions of $u(x,t)$, which we denote by $u_s(x,t)$,
	plotted in dashed curves.
	Panel C displays realizations of $Y_t$, the process generating the measure under which the 
	solution is calculated.
	The analytical control policy at $t = T$ is given by 
	$u(t) = -\frac{1}{2}[\delta(x - 1) - \delta(x + 1)]$. 
		}
	\label{fig:pimc-example}
\end{figure}
In this figure we display the approximate value functions $V(x,t)$
along with their corresponding approximate control
policies $u(x,t) = -\frac{1}{2}\frac{\partial V(x,t)}{\partial x}$.
We calculated these approximations on a grid of $N_x = 500$ linearly spaced $x_n \in [-2, 2]$.
Since the approximated $u(x_n,t)$ are noisy stochastic functions we also plot smoothed versions of them
in Fig.\ \ref{fig:pimc-example}.
These smoothed versions are defined by 
\begin{equation}
	u_s(x_n,t) = \sum_{n'=-k}^k u(x_{n + n'}, t).
\end{equation}
We set $k = 7$ and hence $u_s(x_n, t)$ are $2k + 1 = 15$-step moving averages of the more noisy $u(x_n, t)$.
As $t \rightarrow T$, $u_s(x_n, t)$ approaches the analytical solution of the control problem at $t = T$, 
which is given by $u(x, T) = -\frac{1}{2}[\delta(x - 1) - \delta(x + 1)]$.
\medskip

\noindent
In the further restricted case where there is a credible commitment by one party to play 
a constant control policy $v(t) = v$, we can derive further analytical results.
Under this assumption, the probability law corresponding with Eq.\ \ref{eq:abm-const-control} is given by 
\begin{equation}
	u(y,t) = 
	\frac{1}{\sqrt{2\pi \sigma^2t}}\exp\left\{ \frac{1}{2\sigma^2 t}[(y - x) - vt]^2 \right\},
\end{equation}
so that the (exponentially-transformed) value function reads
\begin{align}\label{eq:exp-value-function}
	&\varphi(x,t) = E_{u(y,T - t)}\left[ 
		\exp\left\{ - \frac{\lambda v^2}{2\sigma^2}(T - t)\Phi(Y_T) \right\}
	\right]\\
	&\quad= \frac{ \exp\left\{ - \frac{\lambda v^2}{2\sigma^2}(T - t) \right\} }
	{\sqrt{2\pi \sigma^2 (T - t)}}
	\times\\
	&\int\displaylimits_{-\infty}^{\infty}
	\exp\left\{-\frac{1}{2\sigma^2}\left[ \Phi(y) 
	+ \frac{((y - x) - v(T - t))^2}{T - t} \right]  \right\} \dee y \notag.
\end{align}
This integral can be evaluated exactly for many $\Phi(y)$ and, for many other final conditions, 
it can be approximated using the method of 
Laplace.
When $t \rightarrow T$  so that the denominator of the argument of the exponential in Eq.\ \ref{eq:exp-value-function}
approaches zero,
Laplace's approximation to the integral reads
\begin{equation}\label{eq:laplace-approx-integral}
\begin{aligned}	
	&\int\displaylimits_{-\infty}^{\infty}
		\exp\left\{-\frac{1}{2\sigma^2}\left[ \Phi(y) 
		+ \frac{((y - x) - v(T - t))^2}{T - t} \right]  \right\} \dee y\\
		&\quad\approx \sqrt{2 \pi \sigma^2 (T -t)}
		\exp\left\{ -\frac{1}{2\sigma^2}\Phi(x +(T - t)v) \right\}.
\end{aligned}
\end{equation}
Inverting the transformation $\varphi$, the value function is approximated by 
\begin{equation}\label{eq:laplace-value}
	V(x,t) = \lambda v^2 (T - t) + \Phi(x + (T - t)v),
\end{equation}
and the control policy by 
\begin{equation}
	u(x, t) = -\frac{1}{2}\Phi'(x + (T - t)v).
\end{equation}
\begin{figure}
\centering
	\includegraphics[width=\columnwidth]{laplace-value-approximation.pdf}
	\caption{When player $\lnot i$ commits to playing a constant strategy profile $v(t) = v$ for a
	fixed interval of time, 
	an analytic approximate form for player $i$'s value function $V(x,t)$ is given by 
	$V(x,t) \approx \lambda v^2 (T - t) + \Phi(x + (T - t)v)$. 
	We show the numerically-determined value functions 
	at time $t=0$ in solid black curves,
	We display the Laplace approximations at $t=0$ in dashed black curves. 
	The lighter-hue curves are the value functions at the final time $T$, i.e., the 
	respective final conditions.
	The top panel demonstrates results for the final condition $\Phi(x) = \frac{1}{2}x^2$, 
	while the bottom panel has $\Phi(x) = \tanh(x)$.
	}
	\label{fig:laplace-value-approximation}
\end{figure}
We display the results of approximating the value function with 
Eq.\ \ref{eq:laplace-value} at $t = 0$ in Fig.\  \ref{fig:laplace-value-approximation},
along with the actual numerically-determined value function at both $t = 0$ and, for reference, $t = T$.

\subsubsection{Dependence on a free parameter}
The Laplace-approximated value function may depend on a free parameter $a$ that can be used as a
``control knob'' to adjust the approximation.
For example, player $i$ might use $a$ to tune the sensitivity of the approximation to the 
electoral process's distance from a dead-heat.
Ideally, the approximated control policy
should have similar scaling and asymptotic properties as the true control policy.
Solving an optimization problem for optimal values of $a$ is
be one approach to satisfying this desideratum.
\medskip

\noindent
As a case study we consider the behavior of the Laplace-approximated value function 
$V^{(a)}(x,t)$ and its corresponding control policy $u^{(a)}(x,t)$ when we set 
$\Phi^{(a)}(x) = \tanh(ax)$.
We consider this specific example because $\Phi^{(a)}(x) \rightarrow \Theta(x) - \Theta(-x)$
as $a \rightarrow +\infty$.
This limit can be the source of complicated behavior in 
a variety of fields such as piecewise-smooth dynamical systems
(both deterministic and stochastic) \cite{