  We consider one-dimensional excited random walks with finitely many cookies
at each site. There are certain natural monotonicity results that are known for
the excited random walk under some partial orderings of the cookie
environments. We improve these monotonicity results to be strictly monotone
under a partial ordering of cookie environments introduced by Holmes and
Salisbury. While the self-interacting nature of the excited random walk makes a
direct coupling proof difficult, we show that there is a very natural coupling
of the associated branching process from which the monotonicity results follow.
