This paper establishes a fundamental difference between
$\mathbb{Z}$
subshifts of finite type and
$\mathbb{Z}^{2}$
subshifts of finite type in the context of ergodic optimization. Specifically, we consider a subshift of finite type
$X$
as a subset of a full shift
$F$
. We then introduce a natural penalty function
$f$
, defined on
$F$
, which is 0 if the local configuration near the origin is legal and
$-1$
otherwise. We show that in the case of
$\mathbb{Z}$
subshifts, for all sufficiently small perturbations,
$g$
, of
$f$
, the
$g$
-maximizing invariant probability measures are supported on
$X$
(that is, the set
$X$
is stably maximized by
$f$
). However, in the two-dimensional case, we show that the well-known Robinson tiling fails to have this property: there exist arbitrarily small perturbations,
$g$
, of
$f$
for which the
$g$
-maximizing invariant probability measures are supported on
$F\setminus X$
.
Ergodic optimization for countable alphabet subshifts of finite type
