  In this paper, we provide new results about an invariance of $p$-harmonic
functions under boundary perturbations by using tug-of-war with noise; a
probabilistic interpretation of $p$-harmonic functions introduced by
Peres-Sheffield in \cite{ps}. As a main result, when $E\subset\bo $ is
countable and $f\in C(\bo)$, we provide a necessary and sufficient condition
for $E$ to guarantee that $H_g=H_f$ whenever $g=f$ on $\bo\setminus E$. Here
$H_f$ and $H_g$ denote the Perron solutions of $f$ and $g$. It turns out that
$E$ should be of $p$-harmonic measure zero with respect to $\Omega$. As a
consequence, we analyze a structure of a countable set of $p$-harmonic measure
zero. In particular, we give some results for the subadditivity of $p$-harmonic
measures and an invariance result for $p$-harmonic measures. In addition, the
results in this paper solve the problem regarding a perturbation point Bj\"orn
\cite{Bjorn} suggested for the case of unweighted $\R^n$.
