  Let $d(x,y)$ denote the length of a shortest path between vertices $x$ and
$y$ in a graph $G$ with vertex set $V$. For a positive integer $k$, let
$d_k(x,y)=\min\{d(x,y), k+1\}$ and $R_k\{x,y\}=\{z\in V: d_k(x,z) \neq
d_k(y,z)\}$. A set $S \subseteq V$ is a \emph{distance-$k$ resolving set} of
$G$ if $S \cap R_k\{x,y\} \neq\emptyset$ for distinct $x,y\in V$. In this
paper, we study the maker-breaker distance-$k$ resolving game (MB$k$RG) played
on a graph $G$ by two players, Maker and Breaker, who alternately select a
vertex of $G$ not yet chosen. Maker wins by selecting vertices which form a
distance-$k$ resolving set of $G$, whereas Breaker wins by preventing Maker
from winning. We denote by $O_{R,k}(G)$ the outcome of MB$k$RG. Let
$\mathcal{M}$, $\mathcal{B}$ and $\mathcal{N}$, respectively, denote the
outcome for which Maker, Breaker, and the first player has a winning strategy
in MB$k$RG. Given a graph $G$, the parameter $O_{R,k}(G)$ is a non-decreasing
function of $k$ with codomain $\{-1=\mathcal{B}, 0=\mathcal{N},
1=\mathcal{M}\}$. We exhibit pairs $G$ and $k$ such that the ordered pair
$(O_{R,k}(G), O_{R, k+1}(G))$ realizes each member of the set $\{(\mathcal{B},
\mathcal{N}),(\mathcal{B}, \mathcal{M}),(\mathcal{N},\mathcal{M})\}$; we
provide graphs $G$ such that $O_{R,1}(G)=\mathcal{B}$, $O_{R,2}(G)=\mathcal{N}$
and $O_{R,k}(G)=\mathcal{M}$ for $k\ge3$. Moreover, we obtain some general
results on MB$k$RG and study the MB$k$RG played on some graph classes.
