  A $k$-majority tournament $T$ on a finite set of vertices $V$ is defined by a
set of $2k-1$ linear orders on $V$, with an edge $u \to v$ in $T$ if $u>v$ in a
majority of the linear orders. We think of the linear orders as voter
preferences and the vertices of $T$ as candidates, with an edge $u \to v$ in
$T$ if a majority of voters prefer candidate $u$ to candidate $v$. In this
paper we introduce weighted $k$-majority tournaments, with each edge $u \to v$
weighted by the number of voters preferring $u$.
  We define the maximum approval gap $\gamma_w(T)$, a measure by which any
dominating set of $T$ beats the next most popular candidate. This parameter is
analogous to previous work on the size of minimum dominating sets of
(unweighted) $k$-majority tournaments. We prove that $k/2 \leq \gamma_w(T) \leq
2k-1$ for any weighted $k$-majority tournament $T$, and construct tournaments
with $\gamma_w(T)=q$ for any rational number $k/2 \leq q \leq 2k-1$. We also
consider the minimum number of vertices $m(q,k)$ in a $k$-majority tournament
with $\gamma_w(T)=q$.
