  Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a
balanced copy of $G$ if we can find a copy of $G$ such that half of its edges
is in each color class. If there exists an integer $k$ such that, for $n$
sufficiently large, every 2-coloring of $K_n$ with more than $k$ edges in each
color contains a balanced copy of $G$, then we say that $G$ is balanceable. The
smallest integer $k$ such that this holds is called the balancing number of
$G$.
  In this paper, we define a more general variant of the balancing number, the
list balancing number, by considering 2-list edge colorings of $K_n$, where
every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the
color set $\{r,b\}$. In this case, edges $e$ with $L(e) = \{r,b\}$ act as
jokers in the sense that their color can be chosen $r$ or $b$ as needed. In
contrast to the balancing number, every graph has a list balancing number.
Moreover, if the balancing number exists, then it coincides with the list
balancing number.
  We give the exact value of the list balancing number for all cycles except
for $4k$-cycles for which we give tight bounds. In addition, we give general
bounds for the list balancing number of non-balanceable graphs based on the
extremal number of its subgraphs, and study the list balancing number of $K_5$,
which turns out to be surprisingly large.
