  We introduce the \textit{$(m,1)$-edge coloring game}, which is alternatively
played by two players Maker and Breaker on a finite simple graph $G$ with a set
of colors $X$. Maker plays first and colors $m$ uncolored edges on each turn.
Breaker colors only one uncolored edge on each turn. They make sure that
adjacent edges get distinct colors. Maker wins if eventually every edge is
colored; Breaker wins if at some point, the player who is playing cannot color
any edge. We define the \textit{$(m,1)$-game chromatic index} of $G$ to be the
smallest positive integer $k$ such that Maker has a winning strategy with
$|X|=k$. We determine the exact $(m,1)$-game chromatic indices of some
caterpillars and all wheels.
