  In a $(1:b)$ biased Maker-Breaker game, how good a strategy is for a player
can be measured by the bias range for which its rival can win, choosing an
appropriate counterstrategy. Bednarska and {\L}uczak proved that, in the
$H$-subgraph game, the uniformly random strategy for Maker is essentially
optimal with high probability. Here we prove an analogous result for the
$H$-graph minor game, and we study for which choices of $H$ the random strategy
is within a factor of $1+o(1)$ of being optimal.
