  The Maker-Breaker connectivity game and Hamilton cycle game belong to the
best studied games in positional games theory, including results on biased
games, games on random graphs and fast winning strategies. Recently, the
Connector-Breaker game variant, in which Connector has to claim edges such that
her graph stays connected throughout the game, as well as the Walker-Breaker
game variant, in which Walker has to claim her edges according to a walk, have
received growing attention.
  For instance, London and Pluh\'ar studied the threshold bias for the
Connector-Breaker connectivity game on a complete graph $K_n$, and showed that
there is a big difference between the cases when Maker's bias equals $1$ or
$2$. Moreover, a recent result by the first and third author as well as Kirsch
shows that the threshold probability $p$ for the $(2:2)$ Connector-Breaker
connectivity game on a random graph $G\sim G_{n,p}$ is of order
$n^{-2/3+o(1)}$. We extent this result further to Walker-Breaker games and
prove that this probability is also enough for Walker to create a Hamilton
cycle.
