Creating cycles in Walker–Breaker games

We consider biased (1:b)(1:b) Walker–Breaker games: Walker and Breaker alternately claim edges of the complete graph KnKn, Walker taking one edge and Breaker claiming bb edges in each round, with the constraint that Walker needs to choose her edges according to a walk. As questioned in a paper by Espig, Frieze, Krivelevich and Pegden, we study how long a cycle Walker is able to create and for which biases bb Walker has a chance to create a cycle of given constant length.