  Consider two random walks on $\mathbb{Z}$. The transition probabilities of
each walk is dependent on trajectory of the other walker i.e. a drift $p>1/2$
is obtained in a position the other walker visited twice or more. This simple
model has a speed which is, according to simulations, not monotone in $p$,
without apparent "trap" behaviour. In this paper we prove the process has
positive speed for $1/2<p<1$, and present a deterministic algorithm to
approximate the speed and show the non-monotonicity.
