  An analytical method based on the classical ruin problem is developed to
compute the mean reaction time between two walkers undergoing a generalized
random walk on a 1d lattice. At each time step, either both walkers diffuse
simultaneously with probability $p$ (synchronous event) or one of them diffuses
while the other remains immobile with complementary probability (asynchronous
event). Reaction takes place through same site occupation or position exchange.
We study the influence of the degree of synchronicity $p$ of the walkers and
the lattice size $N$ on the global reaction's efficiency. For odd $N$, the
purely synchronous case ($p=1$) is always the most effective one, while for
even $N$, the encounter time is minimized by a combination of synchronous and
asynchronous events. This new parity effect is fully confirmed by Monte Carlo
simulations on 1d lattices as well as for 2d and 3d lattices. In contrast, the
1d continuum approximation valid for sufficiently large lattices predicts a
monotonic increase of the efficiency as a function of $p$. The relevance of the
model for several research areas is briefly discussed.
