  Given a directed forest-graph, a probabilistic \emph{selection mechanism} is
a probability distribution over the vertex set. A selection mechanism is
\emph{incentive-compatible} (IC), if the probability assigned to a vertex does
not change when we alter its outgoing edge (or even remove it). The quality of
a selection mechanism is the worst-case ratio between the expected progeny
under the mechanism's distribution and the maximal progeny in the forest. In
this paper we prove an upper bound of 4/5 and a lower bound of $
1/\ln16\approx0.36 $ for the quality of any IC selection mechanism. The lower
bound is achieved by two novel mechanisms and is a significant improvement to
the results of Babichenko et al. (WWW '18). The first, simpler mechanism, has
the nice feature of generating distributions which are fair (i.e., monotone and
proportional). The downside of this mechanism is that it is not exact (i.e.,
the probabilities might sum-up to less than 1). Our second, more involved
mechanism, is exact but not fair. We also prove an impossibility for an IC
mechanism that is both exact and fair and has a positive quality.
