Motivated by the role of incentives in large-scale information
systems, Kleinberg and Raghavan (FOCS 2005) studied strategic games in
decentralized information networks.  Given a branching process that
specifies the network, the rarity of answers to a specific question,
and a desired probability of success, how much reward does the root
node need to offer so that it receives an answer with this
probability, when all of the nodes are playing strategically?  For a
specific family of branching processes and a constant failure
probability, they showed that the reward function exhibited a
threshold behavior that depends on the branching parameter $b$.

In this paper we study two factors that can contribute to this
transition behavior, namely, the branching process itself and the
failure probability.  On one hand we show that the threshold behavior
is robust with respect to the branching process: for \emph{all}
branching processes and any constant failure probability, if $b > 2$
then the required reward is linear in the expected depth of the search
tree, and if $b < 2$ then the required reward is exponential in that
depth.  On the other hand we show that the threshold behavior is
fragile with respect to the failure probability~$\sigma$: if $\sigma$
is inversely polynomial in the rarity of the answer, then {\em all}
branching processes require rewards exponential in the depth of the
search tree.