  Inspired by distributed resource allocation problems in dynamic topology
networks, we initiate the study of distributed consensus with finite messaging
passing. We first find a sufficient condition on the network graph for which no
distributed protocol can guarantee a conflict-free allocation after $R$ rounds
of message passing. Secondly we fully characterize the conflict minimizing
zero-round protocol for path graphs, namely random allocation, which partitions
the graph into small conflict groups. Thirdly, we enumerate all one-round
protocols for path graphs and show that the best one further partitions each of
the smaller groups. Finally, we show that the number of conflicts decrease to
zero as the number of available resources increase.
