  The basic idea of voting protocols is that nodes query a sample of other
nodes and adjust their own opinion throughout several rounds based on the
proportion of the sampled opinions. In the classic model, it is assumed that
all nodes have the same weight. We study voting protocols for heterogeneous
weights with respect to fairness. A voting protocol is fair if the influence on
the eventual outcome of a given participant is linear in its weight. Previous
work used sampling with replacement to construct a fair voting scheme. However,
it was shown that using greedy sampling, i.e., sampling with replacement until
a given number of distinct elements is chosen, turns out to be more robust and
performant.
  In this paper, we study fairness of voting protocols with greedy sampling and
propose a voting scheme that is asymptotically fair for a broad class of weight
distributions. We complement our theoretical findings with numerical results
and present several open questions and conjectures.
