  Though deliberation is a critical component of democratic decision-making,
existing deliberative processes do not scale to large groups of people.
Motivated by this, we propose a model in which large-scale decision-making
takes place through a sequence of small group interactions. Our model considers
a group of participants, each having an opinion which together form a graph. We
show that for median graphs, a class of graphs including grids and trees, it is
possible to use a small number of three-person interactions to tightly
approximate the wisdom of the crowd, defined here to be the generalized median
of participant opinions, even when agents are strategic. Interestingly, we also
show that this sharply contrasts with small groups of size two, for which we
prove an impossibility result. Specifically, we show that it is impossible to
use sequences of two-person interactions satisfying natural axioms to find a
tight approximation of the generalized median, even when agents are
non-strategic. Our results demonstrate the potential of small group
interactions for reaching global decision-making properties.
