  We consider a sharing economy network where agents embedded in a graph share
their resources. This is a fundamental model that abstracts numerous emerging
applications of collaborative consumption systems. The agents generate a random
amount of spare resource that they can exchange with their one-hop neighbours,
seeking to maximize the amount of desirable resource items they receive in the
long run. We study this system from three different perspectives: a) the
central designer who seeks the resource allocation that achieves the most fair
endowments after the exchange; b) the game theoretic where the nodes seek to
form sharing coalitions within teams, attempting to maximize the benefit of
their team only; c) the market where the nodes are engaged in trade with their
neighbours trying to improve their own benefit. It is shown that there is a
unique family of sharing allocations that are at the same time most fair,
stable with respect to continuous coalition formation among the nodes and
achieving equilibrium in the market perspective. A dynamic sharing policy is
given then where each node observes the sharing rates of its neighbours and
allocates its resource accordingly. That policy is shown to achieve long term
sharing ratios that are within the family of equilibrium allocations of the
static problem. The equilibrium allocations have interesting properties that
highlight the dependence of the sharing ratios of each node to the structure of
the topology graph and the effect of the isolation of a node on the benefit may
extract from his neighbours.
