  We study trade networks with a tree structure, where a seller with a single
indivisible good is connected to buyers, each with some value for the good, via
a unique path of intermediaries. Agents in the tree make multiplicative revenue
share offers to their parent nodes, who choose the best offer and offer part of
it to their parent, and so on; the winning path is determined by who finally
makes the highest offer to the seller. In this paper, we investigate how these
revenue shares might be set via a natural bargaining process between agents on
the tree, specifically, egalitarian bargaining between endpoints of each edge
in the tree. We investigate the fixed point of this system of bargaining
equations and prove various desirable for this solution concept, including (i)
existence, (ii) uniqueness, (iii) efficiency, (iv) membership in the core, (v)
strict monotonicity, (vi) polynomial-time computability to any given accuracy.
Finally, we present numerical evidence that asynchronous dynamics with randomly
ordered updates always converges to the fixed point, indicating that the fixed
point shares might arise from decentralized bargaining amongst agents on the
trade network.
