Allocation in Networks by Hougaard Jens Leth

Author:Hougaard, Jens Leth
Language: eng
Format: epub
Tags: design; regulation; implementation; axiomatic characterization; mechanism design; economic design; tree structure; organizational hierarchies; decentralized networks
Publisher: MIT Press

4.1.1  The Model

A joint control revenue-sharing problem is defined as a triple (N, r, D), where N is a nonempty finite set of agents, r is a profile specifying the revenue ri of each agent i ∈ N, and D is a dominance structure mapping each agent i ∈ N to her immediate superiors D(i) ⊂ N (the predecessors of i in the digraph) with the convention that D(i) = ∅ if i is a boss. We assume that the digraph induced by D is connected and (undirected) cycle free (i.e., the associated undirected graph contains no cycles). Let 𝒥 denote the set of such problems.

Since the hierarchy is cycle free, deleting any edge ij leads to two components dubbed the i- and the j-components and denoted by and , respectively.

Given a problem (N, r, D), an allocation is a vector x ∈ ℝN satisfying budget balance. An allocation rule ζ is a mapping that, for each problem (N, r, D), assigns an allocation x = ζ(N, r, D). As in the linear hierarchy model, the allocation rule is assumed to be anonymous, that is, for each bijective function g : N → N′, ζg(i)(N′, r′, S′) = ζi(N, r, S), where and S′(g(i)) = g(S(i)) = {g(s): s ∈ S(i)} for each i.

The family of fixed-fraction transfer rules (defined in (2.3) for linear hierarchies, and in (2.18) and (2.19) for branch hierarchies) generalizes easily to the joint-control setting by transferring an equal split of the accumulated surplus of a given agent i to all immediate superiors j ∈ D(i). That is, a lowest-ranked agent (i.e., i for which D−1(i) = ∅) gets payoff λri, while a predecessor j ∈ D(i) gets payoff

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