Altruism in Networks

Abstract

THIS SUPPLEMENT contains proofs and additional results that complement the paper "Altruism in Networks." The first section studies the transfer cost minimization problem that underlies the potential maximization problem of the main paper, and uses it to derive additional properties of equilibrium transfers, and provide different proofs of some results. The generic uniqueness of equilibrium transfers is a consequence of this analysis. We also use this section to explain the connections between our altruistic transfer game and two classical transportation problems: the minimum-cost flow problem, and the MongeKantorovich optimal transportation problem. In the second section, we show convergence of best-response dynamics in the transfer game. In the third section, we look at conditions for the presence or absence of transfer intermediaries, and provide a proof of Theorem 2 of the paper. Finally, in the fourth section, we consider comparative statics with respect to initial income profiles and altruism. We prove the genericity result used in the proofs of the paper, and provide some additional comparative statics results. APPENDIX A: THE COST MINIMIZATION APPROACHIn this section, we analyze in detail the cost minimization problem and use it to draw connection with classical linear programming problems, and to prove generic uniqueness of equilibrium transfers. We also exhibit some additional properties of optimal transfer networks such as cyclical monotonicity. The presentation of the first results borrows from Galichon (2011). For an overview of the use of optimal transport methods in economics, see Galichon (2016).Recall that the maximization of the potential is related to the cost minimization problemwhere A = {(i j) : α ij > 0} is the set of arcs of the altruistic network α, and S(y) = {T ∈ S : ∀i y i = y 0 i − j t ij + j t ji } is a closed convex polytope since it is defined by a finite number of weak inequalities. Note that S(y) is unbounded if the altruism network α admits a directed cycle, since one can then indefinitely increase the transfers of any T ∈ S(y) along the cycle while still reaching y from y 0 . This problem is a classical linear programming problem known as the Minimum Cost Flow problem. Indeed, if each c ij is interpreted as the marginal transportation cost between i and j, this problem consists of minimizing transportation cost over the network of agents with the constraint of reaching distribution y from distribution y 0 . In network flow problems, a transfer profile T is called a flow, and we will sometimes use this terminology. A first useful result from the network flow literature (see Galichon (2011)) says that any flow can be decomposed into paths and cycles. Before we do that, we partition the set of agents into three sets: the set of net givers I G = {i : y i < y 0 i } (or sources), the set of net receivers I R = {i : y i > y 0 i } (or sinks), and the remaining agents. We let P ij be the set of paths between i and j in the altruism network, P = (i j)∈I G ×I R P ij , be the set of paths ...

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