A connections model with decreasing returns link-formation technology

Abstract

We study a connections model where the strength of a link depends on the amount invested in it and is determined by an increasing strictly concave function. The revenue from investments in links is the value (information, contacts, friendship) that the nodes receive through the network. First, assuming that links are the result of investments by the node-players involved, there is the question of stability. We introduce and characterize a notion of marginal equilibrium, where all nodes play locally best responses, and identify different marginally stable structures. This notion is based on weak assumptions about node-players’ information and is necessary for Nash equilibrium and for pairwise stability. Second, efficient networks in absolute terms are characterized. Efficiency and stability are shown to be incompatible, but partial subsidizing is shown to be able to bridge the gap.

Efficient Networks in Connections Models with Heterogeneous Nodes and Links

“…I nt h i sc a s et h e r ei sad r a wb e t w e e n the all-encompassing star, the complete network and a whole range of a particular type of nested split graph structures intermediate between them. In Olaizola and Valenciano (2022) it is shown that this cannot occur if  is strictly concave, differentiable, and s.t. and  0 ()  0,f o ra l l ≥ 0…”

“…If there are still disconnected nodes and available links, which of the two options is preferable depends on the subsequent choices of which among the not directly connected pairs of nodes to connect with the subsequently available links. 8 We describe an algorithm to produce a network with the available links (i.e. some of those which form )t h a td o m i n a t e s.…”

“…whose reflexive closure º is a total order in S. 10 A similar refinement of the order in set D is defined breaking ties in the following w a y . I nt h ec a s eo fd i r e c tc o n n e c t i o n sj u s td e fine  1 Â  2 Â  Â   ; in the case of 8 For instance, if there are 4 nodes with  1 =  2 =  3 = 100 and  4 =4 , and 3 available links…”

“…Olaizola and Valenciano (2021) consider a notion of pairwise stability "under free cost sharing" in a model with heterogeneous nodes and homogeneous links and show that efficiency and stability in this sense are compatible. Olaizola and Valenciano (2022) introduce a notion of "marginal equilibrium", a weak form of local Nash equilibrium, in a model with homogeneous nodes and heterogeneous links, and show that efficiency and stability in this sense are incompatible. It may be interesting to apply both notions of stability and see whether those results extend in the more general model considered in this paper.…”

“…Olaizola and Valenciano (2020a, 2020b, 2021and 2022). 2 Bloch and Dutta (2009) introduce endogenous link strength in a connections model by replacing Jackson and Wolinsky's discrete technology by a non-decreasing returns technology.…”