A Degree‐Distance‐Based Connections Model with Negative and Positive Externalities

• DOI: http://doi.org/10.1111/jpet.12183
• Published date: 01 April 2016
• Date: 01 April 2016
• Author: AGNIESZKA RUSINOWSKA,PHILIPP MÖHLMEIER,EMILY TANIMURA

ADEGREE-DISTANCE-BASED CONNECTIONS MODEL

WITH NEGATIVE AND POSITIVE EXTERNALITIES

PHILIPP M ¨

OHLMEIER

Bielefeld University

AGNIESZKA RUSINOWSKA

Paris School of Economics - CNRS

EMILY TANIMURA

Universit´

e Paris I Panth´

eon-Sorbonne

Abstract

We develop a modification of the connections model by Jackson and

Wolinsky that takes into account negative externalities arising from the

connectivity of direct and indirect neighbors, thus combining aspects

of the connections model and the coauthor model. We consider a gen-

eral functional form for agents’ utility that incorporates both the ef-

fects of distance and of neighbors’ degree. Consequently, we introduce

a framework that can be seen as a degree-distance-based connections

model with both negative and positive externalities. Our analysis shows

how the introduction of negative externalities modifies certain results

about stability and efficiency compared to the original connections

model. In particular, we see the emergence of new stable structures,

such as a star with links between peripheral nodes. We also identify

structures, for example, certain disconnected networks, that are effi-

cient in our model but which could not be efficient in the original

connections model. While our results are proved for the general util-

ity function, some of them are illustrated by using a specific functional

form of the degree-distance-based utility.

Philipp M¨

ohlmeier, Bielefeld University, BiGSEM, Center for Mathematical Economics, Bielefeld,

Germany (philipp.moehlmeier@uni-bielefeld.de). Agnieszka Rusinowska, Paris School of Economics

- CNRS, Universit´

e Paris I Panth´

eon-Sorbonne, Centre d’Economie de la Sorbonne (agnieszka.

rusinowska@univ-paris1.fr). Emily Tanimura, Universit´

e Paris I Panth´

eon-Sorbonne, Centre

d’Economie de la Sorbonne, 106-112 Bd de l’Hˆ

opital, 75647 Paris, France (Emily.Tanimura@univ-

paris1.fr).

We would like to thank an anonymous referee, the editor, and participants of the conference “The

Role of Externalities in Networks” (Baton Rouge, February 23–24, 2013), in particular Hans Haller,

Ahmed Saber Mahmud, and Sudipta Sarangi, and also Herbert Dawid and Tim Hellmann, for help-

ful comments. Agnieszka Rusinowska and Emily Tanimura acknowledge the support of the National

Agency for Research (Agence Nationale de la Recherche), Project DynaMITE (ANR-13-BSH1-0010-

01). Philipp M¨

ohlmeier carried out this research within the International Research Training Group

Economic Behavior and Interaction Models (EBIM) financed by the German Research Foundation

(DFG) under contract GRK 1134/2 and within the Bielefeld Graduate School of Economcis and Man-

agement (BiGSEM). Financial support by EBIM, BiGSEM, and the Center for Mathematical Economics

is gratefully acknowledged.

Received April 28, 2013; Accepted June 5, 2015.

C2016 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (2), 2016, pp. 168–192.

168

A Degree-Distance-Based Connections Model 169

1. Introduction

The connections model, introduced in the seminal paper of Jackson and Wolinsky

(1996) is a setting in which only direct contacts are costly but discounted benefits spill

over from indirect neighbors. A natural interpretation is that benefits result from access

to a resource conveyed by the network, such as information or knowledge provided by

indirect contacts.

An appealing feature of networks is that they capture the externalities that “occur

when the utility of or payoff to an individual is affected by the actions of others, al-

though those actions do not directly involve the individual in question” (Jackson 2008),

p. 162). In the connections model, network externalities are positive. An additional link

formed by some pair of individuals (weakly) benefits all other agents by providing access

to new indirect contacts or by reducing the distance that information has to travel. Such

positive aspects of increased connectivity are certainly important. However, in many sit-

uations increased connectivity can also have negative side effects. Studying such cases is

what motivates the analysis in this paper: we consider a model in which agents benefit

from indirect contacts as in the original connections model but in which the connectiv-

ity of an agent may also exert a negative externality on his direct and indirect neighbors.

Contexts where this is the case abound. For example, learning of a job opening may

be less useful if the information has been communicated to many others. When there

is competition for some resource transmitted by the network, the benefits from indirect

contacts are reduced when the latter have many connections. In our model, the utility

an agent derives from an indirect contact, viewed as the initial sender of an information,

is reduced when the latter has a high degree and thus sends the information to many

others. However, this might not fully account for the negative impact of all other indi-

viduals in the communication chain who receive the information. Hence, our model

should be viewed only as a simplified or approximate description of the negative effects

of connectivity when there is competition for information.

Another negative effect of high connectivity that our model perfectly captures arises

because the busyness of agents reduces their availability or productivity. The connec-

tions model of Jackson and Wolinsky (1996) could also be interpreted as follows: nodes

generate output by themselves but also forward output from others. Now interpret this

as a situation in which individuals are involved in projects and generate knowledge by

themselves but also receive knowledge from others. Then, a person involved in many

projects will have less time to generate output. On the other hand, the more well con-

nected he is, the more knowledge he will receive and forward to his neighborhood.

Stated in a provocative way, “well connected people are often great talkers, but network-

ing is time-consuming and reduces one’s productive time so that the main work is done

by others.” Nevertheless, the role of such well-connected agents is very important: not

that they contribute a lot by their own knowledge production, but they provide access

to the output of many others.

By integrating the negative effect of the busyness of agents, at first sight our model

looks similar to the well known coauthor model (also introduced in Jackson and Wolin-

sky 1996) where the time devoted to a single project decreases with the total number of

projects a coauthor is involved in. In our version, the nature of the negative externality

is similar but information spills over from indirect connections. The coauthor model

only considers direct collaborations and thus conveys negative effects solely through

the busyness of direct coauthors. Hence, our model combines aspects of the connec-

tions model and the coauthor model. Externalities resulting from additional links can

be both positive and negative. New links are useful for reaching indirect partners, but

the latter will be more busy, less productive, and thus less valuable per se although more