Network Formation under Multiple Sources of Externalities

• DOI: http://doi.org/10.1111/jpet.12177
• Author: SUMIT JOSHI,AHMED SABER MAHMUD
• Date: 01 April 2016
• Published date: 01 April 2016

NETWORK FORMATION UNDER MULTIPLE SOURCES

OF EXTERNALITIES

SUMIT JOSHI

George Washington University

AHMED SABER MAHMUD

Johns Hopkins University

Abstract

The architecture of social and economic networks is often explained in

terms of the externalities shaping the link-forming incentives of play-

ers. We make two contributions to this literature. First, we bring into

its ambit the linear-quadratic utility model. Since players’ utilities are

now a function of their network centralities, this permits endogenizing

their locational incentives in a network. Second, we show that the mode

of transmission of externalities can be crucial in dictating the topology

of equilibrium networks. We consider two alternative modes for chan-

neling externalities. Both have the same primary source (direct links)

but a different secondary source (global effects vs. indirect links). We

characterize equilibrium networks for different positive–negative com-

binations of primary–secondary externalities for both modes. We show

that the mode of transmission influences the equilibrium architecture

when externalities from the primary source are positive;whentheseex-

ternalities are negative, the equilibrium network is empty.

1. Introduction

The equilibrium configuration of interconnections among players that emerges en-

dogenously as players evaluate the benefits and costs of establishing bilateral links has

occupied an important strand in the network literature.1The existing literature has at-

tempted to explain the equilibrium network architecture in terms of the externalities that

1Please see Goyal (2007) and Jackson (2008) for a survey of this literature in the context of both

directed and undirected networks.

Sumit Joshi, Department of Economics, George Washington University 2125 G Street NW, Washing-

ton, DC 20052 (sumios@gwu.edu). Ahmed S. Mahmud, Applied Economics Program, Johns Hop-

kins University, Suite 104, 1717 Massachusetts Avenue NW, Washington DC 20036 (amahmud2@

jhu.edu).

We would like to thank participants at the Networks and Externalities conference (February 2013)

at Louisiana State University. Wewould also like to thank an associate editor and two referees for their

detailed comments and suggestions that have significantly improved the paper. Weremain responsible

for any errors.

Received April 30, 2013; Accepted June 5, 2015.

C2016 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (2), 2016, pp. 148–167.

148

Multiple Sources of Externalities 149

bear upon players by virtue of their position in the network architecture. Our objective

in this paper is twofold. First, we bring into the ambit of the endogenous network forma-

tion literature the linear-quadratic specification of utility which now forms the bedrock

for many games on an exogenously fixed network.2Second, exploiting the tractability

offered by the linear-quadratic model, we demonstrate that the mode of transmission of ex-

ternalities is an equally important determinant in shaping the topology of equilibrium

networks. We now elaborate on each of these objectives.

The linear-quadratic specification of utility has an important feature that explains

its popularity for Nash games on a fixed network: the Nash action and reduced util-

ity of players is a function of their Katz–Bonacich centrality in a network.3In some

games (e.g., the oligopoly game of Example 1 in Section 3 below), Nash actions and

reduced utility are increasing in network centrality. In other games (e.g., provision of

public goods in Bramoull´

e, Kranton, and Amours 2014), higher Nash action levels are

no longer associated with the most central players; in fact players with greater centrality

can free ride on the actions of those less central (Ballester and Calv´

o-Armengol 2010,

example 11). These results lend special impetus toward endogenizing the network for-

mation process and systematically examining the motivation of players to assume, or not

assume, a central position in the network with an eye toward the subsequent game to

be played on the ensuing network. Accordingly, we posit a two-stage game where in the

first stage players form a network and in the second stage they engage in a Nash game

contingent on the network. Inducting backward from the Nash game in actions, players

weigh the benefits and costs of linking with each other. Our first objective is to charac-

terize the network architecture that ensues with particular emphasis on the centrality of

the locations occupied by players in the network.

Our second objective is to investigate within the context of the linear-quadratic util-

ity model two alternative modes of channeling externalities. Our main thesis is that the

mode of transmission of externalities can dictate the architecture of equilibrium net-

works. The two modes that we consider have a primary source and a secondary source

for transmitting externalities. We present examples to show that externalities from ei-

ther source can be positive or negative. We characterize the equilibrium networks for

different positive–negative combinations of primary–secondary externalities for both

modes. In the process, we demonstrate that the mode of transmission has definite im-

plications for the equilibrium network topology when externalities from the primary

source are positive. When these externalities are negative, the equilibrium architecture

is always empty and thus robust to the mode of transmission.

The first mechanism for transmitting externalities is based on Ballester , Calv´

o-

Armengol, and Zenou (2006). The primary source is local interaction (flow of exter-

nalities through direct links or neighbors within a network) while the secondary source

is global interaction (externalities transmitted without the aegis of the network). We

say that strategic local externalities exist and are negative (respectively, positive) when

an increase in the actions of neighbors in a network decreases (respectively, increases)

the utility of a player. Similarly, strategic global externalities are negative (respectively,

2This utility specification was introduced by Ballester, Calv´

o-Armengol, and Zenou (2006) and has

found numerous applications in the literature on games on networks. These include education (Calv´

o-

Armengol, Patacchini, amnd Zenou 2009), crime (Ballester, Calv´

o-Armengol, and Zenou 2010), con-

formity (Patacchini and Zenou 2012), and public goods provision (Bramoull´

eet al. 2014). Please see

Jackson and Zenou (2012) for a survey and Ballester and Calv´

o-Armengol (2010) for extensions.

3Please see Section 3 for the Katz–Bonacich measure, or Bonacich (1987) and Jackson (2008, section

2.2.4).