STRATEGIC COMPETITIONS OVER NETWORKS. A …dg223bd6609/dissertation-augmented.pdfFrom Doug I also learned to be extremely focused on delivering the point of the paper. Extremely, extremely,

STRATEGIC COMPETITIONS OVER NETWORKS.

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ECONOMICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Carlos R. Lever

May 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/dg223bd6609

© 2010 by Carlos Rodrigo Lever Guzman. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/dg223bd6609

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Matthew Jackson, Primary Adviser


Manuel Amador


B. Bernheim

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

I present three applications of network theory to economic applications. The first chap-

ter studies strategic spending in voting competitions with social networks. It finds that

equilibrium spending targets voters whose position in the network has a high eigenvector

influence. The second chapter studies how eigenvector influence changes when disconnected

components of a network begin interacting. The result has implications for inequality in in-

vestment with social spill-overs and for consumption decisions with social influence. The final

chapter studies pricing competitions in infrastructure networks. It finds how the structure

of the network influences pricing behavior and market surplus.

iv

Preface.

This dissertation studies three applications of network models in economic settings. Network

models in economics seek to incorporate insights and techniques from social network analysis

and graph theory into economics. Simultaneously, economists seek to improve these models

by using our models of decision-making and strategic reasoning to explain how individuals

chose to form their network of relationships.

Chapter 1 analizes the effect of social networks on voting competitions. When political

parties carry out a campaign they know that people are influenced by the opinion of their

social neighbors. With the growth of social networking sites we are collecting more detailed

information than ever before on who talks with whom, but currently we don’t have a theory

to understand how this will alter political campaigns. To address this I propose a model of

strategic spending where two parties compete by targeting resources to convince voters to

chose them. I assume that parties can use the information on the structure of social rela-

tionships to target their resources more effectively and I study the implications for spending

strategies.

The same model can be applied to study of lobbying. Competing interest groups in

Congress have to decide how to allocate their time and resources over different legislators. On

the other hand, legislators develop relationships with each other over repeated interactions

across different legislatures. Understanding the role of each legislator in the structure of

relationships is valuable to decide whom to lobby. With this interpretation in mind I test

the predictions of the model using data on campaign contributions by interest groups in

the US Congress. Consistent with the model, I find that the influence of the position of a

legislator in the network is a significant predictor of the campaign contributions by lobbies

even after controlling for possible confounds. In contrast, the prediction of previous models

of strategic spending is not supported by the data.

In joint work with Benjamin Golub, Chapter 2 develops the theory of networks to un-

derstand how the influence of each member changes when the network changes. Through

our previous research, both Ben and I had became interested in self-referential measures of

v

influence. We were interested in understanding these measures changed when you changed

the structure of the network. As we studied the problem we found that there was a gap in

the literature. Previous papers by the markov chain literature had expressions that worked

as long as the network remained connected, but disconnected groups feature prominently in

the study of trade networks and strategic network formation games, so we decided to expand

the set of tools.

The main contribution of the chapter is that we derived a closed-form solution for how

influence changes when disconnected groups start interacting. Ben deserves the credit for

finding the main insight for the proof and writing the main proposition. We then jointly

worked in correcting the proofs and in finding applications of the result to economic settings.

We came up with two important applications: investment decisions with strong social spill-

overs and consumptions decisions with strong social influence.

Finally chapter 3 analyzes pricing competitions in infrastructure networks. In the short-

run, infrastructure networks severely limit which buyers can interact with which sellers,

inducing an imperfect competition. This chapter seeks to understand how the structure of

the network influences pricing behavior and welfare. It provides an important lesson for

network models that are not explicit on how agent are competing through the network: if

competition is very aggressive, having access to more people through the network can be

harmful: the increased competition offsets any possible benefit. Therefore rational players

refrain from getting as much connections as they can.

vi

Acknowledgements

I have benefitted greatly from my advisors.

Doug Bernheim provided very high quality advice in key moments. His ability to frame

an issue in clear and concise terms is nothing short of awe-inspiring. A typical experience

with him would involve me pitching an idea that I had been working on for months, only to

have him describe it back to me in a way that completely transformed the way I had been

thinking about it, and yet immediately seemed as the absolutely obvious way to view things.

From Doug I also learned to be extremely focused on delivering the point of the paper.

Extremely, extremely, focused. I tend to resist any change in my work, but Doug’s sharp

comments stick with you for a long time. Sometimes I would mull over them for months

before I was ready to detach myself from work and accept I had to drop the non-essential

features. In the end it was always for the best.

After having two senior advisors on my committee, I needed someone as approachable as

Manuel Amador to go over the myriad concerns that happen int he day-to-day of research.

Finding him was a stroke of luck, as he was not in my main fields.1 But Manuel is so

sharp that there wasn’t a noticeable language-barrier. Besides giving sharp advice he was

always available, sometimes on a daily basis (albeit he wasn’t necessarily punctual). He also

provided encouragement and motivation in key moments.

My primary advisor, Matt Jackson, deserves a special mention. He played an immense

role in forming me as a researcher. Matt is extremely generous with his time, patient with

ideas in early stages and receptive to methodologies that are not conventional. He always

pushed me to judge my ideas by their economic implications, not by the complexity of the

methods behind them. I learned from him in a very wide range of settings: by taking courses

with him, by being his TA, through his reading group and through interacting with him at

conferences.

Matt has a unique style of advising. There are several episodes worth mentioning. In the

1It’s a shame that the Stanford theory group does not currently have young professors to perform such arole.

vii

beginning of our relationship he supervised my second year paper. He made me go through

so many rounds of revisions that I ended up submitting it six months late. In hindsight

this was one of the best investments I have made. Irrespective of the content of my paper,

investing to write clearly in my first work has had a big return.

Matt also did an incredible job in putting together a reading group where students

actively give feedback to each other and there is a lot of horizontal interaction. Matt has

managed to establish an informal environment where ideas under development can be freely

discussed. To achieve this, Matt seemed to actively hold back in giving feedback to force

the students to participate. At the beginning this generated after a couple of sessions of

uncomfortable silence, until the students in the group became more and more proactive in

giving feedback. Eventually this lead to a self-sustaining path where the students share the

weight of the work in the group.

Through his example, Matt also taught me how to defend my ideas without taking

criticism personally. As with all new fields in research, especially fields that abandon the

established methodological paradigm, network models attract a passionate backlash. Both in

his networks course and in his seminar presentations, I saw Matt field visceral questions that

went beyond questioning the substantive question and sometimes got downright offensive.

Matt always reacted with a relaxed demeanor, separating his work from his person. He

would accept the comments that improved the ideas, ignore any implications unrelated to

the work and relentlessly push back on comments that were incorrect. Over these years I

have seen more than one of his students adopt this relaxed but firm approach to presenting.

Finally I must mention that behind every great man there is a great woman, or in Matt’s

case, three: his wife and two daughters. In this area Matt also teaches a powerful lesson by

example. Matt is always open about the role of his family in his life, and makes a strong

impression by keeping a balance between work and family. Grad school is full of subtle and

not-so-subtle messages that pressure people into being very narrow-minded about work. In

discussions among students we would always point to Matt as a guiding counterexample.

Beside my advisors I spent a valuable time learning from my professors: Jon Levin, Andy

Skrzypacz, Paul Milgrom, Ilya Segal, Monika Piazzesi, Matt Harding, Giacomo deGiorgi,

Brian Knutson and Antonio Rangel.

I was incredibly lucky to have a strong cohort of students at Stanford. I learned as much

from them as from the faculty. In the theory field, I had endless conversations on all sort

of topics with Aaron Bodoh-Creed, Alex Hirsch, Juuso Toikka and Marcello Miccoli. Their

tough questions in the gradlloquium have been the harshest I have ever faced, but it was

all “creative destruction”. Beyond the theorists, I benefitted from other extremely smart

viii

people in my cohort: Albie Bollard, Alessandra Voena, Neale Mahoney and Max Floetotto.

Matt’s research group allowed me to interact with a large quantity of students interested

in networks and political economy, both from Stanford or visiting. There are too many to

mention all, but Matt Elliot, Ben Golub and Jeanne Hagenbach deserve special mention.

Working on a paper with Ben Golub was always a pleasurable if challenging experience, as

I had to keep catching up with his physics intuitions.

I was also lucky to have one of my closest friends, Gaby Calderon, with me at Stanford.2

She kept me in balance by providing a connection to home while being in a unique position

to understand my present-day dilemmas. The other members of the Mexican community

did a great job in welcoming and mentoring me: Quique Seira, Alex Ponce, Rodrigo Barros,

Alejandrina Salcedo, Luis Fernando Perez, Carlos Mery, Santiago Ocejo, and Carlos Gamez.

I have tried to repay them by greeting the new generations of Mexicans at Stanford.

Most important of all, this long and consuming project would not have been possible

without the support of my family: my parents, Delia and Carlos, and my three amazing

sisters, Mariana, Paloma and Victoria. All foreign student bear a substantial cost from being

away from his family and society. Those that stay behind share the cost while receiving a

negligible part of the benefits.

Finally I want to mention that my responsibility to my country has never been far from

my mind. I come from a place where half the population lives in an unacceptable poverty.

Were it not from the extreme inequality in Mexico, everybody could have the opportunity to

seek the good life in the pursuit of happiness. This is the reason I decided to enter economics

and I haven’t given up on it. I think the best way to pay what I owe to my family, friends

and advisors, is through being a productive member of society, using my gifts to give back

to the dispossessed.

2I like to think I played some part in that luck by lobbying her during the flyout.

ix

Contents

Abstract iv

Preface. v

Acknowledgements vii

1 Strategic spending in voting competitions 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 The persuaders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 The voters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 The timing of the game . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.4 The persuasion stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.5 The network stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.6 Some network definitions . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Solving for equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Solving the model with consensus. . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Discussion: Where are the pivotal voters? . . . . . . . . . . . . . . . 14

1.3.3 A Parent-Child example. . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.4 Solving the model without consensus: targeting disconnected groups. 16

1.3.5 Solving the model without consensus: adding ideology to the model. . 17

1.4 Competitions in proportional representation systems . . . . . . . . . . . . . 19

1.5 Competition with fundraising . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.6 Testing the model with data on lobbying. . . . . . . . . . . . . . . . . . . . . 22

1.6.1 The House Specification . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.6.2 The Senate specification . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.6.3 Separating the Degroot weights into direct and indirect effects. . . . . 29

x

1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 The leverage of weak ties 39

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 How linking groups affects centralities. . . . . . . . . . . . . . . . . . . . . . 42

2.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.2 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.3 The intuition for the formula . . . . . . . . . . . . . . . . . . . . . . 46

2.3 Motivating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1 Investment decisions with strong social spillovers . . . . . . . . . . . 47

2.3.2 Consumption with strong social influence . . . . . . . . . . . . . . . . 52

2.4 Empirical implications:

the sensitivity of centrality measures . . . . . . . . . . . . . . . . . . . . . . 54

2.5 Several links between groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Price competition on a buyer-seller network. 63

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 The duopoly model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2.2 Solving the duopoly model . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.3 Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.4 Price discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3 Which networks are likely to form? . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.1 Pairwise-stable networks . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.2 Entry game 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.3 Entry game 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4 Extending the results for oligopolistic competition . . . . . . . . . . . . . . . 79

3.4.1 An arbitrary number of firms competing in a single market . . . . . . 79

3.4.2 Remarks on the general oligopoly model . . . . . . . . . . . . . . . . 81

xi

List of Tables

1.1 House of Representatives campaign contributions regressions . . . . . . . . . 34

1.2 Senate campaign contributions regressions . . . . . . . . . . . . . . . . . . . 35

1.3 Robustness checks for the House . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.4 Robustness checks for the Senate . . . . . . . . . . . . . . . . . . . . . . . . 37

1.5 Decomposition the DeGroot weights. . . . . . . . . . . . . . . . . . . . . . . 38

xii

List of Figures

2.1 A two-way bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 A bridge that is not two-way. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 An example of buyer-seller network. . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 A monopolistic network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 A competitive network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4 An imperfect competition network . . . . . . . . . . . . . . . . . . . . . . . . 68

xiii

Chapter 1

Strategic spending in voting

competitions with social networks.

1.1 Introduction

This paper studies political competitions when voters influence each other’s opinion. When

people are deciding how to vote or which product to buy, they discuss their decision with

people in their social environment. Studying the pattern of social relationships is important

in understanding how individuals are influenced directly and indirectly by the opinion of

others. Currently we do not have a model of competition that takes these effects into

account.

Using techniques from social network analysis, I propose a model where persuaders strate-

gically assign resources across voters based on thier position on a social network. My model

allows a rich structure of influence between individuals. For example, I allow for influence

to be asymmetric between individuals and put no restriction on the number of people they

talk to.

My main finding is that there is a unique equilibrium in pure-strategies. In that equi-

librium, persuaders spend on each voter in proportion to the influence of his position in the

network as measured by the eigenvector centrality of the position. These measures had been

found in the sociology literature.1 Persuaders also adjust to spend less on voters who in the

margin are harder to persuade.

Previous papers on strategic spending in political campaigns and lobbying have found

that resources should be targeted toward voters who have a higher probability of casting a

1See Wasserman and Faust (1994a); Bonacich (1987); Bonacich and Lloyd (2001).

1

CHAPTER 1. STRATEGIC SPENDING IN VOTING COMPETITIONS 2

pivotal vote.2 In contrast, pivotal voters are not targeted at all in my model, only influential

voters are. This yields new predictions on campaign spending. For example, my model

predicts that resources will be spent on voters who have an influential position on the social

network even if they are very unlikely to swing their vote.

Adding the network influence switches spending from pivotal voters to influential voters

because the network spillovers undermine targeting. Pivotal voters are important for elec-

tions because they have the maximum impact in the outcome of the election. This is true

even with network spillovers, but persuaders can no longer effectively persuade individual

voters because their opinions mix with the opinions of their neighbors.

To test the model I match data on campaign contributions by lobby groups with data

on cosponsorships networks in the US Congress. I identify the effect of network influence by

analyzing how the contributions of each legislator vary from one electoral year to the next.

I find that changes in network influence are a significant predictors of changes in campaign

contributions for the House of Representatives, while pivot probabilities are not. After

controlling for several confounds I find that increasing network influence by one standard

deviation increases the campaign contributions by 44,834 US dollars (10.5% of the average

contributions received by a Representative).

In the Senate the result is reversed, network influence is not a significant predictor but

being pivotal to break the filibuster is. These finding have to be taken with caution because

the Senate generates very few observations on political campaigns due to the low number of

legislators and their long office tenures. For example, we cannot statistically reject that the

DeGroot weight in the Senate has the same magnitude that in the House.

My paper brings together two literatures. On the social networks side there has been

much work on identifying the influential members in networks but almost no work has been

done on how this information would be used in competitions. There are a vast number of

measures of network influence.3 But my model predicts that eigenvector measures are the

correct measure.

On the strategic persuasion side, there is a literature on counter-active lobbying4 and

strategic spending in presidential elections,5 but these papers do not allow for voters to

influence one other.

The only previous works on political competitions with network effects is the work in

Galeotti and Mattozzi (2008), which build a model of information disclosure revealed when

2See Shubik and Weber (1981); Snyder (1989).3See Jackson (2008a); Wasserman and Faust (1994a).4Austen-Smith and Wright (1994, 1996).5Merolla et al. (2005).


voters inform themselves through a social network. Their work focuses on the amount of

information revealed when political parties have an incentive to hide their platforms and on

the types of candidates selected to run. Their work puts much less emphasis on the structure

of the network.

Social networks will be increasingly more important for future political campaigns because

growth in social networking sites is generating more data than ever before on the structure

of social networks. This will allow a level of targeting that would have been inconceivable

a decade ago. Furthermore younger are receiving a larger amount of information through

social networking sites. In a survey by the Pew Research Center on the 2008 presidential

election, 27% of people under 30 reported getting information on the campaign through

social networking sites. The number rose to 37% if you only consider those between 18 and

24 years. This drastically differed from the 4% of people in their 30s and the less than 1%

of people above 40 who reported using these sites.

This chapter is structured as follows: Section 3.2 sets-up the model. Section 1.3 solves

the model. Section 1.4 extends the model for competitions in proportional representation

systems. For most of the chapter I assume persuaders have a fixed amount of resources,

Section 1.5 solves the model when persuaders have to raise their resources at a cost. Section

1.6 tests the model with data on legislative cosponsorship networks and data on campaign

contributions in the US Congress. Section 1.7 concludes. All tables are included at the end

of the chapter.

1.2 The Model

1.2.1 The persuaders

Two persuaders, A and B, have to decide how to spend resources over a group of voters

who will choose between them. The persuaders A, B can be thought of as political parties

or competing lobbies. Interpreting A, B as political parties is straightforward: the parties

have to convince voters to chose them and whoever gets a majority wins.6

To interpret the model as a lobbying competitions, A and B have to be taken as two

opposing lobbies who are fighting over a bill in Congress. One lobby wants the bill to pass

and the other wants it to fail. The voters the must convince are the legislators that vote on

the bill. Lobbies must then target their resources over different legislators to convince them

to vote in their preferred direction. Without loss I will assume lobby A wants the bill to

6Except in the US presidential system, where parties need a majority of the electoral college.


pass while lobby B prefers the status quo be sustained.

I will assume that A and B only care about winning the election. They do not care by

how many votes. In the model there will be uncertainty on the votes, so A and B will seek to

maximize their probability of winning. Since bills need a qualified majority of votes to pass,

I will also solve the model for supermajority rules. In these cases, I will assume without loss

that A needs a qualified majority and that B wins whenever A fails to obtain it.

Section 1.4 solves an alternative model where A and B wish to maximize the percentage

of votes they receive. This is particularly important for political systems with proportional

representation where the number of seats in congress depends on the share of the vote. The

results are qualitatively similar, but competitions with a majoritarian rule illustrate better

the difference of adding network influence to a model of strategic spending.

1.2.2 The voters

There is a finite number N of voters that select between A and B. A subscript i denotes

voter i. All voters have to chose A or B, so turn-out is not an issue.

Each voter will have an opinion vi of the relative value of A relative to B. A larger vi will

be more favorable to A. These opinions are a summary statistic of the relevant information

required to chose between A and B. For example, vi could capture the perception on which

candidate is more able to deal with a financial crisis, which candidate is more determined to

carry out difficult reforms or which candidate has more charisma.

Games of strategic spending frequently only have strategies in complicated mixed strate-

gies and characterizing all of them is hard.7 Solving for these along with network influence

would be intractable. To avoid it I will assume voters chose probabilistically. Increasing the

opinion vi will only increase the probability that voter i choses A over B.

I will assume that increasing the relative value vi smoothly increases the probability of

voting for A. The easiest way to model this is to reparametrize opinions so that vi correspond

exactly to the probability i choses A. To be concrete, I assume that votes are cast to maximize

the following utility function.

U(voting for A)− U(voting for B) = vi − ηi

Where vi ∈ (0, 1) and each ηi is distributed Uniform[0, 1] and drawn independently across

agents. Voter i choses A if vi is greater than ηi which occurs with probability vi. Voter i

choses B with probability (1− vi). For technical reasons I assume everybody has a positive

7These are called Colonel Blotto games by the literature. See Roberson (2006) for a great reference.


probability of choosing both candidates, although the probability a voter swings his vote can

be arbitrarily small.

The stochastic element ηi represents uncertainty about the elements that determine a

vote. This shock need not be random from the point of view of the voter, it only matters

that it’s unknown by the persuaders. There can be all sorts of elements that make voters

have a change of heart when they make their decision. For example, a voter might decide to

change his vote because his happened to shake hands with a candidate at a rally; it has even

been reported that that bad weather can change the outcome of an election by influencing

turnout differently for Democrats than Republicans.Gomez et al. (2008) From the persuaders

perspective, these elements are hard to forecast or control.

To model persuasion and network influence I will assume that persuaders can spend

money to change opinions and that a voter’s opinion is influenced by the opinions of his

social neighbors on the network. I proceed to explain how this happens next.

1.2.3 The timing of the game

The game is divided in several stages which are qualitatively different. Inside these stages

are periods which repeat similar actions.

Let vti represent the opinion of voter i at period t.

• The initial stage: (Round 0) Voters begin with an opinion v0i .

• The persuasion stage: (Round 1) Persuaders simultaneously spend resources to

influence the decision of the voters. Section 1.2.4.

• The network stage: (Rounds 2 through T ) After persuaders spend all their budget,

voters update their opinion parameter through the social network. Section 1.2.5.

• Final stage. After updating their opinion several times through the network, the ηi’s

are realized. Voters pick A with probability vTi .

1.2.4 The persuasion stage

During the persuasion stage, persuaders simultaneously spend resources to influence opin-

ions. Every persuader has a fixed amount RA, RB of resources to spend. I solve the model

when persuaders have to raise resources at a cost in Section 1.5.

Let (ai, bi) be, respectively, the percentage of resources persuader A and persuader B

spend on voter i, so (aiRA, biRB) are the amounts in units of resources.


I assume persuaders can change opinions through the following contest success function.

v1i : R2

+ → [0, 1]

v1i (0, 0) = v0

i

v1i (aiRA, biRB) =

v0i (aiRA)γ

v0i (aiRA)γ + (1− v0

i )(biRB)γ; γ > 0

I chose this functional form for tractability. Below are it’s main characteristics.

• It takes values in [0, 1] and varies smoothly with the amount of resources each persuader

spends.

• A persuader can only completely convince a voter by spending and infinite amount of

resources. Opinion v1i tends to 1 as aiRA →∞ and v1

i tends to 0 as biRB →∞.

• If both persuaders spend the same amount of resources, aiRA = biRB, then the opinion

of voter i doesn’t change: v1i = v0

i . This is a symmetry assumption.

• If both persuaders scale the amount they are spending on voter i by any positive factor,

the opinion v1i is left unaffected.

v1i (λx, λy) = v1

i (x, y);∀λ > 0

This occurs because the contest-success function only depends on the ratio of resources

spent on each voter, (aiRA)/(biRB).

• The marginal persuadability decreases when A, B scale-up their resources.

∂v1i

∂x(λx, λy) <

∂v1i

∂x(x, y);∀λ > 1

This is crucial to get equilibria in pure-strategies.

Contest-success functions have been used in the economics literature to study strategic

spending in tournaments, arms races and competitions.8 Skaperdas provides axiomatizations

for this and other contest-success functions.9

8See Hirshleifer (1991); Skaperdas (1992); Siegel (2009, Forthcoming).9Skaperdas (1996)


Shubik & Weber use the contest-success function above to solve a smooth Colonel Blotto

game.10 Snyder uses a slightly more general function that is does not depend on the ratio

of resources but has all the other characteristics above. The results are qualitatively similar

but the ratio formula gives convenient analytical solutions that depend on percentage of

resources spent. My model is different in that it allows influence through the network. 11.

The parameter γ measures determines the impact of resources on opinions. For a large

γ, a small difference in the level of spending between A and B dramatically swings opinions

in one direction or the other. As γ tends to infinity, the game becomes a standard Colonel

Blotto game.

1.2.5 The network stage

After persuaders have spent all their budget, voters update their opinion by taking a weighted

average of the opinion their neighbors on a social network. The network is exogenous and

common-knowledge by the persuaders.

Every round of updating, the opinion vti evolves according to

vt+1i =

N�

j=1

Mijvtj; with

N�

j=1

Mij = 1 and Mi,j � 0

The parameter Mij represents the weight voter i puts on voter j’s opinion and Mii

represents how much voter i keeps his previous opinion. Each voter has a unit of attention

his divides between the opinions of his neighbors and his own. The weights are non-negative.

A network is a row-stochastic matrix M with non-negative entries whose rows sum-up

to 1. It summarizes all the structure of who listens to whom and how voters influence each

other. Voters can have asymmetric weights on each other’s opinion; Mij can be different

than Mji. It can even be that voter i influences j but the converse is not true. For example,

bloggers can influence the opinion of their readers without reciprocating by following their

readers’ twits.

The biggest challenge of models with network influence is keeping track of the evolution

of opinions when the structure of influence is complex. This is even more complicated for

strategic spending in majoritarian competitions because it’s hard to calculate how opinions

determine the probability of winning.

10Shubik and Weber (1981)11Snyder (1989)


By assuming the process above we can set-up the evolution of opinions as a linear-

transition system. That allows us to apply powerful tools from linear-algebra and markov-

chain theory to study the problem.

Let vt be the vector of opinions at time t. This vector evolves according to:

vt+1 = Mv

t = M tv

1

There are two ways of interpreting this influence. It can be interpreted as a model of

information processed in a bounded rational way or as a model of social preferences.

In a an informational world, there is a common-value v that captures the true difference

in value between A and B, but voters do not know it. Instead they have disaggregated

information, or opinions, and they try to update their assessment of v through the opin-

ion of their neighbors. As people update their opinion, the information gets disseminated

through society. Voters update their opinion many times using their neighbors’ opinions to

incorporate the new information that propagates through the network.

Updating opinions through a linear-process with the same weights is not the optimal

bayesian way of processing information. For one, linear-updating assumes that the marginal

influence of voter j’s opinion over voter i is independent of the value of v1j and of the

initial opinion v0j . In the actual political campaigns world, it’s perfectly plausible that

seeing a republican express support for a democratic candidate is more informative than

seeing a moderate democrat express a strong democratic opinion, although both carry some

information.

Linear-updating would be optimal in a world with a normal prior on the true v and with

signals vi that are normally distributed. But even in this world, voters should change the

Mij weights every period to adjust for redundancies in information.12

In spite of these drawbacks, the model can be justified because in general the optimal

bayesian can be quite cumbersome to solve. Therefore boundedly-rational agents might

restrain themselves using simple linear-updating rule. This interpretation was put forward

by DeGroot in the statistics literature and DeMarzo, Vayanos and Zwiebel in the economics

literature. 13

Furthermore, under appropriate conditions on the network structure, this myopic linear-

updating process still processes information in a reasonable way. The work in Golub and

Jackson (2010) shows that myopic linear-updating provides a consistent estimate of the true

v for a large network, as long as the influence of any individual and of any finite group of

12See DeMarzo et al. (2003); Golub and Jackson (2010); Acemoglu et al. (2010) for more on this issues.13DeGroot (1974); DeMarzo et al. (2003)


individuals is not bounded away from zero.

In the second interpretation of network influence, there is no true parameter v. Instead

voters have a preference for choosing A or B and vi measures the intensity of their preference.

Furthermore, voters have a social preference to vote like their neighbors.14 The problem is

that different voters want to imitate different people, so every voters have to continuously

update their preferences to match their neighbors. The linear-process assumes agents are

myopic in doing so.

For now I will assume that as the network stage advances, voters are willing to completely

change their initial preferences to imitate those around them, but in section 1.3.5 I will extend

the model so that voters chose based on two criteria: which candidate is better according to

an unchangeable ideological dimension; and according to which candidate is better along a

persuadable social dimension. The second dimension corresponds to vi.

1.2.6 Some network definitions

I will refer to the voters as the nodes of the network and will say there is a link from i to

j if Mij > 0. A network is directed if there can be a link from node i to node j without a

link from j to i. A directed network is path-connected if for every pair of nodes i, j there is

a directed path from i to j and a directed path back. That is, either i is connected to j or

there exists a series of nodes {k1, . . . , kn} such that {Mi,k1 , Mk1,k2 , . . . ,Mkn−1,kn , Mkn,j} > 0.

I will also need to assume that my network is aperiodic. Aperiodicity is a technical

condition that is verified if at least one voter places a positive weight on his previous opinion.

I will assume this throughout. See Jackson (2008a) for more details on the definitions.

For any network M that is path-connected and aperiodic, we can find it’s associated

DeGroot weights.

Definition 1 (The DeGroot Weights). Let M be a weighted-directed network which is

row-stochastic. Suppose the network is path-connected and aperiodic. Define the DeGroot

weights of network influence, or simply the DeGroot weights, as the unique left-

eigenvector of matrix M that corresponds to the eigenvalue 1 and whose entries have been

normalized to one. I denote it by s.

14You could also call it an altruistic preference in the sense that the utility of a voter is a weighted averageof his utility vi and the utility of his neighbors.


In math, s is the unique vector such that

sM = s with�

si = 1

This completes the set-up of the model. I now proceed to solve for the equilibria.

1.3 Solving for equilibria.

The following result by DeGroot (1974) is important to solve for equilibria. Under mild con-

ditions on the network, in the long-run all opinions converge to a consensus. This consensus

is a weighted-average of the initial opinion of every voter. The weight each voter receives is

given by the DeGroot weight of the voter.

Theorem 2 (DeGroot 1974). Let M be a path-connected, aperiodic network which is row

stochastic. For any initial vector of opinions v1 ∈ RN we have:

limt→∞

M tv = v∗

1...

1

Where v∗ is

v∗ =�

siv1i

The DeGroot weight of a voter only depends on his position in the network. It does not

depend on his actual opinion nor on the opinion of his neighbors. Therefore we can identify

the influential voters without having to solve for equilibrium spending.

I now present the main result of the paper. If there is a large-enough number of rounds of

network-updating, there is at most one unique pure-strategy equilibrium. In this equilibrium

A and B spend the same percentage any given voter i and spend in proportion to the DeGroot

weight of that voter.

This is stated formally in Proposition (3). Proposition (4) shows that the equilibrium

exists and is the unique as long as the opinion of voters is not too responsive to campaign

spending. Section 1.3.3 explains the example through a stylized example.


It is hard to believe that voters reach a consensus voting probability during political

campaigns. This is not essential to solve for equilibria, it’s just easier to solve the model. In

Section 1.3.4 and 1.3.5, I solve two extensions of the model without consensus voting.

In Sections 1.3.4, I assume the network is not path-connected but composed of disjoint

groups that are path-connected. The DeGroot consensus result holds inside each group, but

not for society as a whole. In Section 1.3.5, I allow individuals to have an unchangeable

“ideology”.

1.3.1 Solving the model with consensus.

Remark 1. Suppose T → ∞. A strategy profile (a,b) constitutes pure-strategy nash

equilibrium if and only if (a,b) solves

max(a1,...,an)

�siv

1i (aiRA, biRB)

s.t.�

ai = 1

and

min(b1,...,bn)

�siv

1i (aiRA, biRB)

s.t.�

bi = 1

To be precise, for a large enough T , the maximizer of the DeGroot consensus is an

epsilon-optimum of the probability A wins. This follows from the uniform convergence of

opinions to the DeGroot consensus and from the fact that the probability of winning is

uniformly continuous in (a1, . . . , aN). I will not dwell in this point. The details for proving

this are well-established but cumbersome. I will directly assume that persuaders maximize

a monotone transformation of the DeGroot weights.

Proposition 3 (On the structure of equilibria). Let M be a path-connected, aperiodic

network. Suppose T = ∞. Then the unique pure-strategy nash equilibrium in spending, if

there exists such an equilibrium, is:

(a∗i , b∗i ) =

siv1i (RA, RB)

�1− v1

i (RA, RB)�

�sjv1

j (RA, RB)�1− v1

j (RA, RB)�

Proof. This proof is an adaptation of the Shubik & Weber proof to my environment.

I first prove that a pure-strategy equilibrium must be in the interior by the contrapositive.

Suppose that bi = 0. A can spend an arbitrarily small quantity on i to obtain v1i = 1. Since


v1j (·, bj) is continuous for aj > 0, persuader A can increase v1

i by a discrete amount while

decreasing v1j for some j by an arbitrarily small amount. This increases the DeGroot consen-

sus by a discrete amount. Since this is true for an arbitrarily small change in expenditure,

persuader A has no best-response and the strategies cannot constitute an equilibrium.

Next I show that persuaders spend the same percentage on each voter. From the First

Order Conditions (FOCs) I obtain:

si∂v1

i

∂ai= sj

∂v1j

∂aj

si∂v1

i

∂bi= sj

∂v1j

∂bj

⇒ ∂v1i /∂ai

∂v1i /∂bi

=∂v1

j /∂aj

∂vji /∂bj

;∀i, j

By homogeneity of v1, I apply Euler’s law to get

ai∂v1

i

∂ai+ bi

∂v1i

∂bi= 0

−∂v1i /∂ai

∂v1i /∂bi

=bi

ai

From the FOCs we know that the left-hand side must be the constant across i. Therefore

ai/bi must be constant for all voters. This means both A and B must be spending the same

fraction of their resources on each voter: a∗i = b∗i .

We now know both persuaders spend the same percentage on a given voter, but we don’t

know what this percentage is. To find out I use the FOCs.

∂v1

∂a(a∗i RA, b∗i RB) =

∂v1i

∂a(b∗i RA, b∗i RB) =

1

b∗i

∂v1i

∂a(RA, RB) =

γ

b∗iv1

i (RA, RB)�1− v1

i (RA, RB)�

Where the second equality comes from the fact that the partial derivative of v1 is ho-

mogenous of degree -1. I now substitute this in the first order condition for A.

siγ

b∗iv1

i (RA, RB)�1− v1

i (RA, RB)�

= sjγ

b∗jv1

j (RA, RB)�1− v1

j (RA, RB)�


⇒b∗jb∗i

=sjv1

j (RA, RB)�1− v1

j (RA, RB)�

siv1i (RA, RB)

�1− v1

i (RA, RB)�

Since this is true for any two voters and�

ai =�

bi = 1, I conclude that

a∗i = b∗i =siv1

i (RA, RB)�1− v1

i (RA, RB)�

�sjv1

j (RA, RB)�1− v1

j (RA, RB)�

Proposition (3) does not prove a pure-strategy equilibrium exists. It only shows that con-

ditional on one existing, it must have the stated strategies. To complement this Proposition

(4) shows such an equilibrium exists and is the unique equilibrium of the game is voters are

not too responsive to persuasion. Specifically, if γ < 1 the persuader’s objective function is

strictly quasi-concave and FOCs are necessary and sufficient to find an a best-response.

Uniqueness follow because equilibria in zero-sum games are interchangeable. Since the

best-responses are unique, there can be no other strategy profile that constitutes an equilib-

rium.

If γ > 1 the stated strategies might still be an equilibrium, but there might be other

equilibria in mixed strategies. In those situations all equilibria would be payoff equivalent

because this is a zero-sum game.15 We also know that as γ → ∞, the stated strategies

cannot be an equilibrium, because the game approaches a standard Colonel Blotto game

that has no pure strategy equilibria. (And the equilibrium correspondence as γ → ∞ is

upper-hemicontinuous).

Proposition 4 (Existence and uniqueness of an equilibrium). Take the same assumptions

as in Proposition (3). If γ < 1, the stated strategies are the unique equilibrium of the game.

Proof. Because the probability of winning is a monotone transformation of v∗, they share

the same maximizers. Taking the derivative of v∗ I get

∂2v∗

∂2a= si

�γ

a

�2v1

i (a, b∗)�1− v1

i (a, b∗)��

1− 2v1i (a, b∗)− 1

γ

�

Which is strictly negative whenever γ < 1. Therefore a∗ is the unique best-response to

b∗. Mutato mutandis, b∗ is the unique best-response to a∗. This proves existence.

Uniqueness follows because equilibria for zero-sum games are interchangeable. Take any

equilibrium of the game: (σa, σb). (These are potentially mixed-strategies.) It must be

15See the minimax theorem in Mas-Colell et al. (1995).


that (σa, b∗) and (a∗, σb) are also equilibria. Since a∗ is the unique best-response to b∗, and

viceversa, I conclude that (σa, σb) = (a∗, b∗).

1.3.2 Discussion: Where are the pivotal voters?

Previous models of strategic spending had found that persuaders should target their resources

on pivotal voters. 16 A voter is pivotal for the election if, conditional on the votes of the

others, his choice changes the outcome. Let N be the number of votes A needs to win. The

the probability voter i is pivotal under v1(RA, RB), represented by qi, is

qi =�

S⊂N\{i}|S|=N−1

�

j∈S

v1j (RA, RB)

�

j� /∈Sj� �=i

�1− v1

j�(RA, RB)�

Pivotal voters are important because persuaders only care about winning, which means

pivotal voters have the highest expected marginal benefit. Spending money to change a vote

that is not pivotal is a waste of resources. This reasoning carries through to elections with

networks, yet pivotal voters play no role in equilibrium spending. Why is this so?

The shift from pivotal to influential voters happens because the network prevents re-

sources from being targeted. When T = ∞ persuaders cannot change the opinion of a voter

in isolation, their opinions mix with the opinion of their neighbors. Persuaders respond by

working to convince society as a group by focusing on the DeGroot consensus. The most

effective way to change this is to target the influential voters.

Even though society only reaches a consensus in the limit, the network blunts targeting

starting on the the first round of updating. A similar shift in resources must occur even

without a consensus. For example, in Section 1.3.5 voters have an ideology parameter that

prevents a consensus in the voting probabilities, but persuaders still target the influential

voters instead of the pivotal voters.

It’s likely that the equilibrium spending with a finite T would involve some combination

pivot probabilities and network influence. Unfortunately, solving for this can be quite cum-

bersome. Section 1.3.4 shows one way of approaching the problem by assuming that society

is composed of disconnected groups. As T tends to ∞, each group reaches a consensus, but

disagreements can persist across groups. Persuaders respond by spending more on pivotal

groups but targeting the influential voters inside the group.

Is there a systematic relationship between pivotal and influential voters? Theoretically

no, these two concepts are orthogonal. One can always construct a network where pivotal

16See Shubik and Weber (1981); Snyder (1989).


voters are the same as influential voters, or construct a network where influential voters are

completely different than pivotal voters. This is a consequence of linear-updating. Under

linear-updating influence is independent of opinions, but the probability of being pivotal

crucially depends on them. This is shown in the following example.

1.3.3 A Parent-Child example.

Two voters, a parent and a child, have to decide between two (almost) identical products: A

and B. The main difference is that product A is sponsored by a popular cartoon character.

The child is very much convinced that A is better than B so v0child = p ≈ 1. The parent is

of the opposite state of mind. For symmetry, assume v0parent = 1− p.

To decide which product they want, the parent and the child are going to take a vote.

Product B is the status quo object, both the parent and the child have to vote for A to buy

it. Suppose the persuaders, firms A and B, have the same amount of resources to spend on

advertising.

Because of the unanimity rule, a voter is pivotal only if the other voter choses for A. The

parent will be pivotal with probability p and the child with probability 1−p. It’s much more

likely that the parent’s vote will be decisive for the election. Both firms react rationally by

heavily targeting the parent. In equilibrium, both firms spend a fraction p of their budget

on persuading the parent and a fraction 1− p on persuading the child.

Suppose instead that before taking the decision the parent and the child will deliberate

about the decision. The parent feels it’s important to give an equal weight in the decision

to his child’s opinion. The child, being a childish, pays very little attention to the parent.

She places ξ/2 ≈ 0 weight on the parent’s opinion and 1− ξ/2 on her own.


The matrix representation of the network is

M =

�Mparent,parent Mparent,child

Mchild,parent Mchild,child

�=

�1/2 1/2

ξ/2 1− ξ/2

�

The corresponding DeGroot weights are

s =

�sparent

schild

�=

�ξ

1+ξ1

1+ξ

�

Given this, if the parent and the child talk for long enough, the opinion of the child will

almost completely prevail. Knowing this, the firms would spend a large fraction of their

resources on the child, ξ/(1+ξ).

Which is the right model? Different products might have different levels of communica-

tion. The parent might not be willing to discuss with the child what is the right type clothes

for playing in the snow. On the other hand, the car drive from San Francisco to LA will give

the child ample time to convince the parent they should go to Disneyland instead of the LA

Museum of Contemporary Art.

1.3.4 Solving the model without consensus: targeting disconnected

groups.

The previous analysis focused on networks that were path-connected. This section will

extending the analysis to disconnected groups to shows how pivot probabilities and network

influence interact with each other.

Assume the voters can be partitioned into m disjoint groups such that each group is

path-connected and aperiodic. Label them {I1, . . . , Im, . . . , Im}.Theorem (2) implies each group will reach a “consensus” the long-run, but different

groups might end up with different opinions. The DeGroot weights can be constructed for

each group. Let s be the eigenvector of stacked DeGroot weights for each group. (The entries

of the vector belonging to the same group must sum-up to 1.)

Let Nm be the group size of Im and let qm be the average pivot probability in Im.

qm =1

Nm

�

i∈Im

qi


The unique pure-strategy equilibrium of the game is

a∗i = b∗i =Nmqmsiv1

i (RA, RB)�1− v1

i (RA, RB)�

�m� Nm� qm�

�j∈Im� sjv1

j (RA, RB)�1− v1

j (RA, RB)� ; i ∈ Im

From this I can rewrite the relative spending across voters and across groups as

a∗ia∗j

=siv1

i (RA, RB)�1− v1

i (RA, RB)�

sjv1j (RA, RB)

�1− v1

j (RA, RB)� ; if i, j ∈ Im.

�i∈Im

a∗i�j∈Im� a

∗j

=qmNm

�i∈Im

siv1i (RA, RB)

�1− v1

i (RA, RB)�

qm�Nm��

j∈Im� sjv1j (RA, RB)

�1− v1

j (RA, RB)�

Spending across groups is proportional to the average pivot probability, the size of the

group and a network-average of the marginal persuadability. Spending across voters inside

each group is proportional to the DeGroot weights inside the group.

Even if the real network is path-connected, reaching a consensus could take an arbitrary

long-period of time. This is especially true of a network with high homophily. Homophily

refers to the tendency of individuals to interact with individuals who are similar to them.17

The most persistent disagreements in societies with homophily are disagreements across

different groups. Homophily decreases the speed of convergence of opinions across groups,

but increases the convergence within groups, as shown in Golub and Jackson (2009). For

these societies the disconnected network might be a better approximation to model campaign

spending.

1.3.5 Solving the model without consensus: adding ideology to

the model.

To show that convergence in voting probabilities is not crucial for network effects to dominate

pivot probabilities, I will add ideology to the model. Let θi ∈ [0, 1] be an ideology parameter

for voter i. The closer θi is to 1 the more inclined the voter is to support A. These parameters

are common-knowledge between A and B.

17See McPherson et al. (2001) for many examples of groups that exhibit homophily.


Voters maximize the following utility function.

U(voting for A) = u(vi; θi)− ηi

U(voting for B) = 0

Where u : [0, 1]2 → (0, 1) is continuous function and strictly increasing in vi and in θi.

With this parametrization the probability that voter i choses A is u(vi, θi).

For example, u could be

u(vi; θi) =vi + θi

2

Everything else remains as before. Persuaders can spend to change vi through the contest

success function and voters update vi through the network. The ideology is fixed cannot be

changed by spending nor by the opinions of other people.

In the information interpretation of the model, the parameter θi represents preferences,

while vi represents information. The underlying assumption is that when voters interact

through the network they are able to separate information from ideologies.

In the preference interpretation of the model, the θi represent the private aspects of choice,

those that cannot be influenced by other people, while vi represents the social dimensions of

choice, those aspects that voters wish to match with their social neighbors.

In a pure-strategy equilibria with T = ∞, equilibria of this model are identical to a

model without ideology. Since persuaders cannot change ideologies they focus instead on

influencing the consensus. This is spelled out in detail in Remark 2.

Remark 2. In a pure-strategy equilibrium with T → ∞, persuader A maximizes

v∗ while persauder B minimizes v∗. To see this note that if all probabilities are between

zero and one, the probability persuader A will win the election is a strictly increasing in vi

for any i. Now let v1, v2 be two possible values for v∗ such that v1 > v2. Since the probability

voter i choses A is strictly-increasing in vi, the distribution of votes under a consensus of v1

first-order stochastically dominates the distribution under v2. Therefore the probability A

wins is just a monotone transformation of v∗.18

Therefore the unique pure-strategy equilibrium is as stated in Proposition 3.

18This argument is only true for pure-strategy equilibria.


1.4 Competitions in proportional representation sys-

tems

In this section I solve for equilibria when persuaders want to maximize the share of voters

who select them. This model can also be applied to electoral systems with proportional

representation, where parties get seats in parliament in proportion to the share of votes they

get in the election.

The main result is qualitatively the same as before: persuaders spend over voters in

proportion to another eigenvector based measure of network influence: Bonacich influence.19

Pivotal voters do not matter in proportional representation systems because there is no

threshold to win the election.20

Solving for equilibria in proportional competitions requires less assumptions. For ma-

joritarian competitions, I could solve for equilibria with T = ∞. Proportional competitions

allows me to solve for competitions with a finite T . Instead I will assume T is random

and follows a geometric distribution. This allows me to relate my result with the Bonacich

measure, an important influence measure in the sociology literature.21

The random variable T follows a geometric distribution if the probability the game moving

to t + 1 conditional on reaching round t is constant for all t.

Prob(T � t + 1|T � t) = δ; for δ ∈ (0, 1)

When voters stop deliberating they vote for A with probability vTi .

Conditional on B’s strategy, persuader A solves

max(a1,...,aN )

ET

� �vT

i

�= max

(a1,...,aN )(1− δ)

∞�

t=1

δt−1�

i

vti

As δ → 1 the game approaches the game with T = ∞, so this game is more general.

I can get a stronger result because in proportional competition the objective function of

the persuaders is linear, while in majoritarian competition it was highly non-linear near the

threshold of votes required to win. The non-linearity doesn’t matter in the limit, but it’s

hard to analyze for any finite time-horizon.

19This measure is known as Bonacich centrality in the sociology literature, but to be consistent with myapplication I call it influence.

20Indeed, there is no such thing as a pivotal voter.21See Ballester et al. (2006); Bramoulle et al. (2009) for the relationship between Bonacich influence and

Nash equilibria in games with linear-best responses.


Definition 5. Fix δ ∈ (0, 1). The vector s of Bonacich influence weights for a matrix

M is

s = (1− δ)(1/N, . . . , 1/N)[I − δM ]−1

Proposition 6. Suppose each persuader wants to maximize the percentage of voters that

selects him. Then the unique pure-strategy nash equilibrium, if it exists, is:

a∗i = b∗i =siv1

i (RA, RB)�1− v1

i (RA, RB)�

�sjv1

j (RA, RB)(1− v1j (RA, RB))

If γ < 1, this is the unique equilibrium of the game.

Proof. Take (a,b) ∈ (0, 1)n. I simply show that the objective function of each persuader is

equal to�

siv1i . The rest of the proof follows the logic in Proposition (3) and Propostion

(4) . Setting-up the persuader A’s maximization problem we have

maxa1,...,aN

(1− δ)∞�

t=1

δt−1�

i

vti ∼ max

a1,...,aN

(1− δ)(1, . . . , 1)∞�

t=1

δt−1M t−1v

1

= maxa1,...,aN

(1− δ)(1, . . . , 1)[I − δM ]−1v

1

∼ maxa1,...,aN

s · v1

1.5 Competition with fundraising

Until now I have assumed that the amount of resources was exogenous. In this section I

analyze the possibility that persuaders have to raise resources at a cost.

I find that in equilibrium the ratio of resources raised is independent of the network

influence, of the specific campaign rules and of the initial distribution of opinions. The ratio

of resources only depends on the relative costs each persuader has for raising resources. The

absolute level of resources raised does depend on the rules and the distribution of opinions,

but in ways that are hard to characterize.

I assume each persuader has to pay an cost cj(Rj)k to raise resources with k > 1. The

parameter cj determines the marginal cost of raising resources.

In the first stage of the game, persuaders simultaneously collect resources and the amounts

they raise become common-knowledge. In the second stage, persuaders decide where spend


it. By backward induction the spending patterns in the second stage have to be the same as

in Proposition (3).

The second stage pay-offs only depend on the ratio of resources collected. Let r = RA/RB

be such ratio and let π(r) be the second-stage pay off for persuader A. I can write each

persuader’s maximization problem as one that only depends on r.

maxRA

π(r)− cARkA = max

rπ(r)− cA(rRB)k

maxRB

�1− π(r)

�− cBRk

B = maxr

(1− π(r))− cB(RA/r)k

To solve for the equilibrium r∗ I again appeal to the ‘hidden symmetry’ of the game. The

FOCs for the problem are

dπ

dr− kcArk−1Rk

B = 0

−dπ

dr+ kcBr−k−1Rk

A = 0

Solving this yields a solution that is independent of π.

r∗ =�cB

cA

�1/k

If cA = cB both persuaders will raise the same amount of resources and their probability

of winning will not change from that determined by the initial opinion of voters plus the

network updating.

Since marginal benefit only depends on r∗ we can find the absolute level of resources by

equate the marginal cost to the marginal benefit in the FOCs above. From this I can derive

two easy comparative statics.

• Constants everything else, if voters are less persuadable, γ decreases, the total amount

of resources raised by each persuader decreases.

• Suppose the marginal cost parameters increase proportionally. That is, (cA, cB) changes

to (λcA, λcB) with λ > 1. Then the total amount of resources raised by each persuader

decreases.

For majoritarian elections the marginal benefit of resources increases with the probability

the election will be decided by a pivotal vote. Persuaders spend more money on elections

that are likely to be close.


The network has an ambiguous effect on campaign spending because it can make the

election more or less close. For example, if everybody is very likely to choose for A except

for one very influential voter, the competition with the network will be more close than

without it. On the contrary, one very influential voter can tilt a large number of undecided

voters, making the competition less close.

1.6 Testing the model with data on lobbying.

My model puts stringent conditions on the behavior of voters and requires a large number

of rounds of network-updating to solve for equilibria. To show if my model is still a useful

approximation to the real world I will test it using data on lobbying expenditures in the U.S.

Congress. My main aim is to test if lobbyists spend more on legislators who have a larger

network influence.

The estimation proceeds in three steps. First I measure the bilateral influence between

each pair of legislators: the weights of the links. Next I calculate the global influence of

each legislator by calculating his DeGroot weights. Finally I regress campaign contributions

on network influence and on a measure of the pivotality of each legislator to see which is

a better predictor. In my data I will observe the same legislator several times, so I will be

able to use the time-variation of campaign contributions and network influence to control

for confounds.

To build the network I use data on the cosponsorship structure of bills in the U.S.

Congress. Every time a bill is proposed in Congress it must have a sponsoring legislator.

Other legislators can sign up as cosponsors of the bill. I will use the frequency of cosponsor-

ship as a proxy for bilateral influence. Every time legislator j cosponsors one of legislator’s

i’s bills, I interpret that i has some influence over j. This data is very convenient because

there is a direction of influence (from cosponsor to sponsor) and because legislators cospon-

sor together many times in the same congress and across different congresses. This allows

me to build a weight for each link.

The cosponsorship data I use has all the bills, resolutions and amendments between 1972

and 2006, from the 93rd to the 109th Congress. The data was collected from the library of

Congress by Fowler (2006a,b).

To construct the network for electoral year t, I take each pair of legislators i, j who

served in year t and construct the entry Mji by counting the number of times that legislator

j cosponsored a bill sponsored by i in any congress where they both served together before

year t. I do the corresponding thing to measure Mji.


After I counted all the cosponsorships, I divide the cosponsorships from j to i by the

total number of times j cosponsored with anybody else. Therefore the influence from i to

j is measured by the frequency with which j cosponsors i’s bills relative to how often j

cosponsors with anybody else. This makes the network matrix row-stochastic.

Links in the network accumulate over time for legislators that remain in Congress. Since

this strongly biases the DeGroot weights in favor of more senior legislators, I control for

seniority in the regression.

A problem with the data is that some bills are cosponsored by a almost everybody in

congress. This probably has more to do with the content of those bill rather than the influence

of the sponsoring legislator. The distribution of cosponsors per bill decreases exponentially

but spikes up when the number of cosponsors approaches the half the chamber. These peaks

hint that these cosponsored bills involve position signaling by the majority party instead of

influence by the sponsoring legislator.

To deal with this I do two things: I drop the bills that have more legislators than the

threshold where the distribution peaks up (215 legislators for the House and 49 for the

Senate). I then weigh down the links between cosponsors and sponsors by the number of

cosponsors in each bill. If j cosponsored i’s bill along with 9 other legislators, I assign a

weight of 1/10 from j to i. Running the regression without these adjustments yields similar

coefficients but higher standard errors.

The model assumes legislators also place weight on their own opinion, but I do not observe

self-links in the data. To identify the DeGroot weights I assume that all individuals put the

same weight on their own opinion. As long as this weight is positive, the DeGroot weights

will be the same regardless of what value one choses.22

To measure lobbying expenditures I use the campaign contributions by Political Action

Committees (PACs) using data from the Federal Elections Committee from 1990 to 2006.

The data is made available by the Center for Responsive Politics.23

It’s generally considered that PACs donate to get access to legislators and to influence

their vote. But PACs also donate to help elect legislators who are affine to their positions.

Therefore these expenditure do not correspond exactly to the lobbying expenditures in my

model. But this is just an extraneous source of variation in my dependent variable. It

should not bias my estimate for network influence, but will increase the standard errors in

my regression. As such it only makes it harder to test if network influence significantly

22To see this let α > 0 be the weight each legislator puts on himself and M be the network matrix whosemain diagonal is zero and whose rows sum to one. The true network would be αI +(1−α)M but the largestleft-eigenvector of αI + (1− α)M is also the largest left-eigenvector of M .

23http://www.opensecrets.org


predicts campaigns contributions.

PAC contributions are not the only way interest groups spend resources on legislators.

Interest groups can also hire full time lobbyists. The predictions of the model refer to the

total resources lobbies spend on a legislator, both through PAC contribution and through

lobbyists. As long as these two expenditures are positively correlated, PAC contributions

works as a proxy for lobbying spending.

In each electoral year, many bills are presented and many different lobbies compete over

separate issues. I interpret each bill as an independent realization of my model, with an

interest group on each side of the issue that spends according to network influence or pivot

probabilities. Even if there are many lobbies influencing a single bill, once the content of the

bill is fixed there are only two sides to the issue: those who are in favor and and those who

are not. Groups of lobbies spending in a coordinated matter should spend as in my model. If

on each bill lobbies spend as my model predicts, the total contributions a legislator receives

in a year should also be proportional to the influence of each legislator.

Do lobbies target the same legislators? Unfortunately I cannot test this directly, because

I cannot match PAC contributions to specific bills.

Are lobbies strategic in their campaign contributions? Do they target legislators with

opposing views? There is evidence that some PACs are very strategic in their contributions.

In the 2006 electoral cycle, the top contributing PAC was the National Association of Realtors

which gave to 49% to Democratic candidates and 51% to Republican.

A strand of papers on counteractive lobbying have found that for the number of affine

lobby groups engaged in persuading a particular legislator is positively correlated with the

number of rival lobby groups who try to persuade him.24

At same time many interests groups that have a PAC do not have a lobbyist and the

majority of groups that have a lobbyist do not have a PAC. Nevertheless, as reported in

Tripathi et al. (2002), the groups that have both account for a large share of the total

contributions. “[Although] groups that have both a lobbyist and a PAC account for only

one-fifth of all groups in our sample, these groups account for fully 70% of all interest group

expenditures and 86% of all PAC contributions.”

24Austen-Smith and Wright (1996, 1994)


1.6.1 The House Specification

To test out hypothesis I run two specifications, ordinary least-squares (OLS) and OLS with

fixed-effects. The specifications for the House are

Contributionsi,t+1 = α + β1DeGrooti,t + β2RelativeP ivoti,t + βXXi,t + ei,t (OLS)

Contributionsi,t+1 = αi + β1DeGrooti,t + β2RelativeP ivoti,t + βXXi,t + ei,t (OLS FE)

I include the relative probability that a legislator is pivotal to compare my theory

with the predictions of the Shubik and Weber model.

To calculate the average time a legislator was pivotal probabilities I had to simulate the

voting on the bills because empirically bills are almost never passed by a single vote, so

legislators it’s hard to estimate average pivot probabilities from the data. I used Poole and

Rosenthal’s DW-Nominate scores to predict the probability each legislator would vote in

favor or against a bills presented in the last congress.25 I then simulated a vote on each bill

many times using independent draws for each legislator. After running the votes for tens of

thousands of times I can estimate how often a legislator would have been pivotal for a given

bill. I then average across all bills to measure his average pivot probability.

Consistent with my model I normalize the pivot probabilities to sum to one, because only

the relative probabilities matter when deciding where to spend money.

Theoretically, pivot probabilities and DeGroot weights are completely unrelated. Legis-

lators can be influential while being firmly grounded on one side of an issue. Empirically,

the pivotality measure is not correlated with DeGroot weights. The correlation is −0.08 for

the House and 0.05 for the Senate.

The matrix the Xi,t is a group of controls that includes the following variables:

1. Seniority and seniority squared. Measured from the first time a legislator entered the

House.26 It’s particularly important to control for seniority because the measure of

network influence accumulates over time, so network influence is strongly correlated

with seniority.

2. Leadership dummies: I include dummies for the House Speaker, the Majority and

the Minority leaders and whips, as well as for the chairmen of two influential commit-

tees: “Ways and Means” and Appropriations.

25Data available at www.voteview.com.26This is almost identical as a number of years a legislator has served. In general, legislators leave Congress

only once.


3. Party: When PACs spend to get legislators re-elected, Republicans and Democrats

receive funding from different sources. I add a dummy to reduce the noise in the data.

4. Majority dummy: Previous work by Cox and Magar had found that being in the

majority party is a significant predictor of campaign contributions. I include it to

reduce noise, but’s it’s not significantly correlated with the variables of interest. See

Cox and Magar (1999)

5. Congress year dummies. My theory of lobbying spending is a theory on the relative

contributions each legislator receives with respect to the other legislators. In my model

network influence does not predict the total amount lobbies would spend. In the data

I observe a lot of year to year variation in total contributions. The average year-to-

year standard deviation per legislator is 35 per cent of the average contributions per

legislator. These could be driven by the economic activity or how much lobbies care

about the issues presented in a given year. Adding these dummies helps control the

year to year variations.

The OLS estimates for network influence might suffer a serious endogeneity problem

because each legislator might has an unobserved ability to raise resources that might be

correlated with his ability to get cosponsors. As such, in the DeGroot weights in the OLS

might be a proxy for persuasiveness instead of measuring the effect of a legislator’s position

in the network.

To control for this I will focus on the second specification, an OLS regression with leg-

islator fixed-effects. This specification exploits the multiple observations I get for each leg-

islator. Assuming a legislator’s intrinsic persuasiveness is constant from one electoral year

to the next, the fixed-effects regression will measure how changes in a legislators’ DeGroot

weight predicts changes in his contributions.27

The DeGroot weights of a legislator change over time for two reasons: legislators spon-

sor and cosponsor new bills, and the previous cosponsors of a given legislator might leave

congress. Since legislators get a higher DeGroot weight if their cosponsors have a high De-

Groot weight, having an influential cosponsor leave can significantly diminish the influence

of a legislator. For example, during the 1994 Republican take-over of Congress, the Demo-

crat Representative Dick Gephard lost 28% of his DeGroot influence falling 16 places in

the ranking of legislators by DeGroot influence. (He also stopped being the Speaker of the

House).

27The regression also includes seniority and seniority squared which allows for quadratic time trends.


Pivot probabilities also change from year to year because the composition of the cham-

ber changes. Before the Republican takeover of 1994, the sum of pivot probabilities for

Democrats and Republicans were respectively 0.83 and 0.56, while afterwards they became:

0.54 and 0.70. After the change in membership, it became much more likely that a Repub-

lican legislator would be pivotal for the issues presented.

Table 1.1 presents the results for the House. After controlling for the other potential con-

founds, the DeGroot weights are a statistically significant predictor of changes in campaign

contributions (p = 0.01), while the probability of being pivotal is not (p = .69). Since the

DeGroot units are hard to interpret, I also calculate the effect semi-elasticity at the mean

of increasing a variable one standard deviation units (column 4). My specification predicts

that an increase in the DeGroot weight of one standard-deviation in time is associated with

an increase of 44, 834 dollars, or 10.5% of the average campaign contributions in my sample.

In terms of it’s standard-deviation, the DeGroot are one of the largest largest predictor

in the variation in campaign contributions. The largest effect is seniority which has a large-

nonlinear effect through the quadratic term. The variation in DeGroot weights explain

a similar amount of variation than the leadership variables. Becoming a House Speaker

increases campaign contributions by 1,097,000 dollars (258% of the average contributions)

but only predicts an increase of 12.5% per unit of standard deviation. The effect of becoming

the Speaker is huge, but it happens to very few legislators.

Using the structure of relationships in Congress provides a new theory of influence in

Congress where a legislator’s depends on the relationships they develop. This is not to say

that the formal positions of power don’t matter. As seen in the regression, the effect of

getting one of the top leadership positions can be huge. The DeGroot weights complements

the previous work on formal institutions by providing a way to measure informal influence.

Do we really need a complicated measure like the DeGroot weights to measure influence?

Could we instead simply measure the number of cosponsors legislator has? This is called

the degree of a legislator in network terms. In a sense the degree of a legislator measures

the number of people he can directly influence. The DeGroot weights incorporate all direct

and indirect influences of the links: i.e. the number of cosponsors, the number of cosponsors

of the cosponsors’ has, the number of cosponsors of the cosponsors of the cosponsors, etc.

In the limit this can be summarized by a recursive formula: a legislator will have a large

DeGroot weight if his cosponsors have a high DeGroot weight. DeGroot weights count the

quality rather than the quantity of cosponsors.

DeGroot weights are strongly correlated with degrees (ρ = 0.66 for the House and ρ =


0.41 for the Senate), but we can test which one is a better predictor by running the fixed-

effects regression with both variables. Doing this yields a significant coefficient for the

DeGroot weight, but a non-significant coefficient for the degree. (See Table 1.3) The DeGroot

measure is a better predictor of changes in campaign contributions.

Could it be that DeGroot weights are just a proxy for productivity? Legislators who

sponsor more bills are likely to get more cosponsors and a higher DeGroot influence. Table 1.3

shows shows the regression controlling by the number of bills a legislator sponsors (columns

5 and 6). The point estimate is similar to the original regressions (column 1 and 2), but the

standard errors increase. The new p-value for the DeGroot weight is p = 0.052 for the OLS

with fixed-effects.

To control for legislator’s ideological moderation, Columns 7 and 8 of Table 1.3 includes

the distance from the DW-Nominate ideological parameter to the median opinion in the

respective Congress. This new variable is not a significant predictor and does not change

the significance of the relevant coefficients.

1.6.2 The Senate specification

Testing the model with data from the U.S. the Senate is much more difficult. There are

much less legislators, 101 versus 450 in the House, and they are elected for 6 years, so they

run for office much less often than the Representatives. Since I only have 18 years of data,

the best I can hope for is to observe a Senator running 3 times, and that only if he happened

to serve the full 18 years in my sample. Serving 18 years is not a small amount of time in

the career of a politician.

To further complicate matters, Senators are elected in a staggered fashion, so every

electoral year I only observe a third of the chamber getting reelected. This means I cannot

observe the full initial cycle of contributions for two-thirds of the legislators who start in

my panel. To avoid dropping too many observations I focus on the contributions a Senator

receives on the biennial he runs for reelection and ignore the contributions they receive

during their first four years. This can be justified because Senators receive most of their

contributions during this period, about 70% during the last two years compared to roughly

15% for each of the first two biennials in their cycle.

One important difference of the Senate with respect to the House is it’s use of the filibuster

rule to close the debate on a bill. This effectively requires a super-majority to pass important

legislation. I expand my specification from the House to incorporate the relative probability

a Senators is pivotal to break the filibuster rule.


The new specifications are

Contributionsi,t+1 = α + β1DeGrooti,t + β2RelativeF ilibusteri,t

+ β3RelativeP ivoti,t + βXXi,t + ei,t (OLS)

Contributionsi,t+1 = αi + β1DeGrooti,t + β2RelativeF ilibusteri,t

+ β3RelativeP ivoti,t + βXXi,t + ei,t (OLS FE)

The results are presented in Table 1.2.

The DeGroot weights are not a significant predictor of changes in campaign contributions,

although the point estimate is similar to the one in the House. Increasing the DeGroot weight

of Senator by one standard deviation is associated with an increase of 15.9% of the average

campaign contributions. The standard errors are large enough (σ = 10.9) that we cannot

reject that the coefficient for the House is the same than for the Senate.

On the other hand, the filibuster pivot measure is a significant predictor of changes in

campaign contributions, supporting the prediction of pivot probability models. My regression

predicts that if we observe an increase of one deviation the probability a Senator is pivotal we

should observe an increase in 208,364 dollars, of 16.9% of the average campaign contributions

for the Senate. The filibuster pivot probability has one of the largest coefficients between

significant variables, with a larger coefficient than most of the leadership variables.

1.6.3 Separating the Degroot weights into direct and indirect ef-

fects.

One important implication of the DeGroot model is that the influence of a legislator depends

not only on the number of his cosponsors, but on the number of cosponsors of his cosponsors,

the number of cosponsors of the cosponsors of the cosponsors, etc. In the data, I can test

if these higher order effects really matter in explaining the pattern of contributions. I do

by decomposing the DeGroot weights into direct and indirect effects and then running the

regression on both.

I do this in two different ways. Table 1.5 shows the results. In the first regression, the

influence of legislator i is the total number of legislators who have cosponsored at least one

bill with i in any legislature, divided by the total number of cosponsors for all legislators

in the chamber.28 This create an influence index which can be compared to the DeGroot

weights. To get the indirect effect I subtract the direct effect from the DeGroot weights.

28This only uses the cosponsors of bills that had less than half the chamber.


The second regression measures the direct effect by weighing each cosponsor with the

frequency of cosponsorship. The weights assigned to every cosponsor are the same weights

used in calculating the DeGroot measure.29 The influence of legislator i is the weighted

sum of his cosponsors divided by the weighted sum of the cosponsors of everybody in the

chamber. The indirect effect is again the DeGroot measure minus the direct effect.

As can be seen in Table 1.5, the results are mixed. Measuring the direct effect with the

number of cosponsors yields a significant coefficient for both the direct and indirect effect.

This means that adding the indirect effect improves the prediction. Furthermore, both the

direct and indirect coefficients are of similar magnitude. Using the second decomposition

yields an indirect effect that is not significant. This could be because the approach is so

similar to the DeGroot weights that the residual is quite small.

Which one is the correct decomposition? Using the weights is justified in the context

of the DeGroot model, but it’s an ad hoc approach otherwise. Therefore I believe that the

first regression is the right benchmark for a theory of direct influence in the absence of the

DeGroot model. Nevertheless, since this regression provides a stronger support to my model

I decided to include both regressions in the spirit of full disclosure. The reader can make up

his own mind.

1.7 Conclusion

I proposed a model of strategic persuasion over social networks. This is one of the first models

to address the role of network influence under competitions to persuade public opinion. The

model is tractable and allows me to solve for the equilibrium spending across voters in

arbitrarily large networks with few restrictions on the pattern of influence.

In equilibrium, the expenditure on a voter is proportional to his network influence. The

result contrasts with previous findings on strategic spending for majoritarian competitions,

which found that in equilibrium, spending targets voters who are more likely to be pivotal

for the outcome of an election.

Network influence displaces pivot probabilities because the network spillovers preclude

targeting. When opinions are frequently updated through a social network, it’s impossible

to change the opinion of a single voter because the network moves all opinions toward a

consensus. Therefore persuaders decide instead to spend on voter with influential positions

on the network.29The weight of cosponsor j for i is the number of times j cosponsored i’s bills divided by the number of

times j cosponsored with anybody else.


When the network is not path-connected, disconnected groups the opinion only converge

within each group. In this case persuaders spend more groups that are more likely to be

pivotal for the election, but spend on the influential members inside of each group.

The model predicts that the relevant measure of network influence is an eigenvector

measures of influence, the DeGroot weights. Eigenvector measures of influence are self-

referential: individuals are influential if influential individuals listen to them. This measure

highlights the quality rather than the quantity of connections.

To test my model I put together data on lobbying expenditures by Political Action

Committees with data on cosponsorship networks in the US Congress for the electoral cycles

from 1990 to 2006. After controlling for several confounding variables, I find that network

influence is a significant predictor of campaign contributions for House of Representatives.

An increase of network influence by one standard deviation from one electoral year to

the next predicts an increase of 44,834 dollars (p = 0.01) in the campaign contributions of a

Representative. This amount corresponds to 10.5% of the average campaign contributions.

Pivot probabilities are not a significant predictor of campaign contributions.

Network influence is not a statistically significant predictor of campaign contributions

in the Senate, although the point estimate is similar: contributions increase 15.9% of the

average per each standard deviation of network influence. The higher standard errors could

be due to the low number of observations to small number of observations.


Tables

Summary Statistics for the House of RepresentativesVariable Mean Std. Dev. Min MaxCampaign contributions* 427 352 -6 4,480DeGroot Weights 0.23 0.24 0.00 1.79Relative Pivot Probability 0.23 .03 0.01 .27Seniority 10.8 8 2 52Bills Sponsored 17.6 14 0 158Number of Cosponsors 205 103 0 429

*In thousands of 2006 dollars

Most Influential Representatives Most Pivotal RepresentativesDeGroot Relative

Representative Con- weights Representative Con- pivot probgress (sum=100) gress (sum=100)

1 Solomon Gerald 105 1.792 Skeen Joe 102 0.27222 Stark F. Pete 103 1.734 McHenry Patrick 109 0.27203 Gilman Benjamin 106 1.693 Taylor Gene 106 0.27194 Stark F. Pete 104 1.657 LaHood Ray 105 0.27185 Gilman Benjamin 105 1.583 Barcia James 104 0.27166 Crane Philip M. 105 1.556 Davis Danny 105 0.27147 Stark F. Pete 105 1.550 Blunt Roy 105 0.27138 Crane Philip M. 106 1.450 Foxx Virginia 109 0.27139 Gilman Benjamin 107 1.398 Clement Bob 106 0.271110 Crane Philip M. 104 1.371 Ganske Greg 105 0.271111 Solomon Gerald 104 1.354 Matheson Jim 109 0.270912 Stark F. Pete 102 1.348 Miller Dan 105 0.270713 Stark F. Pete 106 1.338 Brown Jr. Henry 109 0.270614 Gilman Benjamin 104 1.317 Pappas Michael 105 0.270615 Shaw E. Clay Jr. 106 1.311 Nussle Jim 103 0.2705


Summary Statistics for the SenateVariable Mean Std. Dev. Min MaxCampaign contributions* 1,255 811 -6 3,910DeGroot Weights 1.08 0.72 0.09 4.30Relative Majority Pivot Prob 1.00 0.12 0.24 1.18Relative Filibuster Pivot Prob 1.00 0.08 0.31 1.09Seniority 14.1 8.7 2 48Number of Cosponsors 84.3 45.4 9 234Bills Sponsored 88.7 7.2 34 100

*In thousands of 2006 dollars

Most Influential Senators Most Pivotal SenatorsDeGroot Relative

Senator Con- weights Senator Con- filibustergress (sum=100) gress pivot prob

1 Dole Robert 103 4.63 Wyden Ron 108 1.0922 Dole Robert 102 4.30 Bingaman Jeff 108 1.0913 Dole Robert 101 4.28 Reid Harry 108 1.0904 Bingaman Jeff 108 3.70 Durbin Richard 108 1.0895 Bingaman Jeff 109 3.67 Mikulski Barbara 108 1.0876 Kennedy Ted 103 3.48 Cantwell Maria 108 1.0867 McCain John 107 3.44 Feingold Russell 108 1.0868 Kennedy Ted 104 3.36 Domenici Pete 103 1.0869 Kennedy Ted 108 3.30 Lautenberg Frank 108 1.08510 Kennedy Ted 107 3.28 Levin Carl 108 1.08511 Hatch Orrin 106 3.24 Coats Daniel 102 1.08412 Kennedy Ted 109 3.18 Stabenow Debbie 108 1.08413 McCain John 106 3.17 Schumer Charles 108 1.08314 Kennedy Ted 102 3.17 Byrd Robert C. 108 1.08315 Kennedy Ted 105 3.16 Murray Patty 108 1.081


House Regressions.(1) (2) (3) (4)

OLS OLS OLS fixed-effects OLS FESeimielasticity Seimielasticity

VARIABLES per Std Dev per Std Dev

DeGroot Weights 141.8*** 7.9*** 189.2** 10.5**(33.0) (1.8) (73.3) (4.1)

Rel Pivot Probability 458.6* 3.2* -113.3 -0.8(277.9) (1.9) (287.0) (2.0)

Seniority -13.2*** 64.9*** 38.3 114.3**(2.5) (20.8) (26.0) (55.0)

Seniority Squared 0.27*** - 0.13 -(0.08) - (0.12) -

Majority dummy 48.0*** 5.6*** 50.6*** 5.9***(12.1) (1.4) (13.4) (1.6)

Republican dummy -42.1*** -4.9*** -186.3*** -21.8***(10.3) (1.2) (70.5) (8.3)

House Speaker 1,115.8*** 12.7*** 1,097.0*** 12.5***(172.1) (2.0) (96.2) (1.1)

Majority Leader 1,097.9*** 12.5*** 856.1*** 9.7***(172.3) (2.0) (243.5) (2.8)

Minority Leader 361.6*** 4.1*** 514.6** 5.8**(138.0) (1.6) (212.2) (2.4)

Majority Whip 701.2*** 8.4*** 438.0*** 5.2***(102.6) (1.2) (22.8) (0.3)

Minority Whip 816.0*** 9.3*** 600.1*** 6.8***(87.9) (1.0) (63.2) (0.7)

Observations 3823 3823 3823 3823R2 0.136 0.136R2 (with-in) 0.115 0.115Number of 929 929Representatives

Robust standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1

Table 1.1: Campaign contributions in thousands of 2006 dollars.


Senate regressions.(1) (2) (3) (4)

OLS OLS OLS fixed-effects OLS FESeimelasticity Seimelasticity

VARIABLES per Std Dev per Std Dev

DeGroot Weights 201.5** 11.5** 278.4 15.9(84.0) (4.8) (191.4) (10.9)

Rel Pivot Probability 12.7 0.1 -257.1 -2.4(973.7) (9.1) (870.4) (8.1)

Rel Filibuster Prob 858.8 5.3 2,676.2** 16.6**(960.4) (5.9) (1,094.8) (6.8)

Seniority -37.2* 4.9 -5.4 -84.2(21.4) (55.5) (30.8) (71.31)

Seniority squared 0.2 - 0.6 -(0.5) - (0.5) -

Majority dummy 274.8 11.0 424.9** 16.9**(197.4) (7.9) (169.3) (6.8)

Republican dummy 38.3 1.5 269.0 10.7(126.7) (5.1) (224.2) (8.9)

Majority leader -1,047.5* -7.3* -1,001.8* -7.0*(589.4) (4.1) (533.6) (3.7)

Minority leader 1,151.9*** 9.8*** 1,010.7*** 8.6***(344.3) (2.9) (173.6) (1.5)

Majority whip 580.0*** 4.0*** 1,083.2** 7.5**(145.3) (1.0) (522.2) (3.6)

Minority whip 171.9 1.5 446.7 3.8(461.9) (3.9) (305.6) (2.6)

Observations 262 262 262 262R2 0.162 0.162R2 (with-in) 0.306 0.306Number of Senators 143 143


Table 1.2: Campaign contributions in thousands of 2006 dollars.


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Decomposing the DeGroot weights into direct and indirect effects.(1) (2)

VARIABLES Percentage of Percentage ofAverage Contributions Average Contributions

Direct network influence1 56.0**(25.3)

Direct network influence2 66.5***(21.3)

Indirect network influence 42.4** 25.7(17.8) (21.4)

Observations 3822 3822R2 0.115 0.116Number of Representatives 928 928


Table 1.5: The table shows thew regression with the decomposition of the DeGroot weightsinto direct and indirect influence. In column 1, the direct influence is measured by thenumber of legislators who cosponsored with each legislator. In column 2 the direct effectweighs each cosponsor by the number of bills they cosponsored. In both cases the indirecteffect is the DeGroot weights minus the direct effect.

Chapter 2

The leverage of weak ties:

how linking groups affects inequality.

2.1 Introduction

Why do some groups in society invest so much in certain activities — such as education or

crime — compared with other groups? How does the distribution of investment depend on

social structure? In particular, how does it depend on the nature of the interactions between

mostly separate, weakly interlinked groups in society?

Several studies — most prominently Ballester, Calvo-Armengol, and Zenou (2006) and

Calvo-Armengol, Patacchini, and Zenou (2009) — have found that network centrality relates

to investment levels in social games with linear externalities over a network structure, both

in theory and in the data. Thus, we address the above questions and others by studying

how network centrality is affected when previously separate groups are first connected.

Our results show that, within the Calvo-Armengol, Patacchini, and Zenou model of

investment choices in the presence of social network complementarities, asymmetries in the

weak ties that link homophilous groups strongly affect group-level asymmetries in investment.

The details of the slight intergroup connections matter in subtle and extremely stark ways for

the distribution of investment; the weak ties have a lot of leverage. In particular, even when

the intrinsic productive capacities of all individuals are identical, the shares of investment

undertaken by different equally-sized groups can be arbitrarily asymmetric as a result of

arbitrarily weak intergroup interactions.

We identify two main effects. Suppose we begin with two previously non-interacting

groups X and Y and create an intergroup connection between a member x of group X and

39

CHAPTER 2. THE LEVERAGE OF WEAK TIES 40

a member y of group Y . If x was making a small investment relative to his own group and

y was making a large one relative to her group, then group X will, as a result, have a larger

share of the overall equilibrium investment in the newly connected society. Indeed, the ratio

of the overall effort level of group X to that of Y will be approximately proportional to the

ratio of the pre-connection effort level of y to to that of x. A second, different effect is that if

x benefits more from the connection to y than vice versa, then group X will, as a result, have

a proportionally larger share of the overall investment. Both effects remain strong no matter

how weak the link between groups is; only the asymmetry in the relationship matters.

The two effects complement each other, leading to even greater inequality when both

types of asymmetry are present; we fully characterize the impact of both phenomena on

investment shares with a simple formula. We also fully characterize how each individual’s

investment changes and find that the agents with the intergroup connections increase their

shares of investment relative to their groups, while the ratios of investments of other agents

within a given group do not change.

There are two ways to interpret our results: as a warning about how inequality can depend

crucially on the structure of social relationships on inequality across sparsely connected

groups; or as a warning against our modeling approach for sparsely connected groups.

From the modeling perspective, network models with strong linear spill-overs have become

very common in the study of social networks.1 These models are popular because they give

a tractable framework with very little assumptions on the size or structure of the network.

Currently the biggest challenge of the theoretical models of economic networks is finding

tractable models that allow for complicated patterns of influence across large networks.

Our results give a dire warning for any attempt to bring these models to the data. The

relative centrality across groups that are sparsely connected is very sensitive to measurement

error in the strength of the links across groups. Unless the researcher is very confident of

these measures, the measured centralities must be treated with extreme care.

Our finding isn’t completely negative. Our results also show that the relative centrality

inside groups that are well connected is robust to perturbations that connect the group to

other parts of the network and to perturbations in the link structure inside the group.

From the inequality perspective, an important characteristic we observe in many social

networks is that people tend to form relationships with people who are similar to them

by race, gender, age, class, geographical location and other factors.2 This effect is called

homophily in the sociological literature and while it is widely believed that it matters for

1Ballester et al. (2006); Calvo-Armengol et al. (2009); DeMarzo et al. (2003); Bramoulle et al. (2009);DeGroot (1974); Golub and Jackson (2010, 2009) and see Chapter 1.

2See McPherson et al. (2001) for a comprehensive survey.


social outcomes, there is not much of a theoretical understanding of exactly how it matters.

Our results suggests that the details of the links bridging homophilous groups can be very

important to determine relative outcomes of groups in societies. These details include the

identity of the individuals that have relationships across groups and the position they have

in the network of their own group, as well as the relative direct influence between bridge

agents of different groups.

Inequality across different social groups in society has been widely documented. For in-

stance, in data from the National Longitudinal Survey of Youth (1971–91 waves), 52 percent

of white American males have completed high school at age 24, but only 38 percent of Latin

Americans males have done so.3 As such, we cannot rule out that the structure of social

relationships is influencing the outcomes.

On the normative side, there are many social interventions and programs whose explicit

purpose is to link separate social groups. This includes cultural exchange between coun-

tries as well as charities such as the Big Brothers and Big Sisters of America, which pairs

vulnerable youth with mentors from different social groups. In an equilibrium model, such

interventions have indirect effects on many agents beyond the ones they directly involve, and

ultimately on inequality of investment within a society. Understanding these externalities

is important to designing such of interventions, and our work takes a step toward such an

understanding.

On the way to our results, we make two general contributions that are of independent

interest for the study of centrality measures in networks.

First, we show that eigenvector centrality is the limit of Bonacich centrality in an ap-

propriate sense. This extends a result proved in Bonacich (1991) to a much larger class

of interaction matrices. It also establishes a broad connection between two of the most

important centrality measures used in practice: Bonacich centrality, on the one hand, and

eigenvector centrality, on the other.4

Second, we derive a characterization of the eigenvector centrality of a network obtained

by connecting two previously disconnected networks. Beyond the application to a model of

investment with complementarities, this characterization applies to eigenvector centralities

in other settings such as the Google PageRank measure, the theory of Markov chains (where

the eigenvector centrality vector is the stationary distribution) and numerous applications in

3See (Cameron and Heckman, 2001) for this and more statistics on racial gaps in school achievement andan overview of some of the literature.

4Bonacich centrality was introduced in Bonacich (1987). Eigenvector centrality is also known as Katzprestige (Katz, 1953). It measures influence in an important model of belief evolution (French, 1956; DeGroot,1974), and is also used in the computation of the Google PageRank measure (Langville and Meyer, 2006).


sociology. In these settings, too, arbitrarily weak intergroup links can have arbitrarily large

effects on the distribution of centrality across groups. We comment on some of the broader

implications of this in Section 2.4.

The intergroup connections that we study have been a focus of investigation across several

fields. Granovetter’s (1973) seminal analysis of how social networks mediate learning about

jobs emphasized the informational importance of weak ties connecting otherwise separate

communities. Burt (1992) developed an influential theory of structural holes, exploring the

advantaged positions enjoyed by the agents incident to a bridge — a feature that is also

present, for different reasons, in our analysis. Both works have spawned large literatures in

sociology. In a recent paper on the spread of misinformation in social networks by Acemoglu,

Ozdaglar, and ParandehGheibi (2010), bridges are important in characterizing when agents

can successfully spread their potentially biased beliefs. Bridges in graphs are also well-

studied objects in graph theory (see, for example Diestel (2005)). Our work links the notion

of bridges with the study of centrality which, as mentioned above, has been an important

area in sociology and network analysis more generally (Wasserman and Faust, 1994b).

The paper is organized as follows. Section 2.2 sets up the model, presents the main

definitions on networks and provides our main result: a formula for how connecting groups

changes eigenvector centrality. Section 2.3 then uses the result to derive new predictions

for two economic applications, investment decisions with positive spillovers through a so-

cial network (Section 2.3.1) and consumption decisions with social influence (Section 2.3.2).

In Section 2.4, we summarize the main insights and comment on some implications and

extensions of the results.

2.2 How linking groups affects centralities.

2.2.1 Setup

Let there be n agents (also called nodes) N = {1, . . . , n}. A weighted social network is a

matrix A with non-negative entries aij. An entry aij represents the weight of the link from i

to j. We will assume that A is a column-stochastic matrix,�

i aij = 1 for all j. (See Section

2.3.1 and Section 2.3.2 for an economic justification.) We assume that aii = 0 for all i, which

means we don’t allow self-links.

Below are some common definitions for networks.5

A walk in A is a sequence of nodes i1, i2, . . . , iK , not necessarily distinct, such that

5See Jackson (2008b) for these and other definitions relating to networks.


aikik+1> 0 for each k ∈ {1, . . . , K − 1}. We say that a walk i1, i2, . . . , iK goes from i1 to iK .

The length of the walk is defined to be K − 1. The weight of the walk is defined to be the

product�K−1

k=1 aikik+1. A walk is closed if it starts and ends at the same node, i.e. i1 = iK .

A path is a walk whose nodes are distinct. A network A is path-connected (also called

irreducible) if for every pair of agents i, j, there is a directed path from i to j and back.

We are interested in studying changes in eigenvector centrality, as defined below.

Definition 7 (Eigenvector centrality). Given a nonnegative, path-connected matrix A, the

eigenvector centrality e(A) is the right-hand (column) eigenvector of A with nonnegative

entries. This eigenvector is unique up to scale; we will always assume its entries sum to 1.6

We may also speak of the eigenvector centrality of a group X ⊂ N , written eX(A), which

is just the sum of the eigenvector centralities of its members.

In many economic applications, we are interested in studying how the centrality of an

agent changes when we change the weight of a link or when we connect disconnected groups.

When a network is path-connected, there are formulas for the derivative of the eigenvector

centralities with respect to the weight of any link; we discuss some of these in Section 2.5.

Unfortunately these formulas do not apply for disconnected networks. Since linking discon-

nected groups features prominently in studies of trade, peer effects and strategic network

formation, it’s important to have tools to address this scenario.

Our main result below bridges this gap and provides a closed-form solution for the change

in eigenvector centrality when we link disconnected groups. To get closed-form solutions we

focus on a single link connecting two segments of society. This two-way bridge is defined

formally below.

Definition 8 (A two-way bridge). Let X, Y be a partition of the agents. We say a path-

connected matrix A has a two-way bridge if there exist x ∈ X and y ∈ Y so that

axy > 0, ayx > 0

and

aij = 0;∀(i, j) ∈ (X × Y ) \ {(x, y)}

We call x and y the bridge nodes.

6All vectors in the paper are column vectors unless otherwise noted. The existence and uniqueness of theeigenvector centrality are a consequence of the Perron-Frobenius theory of nonnegative matrices, and arediscussed in (Meyer, 2000, Section 8.3).


Figure 2.1: The nodes (x, y) form a two-waybridge between groups X and Y .

Figure 2.2: This is not a two-way bridge, be-cause the node that connects X to Y is notthe same that connects Y to X.

In words, A has a two-way bridge if it can be divided into two path-connected groups

X and Y such that there is a single pair of agents (x, y) who have influence across groups.

Having a single link that connects the groups will be crucial to get a clean closed-form

solution, but we believe most of our qualitative results hold if the groups have many bridge

links but are sparsely connected. We discuss more on these matters in Section 2.5.

It will be important for our purposes to talk about the restriction of the network to a

subgroup. Given a matrix A with a two-way bridge between X and Y and bridge nodes

x ∈ X and y ∈ Y , let AX denote the matrix with index set X so that

aXij =

aij if j �= x

aij/(1− ayx) if j = x.

The entries are normalized to make AX column-stochastic while preserving the relative

influence of x over the other members of X. We define AY analogously. These restrictions

of A capture the interactions that would exist if there were no bridge between X and Y .

Intuitively, we can think that in the beginning X and Y belong to separate networks

each with column-stochastic interaction matrices AX and A

Y . Then we connect them by

creating the link between x and y, giving rise to a new combined interaction matrix A.


2.2.2 The main result

We now present our main result. The following theorem characterizes the eigenvector cen-

tralities of all agents in the connected network based on their eigenvector centralities in the

restricted networks.

Theorem 9. Assume A is path-connected and has a two-way bridge between X and Y with

bridge nodes x ∈ X and y ∈ Y . Let

cx = 1− ayx(1− ex(AX)) and cy = 1− axy(1− ey(A

Y ))

and

r =axy

ayx· ey(AY )

ex(AX)· cx

cy. (2.1)

Then

ei(A) =1

1 + r·

ei(AX) · rc−1x if i = x

ei(AX) · rc−1x (1− ayx) if i ∈ X \ {x}

ei(AY ) · c−1y if i = y

ei(AY ) · c−1y (1− axy) if i ∈ Y \ {y}.

The formulas are simple to interpret. The centrality of an agent in the connected net-

work is proportional to the his centrality in the disconnected network multiplied by certain

correction factors that depend on the weights of the two-way bridge. The agents in in X

have their centralities multiplied by a factor of r, and then adjusted by the correction factor

c−1x . The agents in in Y have an analogous correction term. The fraction (1 + r)−1 on the

outside is merely a normalization.

Notice that the ratios of centralities between members of the same group remain the same

except when a bridge node is involved. Relative to the other agents in his group, the bridge

agent x has his centrality increased by a factor of (1 − ayx)−1. This generates a perverse

incentive: when agent x increases the benefits he gives to y in the other group, all members

of his group lose centrality, but x gains a higher relative position inside of X.

The logic for the proof is straightforward. Since we know the eigenvector is unique we

guess it has the form above and verify it. The real challenge was coming up with the candidate

guess. We present the intuition below. Reviewing this will clarify why our approach requires

connecting X and Y through a single two-way bridge.


Extending our approach to multiple links is difficult, but once the groups have become

connected by single-link, we can apply derivative formulas for path-connected networks. See

Section 2.5.

2.2.3 The intuition for the formula

The key to guess the formula for eigenvector centrality is to use the fact that the eigenvector

centrality of A is the same as the stationary distribution of a Markov chain with a transition

matrix which is the transpose of A. This Markov chain is one in which a particle randomly

hops around the nodes N with a probability aji of moving from node i to node j. While the

intuitions of this process are rather far from the economic applications we present below, the

advantage of the Markov chain approach is that we can use probabilistic results to study our

problem. This provides natural intuitions that motivate the expression.

Suppose at time n = 0 the particle starts a random walk from node x. Let Wn be the

position of the particle at time n, which is a random variable. Analogously define a Markov

chain starting at x corresponding to AX and denote the position of that particle at time n

in that process by WXn . This process corresponds to a particle that walks only in X, with

the transition probabilities prescribed by AX .

Define the random variable

Ti = inf{n ≥ 1 : Wn = i}

This is the time of the first visit to i after time 0. A well-known formula (Durrett, 2005,

Chapter 5, (4.3)) states that for any M ⊆ N

eM(A) =Ex

��Tx−1n=0 1{Wn∈M}

�

Ex(Tx). (2.2)

Here, the subscripts on the expectations remind us that the particle starts at x and 1{Wn∈M}

is the indicator random variable which takes value 1 if and only if the particle is in M at

time n.

This tells us, for example, that:

Claim 1. For i ∈ X such that i �= x

ei(A)

ex(A)= (1− ayx) ·

ei(AX)

ex(AX).


This is because (2.2) applied to M = {x} yields:

ex(A) =1

Ex(Tx). (2.3)

When the formula is applied to M = {i}, we reason as follows. After leaving x, the particle

does one of two things. (1) It may go to y with probability ayx, in which case it cannot

possibly hit i before its first return to x (recall that there is a bridge from x to y which,

by definition, is the only way to get between the two groups). (2) Or it may go somewhere

within X with probability (1− ayx), and conditional on this event its travels throughout X

before its next return to x have the same probability distribution as they did without the

bridge. Thus we may write

ei(A) =(1− ayx)EX

x

��Tx−1n=0 1{W X

n =i}

�

Ex(Tx). (2.4)

The superscript on the expectation indicates that it refers to the chain conditional on staying

in X, whose transition matrix is given by the transpose of AX . Dividing (2.4) by (2.3) and

using (2.2) for the chain corresponding to AX yields the claim. An analogous claim holds

for Y .

This pins down the new eigenvector centralities inside each group. To finish we only

have to determine the relative influence across groups. For this it suffices to consider the

eigenvector condition

ex(A) =�

j

axjej(A).

2.3 Motivating examples

We know use our formula to provide new results for two economic problems. In both ex-

amples, eigenvector centrality will characterize the equilibrium decisions in the limit with

strong social influence.

2.3.1 Investment decisions with strong social spillovers

We work with a special case of the model of Ballester, Calvo-Armengol, and Zenou (2006)

and define the following investment game. Each agents i selects an effort level zi ≥ 0, which

can be interpreted as human capital investment. Agent i selects zi to maximize:


ui(z1, · · · , zn) = zi −1

2cz2

i + γ�

j �=i

aijzizj. (2.5)

Here, c > 0 is a marginal cost parameter; γ > 0 is a social spillover parameter and aij is an

entry of the column-stochastic matrix A.

The interpretation is that each person’s utility is a combination of three things. First,

there is a linear own-effort effect which we normalize to have a unity coefficient. Second, there

is a convex cost in own effort introduced by the quadratic second term and parametrized

by c. We assume everybody has the same marginal cost of effort. Finally, there are social

complementarities. An agent j spends an amount of time aij “teaching” agent i. The

teaching has complementarities in the level of knowledge of the teacher and of the student.

The benefit i receives from j is increasing in zi and in zj. The total benefit i receives from

other people’s “lessons” is given by γ�

j aijzizj. Agent i’s marginal benefit of investing in

education is increasing in the investment level of the people connected to her through the

social network.

We assume that each agent j spends the same amount of time teaching other agents.7

With an appropriate choice of γ we normalize this so that�

j aij = 1 and the matrix A

is column-stochastic. This amounts to assuming that no agent has more potential to teach

than others, and corresponds to our focus on the network creating inequality as opposed

to ex-ante differences between agents. Note, however, that this is not a strong symmetry

assumption. The weight from i to j might be different from the weight from j to i. The

returning link might even not exist at all.

In order to focus on the consequences of changing the network, we will treat the values

for aij as exogenous.

Theorem 11 below characterizes the equilibria of the game. It is almost follows from

Theorem 1 in Ballester, Calvo-Armengol, and Zenou (2006). Before the statement, we need

one definition.

Definition 10. Given a nonnegative matrix M, provided that [I − αM]−1 is well-defined

and nonnegative, the vector of Bonacich centralities of M with parameter α is

b(M, α) = [I− aM]−11 =

∞�

k=0

αkM

k1, (2.6)

where 1 is the n× 1 vector of ones. Let B(M, α) be the sum of entries of b(M, α).

7A model where every agent spends the same fixed-amount to “learn” rather than “teach” would bestrategically equivalent to out second motivating example. See Section 2.3.2.


Theorem 11. The investment game has a Nash equilibrium if and only if γc < 1. When it

exists, the Nash equilibrium is unique and is given by:

z∗ =

1

cb

�A,

γ

c

�.

All proofs appear in the Proof Sections.

The intuition for this result is that investment is proportional to network position as

measured by Bonacich centrality. The agents who invest the most are the ones who most

benefit from feedback loops of social complementarities. This can be seen in equation (2.6)

which shows that the Bonacich centrality of agent i is a weighted sum of the walks in A that

start at i. This is just a multiplier effect. The size of the multiplier is determined by the

structure of the network and each agent’s position determines how much he or she benefits.8

Bonacich and eigenvector centrality are closely related: as the network feedback-loops

become large, Bonacich centrality converges to eigenvector centrality. This is stated generally

in Theorem 12, which extends a theorem in Bonacich (1991). Bonacich shows the result for

symmetric matrices using a diagonalization argument. Our proof is quite different and does

not require symmetry or even diagonalizability.

To state the result precisely and in full generality, we need one more technical notion —

that of aperiodicity, which is standard in the study of Markov chains. It is a mild technical

condition that holds generically and requires that the greatest common divisor of the lengths

of all closed walks be 1. For example, this condition holds if there is at least one closed walk

with two agents and at least one closed walk with three agents.

Theorem 12. Given a nonnegative path-connected aperiodic matrix M whose largest eigen-

value in magnitude is µ, we have

limα↑µ−1

b(M, α)

B(M, α)= e(M).

The relationship between Bonacich centrality and eigenvector centrality can be under-

stood in the following way: the Bonacich centrality of i is computed by starting with a

baseline centrality of 1 (which corresponds to the linear own-effort term in our investment

game) and sums all walks starting at i, with walks of length k getting weight αk.

8When γc � 1, the social spillovers are so big that there is no equilibrium, because agents would always

want to invest more. One way to see this is to note that this is a supermodular game, so the best-responsemapping converges to the lowest Nash equilibrium when the mapping starts from the lowest action. Whenγc � 1, this dynamic is explosive, so no equilibrium can exist.


Eigenvector centrality measures influence by giving equal weight to all walks starting at

i. The number of such paths is infinite, but by taking appropriate limits, we can still make

comparisons between nodes. Eigenvector centrality pins down these centralities. Adding a

normalization determines the levels.

The higher α is, the greater is the importance of long walks for Bonacich centrality. In the

limit, the baseline effect and the short-distance walks are completely insignificant. Therefore

as the network feedback become large, the ratio of Bonacich centralities converges to the

ratio of eigenvector centralities.

This allows us to use our result on eigenvector centrality to study the investment game

when there are strong social spillovers, focusing on the distribution of investment across

society. For a given interaction matrix A and parameters γ, c, let the equilibrium investment

share of a set X of agents be defined by

sX(A, γ/c) =

�i∈X z∗i�i∈N z∗i

,

where z∗ is the equilibrium as characterized in Theorem 11. Note that even though z

∗

itself depends on the levels of both γ and c, the equilibrium shares as defined above depend

only on the ratio γ/c, which explains the notation. We will also write si(A, γ/c) instead of

s{i}(A, γ/c), and use the boldface notation s(A, γ/c) for the vector with si(A, γ/c) in the

ith position. We will drop the arguments when they are clear from context.

A corollary of Theorem 12 is that when social spillovers are big, these shares are well

approximated by eigenvector centrality.

Corollary 13. Assume A is path-connected and aperiodic. As γ/c approaches 1 from below,

the investment share of a group X approaches the eigenvector centrality of X with respect

to A. That is:

limγc ↑1

s(A, γ/c) = e(A).

As a consequence of Theorem 9, we can characterize the relative investment shares of the

two groups.

Corollary 14. Assume A is path-connected and has a two-way bridge between X and Y

with bridge nodes x ∈ X and y ∈ Y . Then

eX(A)

eY (A)=

axy

ayx· ey(AY )

ex(AX)· 1− ayx(1− ex(AX))

1− axy(1− ey(AY )).


In view of Corollary 13, this means that the ratio of investment shares is given by the

above equation. A simpler formula can be obtained by taking a limit as the link between X

and Y becomes weak.

Corollary 15. Let A be as above and axy = kayx for some constant k > 0. Then

limayx→0

eX(A)

eY (A)= k · ey(AY )

ex(AX).

This simplified formula contains the main insights about how the bridge affects each

group’s centrality.

First, the ratio of the investment of group X to the investment of group Y is directly

proportional to the ratio of the weights between the bridge-nodes: (axy/ayx). If agent x gets

more benefit from the complementarity with y than vice versa, then group X ends up doing

most of the investment in the society after the connection is made.

Second, the ratio of the investment of group X to the investment of group Y is directly

proportional to the ratio ey(AY )/ex(AX). This is the ratio of the bridge agents’ investment

shares in their own groups before the bridge is created. In other words, connecting a relatively

high-investing member of group Y to a relatively low-investing member of group X is good

for the investment share of group X in the combined network.

Third, and perhaps most surprisingly, arbitrarily weak links can have arbitrarily large

effects on the investment shares in the combined network. Even in the limit as the level of

interaction between the two groups tends to zero, the shares of investment can remain quite

unbalanced. Indeed, the formula in Corollary 15 shows that there is a discontinuity as a link

is added; without the link, two groups may invest the same amounts, but after an arbitrarily

weak link is introduced, they may become very unequal.

Fourth, the investment ratio of the groups X and Y depends only on the link between

them and on the within-group investment shares of the bridge nodes. The relative size of

each group is irrelevant as well as other traditional metrics on network. It does not matter

which group is more densely connected or which group has a larger network diameter.

This inequality in investment does not go away if the link between both groups is strong.

The next theorem shows that the comparative statics for strong links go in the same direction:

the investment share of X relative to Y increases if y increases the amount of help she gives

to x; it is also increasing in the influence of y within his own group Y .

Theorem 16. The ratio of investments eX(A)eY (A) is strictly increasing in axy and ey(AY ). It is

strictly decreasing in ayx and ex(AX).


2.3.2 Consumption with strong social influence

We now study a model of consumption with social influence. Every agent makes a consump-

tion decision zi ∈ R and is influenced by the choice of his neighbors. Agents simultaneously

make their consumption decision to maximize:

ui(zi, z−i) = −1

2

�zi − θi

�2 − β

2

�zi −

�

j

aijzj

�2

The parameter θi represents agent i’s ideal choice in a world without social influence.

It is fixed and common knowledge. We call it the autarky ideal point of i. The weight β

represents the agent’s preference for acting like his neighbors. Each parameter aij is an entry

of a matrix of influence A and represents the influence of j over i. We assume the matrix

of influence A is row-stochastic, so every agent is trying to match a weighted average of the

decisions of his neighbors.

Agents face a trade-off between being closer to their autarkic ideal point and being close

to their neighbors’ decision. Because being away from either point has an the increasing

marginal cost, agents chose something in-between.

The theorem below characterizes the equilibria of the game:

Theorem 17. The unique Nash equilibrium of the game is

z∗ =

1

1 + β

�I− β

1 + βM

�−1�θ1, . . . , θN

��.

Corollary 18. In the limit when social preferences dominate (β →∞), all agents make the

same consumption choice which is a weighted-sum of the autarkic ideal points:

limβ→∞

z∗i =�

j

ej(A)θj;∀i.

The equilibrium of the consumption game is very similar to that of the investment game.

The strategies in Theorem 17 involve a modified version Bonacich centrality. There is one im-

portant difference between the games: in the investment game with strong social spillovers,

each agent invests different amounts in proportion his eigenvector centrality. In the con-

sumption game with strong social spillovers, all agents invest the same amount, which is a

weighted sum of the autarkic ideal points. The weights are the eigenvector centralities.

Using Theorem 14, we see that when group X and Y start interacting (become linked)

the new equilibrium can be very biased toward the ideal-points of one of the groups. In


particular if ayx � axy and ey(AY ) � ex(AX) with one strict inequality, the ideal-points of

group X will have a larger weight in the final consumption decision.9

In the limit when the bridge-links are close to zero as in Corollary 15, the new consump-

tion is

z∗ =r

1 + r

�

i∈X

ei(AX)θi +

1

1 + r

�

j∈Y

ej(AY )θj

where r is the ratio

r =ayxey(AY )

axyex(AX)

We can take the analysis one step further by assuming that consumption preferences

evolve over time. Suppose that the consumption game is played repeatedly many times,

but that the ideal points θi change randomly every period. After the θi’s are realized they

become common-knowledge and all agents play the static Nash equilibrium.

Suppose all agents in the same group draw θi from the same distribution but that the

realizations of θi are independent across agents. Let σ2x, σ

2y be the variances of the distribution

for groups X and Y . Suppose that half the members of society belong to group X and half

to group Y . Before the groups are connected the average of agents’ period-to-period variance

in consumption is

σ2z∗ = 2

��1

2

�2σ2

x

�

i∈X

�ei(A

X)�2

+�1

2

�2σ2

y

�

j∈Y

�ej(A

Y )�2

�

Assume that the variance in the equilibrium-consumption in group X is larger than in

group Y .

σ2x

�

i∈X

�ei(A

X)�2

> σ2y

�

j∈Y

�ej(A

Y )�2

This can happen for two (non-exclusive) reasons: the variance of tastes per agent in X

is larger than in Y : σ2x > σ2

y ; or the eigenvector centralities in Y group are more evenly

balanced than in X:�

j∈Y

�ej(AY )

�2<

�i∈X

�ei(AX)

�2. This second effect occurs because

social influence between members of the same network “averages out” the shocks in θi. But

if the eigenvector centrality is distributed very unevenly there is little “averaging-out”. In

the extreme case, all agents listen to a single member of the group and the group variance

9Here and throughout the section ayx and axy are switched with respect to the formula in Corollary 14because here the matrix A is row-stochastic, so we have to transpose it to apply Theorem 9.


in consumption is identical to the variance of that individual’s θi.

Now assume groups X and Y become connected. The new the variance in consumption

becomes

σ2z∗ = σ2

x

�

i∈X

�ei(A)

�2+ σ2

y

�

j∈Y

�ej(A)

�2,

which in the limit as axy, ayx → 0 becomes

σ2z∗ =

� r

1 + r

�2σ2

x

�

i∈X

�ei(A

X)�2

+� 1

1 + r

�2σ2

y

�

j∈Y

�ej(A

Y )�2

Notice that there is a 2 that dropped out when we connect the two groups. This represents

the drop in variance because society it now twice as large, so it has more members “averages

out” the shocks in θi. This potential reduction is diminished depending on how we connect

the groups. When r → 1 the variance in consumption is just the variance in group y and

there is no “averaging-out” effect across groups, so the period-to-period variance increases.

Therefore connecting the groups makes the consumption decisions more homogenous

inside each period and might increase the average variance in consumption from period-to-

period. This last effect occurs because the tastes of the volatile group has a disproportionate

influence in the equilibrium consumption decisions.

2.4 Empirical implications:

the sensitivity of centrality measures

Our results also point out that the global properties of centrality measures can be very

sensitive to small, local perturbations when there are islands bridged by weak ties, as there

often are in real social networks. Imbalances in these weak links can lead to large global

imbalances of centrality, so that small absolute changes in some network parameters can lead

to large absolute changes in the centralities.

This has implications for the empirical analysis of networks.10 When there is noise in

the measurement of the network and the network has an “island-like” structure, our analysis

suggests that the relative centrality across groups should be treated with caution, since they

may be very sensitive to measurement error in the links across groups.

Suppose, for example, that there is one link between agents x and y bridging two groups,

10We thank Matt Jackson for emphasizing the implications of this to us.


and the estimate of the complementarity axy that agent x gets from agent y is twice the

estimate of ayx; however, in reality, the two are equal. This is quite plausible if both quantities

axy and ayx are small, so that the error is very small in absolute terms but leads to a large

error in the estimate of the ratio axy/ayx. In this case, the measurement error causes all the

computed centralities in the group of agent x to be off by a factor of two! On the other

hand, Theorem 9 shows that the ratios of centralities within a group will be fairly accurate,

because they essentially do not depend on the weak intergroup links.

More generally, this seems to suggest that, in the presence of measurement error, eigen-

vector centrality is not a robust tool for comparing centralities of nodes located in parts

of a network that are only weakly connected to each other. At the same time, it is quite

reasonable for comparing centralities of agents in a thick region of the network. Making this

speculation more precise could be a fruitful direction for further work.

2.5 Several links between groups

Our analysis characterizes how investment and consumptions decisions — or, more generally,

network centralities — change when previously separate groups start interacting. A natural

question is what happens when further links are added beyond the first, so that we are

looking at two islands connected by a few links as opposed to just one bridge. This question

is answered in a beautiful paper of Conlisk (1985). Start with a path-connected column-

stochastic A and consider perturbating a single link: Aji → Aji + � and Aki → Aki − �. All

other entries of A are held fixed. This corresponds to agent i increasing his complementarity

or interaction with j (the “favored node”) at the expense of k (the “disfavored node”). The

decrease is necessary to satisfy the assumption that the total complementarity shared out

by any agent is fixed, so that increasing the benefit to one neighbor requires decreasing the

benefit to another.

Conlisk shows that, under this perturbation, for any �, we have

∂e�

∂�= ei(A) · (wj� − wk�),

where wj� is the mean first passage time from j to � in the Markov chain whose transition

matrix is given by the transpose of A. (See our intuition in Section 2.2.3.) This is a measure

of the network distance between j and �. Thus, the centrality of node � changes in proportion

to the difference wj�−wk�. If � is closer to j (the favored node) she will gain centrality while

if she is closer to the disfavored node k she will lose it. The magnitude of the change is also


proportional to the centrality of the node i, the agent doing the redistributing.

The intuitions of the formula cohere with those of our main result. Adding directed links,

all else equal, helps those who get more weight (and the agents near them) and hurts those

who lose weight (and the agents near them). This effect is stronger when the originating

node is more influential.

Indeed, in combination with Conlisk’s results, our treatment of bridges completes the

picture on the comparative statics of the centralities of normalized network matrices (equiv-

alently, the stationary distributions of Markov chains). For any change one might like to

consider, the results working in tandem can completely characterize the effects of the pertur-

bation in an intuitive way — though an exact computation of the quantities would require

solving a system of differential equations.11 Of course, since the relevant centralities are solu-

tions to a well-studied fixed-point problem, one could always simply compute them explicitly

with a program like MATLAB. The virtue of the comparative statics formulas is that they

explain exactly what matters and how, which sheds more light on positive and normative

economic questions than a computational approach. Indeed, these formulas remove some of

the mystery of network centralities. Not only do we know that they satisfy some desirable

fixed-point property, but we also know, in a reasonably explicit way, how they change when

interesting changes happen in the network.

2.6 Conclusion

We presented a new theoretical result that provides a simple, closed-form solution for how

eigenvector centrality changes when disconnected groups become linked. Previous work had

studied only the derivative of eigenvector centralities in connected networks.

The result is particularly useful for comparing group centralities. We find that depending

on the way the groups are connected, the influence of one group can dominate the influence

of the other. In particular, if a member with a central position in group Y becomes linked

to a member with a noncentral position in group X, the resulting eigenvector centrality of

11One change which may at first seem tricky to treat is that of introducing a connection between twoseparate groups which is not a two-way bridge, as in Figure 2.2. For example, it may be that there isa directed link from x ∈ X to y ∈ Y , but the only link in the reverse direction comes back from somey� �= y in Y to x� ∈ X, which may or may not be the same as x. Using techniques similar to those usedto prove our main result, one can characterize the post-perturbation centralities exactly. The formulas arenot particularly elegant. But one can also characterize them by combining Conlisk’s result with ours. Onewould add a two-way bridge at first (say between x and y), and then use the Conlisk formulas to characterizewhat happens when one of its directions is gradually replaced with the correct return link from y� to x�. Wethank Tomas Rodriguez-Barraquer for pointing out that such a change can be substantively important, forexample when academic disciplines are first linked via two separate one-way bridges.


group X will be proportionally larger than that of group Y . This occurs regardless of the

relative size of the groups and other network metrics: their clustering, the density of their

connections, or their diameter.

We also showed how the result is useful through two economic applications. In one,

individuals invest in human capital and generate positive externalities by teaching others

through a social network. In equilibrium, agents invest in proportion to their eigenvector

centrality, which means that one group may end up with a disproportionate share of the

total investment after the groups become connected.

In the second example, every period agents locate themselves in a one-dimensional con-

sumption space by balancing their private tastes with their desire to match the decisions

of their neighbors. In equilibrium, the group with a higher eigenvector centrality has a dis-

proportionate weight in determining the final location of consumption decisions in society.

Furthermore, connecting two groups, somewhat counterintuitively, increases the variance

in consumption from one period to the next. This happens because a small group has a

disproportionate influence in the final decision.

Our results caution that network models with linear-spill over should be used with caution

when dealing with sparsely connected groups. Empirical measures of the relative centrality

across groups are extremely sensitive to measurement error in the links across groups. From

the modeling perspective, the stark conclusions for investment and consumption decisions

across groups also raise questions about the modeling assumptions. The usual justification

for linear spill-overs is that it is good local approximation to any continuos influence function.

Our results point that this approximation is too sensitive to be of use for groups that are

sparsely connected.

At the same time, our results highlight that centrality inside groups that are well con-

nected can be robustly measured as it does not change with small perturbations across

groups.

One of the most interesting directions for further work is to endogenize the network.

Our approach views the network is exogenous, which is reasonable when constraints like

languages, occupations, and geography determine much of the interaction that goes on in

the short and medium run. In that case, the formation of the new links across groups arises

exogenously because these parameters are varied by external forces. Nevertheless, over longer

time scales agents do have choices in their interactions, though these are constrained and

costly. The main challenge is to formulate a model which can encompass a fairly broad range

of constraints on the network formation problem (such as different costs of linking between

different pairs of agents) and produce tractable and interesting network formation dynamics.


Proof sections for the chapter.

Proof of Theorem 9

The proof amounts to checking that the claimed specification — call it �e(A) — satisfies the

eigenvector condition

�ei(A) =�

j

aij�ej(A)

for every i. To do this, we use the facts that e(AX) is an eigenvector centrality (i.e. an

eigenvector with eigenvalue 1) of AX , and e(AY ) is an eigenvector centrality of A

Y . For

example, to check the condition for i = x, we compute:

�

j

axj�ej(A) = axy�ey(A) + axx�ex(A) +�

i∈X\{x}

axi�ei(A)

= axy · 1

1 + r· 1

1− axy(1− ey(AY ))· ey(A

Y )

+ axx · r

1 + r· 1

1− ayx(1− ex(AX))· ex(A

X)

+�

i∈X\{x}

axi ·r

1 + r· 1− ayx

1− ayx(1− ex(AX))· ei(A

X)

= axy · 1

1 + r· 1


Y ) (Using the definition of AX)

+ aXxx(1− ayx) ·

r

1 + r· 1


X)

+�

i∈X\{x}

aXxi ·

r

1 + r· 1− ayx

1− ayx(1− ex(AX))· ei(A

X)

= axy · 1

1 + r· 1


Y )

+r

1 + r· 1− ayx

1− ayx(1− ex(AX))

�

i∈X

aXxiei(A

X)

= axy · 1

1 + r· 1


Y ) (Using the definition of e(AX))

+r

1 + r· 1− ayx


X)


If we now plug in r from (2.1) and simplify, this is equal to

r

1 + r· 1


X) = �ex(A).

This verifies the eigenvector condition for i = x. The calculations for the other indices are

equally straightforward.

Proof of Theorem 11

Part 1. If γc < 1 a Nash equilibrium exists and is unique.

Both of these statements follow from the same reasoning as Theorem 1 in Ballester,

Calvo-Armengol, and Zenou (2006). The proof works by solving the first order conditions,

which are a linear system of equations. The condition γc < 1 guarantees the linear system

has a solution, given by z∗ below, which proves existence.

z∗ =

1

cb

�A,

γ

c

�.

To show uniqueness, note that payoffs are linear in the actions of the other players,

so when opponents play mixed strategies, only the expectations of their choices matter.

Additionally there is a unique best response to any mixed strategy because the cost of effort

is convex. Therefore all equilibria in this regime are in pure strategies. When γc < 1, only

z∗ solves the first order conditions for a pure-strategy Nash equilibrium.

Part 2. If γc ≥ 1, there is no pure-strategy Nash equilibrium.

Suppose, toward a contradiction, that α := γ/c ≥ 1 and there is an equilibrium z∗. The

first-order conditions imply that

z∗ = αAz

∗ +1

c1,

By recursively substituting the entire right-hand side for z∗ we get, for every natural

number K:

z∗ =

1

c

�K−1�

k=0

αkA

k

�1 + αK

AKz.

Since A is column-stochastic, so is Ak for every k. In particular, column i of

�Kk=0 αk

Ak

sums to�K

k=0 αk ≥ K, and thus some entry of that column exceeds K/n. Choosing K so

that K/(cn) exceeds the maximum entry of z∗ yields a contradiction.


Part 3. If γc ≥ 1, there is no Nash equilibrium.

Take a mixed-strategy Nash equilibrium F, which is a vector of cumulative distribution

functions. Create a pure strategy profile z where each j sets zj to the expectation of his

random investment under Fi. Using linearity of expectation as well as the fact that the

payoff of i is linear in each zj, we find that the function zi �→ ui(zi; z−i) is the same as

zi �→ ui(zi;F−i), and so what was a best response remains a best response. This gives

a pure-strategy Nash equilibrium. Thus, by the previous part, there cannot be a mixed-

strategy Nash equilibrium in the regime γc ≥ 1.

Proof of Theorem 12

Let � · � be the supremum norm on Rn and let � · � also denote the induced matrix norm

when the argument is a matrix. Define T = µ−1M. Aperiodicity of M implies that T

k has

a positive diagonal entry for some high enough k (Durrett, 2005, p. 310). That, in turn,

implies that all eigenvalues of A are smaller than µ (Meyer, 2000, Section 8.3). From this it

follows that G = limk→∞Tk is well-defined and that

G =e(A)e(A�)�

e(A�)�e(A),

where e(A) is as in Definition 7 (Meyer, 2000, Section 8.3). The prime notation denotes

transposition.

Claim 1. lima↑1(1− a)(I− aT)−1 exists and is equal to G.

Proof. Fix δ > 0. Choose K so large that for k > K we have��T

k −G�� < δ/2 and then

choose a < 1 so that��K

k=0(1− a)ak�� < δ/4. The Neumann series

(1− a)(I− aT)−1 =∞�

k=0

(1− a)akT

k


converges (Meyer, 2000, Section 7.10). Now,

��

K�

k=0

(1− a)akT

k −G

�� =

��

∞�

k=0

(1− a)akT

k −∞�

k=0

(1− a)akG

��

≤K�

k=0

(1− a)ak�Tk −G�+∞�

k=K+1

(1− a)ak�Tk −G�

≤ 2K�

k=0

(1− a)ak +∞�

k=K+1

(1− a)ak�Tk −G�

≤ 2 · δ

4+

δ

2

∞�

k=K+1

(1− a)ak

≤ δ.

Here we have used the triangle inequality and the fact that Tk and G are both stochastic,

so have matrix norm at most 1.

Recall that that by Definition 10 we have

(1− a)b(T, a) = (1− a)(I− aT)−11.

Thus, for any � > 0, there is a δ > 0 so that the statement

�(1− a)(I− aT)−1 −G� < δ

implies ��bi(T, a)

B(T, a)−

�j Gij�

j,k Gjk

�� =

��(1− a)bi(T, a)

(1− a)B(T, a)−

�j Gij�

j,k Gjk

�� < � for all i.

Recalling that �j Gij�

j,k Gjk= ei(T)

and putting everything together with Claim 1 shows that for every � > 0, all high enough

a < 1 satisfy ��bi(T, a)

B(T, a)− e(T)

�� < � for all i.

Since b(T, a) = b(M, aµ−1) and eigenvectors are invariant to scale, this completes the proof.


Proof of Theorem 16

We will show that the derivative of the logarithm of the ratio is positive.

First for axy:

∂ log�

eX(A)eY (A)

�

∂axy=

∂

∂axy

�log(axy)− log(ayx) + log

� ey(AY )

ex(AX)

�+

log�1− ayx(1− ex(A

X))�− log

�1− axy(1− ey(A

Y ))��

=1

axy+

1− ey(AY )

1− axy(1− ey(AY ))> 0

Likewise for ey(AY ) we have

∂ log�

eX(A)eY (A)

�

∂ey(AY )=

1

ey(AY ))− axy

1− axy(1− ey(AY ))

=1− axy

ey(AY ))�1− axy(1− ey(AY ))

� > 0

The results for ayx are ex(AX) follow by symmetry.

Proof of Theorem 17

Taking the first-order conditions for each agent’s consumption decision we obtain:

z∗ =

β

1 + βAz

∗ +1

1 + βθ

Which is identical to the FOCs of the investment game with γ = β and c = 1 + β. The

rest of the proof follows from Theorem 11.

Proof of Corollary 18

This follows from Claim 1 of Theorem 12.

Chapter 3

Price competition on a buyer-seller

network.

3.1 Introduction

In standard models of competition, firms can sell to all consumers. Nevertheless, in many

interesting markets, firms have a restricted access to consumers and must compete with other

firms under asymmetric potential consumers. This restriction can come from technological

constraints, for example when consumers have to invest in a technological platform in order

to buy from firms; from geographical constraints, as in land or maritime trade routes; or in-

frastructure constraints, as in the gas, electricity or water markets, where the transportation

infrastructure is fixed in the short run.1

This paper proposes using a network to model the possible transactions between sellers

and buyer by relating the graph-theoretical properties on the network with strategic pricing

and welfare. The network allows for a rich structure of limited interaction between sellers

and buyers. Therefore, we address the age-old question of what lies between the extreme

cases of a perfect monopoly and perfect competition. Networks model are not the first to

allow for local interaction. The closest class of model are the spatial location models, also

called Hotelling models.2 We view network models as providing a tractable alternative for

1In the US, natural gas pipelines are privately owned and offer a a bundled service of purchase anddelivery to to local distribution companies and large industrial buyers. The industry has traditionally beenheavily regulated but in 1996 the Federal Energy Regulatory Commission (FERC) decided to allow market-based transportation rates to allow for a greater supply-demand responsiveness. A necessary condition toderegularize a pipeline was showing it did not have significant market power. For a more detailed analysissee McAfee and Reny (2007).

2For a reference see Tirole (1988).

63

CHAPTER 3. PRICE COMPETITION ON A BUYER-SELLER NETWORK. 64

different environments.

Our main findings are that under price competition, aggregate surplus decreases mono-

tonically as the network becomes less connected but the payoffs for individual firms are

not monotonic. Aggregate surplus is maximizes by a fully connected network which gener-

ates perfect competition. The maximum inefficiency occurs in disconnected networks where

each firm is a local monopolist. We find that stable networks are midway between the two

extremes.

We find that firms do not always want to have access to more consumers because it could

increase the aggressiveness of competition. Stable networks are those where firms balance the

trade-off between more potential consumers and more aggressive competition. We also find

that firms do not want to have the ability to price-discriminate because it makes competition

more aggressive.

Two important previous papers on the matter are Kranton and Minehart (2001) and

Corominas-Bosch (2004). Both papers deal with buyer-seller networks. The main difference

between these papers and ours is the market protocol used to determined prices and alloca-

tions. We model the market through price (Bertrand) competition. Kranton & Minehart use

an ascending-bid auction that simultaneously determines prices for all sellers. Corominas-

Bosch analyzes an alternating-offers bargaining protocol. More recent paper by Blume et al.

(2007) solves a model where traders balance supply and demand by setting a spread be-

tween the price for buyers and for consumers; while Elliot (2010) finds the core allocations

for trades with heterogenous values.

In contrast to our model, all of the cited papers find that the allocation is efficient

conditional on the feasibility constraint imposed the network.3 Even degenerate networks

where firms are monopolists have full efficiency! Therefore their choice of the underlying

economic environment does not allow us to see how the network structure moves welfare

with through it’s influence on the market power of each agent. The only thing we can be

sure of is that the network structure does not add any interesting inefficiency of it’s own.

In our model we allow both full efficiency and for inefficiencies dor the networks that

correspond to the standard cases studied in economics: perfect competition and monopoly

pricing. We then ask how the network mediates the inefficiency as we move get networks

that are more similar to one or the other.

To analyze which networks are likely to form we take two approaches. First we study

the properties of pairwise-stable networks. Pair-wise stable networks are between networks

with local monopolists and the network with perfect competition. Next we study a couple of

3Corominas-Bosch does find inefficiencies in the way the network is formed.


entry games where before the pricing game firms chose which links they want to form with

consumers in the other firm’s niche. It turns out that the equilibrium of the entry games are

always pairwise-stable.

The paper is structured as follows: Section 3.2 sets up the basic duopoly model; Section

3.2.1 solves an example of competition between tollway stations on a road network that

shows our main findings and contains most of the intuition for solving the model; Section

3.2.2 solves for the unique Nash equilibrium of the duopoly model. Section 3.2.3 does some

comparative static exercises for welfare. Section 3.2.4analyze what would change if we allow

for price-discrimination. Section 3.3 analyzes which networks are likely to form. Finally,

Section 3.4.1 extends the results to a case of oligopolistic competition where consumers can

either buy from one firm or all of them and closes with a discussion on the challenges to

solve more general oligopoly models.

3.2 The duopoly model

F1 F2

C1

��

��

C1,2��

��

��

��

C2

��

��

Figure 3.1: A number C1 of consumers can only buy from Firm 1, a number C2 from F2;and a number C1,2 can buy from any firm.

Our model has a finite number consumers and that there are two firms, labeled F1 and

F2. Firms and consumers are represented by nodes on a network. Let C1 denote the number

of consumers who can only buy from Firm 1, C2 the number who can only buy from F2, and

C1,2 the number who can buy from both. Consumers must be linked to a firm to be able to

buy it’s product or service. In Figure 3.1, for example, C1 can only buy from F1 while C1,2

can buy from both F1 and F2.

The network structure is exogenously determined before any transactions are carried out

and is common knowledge between the players. We will discuss how the network is formed

in Section 3.3.

We refer to the C1 and C2-type consumers as locked-in consumers, to the C1,2-type as

mobile consumers and label the firms such that Firm 1 is the one with weakly more


locked-in consumers: C1 � C2.

Consumers can only buy one product. They has a value for the product that is private

information. Products from different firms are perfect substitutes. Values drawn indepen-

dently from a common distribution function. Let Q(p) be the probability that a consumer

has a value higher or equal to p. That is, Q is defined over [0,∞) → [0, 1] and is decreasing.

We assume Q is differentiable.

For tractability we assume that Q has a decreasing marginal revenue in prices.

Q(p)− p∂Q

∂pis decreasing in p

This assumption to gives tractable predictions for firms who only focus on the revenue

from their locked-in consumers. In our model, firms face a trade-off between the gains

from acting more monopolistically and acting more competitively. The decreasing marginal

revenue property ensures there is a unique solution for a firm who acts like a monopolist.

Also, for the price range below this unique solution, any increases in prices lead to increases

in the revenue from “locked-in” consumers. This monotonicity of revenues will be useful in

solving for equilibrium. We discuss this in more detail in Section 3.2.2.

The nodes Cj can also be thought as different locations and we can interpret Q as the

total demand at each location, where we normalized the maximum possible demand at each

location has been normalized to Cj. In this case a sufficient condition to have the decreasing

marginal revenue party is that the demand has a weakly decreasing elasticity, which can be

checked by verifying the demand is log-concave.

Firms engage in Bertrand competition. Each firm simultaneously announces a price it will

charge to any consumer for buying her product. Firms cannot price discriminate. We show

the consequences of allowing price discrimination in Section 3.2.4. Consumers first observe

all prices and then chose to buy from the cheapest firm available to them with probability

Q(p). They randomize equally across firms in case of a tie.

Firms maximize their expected profits. We assume firms have a constant marginal cost

for selling their products which we normalize to zero. Firms have no fixed costs. Therefore

profits are simply the quoted price times the quantity sold, which is determined by the prices

of both firms as above. Let πj(p1, p2) denote the profits for Firm j when firms quote (p1, p2).

A pure strategy for firm j is a simply a price pj she wishes to quote. Firms can quote

any non-negative number.

pj ∈ [0,∞); j ∈ {1, 2}


In general to find equilibria we will need to allow firms to randomly select with a price.

Definition 19. A mixed strategy pricing scheme for Firm j if it is a distribution function

σj : [0,∞) → [0, 1].

Abusing notation, we use πj(σ1, σ2) for the expected profit for firm j when firms are

randomizing according to (σ1, σ2). A subscript −j denotes the firm who is not j. In the

oligopoly section it denotes all firms that are not j.

To solve the game we look for Nash equilibria. Because the behavior of consumers is

completely captured by Q(p), we focus on the strategies for the firms.

Definition 20. A Nash Equilibrium of the pricing game is a strategy profile, (σ1, σ2), such

that no player j can deviate to an alternative mixed strategy σ�j and get a higher payoff:

πj(σj, σ−j) � πj(σ�j, σ−j);∀σ�j �= σj,∀j ∈ {1, 2}

We denote by a star-supercript strategies that constitute an equilibrium, (σ∗1, σ∗2). Let

(π∗1, π∗2) denote the corresponding expected profits.

Next we develop an example to preview how to solve the game and expose some of the

results of the paper. Those interested in going directly to the details of solving the model

can skip to Section (3.2.2).

3.2.1 An Example

Two consumers, A and B, wish to cross a river to go to an Irish pub. To do so, they must

cross through one of two available bridges. Each bridge is independently operated by a firm

that charges a toll to any consumer who wishes to cross. Consumers are willing to pay up to

1 to cross the river. The only cost they incur is that of the toll. In addition, an exogenous

road network limits the choice of bridges for each consumer. For example, in Figure (3.4),

Consumer A can only cross through the bridge operated by F1 while Consumer B can choose

to cross through any bridge. Firms simultaneously post their tolls (prices). After observing

them consumers choose which bridge to cross, if any.

In the Monopolistic Network, Figure (3.2), each firm knows she has a monopoly and

quotes prices to extract all surplus:

p∗1 = p∗2 = 1

π∗1 = π∗2 = 1


The Irish Pub��

��

��

F1 F2

CA

��

CB

��

Figure 3.2: (The Monopolistic Network) Each Firm is a monopoly in their local market

The Irish Pub��

��

��

F1 F2

CA

�� CB

��

Figure 3.3: (The Competitive Network) A network with perfect competition

The Irish Pub��

F1 F2

CA

��

CB

��

Figure 3.4: (The Mixed Case Network) An intermediate case


The Competitive Network, Figure (3.3), is a standard Bertrand competition where firms

undercut each other until they vanish their profits.

p∗1 = p∗2 = 0

π∗1 = π∗2 = 0

In the Mixed Case Network, Figure (3.4), F1 has a locked-in consumer (C1) from where

she can extract all her valuation by charging p1 = 1. Alternatively she can try to compete

against F2 for the mobile consumer, CB, at the cost of reducing the price charged to her

locked-in consumer, CA. No pure strategy Nash equilibrium exists in this game: If any firms

charges a positive price, at least one of the two firms would want to charge slightly below.4 If

both Firms charged zero, F1 could profitably deviate to charging 1, extracting all the surplus

of her locked-in Consumer. This is the same logic we use to rule out pure strategy equilibria

in the duopoly model.

There is unique a mixed strategy equilibrium. This is described below with it’s corre-

sponding expected profits.

F1 : Post p = 1 with probability 12 . With probability 1

2 chose p < 1 and then price according

to:

σ∗1(p|p < 1) = 2− 1

p;∀p ∈ [

1

2, 1)

F2 : Price according to:

σ∗2(p) = 2− 1

p;∀p ∈ [

1

2, 1)

π∗1 = 1; π∗2 =1

2

From the example we can already see that firms are not always better off by having access

to more consumers. In the Mixed Case Network, Firm 2 receives a payoff strictly larger than

in the Competitive Network because the more aggressive competition dissipates profits.

Another result that generalizes to the duopoly model is that in equilibrium Firm 1 receives

a payoff exactly equal to the monopoly rents from her locked-in consumers. It is fairly direct

to deduce her payoff could not be smaller than this. The fact that it’s exactly the same

comes from Firm 1’s has a higher opportunity cost for charging low prices which places it at

a disadvantage to compete for the mobile consumers. This does not mean Firm 1 only sells

4Such a firm would in fact not have a best response because of an openness problem.


to it’s locked-in consumers. As we see in the example she sells to all potential consumers

with some probability.

Equilibrium pricing strategies for our model are also qualitatively similar to those in the

example. Firm 1 quotes the price the maximizes her monopoly payoffs (1 in the example) with

a discrete probability and both firms mix continuously over prices below. In the example,

both firms can be sure to sell to both consumers if they set a price equal to 1/2. This is the

minimum price Firm 1 is willing to quote. Any price lower must forcefully yield an expected

profit lower than what she can assures herself by charging the monopoly price. Since quoting

lower prices is strictly dominated for Firm 1 they are also ruled out for Firm 2.

3.2.2 Solving the duopoly model

Let’s return to analyzing an arbitrary network with two firms and any number of consumers.

To solve the model it’s useful to derive two useful properties from the decreasing marginal

revenue property of the distribution of consumer values, Q(v). First, there is a unique price

pM that maximizes revenue from locked-in consumers. It turns this price is the same price

for all firms because the number of ”locked-in” consumers only change the optimization by

rescaling it. Second, over the price range below below pM any increase in prices yields higher

revenues from the locked-in consumers.

Proposition 21. For any firm with Cj > 0 consider the maximization over the revenue from

only her locked-in consumers:

maxp∈[0,∞)

pQ(p)Cj

If Q(·) has the decreasing marginal revenue property there is a unique price, labeled

pM , that solves the monopolist’s problem. This price is independent of Cj. Furthermore, for

any prices p, p� such that p < p� � pM , p� yields a higher revenue than p from the locked-in

consumers.

Proof. First we show existence. Note that the limit of the objective function as p goes to

infinity is zero. For low values of p it is strictly positive. Therefore we can restrain attention

to a compact subset of the price space. The theorem of the maximum guarantees existence.

Now we show uniqueness. Take the derivative of the objective function:

Cj

�Q(p) + p

∂Q

∂p

�

By the decreasing marginal revenue property derivative of the objective function crosses

zero only once and from above. Call pM the value where it crosses zero. This is the unique


maximum and does not depend on Cj. Furthermore, for values below pM , the derivative is

positive, so any infinitesimal increase in the price increases revenues. By the fundamental

theorem of calculus, the change in revenue between any two prices below pM is equal to

the integral of this derivative, therefore revenue increases monotonically until it reaches it’s

maximum.

A direct implication of Proposition 21 is that any price above pM is strictly dominated:

decreasing the price to pM would strictly increase the revenue from the locked-in consumers,

while weakly increasing the probability of winning the mobile consumers.

Proposition (22) shows that there are no pure strategy Nash equilibria except for two

extreme cases. One is the standard Bertrand competition which corresponds to C1 = C2 = 0;

the other is when each firm is a separate monopolist, C1,2 = 0.

Proposition 22. There is no pure strategy equilibria for the model with C1 > 0 and C1,2 > 0.

Proof. An pure strategy equilibrium with with a positive price can be ruled out by the

standard Bertrand argument as follows. Take a strategy profile (pj, p−j) such that p−j <

pj � pM , then player −j would wish to deviate to pricing arbitrarily closer to pj. Now

assume pj = p−j > 0, then any player could deviate to an arbitrarily smaller price obtaining

a discrete gain from avoiding the probability of a tie while charging essentially the same

price. Finally, assume (pj, p−j) = (0, 0), then F1 could deviate to extracting a positive profit

by quoting pMand selling to her locked-in consumers.

We therefore look for mixed strategy nash equilibria. It will turn out that the equilibrium

is unique and involves mixing over a continuous interval. Proposition 23 solves for the

equilibrium payoffs. To obtain them the proof first solves for some standard properties

of continuous mixed strategies and Bertrand competition: ties cannot occur with positive

probability; strategies have no atoms below the upper bound on prices, pM ; strategies must

have a common support for the two firms; strategies cannot have gaps in their support and

must not have gaps below the upper bound on prices, pM . These are enough to pin down the

equilibrium payoffs. Corollary 24 solves for the unique equilibrium strategies by plugging

the equilibrium payoffs into the mixed strategy indifference condition.


Proposition 23. The unique equilibrium payoffs for the duopoly network pricing game are:

F1:

π∗1 = pMQMC1

F2:

π∗2 = π∗1C2 + C1,2

C1 + C1,2

Proof. See Proofs Section (3.4.2).

Corollary 24. The unique mixed strategy equilibrium for the duopoly network pricing game

are:

For F1: “Stay out” with probability�1− C2+C1,2

C1+C1,2

�by charging the monopoly price;

For F2: Always “go in”: charge below pM with probability 1.

For Both: Conditional on going-in mix according to:

σj(p|p < pM) = 1− C1

C1,2

�1− QMpM

Q(p)p

�

Proof. See Proofs Section (3.4.2).

3.2.3 Welfare

We can use the payoffs in Proposition (23) to do comparative statics on welfare by changing

the number of consumers.

Firm 1’s payoff does not depend on the size of the mobile market, C1,2, nor on the size

of Firm 2’s locked-in market, C2. She gets a profit which is exactly the same as if extracted

the monopoly surplus from her locked-in consumers. This does not mean Firm 1 doesn’t sell

to the mobile consumers. She does with positive probability. Nevertheless, in equilibrium

Firm 1 must be made indifferent between quoting the monopoly price and quoting below.

As such the mixing must be such that any increase in expected sales by lowering prices must

be exactly offset the decrease in price.

Firm 2 has a profit that is strictly increasing the number of her own locked-in consumers.

Her profits are also strictly increasing in the number of mobile consumers, C1,2. Surprisingly

her payoffs are also strictly increasing in the number of locked-in consumers Firm 1 has.


The driving force is the effect on competition. A larger number of C1 makes Firm 1 less

aggressive, which is beneficial for Firm 2. Firm 2’s profits are smaller than those of Firm 1

but larger than the rents from the monopoly rents out of her locked-in consumers.

π∗1 � π∗2 � C2pMQM

The inequalities are strict whenever, respectively, C1 > C2 and C1,2 > 0.

Every consumer is weakly better off by any increase in the number of mobile consumers,

C1,2, because the distribution of prices shifts downwards. The old distribution first-order

stochastically dominates the new distribution.

Aggregate surplus is maximized by equating demand to marginal cost. Because of we

normalized marginal cost to zero this is equivalent to producing to fulfill all demand. Higher

prices create deadweight loss.

3.2.4 Price discrimination

Firms would prefer not to be able to price-discriminate between mobile and locked-in con-

sumers. If both firms could discriminate in prices, profits from the mobile consumers would

completely dissipate. Firm 1 would be indifferent between being able to price discriminate

or not, she doesn’t receive any profit from mobile consumers anyway. Nevertheless, if there

were an arbitrarily low cost from being able to price-discriminate, she would be deterred

from trying. Firm 2 would strictly prefer price discrimination were not possible as long as

C1 > C2.

Each firm would wish to be able to discriminate if the other firm could not. If a firm were

the only one with the ability to discriminate, she could compete for the mobile consumers

without sacrificing any rents from her locked-in consumers. The discriminating firm would

get the same profit from the mobile consumers as Firm 2 in our duopoly model with C2 = 0.

The firm’s total profit would be that plus the monopoly rents from it’s locked-in consumers.

As long as the other firm had some locked-in consumers, the discriminating firm could get a

positive gain from this.

The table below summarizes this. All payoffs have to multiplied by pMQM . This table

is not symmetric for the case where both firms cannot price-discriminate because Firm 2 is

able to use her lower opportunity cost for lowering prices to compete more agressively.


F2 can discriminate F2 cannot discriminate

F1 can discriminate C1,C2 C1 + C2C1,2

C2+C1,2, C2

F1 cannot discriminate C1, C2 + C1C1,2

C1+C1,2C1,C1

C2+C1,2

C1+C1,2

We are not proposing the table above as a game that is played by the firms. Whether

price discrimination is possible or not, and how costly it would be to implement, depends on

the application at hand. For example, in the bridge example it might not be cost-effective to

identify the geographical origin when a large number of consumers is crossing at rush hour.

3.3 Which networks are likely to form?

Our analysis up to now has taken the network as given. Given that the network structure

can have a big influence on welfare it is important to know which networks are more likely

to arise. To do so we will proceed in two different ways.

We will first ignore the precise protocol of how links are formed and and look for a

solution from cooperative game theory. We will look for networks that are pairwise-stable

with respect to the payoffs induced by the pricing game. We find that stable networks are

always between the two extreme cases of networks with local monopolists and networks with

perfect competition.

We will next analyze two “entry” games where we precisely specify how and when firms

can form links. These games will consist of two stages, an entry stage and a pricing stage. The

network will be determined at the entry stage after which the pricing stage will start treating

the network as given. The pricing stage will correspond exactly to our pricing model. It will

turn out that the equilibria of the entry game are a subset of the pairwise-stable networks.

Entry games are more than a device to understand which networks are likely to form.

They are interesting in their own right and have a long tradition in game-theoretic models

of industrial organization.5 Their aim is to understand under what circumstances and by

what actions, if any, can an incumbent firm credibly deter competitors from entering their

market.

The entry games considered here are different from the traditional ones in that entry

can be partial: we allow firms to establish links with some consumers without having allow

access to all consumers in the incumbent’s niche.5Tirole (1988) is the required reference.


3.3.1 Pairwise-stable networks

Fix the total number of consumers,6 for a given network we ask if a firm has an incentive to

add or delete links to the existing consumers. By adding and deleting links, firms change the

network. The payoff for a firm from a network will be her expected payoff of the pricing game

that would be carried out under such a network. When we say firms have incentives to add

or delete links, we are doing a comparative statics across the payoffs from different networks

in our pricing model. There is no cost to forming links. Unlinked firms and consumers get

a payoff of zero. We want to characterize all the networks that are pairwise-stable.

Definition 25 (pairwise-stable networks). A network is pairwise-stable if :

• No unlinked firm-consumer pair can become linked and increase the payoff of both

members.

• No firm or consumer can delete one of their links and increase their individual payoff.

The Competitive Network of the tollway example we analyzed in Section 3.2.1 was not

pairwise-stable. Any of the two firm could increase their profit by deleting a link. Both

the Monopolistic Network and the Mixed Network were pairwise stable, even though Firm

2 received a strictly smaller payoff in the later. This because Firm 1 has no strict incentive

to add or remove the link that connect it to Consumer B.

Only the firms’ incentives determine if a network is pairwise-stable because, as we saw

in the welfare section (Section 3.2.3), adding a link between any firm and any consumer

increases the expected payoffs of all consumers.

Firms would never want to delete links to their locked-in consumers and would always

want to add a link to consumers that are not linked to the other firm.

Firm 1 is always indifferent between adding links or not to consumers that are linked to

Firm 2. These would only increase C1,2 and decrease C2 by one, but this does not affect her

equilibrium payoffs.

Firm 2 would have an incentive to add a link to a consumer of type C1 if and only if:

C2 + C1,2 + 1 < C1

That is, only if her potential consumer once the link is added is not larger than Firm 1’s

locked-in market.6This is one of the only parts of the paper where the fact that the number of consumers are integers

matters. Multiplying the number of all consumers by a constant does not change the the strategic incentives.Thus allowing the number of consumers to be a continuous variable can be thought as an approximation tolarge numbers of consumers with some adequate rescaling.


Except for integer constraints, Firm 2 would want to increase links at low levels of C1,2

but would want to decrease them at high levels. (As long as C2 < C1) In this sense, pairwise-

stable networks are bounded away from the networks with local monopolists and from those

with perfect competition.

3.3.2 Entry game 1

Suppose Firm 1 is the incumbent and initially has links with all potential consumers. Firm

2 enters by forming links with the consumers. She can decide to form any number of links.

After Firm 2 makes her decision, the network is fixed and the pricing game is played out.

How many links would Firm 2 wish to form? What would be the final network formed?

Firm 2 solves:

max0�E�C1

(C1 − E)pMQM E

(C1 − E) + E

Where E is the level of entry measured by the number of links formed to enter.

Using the results in Section 3.3.1 we know that Firm 2 would continue to add links while

the following condition holds:

E + 1 < C1

Therefore Firm 2 would stop forming links much before she takes away all of Firm 1’s

locked-in market, to keep Firm 1 from being too aggressive in the subsequent pricing game.

Roughly speaking she would only enter half of the potential market.

A similar effect has been previously shown in the entry games literature. It was labeled

by Fudenberg and Tirole (1984) as the “fat cat effect”. The “fat cat effect” refers to a

situation of strategic over-investment in capital by the incumbent firm to act less agressive

conditional on an entry she knows she cannot deter. The effect is also present in the multi-

market oligopoly model of Bulow et al. (1985).

Our focus on the entry stage is different than the previous models. There the incumbent

chose to be a “fat cat” by over-investing in capital. In our model it’s the entrant who choses

to keep the incumbent as a “fat cat” by only entering partially.

Would Firm 1, given a chance, wish to delete links to her initial consumers to accomodate

Firm 2 in better terms? Could she perhaps even deter entry by sustaining a “lean and

hungry” stance described by Fudenberg and Tirole (1984)? The answer is negative. If Firm

preemptively broke some links, Firm 2 would form links to any unlinked consumer and then


continue to form links while the condition holds:

C2 + C1,2 + 1 < C1

The final amount of C1 would remain unchanged. 7 Since Firm 1’s payoff only depends

on this amount, she would never wish to break links to accomodate Firm 2.

3.3.3 Entry game 2

Suppose now that initially Firm 1 and Firm 2 have two separate monopolies. That is, at

the initial stage there are only locked-in consumers. We keep the convention that Firm 1

starts out as the dominant firm: C1 > C2. The game starts with the the entry stage where

firms are allowed to simultaneously form links with consumers in the other firms locked-in

market. Once all link are formed the network is fixed and the pricing game is carried out.

This could be the case of two separate regional monopolies held in place by a restriction

to sell across borders. The entry stage occurs after a free trade agreement lifts restrictions

across borders, but firms still have to decide in which locations across the border they want

to set-up a point of sale. It could also represent two similar products that in the pre-entry

stage could not reach the same consumers because of regulation or technological restrictions.

This happened in the phone and cable service markets where technological advances allowed

cable providers to supply phone services and viceversa, although companies still had the

ability to decide in which regions they would operate.

In the entry stage firms simultaneously announce the number of links they will form with

the other firm’s locked-in consumers. This announcement is their level of entry. We seek to

find how many links would each firm wish to form in a Nash equilibrium.

Proposition 26. The set of Nash Equilibria of the Entry Game 2 is:

E∗1 ∈ Integer

�[0, C2]

�; E∗

2 = Integer�(C1−C2

2 − 1, C1−C22 ]

�

Where the Integer function maps intervals to the set of integers inside that interval.

Proof. Their profits from a strategy profile (E1, E2) are π1(E1, E2) as described below.

π1(E1, E2)

pMQM=

�(C1 − E2); if C2 − E1 � C1 − E2

(C2 − E1)(C1+E1)(C2+E2) ; if C2 − E1 > C1 − E2

7Unless Firm 1 broke links with more than half her locked-in consumers in which case she would have aneven smaller payoff!


π2(E1, E2)

pMQM=

�(C1 − E2)

(C2+E2)(C1+E1) ; if C2 − E1 � C1 − E2

(C2 − E1) if C2 − E1 > C1 − E2

First we rule out any strategy profile that makes Firm 2 the dominant firm after entry.

Take any strategy profile such that:

C2 − E1 > C1 − E2

Firm 1’s best response would be to set her entry level as low as possible: E1 = 0. For this

strategy Firm 2’s best response is Integer[C1−C22 ]. This level of entry is not enough to make

Firm 2 the dominant firm after entry. Therefore there are no equilibria where this happens.

If after entry Firm 1 is still going to be the dominant firm, then Firm 2’s unique best

response is:

E∗2 =

�(C1 − C2

2− 1,

C1 − C2

2]�

For this profile Firm 1’s best response set is:

E∗1 ∈ Integer

�[0, C2]

�

In equilibrium Firm 2 always enters partially. The level of entry is always low enough

such that Firm 1 still remains the dominant firm after entry. Firm 1 has multiple best

responses but has no strict incentive to enter Firm 2’s locked-in market. Doing so does not

alter her payoff nor alter Firm 2’s preferences over entry levels. Adding an arbitrarily small

cost to forming links would reduce the set of equilibria to a unique strategy profile where

only Firm 2 enters as described above.8

Consumer surplus and aggregate surplus strictly increase with the partial entry.

In equilibrium the Firms have no incentive to change they’re links. Because only firms’

incentives matter to determine which networks are pairwise stable, equilibrium networks of

the entry games must necessarily be pairwise stable.

8Except for the case where C1−C22 is an integer. In that case adding an arbitrarily small cost would

diminish Firm 2’s entry level by one link.


3.4 Extending the results for oligopolistic competition

For applications it would be useful to extend the model to allow for an arbitrary number

of firms. Unfortunately it quickly becomes intractable. The potential types of consumers

include all possible subsets of firms. These grow exponentially.

Nevertheless, we can solve a special case of oligopolistic competition where consumers

come in two types: locked-in consumers who can only buy from one firm and perfectly mobile

consumers who can buy from all firms. We will find that most of the pricing behavior and

payoffs extend to this model.

There are several applications that fit this type of environment. For example, a situ-

ation where some people are already locked into a specific technology while other people

are waiting for the prices to be determined before they decide which technology to adopt.

Small fixed-costs for adopting technology could be easily included in the model. Another

pertinent environment is one where some consumers buy online and can see the all the price

information, while others just go to their local store and cannot react to prices from other

firms. A third application would treat the network as a model of brand loyalty. Firms face

a trade-off between tendering to completely loyal consumers or competing for extreme price

seeking consumers.

In this model the pricing action will happen between the most agressive firms, those

with the lowest opportunity cost for lowering their prices. All other firms will give up on

capturing the mobile market and stay out by pricing the monopoly price.

We solve this model in Section 3.4.1. We will discuss what we know and what we don’t

for the general oligopolistic model in Section 3.4.2.

3.4.1 An arbitrary number of firms competing in a single market

The model we are considering has a finite number J of firms each with some of locked-in

consumers.9 We label firms such that:

C1 � C2 � . . . � CJ > 0

In addition to their locked-in consumers, all firms can compete to sell in a large global

market that has a number CG of consumers. Consumer values and the pricing game are as

before.9Our main results for strategies and payoffs are still true when some firms might not have any locked-in

consumers. We rule this out for ease of exposition.


There is always an equilibrium where Firms J and J−1 play as described in the duopoly

model while all other firms price pM . We refer to this pricing behavior as “staying out”

because the equilibrium probability that they will capture the global market is zero. Propo-

sition (27) shows proves that this is an equilibrium.

The intuition behind the proof is that firms face a trade-off between extracting all the

surplus from their locked-in consumers or lowering their price to try to capture to the global

consumers. There are two effects that keep firms with more locked-in consumers out. They

face a higher opportunity cost from lowering prices; and they to face an fiercer competition

if they enter because to win the global market they have to quote the minimum price of ALL

firms going in. This two forces move in the right direction to sustain the equilibrium.

These strategies turns out to be the unique equilibrium when we assume that only two

firms have the have the smallest number of consumers. That is, when CJ−2 > CJ−1. This

is proven in Proposition (28). Uniqueness follows from the fact the is a price interval where

firms J and J − 1 must necessarily mix as in the duopoly model. But higher firms cannot

be made indifferent between quoting the monopoly price and a lower one. This is proven in

Proposition 28.

Proposition 27. The following strategies always constitute an equilibrium of the oligopoly

model:

σJ(p) = 1− C1

C1,2

�1− pMQM

pQ(p)

�

σJ−1(p) =

�CJ+CG

CJ−1+CG

�1− C1

C1,2

�1− pMQM

pQ(p)

��for p < pM

1 for p = pM

The strategies for j > J − 1 are: Quote pj = pM with probability 1.

Firms J − 1 and J play as in the duopoly model, all other firms just quote the monopoly

price.

Proof. See Proof Section (3.4.2).

Proposition 28. The strategies in Proposition (27) constitute the unique equilibrium strat-

egy profile if CJ−2 > CJ−1.

Proof. See Proof Section (3.4.2).

The additional assumption in Proposition (28) is necessary for uniqueness. For example,

if it fails we can construct symmetric equilibria where all the firms tied at the bottom

enter. Nevertheless all equilibria are payoff equivalent and only firms with the two lowest Cj

parameter can actively mix.


3.4.2 Remarks on the general oligopoly model

What can we say for networks with any number of consumers who can arbitrarily buy from

any subset of the firms?

Existence of equilibrium is guaranteed by the main the existence theorem for discontin-

uous games in Dasgupta and Maskin (1986).10 Broadly, for the theorem to hold we require

the following:

• The set of pure strategies be a compact set of Rm. We fulfill this by restricting

attention to the game with prices between zero and the monopolist price, because prices

above are weakly dominated for all firms. Any equilibrium of the restricted game will

be an equilibrium of the unrestricted game.

• Payoff functions must be continuous except for subset of a continuous man-

ifold of dimension smaller than the strategy space. This continuous manifold

can be defined in our model as follows. For each j ∈ 1, . . . , J define:

A∗(j) = {(p1, . . . , pJ) ∈ [0, pMonopoly]J : ∃i �= j with pj = pi} (3.1)

• The sum of all payoffs must be upper semi-continuous and each individual

payoff must be weakly lower semi-continuous. See below for the definition of

upper semi-continuity and weak lower semi-continuity.

Definition 29. A function f : RN → R is upper semi-continuous if:

lim supx→x0

f(x) � f(x0)

Definition 30. A profit function πj(pj,p−j) is weakly lower semi-continuous if ∀pj ∈A∗(j),∃λ ∈ [0, 1] such that ∀p−j ∈

�A∗(−j),

λ lim infpj�pj

πj(pj,p−j) + (1− λ) lim infpj�pj

πj(pj,p−j) � πj(pj,p−j)

.

10Indeed, this theorem was motivated by Bertrand competition models like these.


Theorem 31. Dasgupta and Maskin (1986) Existence of equilibria in discontinuous games.[section]

Let Aj ⊂ R (j ∈ {1, . . . , J}) be a closed interval and let πj : ⊗Ai → R (j ∈ {1, . . . , J}) be

continuous except on a subset A∗∗(i) of A∗(i) as defined in (3.1). Suppose�J

j=1 πj is upper

semi-continuous and πj(pj,p−j) is bounded and weakly lower semi-continuous in pj. Then

the game possesses a mixed-strategy equilibrium.

Corollary 32. For any possible network configuration in our model there exists a mixed

strategy Nash equilibrium.

Proof. The sum of all firms’ payoffs is simply the aggregate demand at the lowest available

price to each consumer, it is therefore continuous, a particular case of upper semi-continuous.

The discontinuity points in each players individual payoffs only occur when two or more

players are tied for the lowest offered price for consumers they all have links to. The limit

from above for this utility function is the firm’s payoff at such a price losing those consumers;

the limit from below is payoff at such a price selling to all those consumers. Choosing λ

equal to 1/2 makes the expression exact when two firms are tied and verifies the inequality

it for prices where multiple firms are tied.

Even though existence is guaranteed, the guess-and-verify approach we took in Section

(3.4.1) does not work. The candidate strategy profile is no longer an equilibrium if there

is at least single consumer that can only buy from two or more of the firms who stayed

out (j > J − 1) because such firms would be tying over a shared consumer with positive

probability. This cannot happen in price competition.

It is also not easy to construct new equilibria. To build the equilibrium for the special

case of many firms, we showed that firms with more locked-in consumers were less willing to

lower prices. For arbitrary networks this not true. When consumers can buy from arbitrary

subsets of firms, then those with a higher number of locked-in consumers no longer necessarily

might also have access to larger number of mobile consumers. It is no longer easy to figure

out which firms actively mix at every price range.


Proof sections for the chapter.

Solving for equilibrium payoffs of the duopoly model.

(Proposition 23.)

Proof. Suppose σ = (σ1, σ2) is a strategy profile. We know σ must hold the mixed strategy

indifference condition:

pQ(p)�Cj + C1,2

�1− σ−j(p)

�� πj(σ1, σ2);∀p

With equality almost everywhere with respect to σj. First, note that very low prices can

be ruled out by iterated deletion of strictly dominated strategies: Firm 1 would never charge

a price so low that selling to all consumers would yield less than her monopoly rents. Firm

2 would also never charge prices strictly below this point. By charging arbitrarily close to

Firm 1’s entry point, Firm 2 can always guarantee herself a positive payoff. We now solve

for some standard properties of equilibria that involve mixing over a continuous interval:

The no atoms property Pricing strategies have no atoms below pM . If a Firm j had an

atom in her distribution at some price p < pM , then there would be an � > 0 such that

F−j cannot be optimizing by mixing over [p, p + �). But then Firm j would not be

optimizing because by the decreasing marginal revenue property she could charge p+ �

and increase her profits for her locked-in consumers without affecting the probability

of selling the mobile consumers.

The no ties property Ties cannot occur with positive probability. We already ruled out

ties below pM by the no atoms property. Ties at pM only happen if both firms have an

atom at pM . This cannot happen in equilibrium because any firm could profitably de-

viate to charging arbitrarily close to pM , obtaining a discrete increase in the probability

of winning the mobile consumers.

The common support property The support of both firm’s the mixed strategy must be

the same. On the contrary, suppose a firm is the only one mixing over an interval,

then such a firm is not optimizing because she could shift probability mass to the

upper-bound of such an interval.

The no gaps property In equilibrium there cannot be an interval (p�, p��) with p�� pM

such that some firm mixes below p’ but no firm mixes in (p�, p��). Suppose there is such

an interval, then charging p�� has the same probability of winning the mobile consumers


than p� with a higher price. Therefore charging p�� is strictly better than charging prices

arbitrarily close to p’. Any firm mixing close to p� would not be optimizing.

The mix to the top property σj(p) < 1 for all p < pM . Suppose not. Let p be the upper

bound for the support Firm j’s mixed strategy such that p < pM . From the common

support property we know that Firm −j is also mixing up to p. Firm −j would not

be optimizing because close to p she has an arbitrarily low probability of winning the

mobile consumers and could shift probability mass to charging the monopoly price

from her locked-in consumers. If Firm −j had no locked-in consumers she would be

making profits arbitrarily close to zero, which cannot happen in a equilibrium.

Using the mix to the top property and the mixed strategy indifference condition we know

that the equilibrium profits for both firms must equal the limit of expected profits as prices

tend to pM . From the no ties property we know that this limit must be equal to monopoly

revenues for locked-in consumers for at least one of the two firms:

No ties property ⇒ limpj→pM

πj(pj, σ−j) = pMQMCj

For some j ∈ {1, 2}.As in the example in Section 3.2.1, this must be Firm 1. To see that, note that there is

a unique price, p, that makes Firm 1 indifferent between capturing the whole market and

extracting her monopoly rents. This price solves:

pQ(p)(C1 + C1,2) = pMQMC1

Firm 1 would never charge below this price. By the no atoms property Firm 2 can always

guarantee herself, in equilibrium, a payoff equal to capturing the whole market at this price.

This profit is higher than her own monopoly profits:

pMQMC2 � pMQMC1 = pQ(p)(C1 + C1,2)

We conclude that Firm 1 cannot get a payoff higher than her monopoly rents. This

implies that the support of equilibrium strategies must mix all the way down to p. By the

no atoms property, Firm 2’s profits are the equal to:

pQ(p)(C2 + C1,2) =pMQMC1

C1 + C1,2(C2 + C1,2)


Solving for equilibrium strategies for the duopoly model.

(Corollary 24).

Proof. From the mixed strategy indifference condition we get:

pQ(p)(Cj + C1,2(1− σ−j(p))) � E[πj];∀p

⇒ σ−j(p) � Cj + C1,2

C1,2− E[πj]

pQ(p)C1,2;∀p

With equality almost everywhere with respect to σj. As usual, the payoff for player −j

determines the strategy for player j. From the no atoms property we know this has to hold

with equality at the point where the right hand side expresion zero. From the common

support property we know this is p, the price at which Firm 1 would be indifferent between

capturing the whole market and extracting her monopoly rents. From the no gaps property

and the mix to the top property we know the mixed strategy indifference condition holds with

equality in the range [p, pM). Finally from the monopolist’s problem, Proposition (21), we

know prices higher than pM can never be charged so any excess probability for Firm 1 must

be concentrated at an atom at pM .

Proving that the strategies for the duopoly model are also an equi-

librium of the oligopoly model. (Proposition 27).

Proof. Take any p ∈ [0, pM ]. For each firm will show that the expected profit from quoting p

is smaller than the equilibrium payoff. From the the duopoly model, Section 3.2.2, we know

this is true for firms J and J − 1. Take any other firm j.


πJ(p; σ−J) � πJ(σ)

pQM [CJ + CG(1− σJ−1(p))] � pMQMCJ

pCJ + pCG(1− σJ−1(p)) � pMCJ

pCG(1− σJ−1(p)) � (pM − p)CJ

⇒ pCG(1− σJ−1(p)) � (pM − p)Cj

⇒ pCG(1− σJ−1(p))(1− σJ(p)) � (pM − p)Cj

pCj + pCG(1− σJ−1(p))(1− σJ(p)) � pMCj

pQM [Cj + CG(1− σJ−1(p))(1− σJ(p))] � pMQMCj

πj(p; σ−j) � πj(σ)

Proving that the strategies for the duopoly model are the unique

equilibrium of the oligopoly model. (Proposition 28).

Proof. To solve for the unique equilibrium we must use some of the properties of equilibrium

strategies we found for the duopoly model. Below we provide some clarifications on how

these properties extend to the oligopoly model. The proofs work pretty much as before, we

therefore just sketch them.

• No atoms property: No firm can have atoms in their strategy below their monopoly

price, because it would imply that for a price range above such an atom, no other firm

could mix and be optimizing.

• No ties at the top property: Ties cannot occur with positive probability. Therefore

at least one Firm must quote below pM with probability 1. This guarantees that if

several firms have an atom at pM they will not tie for the mobile consumers with

positive probability.

• Mixing to the top properties: All firms must mix all the way up to the monopoly

price.

• Overlapping support property: For any price range where a Firm is mixing, at least

one other Firm must also be mixing.


Because there can’t be ties at the top, there is at least one firm who doesn’t have an

atom at pM . All other firms must have an expected utility of pMQMCj, by the mixing to the

top property. Define pj as the unique price that makes Firm j indifferent between capturing

the mobile market and extracting the monopoly rents from her locked-in consumers. Since

only Firm J would be willing to price in [pJ , pJ−1) she can always guarantee herself a payoff

higher than her monopoly rents. Therefore she cannot have an atom at the monopoly price.

The equilibrium strategy profile must involve mixing above pj for all j. Firms J and J − 1

are the only one who can be mixing in [pJ−1, pJ−2) and therefore they must do it according

to the strategies in Proposition (27). Repeating the proof in Proposition (27) using the fact

that CJ−2 > CJ−1 we can verify that for any price p such that below it Firms J and J − 1

price with the prescribed strategies, all other firms would strictly prefer to charge pM to p.

Therefore the strategies in Proposition (27) are the unique equilibrium.

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STRATEGIC COMPETITIONS OVER NETWORKS. A …dg223bd6609/dissertation-augmented.pdfFrom Doug I also learned to be extremely focused on delivering the point of the paper. Extremely, extremely,

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