Welfare Implications of Competing Two-Sided Platforms

Karthik Kannan and Thành NguyenKrannert School of Management, Purdue University

Working paper: May 2014

Abstract

We study the welfare implications of two-sided (decentralized) markets involving multiple

intermediaries between buyers and sellers. For the analysis, we employ a stylized model where

the intermediary sets the quantities of ad slots to purchase and sell, and the prices are a conse-

quence of a Cournot model. The mathematics associated with the model is generic enough that

it carries forward to other platform markets as well. Our analysis points to two main results.

First, we find that with Cournot submarkets, the social welfare does not monotonically change

with the number of competing intermediaries. This result is different from the standard Cournot

models. Second, for various interconnected structures of the platform markets, we compute the

price of anarchy as the ratio between social welfare generated in the market and the optima.

We show that when when the marginal cost and marginal value functions are linear, the price

of anarchy is at least 2/3. Lastly, we also present some analysis on network structure and its

implication on the price of anarchy.

1 Introduction

Our interest in studying the welfare implications of competing two-sided platforms was motivated

by the partnership that Google and Yahoo explored in 2008. That partnership would have allowed

Yahoo to use Google’s ad service to intermediate and deliver ads for Yahoo as well as it partners’

sites in the U.S. and Canada; except that the government intervened because of antitrust concerns.

“After four months of review, including discussions of various possible changes to the agreement,

it’s clear that government regulators and some advertisers continue to have concerns about the

agreement. Pressing ahead risked not only a protracted legal battle but also damage to relationships

with valued partners. . . [S]o, we have decided to end the agreement.” (Drummond, 2008). This

incident led us to consider the following question: Are the welfare implications of platform mergers

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any different than mergers involving, say, manufacturing firms?

The aforementioned question continues to be important for various reasons. There are a number

of platform markets where mergers and acquistions are quite common place. Digital ads market that

is observing an explosive growth is also undergoing a transformation with mergers and/or acquisi-

tions of intermediaries (hereafter, interchangeably used with platforms) occurring quite commonly.

A few examples: Amazon began a mobile ad network service in 2013; Microsoft bought aQuantita-

tive in 2012; and Yahoo purchased Interclick in 2011. Similarly, broadband “intermediaries” such

as Comcast and AT&T are also making a large number of mergers and acquistionsy. Yet, there is

little guidance from prior literature (surveyed later in Section 2) to provide insights regarding the

welfare implications.

We began analyzing a marketplace involving multiple intermediaries. At a generic level, the

intermediaries facilitate connections between buyers and sellers (e.g., the delivery of ads from the

advertisers to the publishers’ webpages). An intermediary may serve multiple buyers and sellers.

Also, each of the buyers and sellers may connect with multiple intermediaries. The dependencies

create a networked market structure, which is a key aspect of our analysis.

Our analysis on a decentralized marketplace wherein intermediaries do not fully connect buyers

and sellers is a distinctive contribution to the body of knowledge. Also, we provide a comprehensive

analysis of the social welfare generated in such a context. Specifically, we identify several seemingly

counterintuitive results that provide valuable actionable insights.

We employ a stylized context for our analysis for the ease of the readers. Note that, as alluded

to earlier, ad intermediaries are involved in a decentralized competition as they enable delivery of

ads from advertisers to publishers. We characterize this competition using a Cournot-type model

where the intermediaries set the quantities (of ad-slots) to purchase from the publishers and to

sell to the advertisers. The prices for these transactions are determined by Cournot competitions.

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Our characterization of the Cournot submarket competition is different from classical models of

Cournot competition wherein all the firms compete in one well-defined market. The differing

interconnections mean the welfare implications are also different between the traditional Cournot

model and the Cournot submarket competition analyzed here. In a classical Cournot competition

model, if we increase the number of competitors making quantity decisions, the social welfare and

consumer surplus in the market improves. However, such a monotonic relationship does not always

occur with Cournot submarket competition. We show that having intermediaries and mergers of

intermediaries can be helpful. We also show that the efficiency critically depends not only on the

numbers of the competitors but also on the structure of the interconnections. We define the price of

anarchy as the ratio between social welfare generated and the social optima. Using our model, we

provide bounds on the price of anarchy and also insights about the market structures that generate

the different welfare implications.

2 Literature Review

Our work relates to two main streams of research. The first stream is the literature on platform-

based markets that is quite well established. The second stream is the literature on decentralized

markets that is nascent but growing extensively.

2.1 Platform Economics

In this subsection, we only provide a brief survey of related platform economics literature. Much of

the early focus on this literature was on single-sided markets (see, for example, the seminal work

by Katz and Shapiro, 1985). The literature on two-sided platforms is relatively new (e.g., Parker

and Van-Alstyne, 2005) and there have been quite a few papers that have analyzed the strategies

associated with two sided platforms. Our paper relates to the two-sided markets but we study the

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welfare implications.

In the context of welfare implications, Lee (2014) has studied the problem involving single-

sided networks. Our problem relates to studying welfare implications in two-sided markets. There

have been a few papers that have studied the welfare implications with two-sided markets as well

(Evans and Schmalensee, 2013; Weyl, 2010). Because of our model characterization, we are able to

focus on general heterogeneous network, and study the welfare implication based on the network

structure – a feature not considered in the prior literature. Specifically, our model does not explicitly

characterize the direct externality effects, but we let it naturally arise in our model. Also, our model

involves a market where the buyers and the sell are not fully connected, a feature considered by

the literature on decentralized network markets.

2.2 Decentralized Network Markets

Seminal related work includes Corominas-Bosch (2004), which studies bargaining in networks, and

Kranton and Minehart (2001), which studies the exchange model. Since then, several papers on

decentralized bargaining have been developed (Abreu and Manea, 2012; Manea, 2011; Polanski,

2007). By assumption, all these papers rule out intermediaries, and only focus on the trades

between the buyers and sellers. Recall that our focus is on analyzing the intermediaries

Amongst other papers analyzing equilibrium in the decentralized network context, Bimpikis

et al. (2014) is closely related to ours. They discuss the market power in the context of a cournot

competition model of two-sided markets without intermediaries. As we will discuss later, our convex

program formulation with intermediaries is generic enough to accommodate their model also.

There are a few other papers that have considered the role of intermediaries in the networked

structure and they are closely related to our paper. Nava (2013) does not explicitly account for

intermediaries, but players in that model can choose to intermediate transactions. In an economy

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where the agents decide on the flow of quantities (purchases are negative flows and sales are positive

flows), the paper establishes equilibrium and welfare properties. Blume et al. (2007) consider a

mediated market where the intermediary sets the prices. Their model always implements an efficient

allocation at equilibria. Feldman et al. (2010) study the equilibrium properties in a model where

buyers buy ad slots from a central buyer via a set of competing intermediaries. They demonstrate

how the interaction between the auction design and double marginalization affects the outcomes.

Our paper is different from these prior works in that intermediaries in our model connect multiple

buyers and the sellers, and the prices are determined by the decentralized Cournot (sub)markets.

Our analysis also focuses on the price of anarchy, social welfare, and presents insights about the

competitive market structures.

3 The Model and Equilibrium Characterization

Recall that we characterize the model in the context of ad intermediaries for the ease of reading.

As alluded to earlier, the mathematics associated with the characterization is such that the main

results are valid to a generic platform context as well. Note further that, unlike in a typical platform

economics model, we do not exogenize the network effects. However, the value that one side of

the platform realizes is implicitly allowed to vary with the number of participants through the

intermediaries’ actions.

Now, consider an economy in which there are I ad-intermediaries competing to offer technology

and services that match the K advertisers with advertisement slots available among the J publishers

(e.g., publishers include mobile apps; news websites; social networking sites; blogs; search website).

We use the variables i, j, and k to respectively denote any individual intermediary, publisher,

and the advertiser. Figure 1 shows a possible decentralized competition inter-mediated by ad-

intermediaries that we are seeking to model. Note that to capture the heterogeneity among ad-

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intermediaries we assume a general tripartite network G among I, J and K, where the set of edges

connect the J publishers with I intermediaries and also the I intermediaries with K advertisers

Figure 1: An Example Structure of Networked Competition

We formulate a perfect information, single shot game, where ad-intermediaries are the key

decision-makers. In the first stage, the ad-intermediaries choose the number of ad slots to buy

from the publishers and sell to the advertisers. In the second stage, the publishers and advertisers

simultaneously compete on price. We model the second stage such that the prices, which the

intermediaries pay to the publishers for ad slots as well as the prices per ad-slot they receive from

the advertisers, follow a Cournot-style quantity competition.

To model the interaction with the publisher, let each ad-intermediary i pre-commit to buying

xij number of slots from publisher j in the first stage. Also, let Xj =∑

i xij be the total number of

slots requested from all the ad-intermediaries. Next, let ad-intermediaries simultaneously compete

on price. The unit price for an advertisement slot is given by

Pj = θj + fj(Xj),

where fj is a function that is increasing and (weakly) convex. Without loss of generality, we assume

fj(0) = 0 , For example, a special case is the standard Cournot model Pj = θj + αj(Xj). As in

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the literature of Cournot models Kreps and Scheinkman (1983), see also Nava (2013), Pj can be

interpreted as the inverse supply function of publisher j, which is equal to the marginal cost of

publisher j. Hence, the cost for a publisher j to publish Xj ads is

Cj(Xj) = θjXj +

∫ Xj0

fj(x)dx.

When webpages are cluttered with ads, the disatisfaction it imposes on the viewers can be inter-

preted as a convex cost.

For the interaction between ad-intermediaries and the advertisers, let zijk be the number of

ad-displays (hereafter, simply ads) to be delivered on publisher j’s website that the intermediary

i offers to k. The total ads from k delivered to the publisher j, thus, is Yjk =∑

i zijk. We assume

that the utility that the advertiser obtains is dependent on where the ad is published. Thus, the

inverse demand curve for each advertiser depends a specific targeted publisher. Namely, we assume

given a publisher j, inverse demand curve for k is

Rjk = µjk − gjk(Yjk),

where gjk is also a function that is increasing and (weakly) convex. Without loss of generality,

we assume gjk(0) = 0. Again, a special case is the standard Cournot model Rjk = µjk − βjkYjk.

Similar to the discussion above, these inverse demand curves can be assumed to be the marginal

utility of the advertiser k. Then, the utility of advertiser k is

Uk(~z) =∑j∈J

(µjkYjk −∫ Yjk0

gjk(y)dy.)

Thus, in our model, each ad-intermediary i’s strategy consists of non-negative values xij-s and

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zijk-s if both (ij) and (ik) are connected by an edge in the network G. Furthermore, xij-s and z

ijk-s

need to satisfy the following inventory constraint:

xij ≥∑k

zijk, (1)

which is that the number of ads that i delivers to publisher j cannot be higher than the number of

ad-slots that he purchases from j.

Each ad-intermediary i aims to maximize his payoff, which is the difference between the total

money paid by the advertisers and the total amount paid to the publishers. It is given by

REVi(~x, ~z) =∑jk

Rjkzijk −

∑j

Pjxij =

∑jk

(µjk − gjk(Yjk))zijk −∑j

(θj + fj(Xj))xij . (2)

Recall that

Xj =∑i

xij and Yjk =∑i

zijk.

The game described above is denoted as

Γ(G, I, J,K, θ, µ, f, g).

Formally, we have the following definition of an equilibrium in this game.

Definition 3.1 (~x, ~z) is an equilibrium if they satisfy (1) and no ad-intermediary i can change

~xi, ~zi to improve his pay-off given by (2).

Social welfare and consumer surplus are naturally defined as follows.

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Definition 3.2 Given a strategy profile (~x, ~z), social welfare is

SW (~x, ~z) =∑k∈K

Uk(~z)−∑j∈J

Cj(Xj) =∑k∈K

(∑j∈J

µjkYjk−∑j∈J

∫ Yjk0

gjk(y)dy

)−∑j∈J

(θjXj+

∫ Xj0

fj(x)dx

)

=∑i,j,k

µjkzijk −

∑i,j

θjxij −

∑k∈K,j∈J

∫ Yjk0

gjk(y)dy −∑j∈J

∫ Xj0

fj(x)dx

=∑j,k

µjkYjk −∑j

θjXj −∑j,k

∫ Yjk0

gjk(y)dy −∑j∈J

∫ Xj0

fj(x)dx. (3)

Definition 3.3 Given a strategy profile (~x, ~z), the total consumer surplus is

CS = SW (~x, ~z)−∑i∈I

REVi(~x, ~z),

where SW (~x, ~z) is given by (3) and REVi(~x, ~z) is defined as in (2).

3.1 Equilibrium Characterization

In this section, we show that there always exists a pure equilibrium in the general game. Further-

more, if fj and gk are linear, then the equilibrium is unique and can be characterized by a quadratic

convex program.

We first show that at any equilibrium, all inventory constraints bind.

Lemma 3.4 Given an equilibrium ~x, ~z, then xij =∑

k zijk.

Proof: In any equilibrium, there cannot be any ad-intermediary i that buys more ad slots from

j (xij) than∑

k zijk, because otherwise i improves his payoff by reducing x

ij to

∑k z

ijk.

Because of Lemma 3.4, the payoff of ad-intermediary i (2) can be written as

∑jk

(µjk−gk(Yk))zijk−∑j

(θj +fj(Xj))xij =

∑i,j,k

(µjk−θj)zijk−∑jk

gjk(Yjk)zijk−

∑j,k

fj(Xj)zijk. (4)

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Observe that for every i ∈ I, the utility function above is concave. Furthermore, gk(Yk) and fj(Xj)

are increasing. So, there exists a constant Z such that if zijk > Z then the payoff above is negative.

Therefore, the game we consider is a bounded, concave game. Because of Rosen (1965), such a

game has a pure equilibrium. Formally:

Theorem 3.5 If gk and fj are increasing convex functions, the game Γ(G, I, J,K, θ, µ, f, g) always

has a pure equilibrium.

When fj , gjk are linear functions, in particular,

fj(x) = αjx and gjk(y) = βjky,

the payoff of ad-intermediary i is

Vi(z) =∑i,j,k

(µjk−θj)zijk−∑jk

βjkYjkzijk−

∑j,k

αjXjzijk =

∑i,j,k

(µjk−θj)zijk−∑jk

βjkYjkzijk−

∑j

αjXj∑k

zijk.

The derivative of Vi(z) according to zijk is

(µjk − θj)− βj,kYjk − βjkzijk − αjXj − αj∑l∈K

zijl

Recall that xij =∑

l∈K zijl. Hence, then the first order condition for z to be a Nash equilibrium is

the following.

(µjk − θj)− (βjkYjk + αjXj + αjxij + βjkzijk) ≤ 0; and equality occurs if zijk > 0. (5)

Note that (5) has a special property that allows us to characterize its unique solution by a quadratic

convex program.

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Theorem 3.6 If fj(x) = αjx, gjk(y) = βjky, αj > 0, and βjk > 0, ~z is an equilibrium iff it is the

unique solution of the following convex program with the unknowns ~z, ~x, ~X, and ~Y :

min :∑

jαj2 X

2j +

∑ijαj2 (x

ij)

2 +∑

jkβjk2 Y

2jk +

∑i,j,k

βjk2 (z

ij,k)

2

s.t : αjXj + αjxij + βjkYjk + βjkz

ijk ≥ µjk − θj . (6)

Proof: First, consider the unique solution of (6). By the complementarity and slackness condition,

z, x,X, Y is the solution iff for every i, j, k such that ij and ik are connected in G, there exists a

dual variable λijk such that

αj2 2Xj =

∑ik αjλ

ijk (7)

αj2 2x

ij =

∑k αjλ

ijk (8)

βjk2 2Yjk =

∑i βjkλ

ijk (9)

βjk2 2z

ijk = (βjk)λ

ijk (10)

if αjXj + αjxij + βjkYjk + βjkz

ijk > µjk − θj , then λijk = 0. (11)

Observe that (10) implies that λijk = zijk. Therefore, from (7) , (8) and (9),

Xj =∑ik

zijk;xij =

∑k

zijk and Yjk =∑i

zijk.

Given this, (11) is equivalent to the first order condition in (5).

To see the the reverse direction, given a ~z satisfying (5), we introduce λijk := zijk; Xj =∑

ik λijk;x

ij =

∑k λ

ijk and Yjk =

∑i λ

ijk. It is straightforward to see that z, x, λ,X, Y satisfy (7-11).

Therefore, z, x, λ,X, Y are actually the unique solution of the program (6).

Hereafter, we will focus on the special case where fjand gjk are increasing linear functions. We

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use the convex program technique to investigate the welfare implications.

4 Merging ad-intermediaries: An Illustrative Example

In this section, we begin our analysis using an example to illustrate how the welfare implications

differ between the standard Cournot model and the Cournot decentralized competition involving

intermediaries. In the classical model of Cournot competition, it has been shown that merging of

firms reduces competition and thereby reduces social welfare as well as social surplus. Arguably,

this theory is at the heart of the antitrust policies in the US. However, the example below shows

that the traditional results from Cournot competition does not hold when the competing firms are

intermediaries.

Figure 2: Two different networks: before and after merging of A and B.

We consider the network and its two scenarios as shown in Figure 2. In scenario I, three ad-

intermediaries A, B and C are competing to deliver ads from advertisers 3 and 4 to publishers 1 and

2. In scenario II, A and B merge. Notice that in Senario I, A, and B compete to deliver ad from 4

to 1; but C is the only ad-intermediary between 2 and 3. On the other hand when A and B merge,

the merged firm AB becomes the monopoly that connects 1 and 4; but it becomes a competitor for

C to deliver ad between advertiser 3 and publisher 2.

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Depending on how advertiser 3 values ads placed on publisher 2 relative to valuations of other

advertiser-publisher pairs, the merging of A and B can may or may not improve social welfare.

Interestingly, for a wide range of parameter values, merging A and B improves consumer welfare.

To be more specific, assume the following cost structure. All gjk, fj are linear, with αj = βjk = 1;

furthermore,

µ23 = V ;µ13 = µ24 = 0;µ14 = 1; θj = 0.

With this, we obtain the following comparative result.

Theorem 4.1 If V > 1, then the revenue of AB in after merging is larger than the total revenue

of A and B before merging; furthermore, both social welfare and consumer surplus in scenario II

are also larger than in scenario I.

If 3/7 < V < 1, then the revenue of AB in after merging is less than the total revenue of A

and B before merging; furthermore, social welfare in scenario II is also less than in scenario I, but

consumer surplus is increased when A and B merge.

Proof: The proof involves a straightforward calculation that we omit. Also see Figure 3 for a

plot the welfare, consumer surplus and revenue as V changes for the two scenarios.

The proof is based on the convex program characterization given in Theorem 3.6, which gives

us an easy computation method for characterizing these equilibria in closed forms. Specifically,

because of symmetry and according to Theorem 3.6 in scenario I the equilibrium is characterized

as the solution of the following program:

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Figure 3: Dot lines represent scenario 2, solid lines represent scenario 1

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min : 12X21 +

12X

22 +

12Y

223 +

12Y

214 + (z

A14)

2 + (zB14)2 + (zC23)

2

s.t : X1 + Y14 + 2zi14 ≥ 1, for i ∈ {A,B}. (12)

X2 + Y23 + 2zC23 ≥ V.

In scenario II the equilibrium is characterized as the solution of the following program

min : 12X21 +

12X

22 +

12Y

223 +

12Y

214 + (z

AB14 )

2 + (zAB23 )2 + (zC23)

2

s.t : X1 + Y14 + 2zAB14 ≥ 1 (13)

X2 + Y23 + 2zi23 ≥ V, for i ∈ {AB,C}..

The intuition for the result in Theorem is fairly intuitive. Notice that, in Scenario I, the

path 2-3 is intermediated by C in a monopolistic fashion. In Scenario II, 2-3 connection is not

monopolistically intermediated anymore. This merger also has a cost. In particular, it leads to a

decreased competition connecting 1 and 4. For large values of V, the value from increasing the

competition in the 2-3 connection dominates the decreasing competition in the 1-4 connection.

Using this example, we briefly next consider how our model implications are different from

those in the prior literature. Our result is different from the Bertrand competition model among

intermediaries in Blume et al. (2007). Recall in their model that all equilibria are efficient, and

so merging of intermediaries would not change the price of anarchy. Compared to Bimpikis et al.

(2014), our model impacts social welfare and consumer surplus differently. In our model, the

publishers and advertisers that are connected to the merged firms are directly impacted because of

the merger. These direct impacts occur in addition to the indirect ones. However, because Bimpikis

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et al. (2014) models no intermediaries, mergers of two firms can only lead indirect impacts.

The example also illustrates that the social welfare generated critically depends on the network

structure. It also demonstrates that comparing two arbitrary networks is in general a difficult task.

Further, it shows that mergers can have both positive and negative welfare implications. While

these are idiosyncratic results, policy questions posed earlier require a richer framework that has

to be analyzed. So, we next perform some comparative static analysis using a generic network

structure. Thereby, we are able to provide insights about how the network structures influence the

efficiency. Further, we are interested in bounding the efficiency loss. Additionally, we are interested

in analyzing how parameters of our model have implications on the efficiency-loss bounds.

In the following section, we employ the traditional “price of anarchy” approach from computer

science.1 Later, in Section 6, we evaluate how network structures influence the price of anarchy.

5 Price of Anarchy

First, we will show that with fj and gjk being linear and increasing functions, the price of anarchy

(PoA) is at least 2/3. In other words, the social welfare in the decentralized market is no worse

than 2/3 of what a social planner can achieve. On the other hand, we later provide an example

where PoA is equal to 3/4.

Theorem 5.1 If f and g are linear, with positive coefficients, then the ratio between social welfare

at any Nash equilibrium and the optimal social welfare is at least 2/3.

Proof: Let zijk be the Nash equilibrium. The social welfare with zijk as the decision is

SWNash =∑jk

(µjk∑i

zijk −βjk2Y 2jk)−

∑j

(θjXj +αj2X2j ).

1Recall that given a utility maximizing game, the price of anarchy is the infimum of the ratios between the socialwelfare at an equilibrium over the maximum social welfare.

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Replacing Xj =∑

i,k zijk and Yjk =

∑ij z

ijk, we have

SWNash =∑i,j,k

(µik − θj)ziik −∑jk

βjk2Y 2jk −

∑j

αj2X2j . (14)

Let cijk be the solution generating the optimal welfare, Aj =∑

i,k cijk, Bjk =

∑i cijk and

cij =∑

k cijk. The optimal social welfare is:

SWOpt =∑i,j,k

(µjk − θj

)cijk −

∑jk

βjk2B2jk −

∑j

αj2A2j . (15)

To compare SWNash and SWOpt, we observe that because of (5), we have

cijk((µjk − θj)− (αjXj + αjxij + βjkYjk + βjkzijk)

)≤ 0

Summing over all i, j, k we have

∑i,j,k

(µjk − θj)cijk −∑j

αjAjXj −∑ij

αjxijcij −

∑jk

βjkBjkYjk −∑i,j,k

βjkcijkz

ijk ≤ 0 (16)

We use the following inequalities2

BjkYjk ≤1

3B2jk +

3

4Y 2jk;AjXj ≤

1

3A2j +

3

4X2j

and

cijxij ≤

1

6(cij)

2 +3

2(xij)

2; cijkzijk ≤

1

6(cijk)

2 +3

2(zijk)

2

2These inequalities come from the fact that (γx− 12γy)2 ≥ 0 implies that xy ≤ γ2x2 + 1

4γ2y2.

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to apply to (16) to obtain the following inequality:

∑i,j,k(µjk − θj)cijk −

∑jαj3 A

2j −

∑ijαj6 (c

ij)

2 −∑

jkβjk3 B

2jk −

∑i,j,k

βjk6 (c

ijk)

2

≤ 32(∑

jαj2 X

2j +

∑ij αj(x

ij)

2 +∑

jkβjk2 Y

2jk +

∑i,j,k βjk(z

ijk)

2). (17)

We will show that the Left hand side of (17) is at least SWOpt and the Right hand side of (17)

is 32SWNash. This will prove our theorem.

Focusing on the Left hand side of (17), first observe that

∑ij

αj6

(cij)2 ≤

∑j

αj6

(∑i

(cij))2 =

∑j

αj6A2j ,

and ∑i,j,k

βjk6

(cijk)2 ≤

∑jk

βjk6

(∑i

cijk)2 =

∑jk

βjk6B2jk

Thus,

Left hand side of (17) ≥∑i,j,k

(µjk − θj)cijk −∑j

αj2A2j −

∑jk

βjk2B2jk = SWOpt. (18)

(The last equality follows from (15).)

Focusing on the Right hand side of (17), we will show that the Right hand side of (17) is equal

to SWNash. To see this, observe that from the Nash equilibrium condition (5), we have

∑i,j,k

zijk ·((µjk − θj)− (αjXj + αjxij + βjkYjk + βjkzijk)

)= 0.

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This is equivalent to

∑i,j,k

(µjk − θj)zijk =∑j

αjX2j +

∑ij

αj(xij)

2 +∑jk

βjkY2jk +

∑i,j,k

βjk(zijk)

2 (19)

Replacing∑

i,j,k(µjk − θj)zijk as in (19) to the formulation of the social welfare at Nash (14), we

have

SWNash =∑j

αj2X2j +

∑ij

αj(xij)

2 +∑jk

βjk2Y 2jk +

∑i,j,k

βjk(zijk)

2

This is what we needed to prove.

The following example shows that PoA is at least 3/4. Consider a game consisting of one

ad-intermediary connecting with one publisher and one advertiser.

Example 1 The inverse demand curve for advertiser is 2− z, where z is the number of ads being

delivered. The inverse supply curve for publisher is 1 + x, where x is the number of ad-slots being

sold. The intermediary set x, z such that x ≥ z to maximize (2 − z)z − (1 + x)x. By a straight-

forward calculation, at equilibrium, x = z = 1/4. The Social optimal solution is x = z = 1/2. PoA

of this game is 3/4.

6 The influence of network structure on efficiency

This section uses the equilibrium characterization in Section 3 and the analysis of price of anarchy

to study the influence of the network structure on efficiency. To compute the optimal social

welfare, we abstract away from the set of intermediaries. Let E ⊂ J ×K be the set of node pairs

j ∈ J ; k ∈ K that are connected by at least one node i ∈ I. According to Definition 3.2, the

19

optimal social welfare can be written as:

max~X,~Y

∑j,k

µjkYjk −∑j

θjXj −∑k∈K

∫ Yjk0

gjk(y)dy −∑j∈J

∫ Xj0

fj(x)dx. (20)

s.t : X,Y ≥ 0

Yjk = 0 ∀jk /∈ E

Xj ≥∑k

Yjk

This characterization is intuitive, because only the structure of feasible connection between adver-

tisers and publishers are relevant for computing the optimal social welfare. This gives us a general

way to define social welfare benchmarks. In particular,

Definition 6.1 Let E ⊂ J ×K , the optimal social welfare subject to E is defined as the optimal

value of (20), and is denoted as OPT (E)

Unlike the the optimal social welfare, Nash equilibrium depends heavily on the overall network

structure between I, J and I,K. In the following we show the impact of this network structure on

the efficiency.

Definition 6.2 Given a network G whose nodes are partitioned into three disjoint classes J, I,K.

The edges of G connect nodes between JI and between IK, We define Eτ (G) ⊂ J ×K, which for

simplicity also denoted by Eτ , be the set of node pairs j ∈ J ; k ∈ K that are connected by at least

τ nodes i ∈ I (that is both ji and ik are edges in G).

Our mail result in this section is the following.

Theorem 6.3 Given a network G over the set of publishers, advertisers and ad-intermediaries,

assume fj(x) = αjx; gjk(y) = βjky, for αj , βjk > 0, the social welfare at Nash equilibrium is at

least (1− 12τ+1)OPT (Eτ ).

20

Proof: See Appendix A.1 The proof of Theorem 6.3 is a nontrivial extension of Theorem 5.1.

Notice that for the special case of τ = 1, we recover Theorem 5.1. Intuitively, τ captures the

degree of competition among the ad-intermediaries. For example, if for every pair of publisher

and advertiser jk in G, there are at least τ intermediaries connecting them, the ratio between the

welfare at Nash and the optimal welfare is at least 1 − 12τ+1 . Thus, as τ increases, the system

approaches to full efficiency.

6.1 The role of ad-intermediaries

Next, we provide some discussion and interpretations of Theorem 6.3. In particular, consider an

environment without intermediaries. One can think of this scenario as an environment where each

ad-intermediary is a represented agent for a publisher. This allows us to compare the efficiency of

this game with a scenario where publishers and advertisers are connected by multiple intermediaries.

Figure 4:

Observe that for this network structure E2 = ∅. That is, for every pair of publisher and

advertiser, there is at most one intermediary connected to both of them. Thus, without connecting

to multiple intermediaries, the social welfare at Nash equilibrium is always bounded away from the

optimal social welfare.3

3It is straight forward to construct such an example. For example, for a connected pair j, k let µjk = µ > 0, allother µj′k′ = �

Figure 5: Every pair in the right hand-side network is connected by at least 2 intermediaries.

On the other hand, consider a network on the left hand-side of Figure 5. Every publisher-

advertiser pair are connected by at least 2 intermediaries. The right hand side shows the same

number of publishers and advertisers but each publisher is connected to at least two advertisers.

By Theorem 6.3, the efficiency at Nash equilibrium of the network on the left is at least 45 the

optimal social welfare of network E2 on the right hand side.

Next, observe that all of publisher-advertiser pairs are connected by at least one intermediary,

thus the efficiency at Nash equilibrium of the network on the left is at least 23 the optimal social

welfare of the complete network between publishers and advertisers.

The main take-away from the discussion above is that a system without ad-intermediaries or too

many represented intermediaries can be a barrier for competition and gives rise to inefficiency. The

results are particularly important as FTC mulls the do not track policy. If third parties, which are

the entities targed by the policy, are consequently eliminated, our result shows that the changing

decentralized network structure may have negative implications.

7 Conclusions

We characterize a tractable model of competition among intermediaries, although specified in the

context of ad networks. We show several interesting insights regarding how the network structure

22

impacts the overall social welfare. In conclusion, we wish to highlight two main insights. The first

one is a seemingly counterintuitive one: despite not exogenously modeling externality effects, we find

that merging ad-intermediaries can encourage competition and lead to social welfare improvement.

This result is different from the standard Cournot models. The second one relates to the study of

the price of anarchy. Specifically, we investigate how the network structure influences this measure

of efficiency. We also show that how, from this perspective, intermediaries help increase competition

and improve the social welfare.

A APPENDIX

A.1 Proof of Theorem 6.3

Let Y ∗jk be the amount of ads k shown on publisher j that gives the optimal social welfare OPT (Eτ )

of the τ -subnetwork. That is Y ∗ maximizes

∑j,k

(µjk − θj)Yjk −∑jk

βjk2Y 2jk −

∑j

αj2

(∑k

Yjk)2.

where Y are non-negative and Yjk = 0 if j and k are connected by fewer than τ ad-intermediaries.

Denote

X∗j =∑k

Y ∗jk.

We introduce the following notations. For any pair j, k that are connected by at least τ ad-

intermediaries, let

cjk =Y ∗jkτjk

,

and let cijk = cjk if i is connected to both j and k; and 0 otherwise.

23

To compare SWNash and OPT (Eτ ), we observe that because of (5), we have

cijk((µjk − θj)− (αjXj + αjxij + βjkYjk + βjkzijk)

)≤ 0

Summing over all i, j, k we have

∑i,j,k

(µjk − θj)cijk −∑j

αjX∗jXj −

∑ij

αjcijxij −

∑jk

βjkY∗jkYjk −

∑i,j,k

βjkcijkz

ijk ≤ 0 (21)

We use the following inequalities

Y ∗jkYjk ≤τ

2τ + 1Y ∗jk

2 +2τ + 1

4τY 2jk;X

∗jXj ≤

τ

2τ + 1X∗j

2 +2τ + 1

4τX2i

and

cijzij ≤

τ

2(2τ + 1)(cij)

2 +2τ + 1

2τ(zij)

2; cijkzijk ≤

τ

2(2τ + 1)(cijk)

2 +2τ + 1

2τ(zijk)

2

to apply to (21), and obtain the following inequality.

∑i,j,k

(µjk − θj)cijk −∑j

τ

2τ + 1αjX

∗j2 −

∑ij

τ

2(2τ + 1)αjc

ij2 −

∑jk

τ

2τ + 1βjkY

∗jk

2 −∑i,j,k

τ

2(2τ + 1)βjk(c

ijk)

2

≤ 2τ + 12τ

(∑j

αj2X2j +

∑ij

αj(xij)

2 +∑jk

βjk2Y 2jk +

∑i,j,k

βjk(zijk)

2

)(22)

Similar to the proof of Theorem 5.1 the right hand side of (22) is 2τ+12τ SWNash. In order to bound

the left hand side of (22), we show the following.

τ∑

i,j(cij)

2 ≤∑

j(X∗j )

2. (23)

τ∑

i,j,k(cijk)

2 ≤∑

jk(Y∗jk)

2. (24)

24

Assume that (23) and (24) are true then, the right hand side of (22) is at least

∑i,j,k

(µjk−θj)cijk−∑j

τ

2τ + 1αjX

∗j2−∑j

1

2(2τ + 1)αjX

∗j2−∑jk

τ

2τ + 1βjkY

∗jk

2−∑jk

1

2(2τ + 1)βjkY

∗jk

2

=∑i,j,k

(µjk − θj)cijk −∑jk

1

2βjkY

∗jk

2 −∑j

1

2αjX

∗j2 = OPT (Eτ ).

Hence we conclude that OPT (Eτ ) ≤ 2τ+12τ SWNash, which is what we need to prove.

Thus, it remains to show that (23) and (24) are true. To see (24), observe that because j, k are

connected by τjk ≥ τ ad-intermediaries, and by definition cijk =Y ∗jkτjk

. Thus,

Y ∗jk2 = τjk ·

∑i,k

(cijk)2 ≥ τ ·

∑i,k

(cijk)2.

To see (23), observe that

(X∗j )2 = (

∑k

Y ∗jk)2 = (

∑k

τjkY ∗jkτik

)2 = (∑k

τjkcjk)2 =

∑k

τ2jkc2jk + 2

∑k

we have ∑i

(cij)2 =

∑i

(∑k

cijk)2 ≤

∑k

τjkc2jk + 2

∑k

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27

IntroductionLiterature ReviewPlatform EconomicsDecentralized Network Markets

The Model and Equilibrium CharacterizationEquilibrium Characterization

Merging ad-intermediaries: An Illustrative ExamplePrice of AnarchyThe influence of network structure on efficiencyThe role of ad-intermediaries

ConclusionsAPPENDIXProof of Theorem 6.3