Welfare Implications in Intermediary Networks

Thanh Nguyen∗ Karthik Kannan†

July 2018

Abstract

We study the welfare implications of competing middlemen in a two-sided market, wheregoods are intermediated between providers and purchasers. In our model, the intermediarysets the quantities to purchase and sell, and the prices are a consequence of a Cournot model.Our analysis shows that, unlike markets without intermediaries, mergers of intermediaries cansubstantially improve social and consumer welfare. We also analyze how the underlying networkinfluences the social welfare outcomes. We define parameter wG as the intermediary capacity ofthe network G and show that the price of anarchy is at least 1− 1

2wG+1 . These results suggestan intuitive and simple measure for the level of competitiveness in a networked market involvingintermediaries.

1 Introduction

Because of digital technologies, platform markets – where an intemediary enables connections be-

tween providers and purchasers of services – are becoming popular. In addition to the new age

companies (such as Uber, Google and Facebook), many leading ‘traditional’ ones inlcuding Cum-

mins, Kaiser Permanente, GE, etc. are developing their digital platform strategies. They perceive

the push toward platform business model as even being critical to their survival (Accenture, 2016).

So, there is an extensive and growing literature focused on developing platform strategies from

a single firm’s perspective. However, there are very few prior works studying the implications of

platforms in a competitive environment. Understanding the eco-systems of competing and hetero-

geneous platforms is important because it provides insights for not only individual companies but

also the policy makers. Our interest in this problem was piqued because of the following anecdote.

∗Krannert School of Management, Purdue University, email: nguye161@purdue.edu†Krannert School of Management, Purdue University, email: kkarthik@purdue.edu

1

In 2008, Google and Yahoo proposed a joint partnership that would have allowed Yahoo to use

Google’s ad service to intermediate and deliver ads for Yahoo as well as its partners’ sites in the

U.S. and Canada. The benefits of this agreement were mentioned in Drummond (2008):

“We feel that the agreement would have been good for publishers, advertisers, andusers – as well, of course, for Yahoo! and Google. Why? Because it would have allowedYahoo! (and its existing publisher partners) to show more relevant ads for queries thatcurrently generate few or no advertisements. Better ads are more useful for users, moreefficient for advertisers, and more valuable for publishers.”

However, the agreement never materialized because of antitrust concerns:

“After four months of review, including discussions of various possible changes to theagreement, it’s clear that government regulators and some advertisers continue to haveconcerns about the agreement. Pressing ahead risked not only a protracted legal battlebut also damage to relationships with valued partners. . . [S]o, we have decided to endthe agreement.”

An oft-cited rationale against this kind of proposed agreement has been that mergers and

coalitions among firms increases the monopoly power, and harms both consumer surplus and social

welfare. While this rationale may seem appropriate at first glance because of our understanding of

traditional markets involving only one type of customer, such an analysis does not always extend

to two-sided markets. In fact, as (Drummond, 2008) mentions above, it is possible that such an

agreement allows advertisers of one company to access publishers of the other. By opening up

access to the other market, the coalition may facilitate more transactions between the two sides,

improving social welfare as a whole. To illustrate the complexity of the intermediary networks,

consider the network structure in the two scenarios shown in Figure 1.

In scenario I, three 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. Clearly, trade-offs need to be considered

because of the mergers. Notice that in Scenario I, A and B compete to deliver ads from 4 to 1;

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

2

A

C

B

1

2

3

4

A-B

C1

2

3

4

I: Three intermediary network II: Merging of A and B

Figure 1: Two different networks: before and after merging of A and B.

the merged firm AB becomes a monopoly when serving 1 and 4; however, the merged entity now

competes with C to deliver ads between advertiser 3 and publisher 2. Thus, unlike the traditional

seller-buyer models, the merging of intermediaries can even increase competition. As a result, the

overall efficiency of such an economy depends critically on the underlying network structure. This

means that, in large and complex networks, understanding the implications of antitrust policies

becomes difficult. In this paper, we present this idea more formally. In addition, we ask the

following question. Can we use a simple but intuitive set of network parameters to estimate the

level of efficiency? The answers to this question not only give insights into the nature of competition

among the intermediaries in a networked market but also provide guidelines for conducting policy

analysis when comparing alternate network structures. As the platform markets continue to grow,

one can anticipate more number of partnerships, mergers, and acquisitions. Hence, it is becoming

increasingly important to understand the welfare implications of these markets.

We are already seeing some similar dynamics in platform markets. Consider the transformations

in digital ads market. There are new entrants in this market such as the mobile ad service by

Amazon (began in 2013). Mergers are also quite commonplace in this sector: Microsoft bought

aQuantitative in 2012; and Yahoo purchased Interclick in 2011. Another example of the platform

market are the ride-sharing companies, which is also undergoing a significant transformation. In

that sector, Lyft and Didi Chuxing were initially in a partnership to thwart Uber. Eventually,

3

Uber merged its operation with Didi Chuxing. Yet another example is in the e-commerce space,

where online market-makers are intermediaries connecting sellers and buyers. Amazon and Walmart

competed fiercely to buy the Indian online retailer Flipkart, which Walmart eventually won.

The issues we study in our paper are not just specific to the new age markets, but also are

relevant for other markets with similar structures. The network services segment is transforming

with “intermediaries” such as AT&T acquiring other intermediaries such as DirectTV. Similarly,

in the traditional world, retailers may be viewed as simply intermediating between manufacturers

and consumers. Our analysis can provide insights into mergers and acquisitions in such cases also.

We investigate the welfare implications of a marketplace involving multiple intermediaries.

An intermediary may serve multiple providers (e.g., web publishers with ad-slots, or ride-sharing

drivers) and purchasers (e.g., advertisers, ride-sharing passengers respectively). A provider or a

purchaser may connect with multiple intermediaries. The dependencies create a networked market

structure. We study how the nature of networked structure have welfare implications. In general,

the price clearing mechanisms vary – for example, GSP is used in the ad auctions or dynamic ‘surge

prices’ for ride-shares, etc. We employ a Cournot competition as the price clearing mechanism.

Such a model leads us to identify a unique equilibrium, which is specified by a quadratic program.

With this characterization, we provide several comparative analyses. We show that competition

in an unbalanced market can reduce social welfare. Mergers in sparse markets, however, can create

competition and improve market efficiency. We also study how well the best social welfare obtained

using the networked structure in equilibrium compares with the maximum social welfare by using

the price of anarchy measure. Obviously, the price of anarchy is dependent on the intensity of

competition, which we capture through the term intermediary capacity of the network. Specifically,

we find that the larger the intermediary capacity of the network, the more efficient the equilibrium.

The rest of the paper is organized as follows. In the following section, we survey the related

4

literature. Section 3 formally defines the model, which is analyzed in Section 4. Section 5 provides

some comparative analysis. Section 6 studies the impact of the network structure on the level of

efficiency. Section 8 concludes the paper. Some technical proofs are relegated to the appendix.

2 Literature Review

Our work relates to several streams of research. The first stream is the well-established literature

on platform-based markets. The second stream is the literature on networked markets that is

nascent but growing extensively. We provide a brief survey of the first stream and somewhat more

extended one of the second. Since there is some related work in individual contexts (ad markets,

ride-sharing, etc.) we also provide a brief survey of the related work as the third subsection.

2.1 Platform Economics

Our paper studies intermediaries that connect different sides of a market, as found in the literature

on network-effect based platforms. This domain has been extensively researched. The early focus

on this literature was on single-sided networks (e.g., the seminal work by Katz and Shapiro, 1985)

but has expanded in recent years to include multi-sided platforms (e.g., Parker and Van-Alstyne,

2005, study two-sided networks). Many papers have analyzed the strategic aspects to managing

these network based platforms. For example, they analyze what the pricing strategies should be,

how to launch a network effects based market.

A recent few papers have also studied the welfare implications. Lee (2014) has studied the

problems involving single-sided networks. Others (Evans and Schmalensee, 2013; Weyl, 2010) have

studied it with respect to the two-sided markets. To the best of our knowledge, these papers have not

considered the welfare implications of mergers across the intermediaries. Moreover, the “network”

effects studied in these paper are very different from ours. In particular, most papers in platform

economic literature model symmetric environments and focus on the externality/complementarity

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effect that a platform creates for its members. This is, for example, integrated directly into a

member’s utility that depends on how many others are using the same platform. Our paper, on the

other hand, moves away from such effects to focus on the impact of heterogeneity in the connection

structure. This other type of “network effect” is actively investigated by a relatively new but

fast-growing literature on network markets that we survey next.

2.2 Network Markets

The majority of the early literature on network markets focuses on seller-buyer networks. Kranton

and Minehart (2001), Corominas-Bosch (2004), (Abreu and Manea, 2012; Manea, 2011; Polanski,

2007) and Elliott (2014) are a few examples. By assumption, all these papers rule out intermediaries

and focus only on the trade between buyers and sellers.

Blume et al. (2009) was among the first to investigate a mediated market in a network setting.

The network structure in our paper is similar to that of Blume et al. (2009). However, Blume

et al. (2009) consider Bertrand competition among intermediaries and assume buyers have unit

demand. In two-sided markets, such as ad-networks, considering multi-unit demand and supply

with heterogeneity among the players is more relevant. In the context of ad-intermediaries, agents

target different amounts of impressions, and trades are executed by market-clearing auctions. It

is natural, therefore, to study such a network using a Cournot model, which we do in this paper.

Such a characterization is a unique feature of our model. Because of the differences in the model

characterizations, the equilibrium outcomes are also different. All equilibria in Blume et al. (2009)

are efficient, whereas, that is not the case in our model. Hence, our main aim of measuring the

level of efficiency based on the structure of the underlying network is relevant.

Our paper is closely related to recent models of Cournot competition in networks by Bimpikis

et al. (2018) and Perakis and Sun (2014). However, unlike us, they do not consider intermediaries.

We compare our results with these papers in more detail in the subsequent sections. Bose et al.

6

(2014) also studies intermediaries and market makers in a Cournot game. The main question that

Bose et al. (2014) address is how to modify the objective of market makers to maximize social

welfare. Here, we focus on how the network structure influences efficiency.

Several recent papers have studied intermediaries, including Nguyen (2015, 2017) and Manea

(2018). The settings in these papers, however are quite different from ours. In particular, they study

the incentive of non-cooperative bargaining and assume unit-demand agents. Even though, in the

advertising industry, bargaining is part of the contracts, automated auctions control the majority

of the interaction among the agents in those studies. 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 outcomes. Our paper is different from this work in that intermediaries in

our model connect between multiple buyers and the sellers, and the prices are determined by the

decentralized Cournot (sub)markets.

2.3 Context Specific Literature

Digital and search ads receive significant attention from various domains, including computer sci-

ence, marketing, information systems, and economics. A seminal piece in this regard is from Edel-

man et al. (2007) in which they analyze the equilibrium of the Generalized Second Price (GSP)

auction. Variants of GSP have been implemented and also studied. Feng et al. (2007) using simu-

lations and Balachander et al. (2009) using game-theory compare alternative GSP auction policies.

More generally, papers have also evaluated the welfare implications of the search ads market. Usu-

ally, they are executed in the context of a single ad-intermediary. For example, Chen and He (2011)

evaluated the efficiency of ads on the consumer search process. Similarly, welfare implications are

also studied when considering policy changes for the auctions. As another example, Shin (2015)

study the subject of search engines requiring budget constraints for advertisers in GSP auctions.

7

More generally, computer science has extensively studied mechanism design problems in the ad

network context. In the computer science literature, a number of papers have analyzed the mecha-

nism design problem from the perspective of matching algorithms – specifically, how to match ads

to positions (e.g., Mehta et al., 2007; Caragiannis et al., 2015). Note that we focus on multiple

search ad intermediaries.

The literature on ride sharing is expanding rapidly in recent years. See for example Cachon et al.

(2017); Bimpikis et al. (ming); Banerjee et al. (2017); Fang et al. (2017). The focus of this literature,

however has been on the optimal design of a monopoly’s matching and pricing mechanisms. Our

paper, on the other hand, considers the problem from an industry level perspective. Given that

there are many competing platforms and the connections of these platforms with the two sides of

the markets are heterogeneous, our paper analyzes the impact of such underlying network structure

to the efficiency of the whole ecosystem.

3 The Model

IntermediariesProviders Purchasers

IJ K

Figure 2: An Example Structure of Networked Competition

In the intermediary contexts, the demand and supply significantly change over time and so, the

price changes are highly unpredictable and volatile. Modeling these price variations in general is

quite hard. However, given our interest in studying the role of network structure on outcomes, it

is sufficient to study this using a parsimonious static model.

8

Figure 2 shows a possible decentralized competition among the intermediaries studied in our

context. The general tripartite network G involves I intermediaries, J providers, and K purchasers,

where the set of edges connect the J providers with the I intermediaries and also the I intermediaries

with the K purchasers. In the ad-auction context, webpage publishers would correspond to the

providers, advertisers to the purchasers, and intermediaries to companies similar to Google or

Yahoo. The goods traded in this marketplace are the ad-slots on webpages. In the ride-sharing

context, the good traded is the taxi service; and drivers and passengers in a specific geographic

area are the providers and purchasers, respectively. The ride-sharing companies such as Uber, Didi

Chuxing, and Lyft, choosing to operate in the different regions, are the intermediaries. When we

refer to an individual intermediary, provider, and purhcaser, we denote them by the corresponding

lower-case variables i, j, and k respectively.

Next, consider the bargaining power in this eco-system. One way to view the bargaining power

is to compare the number of providers or purchasers versus the intermediaries in the market.

Given the fewer number of intermediaries (e.g., Google, Uber, Lyft), it should be clear that the

intermediaries hold the power. Another way we argue for the intermediary’s influence in the market

is by considering the recent initiative by Facebook – subsequent to the follow out with Cambridge

Analytica – to require all apps to re-specify the privacy policies. Another supporting fact is that

intermediaries choose their own market-clearing (or price-determining) mechanisms: be it GSP for

ad auctions or surge prices for ride-sharing context. Therefore, for the aforementioned reasons, we

model the purchasers and providers to not have any significant bargaining power relative to the

intermediary and are assumed to simply respond to the offers. In line with that, we next define

the decision variables and the payoff functions for each player.

Let intermediary i obtain xij amount of goods (for example, number of slots) from provider j.

Define yijk as the amount of goods/service provided by j to k via the intermediary i. Thus, in our

9

model, an intermediary i offers a “package” of goods from different providers to a purchaser k.

One may view the packages differently depending on the context. For example, in ad auctions, a

package corresponds to the number of ad slots that the intermediary allocates to a given advertiser

at different publisher sites. Similarly, in the ride-sharing contexts, a specific passenger may obtain

taxi services from different types of providers based on geographical regions. Note that this aspect

where different purchasers may get different “packages” of bundles is a key difference from other

models such as Bimpikis et al. (2018), where they assume that the goods delivered are homogeneous.

Note also that, because the amount of goods that i delivers to provider j cannot be higher than

the amount purchased from j, the following inventory constraint must be satisfied:

xij ≥∑k

yijk. (1)

Define Xj =∑

i xij as the total amount of goods requested from all the intermediaries for

provider j; and Yjk =∑

i yijk to be the total amount of goods provided by j to k. The amount of

goods that are finally allocated to a provider j is Tj =∑

i∈I∑

k∈K yijk. Notice that Xj ≥ Tj .

Given these variables, the provider’s payoff is:

PjTj − Cj(Tj) + fj · (Xj − Tj), (2)

where Pj is the unit price at j determined by a market clearing mechanism that we describe

later. We assume that j faces an increasing convex cost Cj(Tj) with respect to the amount of

goods allocated, Tj . For simplicity, we assume Cj(Tj) = θjTj +αj2 T

2j , where αj and θj are non-

negative coefficients that are exogenous to our model (it also normalized such that the cost is

zero when Tj = 0). In the advertising market, such a convex cost assumption is consistent with

the dissatisfaction that ads impose on webpage viewers. In the ride-sharing context, the cost is

10

reflective of the inconvenience that the drivers face from driving for long hours. We also assume

that the provider charges a fixed unit penalty, fj ≥ 0, for the goods requested by the intermediary

but not allocated to a purchaser. This penalty is imposed on the unused but reserved capacity.

Next, we define the purchaser’s utility. We assume that the utility of the purchaser k from the

goods sold by provider j is Ujk(Yjk), where Ujk is assumed to be concave for all j, k. (Recall that

Yjk =∑

i yijk is the total amount of goods from provider j offered to k by all the intermediaries.)

Specifically, we assume that Ujk(Yjk) = µjkYjk −βjk2 Y

2jk, where βjk and µjk are non-negative

coefficients that are also exogenous to our model (it also normalized such that the utility is zero

when Yj = 0). To generate the initial set of insights, we assume that the payoff for purchaser

k is separable as follows:∑

j Ujk(Yjk). We later extend the model in Section 7 to capture the

substitutability of goods. The purchaser’s surplus is then given by:

∑j

Ujk(Yjk)−∑j

RjkYjk, (3)

where the unit price Rjk that purchaser k needs to pay for goods from j is obtained from a market

clearing mechanism described later.

Lastly, consider the intermediaries. Each intermediary i maximizes its payoff, which is the

difference between the money paid by the purchasers and the amount paid to the providers:

Πi(~x, ~y) =∑jk

Rjkyijk −

∑j

Pj∑k

yijk −∑j

fj · (Xj − Tj). (4)

Next we define how prices are discovered through a market clearing mechanism that varies with

the context. As mentioned earlier, the examples include GSP in ad-auctions and the “surge-pricing”

in ride-sharing contexts. If one ignored for a moment the context-specific pricing mechanism, the

structure is reminiscient of quantities first, then prices models. The quantity-first then prices is best

11

exemplified in the surge prices, which are set by Uber (or Lyft) after considering the supply of the

drivers at a given location. Kreps and Scheinkman (1983) shows that “the SPE outcome of the two-

stage model ‘first quantities, then prices’ also corresponds to the Cournot equilibrium.” This led to

us considering Cournot competition in our context as the price clearning mechanism. It turns out

that prior work has extensively shown similarities between Cournot competition models and non-

Cournot settings. Daughtery (2008) provides a survey of such works. A work cited in their survey

is Klemperer (1986) who shows the conditions when their modeling of a competition resembles

Cournot outcomes. Specifically with respect to auctions and cournot models, the following works

have made the connections. In the FCC spectrum auction, Milgrom (2004) argues that in a model

with positive supply elasticity, the “auction outcomes resemble a Cournot competition among

buyers.” Vasin and Kartunova (2016) studies the electricity markets using a similar structure.

More specifically, in the ad-auctions, it has been used by Itai Ashlagi (2018); Nava (2015) and

Bimpikis et al. (2018). We believe that these pointers justify the use of Cournot competitions.

As regards the quantity decisions, consistent with our discussion on bargaining power, we assume

that the variables xij and yijk will be chosen endogenously in our model by the intermediary in the

first stage. The variable xij is set to zero if intermediary i is not connected to j, and yijk is zero

if either provider j or purchaser k is not connected to i. However, it is possible at equilibrium

that xij = 0 and yijk = 0 even if both (ij) and (ik) are connected by an edge in the network G.

Consistent with the discussion on the previous paragraph, given xij and yijk, the prices and the

revenues for all the participants are generated by Cournot competition in the second stage.

We first characterize the prices and revenues corresponding to the second stage. As a conse-

quence of the Cournot competition, the price Pj for the unit good offered by j is simply its marginal

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cost obtained from considering its payoff in Equation 2:

Pj = C ′j(Tj).

= θj + αj · Tj .

Similarly, for the purchaser, the marginal gain equaling the marginal payment determines Rjk:

Rjk = U ′jk(Yjk)

= µjk − βjk · Yjk,

obtained from considering the purchaser’s surplus in Equation 3. The game described above is de-

noted as Γ(N , θ, µ, α, β). Formally, in this game, we have the following definition of an equilibrium:

Definition 3.1. (~x, ~y) is an equilibrium if they satisfy (1) and no intermediary i can change ~xi, ~yi

to improve his/her pay-off given by (4).

We illustrate the equilibrium with an example for two main reasons. The first is to show how

the variables we defined earlier correspond to supply and demand functions. The second purpose

is to provide a figurative perspective on social welfare, which we define subsequently.

Example 1. Consider the example with one provider, one purchaser, and one intermediary. The

intermediary’s decision variable is the amount, x, to buy from the provider and the amount, y, to

deliver for the purchaser. We assume that the marginal cost for the provider is 1 + x and marginal

gain for the purchaser in the market is 2− y. Then, the intermediary pays the provider P = 1 + x,

and charges the purchaser R = 2− y. So, the intermediary maximizes

maxx,y

R · y − P · x = (2− y)y − (1 + x)x

∣∣ 0 ≤ y ≤ x.

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It is straightforward to see that the optimal solution is x = y = 1/4. Figure 3 shows the equilibrium

obtained as a consequence of plotting the marginal utilities and marginal costs. The intermediary

surplus is represented as the shaded rectangle in the same figure. Note that only 1/4 units of goods

(ad slots) are transferred from the provider to the purchaser. The picture also shows the consumer

surplus generated by the purchaser and the provider separately. The total social welfare obtained is

the sum of all these components and is shaped like a parallelogram ADECs.

A

B

C

Purchasers’ surplus

Providers’ surplus

Intermediaries’surplus

D

E

Figure 3: Illustrating the welfare metrics associated with Example 1

From an efficiency standpoint, we can see that when supply meets demand, that is, y = x and

2− y = 1 + x, we have the maximum welfare obtained when x = y = 1/2. This maximum welfare

corresponds to the area of the triangle ABC in the same figure. That corresponds to when the

intermediary makes zero profit. In this scenario involving only one provider and one purchaser,

increasing the number of intermediaries will always improve social welfare. However, that may not

be the case when the market structure and networks are different, as will be shown later.

In order to be able to evaluate the social welfare generated for any given networked structure,

we define social welfare as follows:

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

SW (~x, ~y) =∑jk

Ujk(Yjk)−∑j

Cj(Xj) =∑jk

(µjkYjk −1

2βjkY

2jk)−

∑j

(θjXj +1

2αjX

2j ). (5)

4 Equilibrium Characterization

In general, equilibria are hard to characterize in games with complex network structure but equilib-

rium always exists in our model. Furthermore, the equilibrium is unique and can be characterized

by a convex program. Hence, we are able to study the sensitivity of the network structure on

various metrics. This section characterizes the nature of the equilibrium.

First, observe that all inventory constraints bind in equilibrium. This means that there cannot

be any intermediary i that buys more from j, which is xij , than∑

k yijk, because otherwise i improves

the payoff by reducing xij to∑

k yijk. Formally:

Lemma 4.1. Given an equilibrium ~x, ~y, then xij =∑

k yijk.

Because of Lemma 4.1, the payoff of intermediary i expressed in (4) can be written as

∑j,k(µjk − βjkYjk)yijk −

∑j(θj + αjXj)x

ij =

∑i,j,k

(µjk − θj)yijk −∑j,k

βjkYjkyijk −

∑j,k

αjXjyijk.

Observe that, for every i ∈ I, the utility function above can be seen as a concave function of ~y.

Furthermore, notice also a constant Z exists such that if yijk > 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.

Given the specific payoff structure in our game, we can provide additional insights. For example,

the equilibrium is unique and can be characterized as follows:

15

Theorem 4.2. If αj > 0, and βjk > 0, ~y is an equilibrium if and only if it is the unique solution

of the following convex program with the unknowns ~y, ~x, ~X, and ~Y :

min :∑

jαj2 (Xj)

2 +∑

ijαj2 (xij)

2 +∑

j,kβjk2 (Yjk)

2 +∑

i,j,kβjk2 (yijk)

2

s.t : αjXj + αjxij + βjkYjk + βjky

ijk ≥ µjk − θj . (6)

The formal proof is given in Appendix A. With network games, for general class of utilities,

it is quite hard to prove the existence of equilibrium, let alone proving uniqueness. Even for this

class of game and a simpler network, for example Bimpikis et al. (2018), it was not known in

the literature that equilibrium can be characterized as a convex program. This limitation exists

even after they make certain unreasonable assumptions – for example, Bimpikis et al. (2018) needs

additional assumptions that positive trades occurs on all links – to characterize equilibrium. In that

regard, the simplicity of the equilibrium characterization of our model using a quadratic program

is a novelty in itself. Because prior literature does not have the convex program characterization

for the equilibrium, existence of the equilibrium is done using fixed point theorems, which do not

have efficient computations. More importantly, though, because our convex program shows the

equilibrium to be unique, it allows us to conduct robust comparative analysis.

5 Merging Intermediaries

Having established the equilibrium, we identify interesting insights about the effect of network

structures on market efficiency. We claim that the welfare implications between the standard

Cournot model and our model using decentralized Cournot competition with intermediaries are

quite different. In the classical Cournot competition, merging of firms always reduces competition

and thereby reduces social welfare. Arguably, this theory is at the heart of the antitrust policies in

the US. However, the traditional results from Cournot competition does not hold when the com-

16

peting firms are intermediaries. This section considers the welfare implications when intermediaries

merge. In Section 5.1, we begin by showing that merging the intermediaries can have a negative

effect on social welfare, a result consistent with our intuition. The same section subsequently shows

a seemingly counterintuitive result that increasing competition among intermediaries can decrease

efficiency. In Section 5.2, we provide insights into how the competition on either side of the market

can dictate the social welfare outcomes.

5.1 The Impact of Increasing Competition on Welfare

To demonstrate the effect of competition among intermediaries on market efficiency, consider the

network shown in Figure 4. In this example, intermediary #1 solely serves purchaser b but purchaser

a is served by all J+1 intermediaries, including #1. Such a scenario can happen when intermediary

#1 is a large incumbent intermediary that has an exclusive connection to a specific market b while

the other intermediaries are new, small firms competing on a smaller market segment, a. Analyzing

such a network structure is useful for gaining insights. It is also convenient because the merger

of any two intermediaries not involving #1 will again result in a network structure that can be

studied with our generic formulation.

#1 #2s s sX x k x= + ⋅

3

1

b

#2say

#1say

#1sby

#2sx

#1sx

#1 #2sa sa saY y k y= + ⋅

#1sb sbY y=

.

.

.

2

J+1

as

Figure 4: J intermediaries

We simplify certain definitions and make additional assumptions for ease of exposition. Because

there is only one provider, we simply denote α = αs, µsa = µa, and µsb = µb. The additional

17

assumptions we make are as follows: θs = 0 for the providers; and β = βsa = βsb for the purchasers.

Because intermediaries 2, 3, . . . , J + 1 are symmetric, y#jsa = y#2sa and x#jsa = x#2

sa for any 1 < j ≤

J + 1. The equilibrium is therefore the optimal solution to the following program:

min : α(

(Xs)2 + J(x#2

s )2 + (x#1s )2

)+ β

((Ysa)

2 + J(y#2sa )2 + (y#1

sa )2)

+ β(

(Ysb)2 + (y#1

sb )2)

s.t : αXs + αx#2s + βYsa + βy#2

sa ≥ µa

αXs + αx#1s + βYsa + βy#1

sa ≥ µa

αXs + αx#1s + βYsb + βy#1

sb ≥ µb.

Notice that α and β capture the market sensitivity of the provider (seller) and the purchasers

(buyers), respectively. We next consider two extreme cases in order to gain intuition about this

network structure. The first one has α = 0, β = 1, and the second has α = 1, β = 0.

Case 1: α = 0, β = 1

Since α = 0, the unit price charged by the provider is a constant and is independent of the amount

of goods sold. On the other hand, the value of goods allocated to purchasers is a diminishing

marginal function. Thus, the most efficient way to allocate goods is when these marginals are 0,

that is, allocate µa and µb amount of goods to purchaser a and b, respectively. The convex program

above defines the equilibrium for this game as:

y#1sb =

µb2

; y#1sa = y#2

sa =µa

J + 2

From this, we can calculate the amount of goods allocated to purchaser a to be J+1J+2µa; and to

purchaser b to be µb2 . Therefore, we obtain the following result:

Corollary 5.1. α = 0, β = 1, Increasing J will make the market more competitive and improve

18

social welfare. However, intermediary #1 remains as the monopoly for purchaser b.

Case 2: α = 1, β = 0

At first glance, increasing J may seem to make the market more efficient. However, as we show

next, this is not always the case. Note that without Intermediaries #2, 3,. . . ,J+1, Intermediary #1

can internalize between the amount of goods allocated to purchasers a and b leading to a reasonably

efficient outcome; but with Intermediaries #2,3,. . . ,J+1 present, the competition for purchaser a

has an indirect effect on the prices for b as well, resulting in a more inefficient outcome.

Suppose µa < µb. The equilibrium, which is the solution of the convex program above is:

y#1sa = 0; y#1

sb =(J + 1)µb − Jµa

J + 2; y#2

sa =2µa − µbJ + 2

if µa < µb < 2µa (7)

y#1sa = 0; y#1

sb =µb2

; y#2sa = 0 if 2µa < µb. (8)

Let us first consider the former case when µa < µb < 2µa. The amount of goods delivered

to a and b are Ysa = y#1sa + Jy#2

sa = JJ+2(2µa − µb) and Ysb = (J+1)µb−Jµa

J+2 = (µb − µa) + 2µa−µbJ+2 ,

respectively, implying that, as J increases, the market for purchaser a becomes more competitive

and Ysa naturally increases. As a consequence, Intermediary #1 now has to pay s more. Now,

consider the total amount of goods delivered Ysa + Ysb = Jµa+µbJ+2 = µa − 2µa−µb

J+2 . Observe that the

total amount of goods allocated to both a and b increase. From the efficiency point of view, there

is clearly a tradeoff. If µb is significantly larger than µa but less than 2µa, then delivering goods to

b is preferred from the social standpoint. When J increases, it simply increases the competition for

purchaser a which further pushes down the amount of goods for purchaser b, resulting in a more

inefficient outcome. Next, we formally establish this result:

Corollary 5.2. α = 1, β = 0,

• µa < µb <54µa as J increases, the welfare will first increase, then will decrease.

19

• 54µa ≤ µb < 2µa, as J increases, the welfare will decrease.

• 2µa ≤ µb the welfare is independent of J .

Figure 5 provides two numerical examples to illustrate Corollary 5.2. In Appendix B, we provide

the proof and use several numerical examples to show that the analysis is robust for a much wider

parameter range. Thus far, we have shown that the mergers of intermediaries can also lead to

0.52

0.522

0.524

0.526

0.528

0.53

0.532

0.534

0.536

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 210.58

0.59

0.6

0.61

0.62

0.63

0.64

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Figure 5: Social welfare as function of k, µa = 1;µb = 1.15 on left and µa = 1;µb = 1.3 on right

decreasing social welfare. This result is valuable when considering the implications of mergers.

Furthermore, the result is not specific to the network structure that we used for the analysis. In

fact, a similar welfare analysis can be conducted on any ad-hoc network, as we show next.

Consider the impact of competing intermediaries on the share of surplus. Notice that, when

µb > 2µa, intermediaries #2, . . . , J + 1 do not participate, and the equilibrium is independent of J .

So, we focus on the more interesting case µa ≤ µb ≤ 2µa, in which we obtain the following result.

Corollary 5.3. α = 1, β = 0 and µa ≤ µb ≤ 2µa,

• as J increases, provider s’s and purchaser a’ payoff increase; but b’s utility decreases.

• The payoff of all the intermediaries decreases as J increases.

See Appendix C for the proof.

20

5.2 Merger in an Ad hoc Network: An Illustrative Example

This section demonstrates how the sensitivity of the competition on either side must be considered

in order to analyze the social welfare implications. For this, we consider the same network and the

two scenarios shown in Figure 1. In scenario I, three intermediaries A, B and C compete to deliver

goods between purchasers 3 and 4 and providers 1 and 2. In scenario II, A and B merge. Notice

that in scenario I, A and B compete to deliver goods from 1 to 4, but C is the only intermediary

between 2 and 3. However, when A and B merge, the merged firm AB becomes the monopoly

between 1 and 4, but becomes a competitor for C between 2 and 3.

Depending on how purchaser 3 values the goods from provider 2 relative to the valuations for

the other purchaser-provider pairs, the merging of A and B may or may not improve social welfare.

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

A more specific analysis is given in the following result.

Corollary 5.4. Consider the network in Figure 1 and the following set of parameters αj = 1,

furthermore, µ23 = V ;β23 = 1, µ13 = µ24 = 0;β13 = β24 = 0;µ14 = 1;β14 = 1; θj = 0.

If V > 1, then the revenue of AB after merger is larger than the combined revenue of A and

B before merger; furthermore, both the social welfare and the consumer surplus in scenario II are

also larger than in scenario I.

If 3/7 < V < 1, then the revenue of AB after merger is less than the combined revenue of A

and B before merger; furthermore, the social welfare in scenario II is also less than that in scenario

I, but not the consumer surplus.

The proof is based on the convex program characterization given in Theorem 4.2, which gives

us an easy way to compute these equilibria. We omit the details of this calculation and provide

only the intuition for the results. Notice that, in Scenario I, the path 2-3 is intermediated by

21

Figure 6: Different welfare measures as functiond of V . Dotted lines represent scenario 2, solidlines represent scenario 1

C in a monopolistic fashion. In Scenario II, the 2-3 connection is no longer monopolistically

intermediated. This merger also has a cost. In particular, it leads to decreased competition

between the intermediaries 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.

We use this example to show how the implications of our model differ from those in the prior

literature. Bimpikis et al. (2018) models no intermediaries. So, mergers of two firms can only lead

to indirect impacts. Compared with Bimpikis et al. (2018), our model impacts social welfare and

consumer surplus differently. In our model, the providers and purchasers connected to the merged

firms are directly impacted because of the merger. These direct impacts occur in addition to the

indirect ones. Therefore, unlike theirs, we can study the implications of mergers of intermediaries.

In conclusion, this section demonstrates that social welfare implications critically depend on

the network structure. Comparing two arbitrary networks is in general a difficult task, and mergers

can have both positive and negative welfare implications. While we have studied interesting policy

22

questions thus far, we are also interested in studying how close we can get to the social optima.

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

analyzing how the parameters of our model may affect the efficiency-loss bounds.

6 Bounding Inefficiency

Our goal in this section is to build further on the previous section to show how network structure

influences efficiency. Such insights can act as a guide for market designers or policy makers to

evaluate alternative network structures from an efficiency standpoint. We specifically study how

the network structures lead the social welfare obtained in the decentralized context compared with

the optimal one (without any such constraint). For this purpose, we use the measure called price of

anarchy which is the ratio between the welfare of a Nash equilibrium and the optimal social welfare

without incentive constraints. This measure is studied in Dubey (1986) and is extensively used in

computer science starting with Koutsoupias and Papadimitriou (1999). If this ratio is close to 1,

then it means that the system is almost optimal.

6.1 Price of Anarchy: Lower Bound

Definition 6.1. Let E ⊂ J × K be the set of node pairs j ∈ J ; k ∈ K. Define OPT (E) as the

optimal social welfare obtainable through the links in E, that is,

OPT (E) := maxX,Y

∑j,k

µjkYjk −∑j

θjXj −∑jk∈E

βjk

∫ Yjk

0ydy −

∑j∈J

αj

∫ Xj

0xdx. (9)

s.t : X,Y ≥ 0

Yjk = 0 ∀jk /∈ E

Xj ≥∑k

Yjk.

If the set of connections (or links) in E constrain the trades to occur only between the connected

23

(or linked) agents, OPT (E) is the maximum level of welfare the system can achieve. In our

environment, a trade between j ∈ J and k ∈ K is possible only if they are connected to at least a

common intermediary. Let E1 ⊂ J ×K be the set of such pairs. Namely,

E1 := (j, k) ∈ J ×K| there exists i ∈ I where both ji and ik are connected.

Thus, at the equilibrium, the welfare is at most OPT (E1). Our first result in this section

provides a lower bound on the welfare of the equilibrium compared with OPT (E1).

Theorem 6.2. The social welfare at the Nash equilibrium is at least 2/3 times the optimal social

welfare, OPT (E1).

The proof of this theorem is provided in Appendix D. There, we establish the equilibrium to

be a series of inequalities. Using those inequalities, we determine the lower bound on the efficiency

of the system. This lower bound of 23 is called the price of anarchy (PoA). It measures the extent

to which selfish behavior affects efficiency. Note that this result differs from that of the Bertrand

competition model among intermediaries in Blume et al. (2009). In their model, all equilibria are

efficient, and so merging of intermediaries would not change the price of anarchy.

6.2 Price of Anarchy: Refined Lower Bound

Even though PoA is an interesting measure, it does not provide information about the role of the

underlying network structure on welfare. For example, in Figure 1, it may be interesting to obtain

a more detailed efficiency comparison of the two networks. To obtain more general bounds that

reveal the structure of networks, the following notions of network connectivity are important.

Definition 6.3. 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 the κ-th layer of G, denoted

24

as Eκ(G) or Eκ for short, to be the set of node pairs jk : j ∈ J ; k ∈ K that are connected by at

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

For example, for each of the networks in Figure 1, the first layer of G, κ = 1, contains the pairs

13, 14, 23, 24. The 2nd layer of network, κ = 2, in scenario I (on the left hand side) is the pair 14,

and in Scenario II is the pair 23. Given this definition, our next result refines the price of anarchy

for these layers of the network.

Theorem 6.4. Given a network G over the set of providers, purchasers and intermediaries the

social welfare at the Nash equilibrium is at least (1− 12κ+1)OPT (Eκ).

The proof of Theorem 6.4 is given in Appendix E. Relative to Theorem 6.2, Theorem 6.4

investigates the level of efficiency in a more refined way. In particular, Theorem 6.4 suggests more

details on the influence of the underlying network structure on the level of efficiency. To illustrate,

continue with the example network in Figure 1. With κ = 2, Theorem 6.4 implies that the welfare

at equilibrium in the network on the left hand side is at least 4/5 times the total trade surplus of

the sub market between 1 and 4, and for the network on the right hand side, its equilibrium welfare

is at least 4/5 times the total trade surplus of the submarket between 2 and 3. This means that

merging A and B is beneficial if the submarket between 2 and 3 has a high trade value.

This theorem can be insightful for a network designer when analyzing policies having substantial

change on the network structure. The designer should consider policy impacts on different layers

of the network, and nurture the subnetworks that have high values and are also highly competitive.

In the remainder of this section, we further discuss the implications of this result.

6.3 Impact of Intermediary Capacity on Price of Anarchy

We start by defining a parameter, which we call the intermediary capacity of the network.

Definition 6.5. For a node pair j ∈ J ; k ∈ K in network G, which are connected by at least one

25

middleman, let wjk ≥ 1 be the number of middlemen that connect j and k. Then, wG, called the

intermediary capacity of G, is defined as the minimum value among all such wjk.

Thus, every node pair j ∈ J ; k ∈ K that is connected by at least one middlemen, is connected

by at least wG middlemen. Further, as a corollary of Theorem 6.4, we obtain the following result:

Corollary 6.6. The price of anarchy is at least 1− 12wG+1 .

Because wG ≥ 1 for any network G, Corollary 6.6 is a generalization of Theorem 6.2, note

that the intermediary capacity of the network intuitively captures the degree of competition among

the intermediaries. As the intermediary capacity of the network increases – i.e., as the economy

becomes more competitive – the system approaches full efficiency. Note that while mergers can

lower the intermediary capacity, it is not always the case, as illustrated by the example in Figure 1.

So, Corollary 6.6 does not contradict with our earlier result that mergers can have ambiguous

implication to welfare.

7 Extension for Subsitute Goods

Previously, we assumed that the utility of k as additive across j i.e.,∑

j Ujk(Yjk). In this section,

we consider the case where the goods are a substitute. We show that a similar characterization

of equilibrium based on a convex program applies but it is even simpler. Recall that yijk is the

amount of goods that intermediaries i provided by j to k; the total amount of good that i sells to

k is Y ik =

∑j y

ijk; the total amount of good that i buys from j is xij =

∑k y

ijk; and Yk =

∑j Yjk

is the total amount of goods that a purchaser k obtains. We assume the utility of a purchaser k is

Uk(Yk) = µkYk − 12βkY

2k . Thus the marginal price at the purchaser k is

∂Uk(Yk)

∂Yk= µk − βkYk.

We further assume that there is a unit cost of cjk ≥ 0 that the intermediary needs to pay i for

26

delivering goods from j to k.1 Then, intermediary i’s payoff function is

Φ(y) =∑k

(µk − βkYk)Y ik −

∑jk

cjkyijk −

∑j

(θj + αjXj)xij

Given an index j∗ and k∗, taking derivative according to yij∗k∗ we obtain:

∑k

(µk − βkYk)∂Y i

k

∂yij∗k∗+∑k

∂(µk − βkYk)∂yij∗k∗

Y ik − cj∗k∗ −

∑j

(θj + αjXj)∂xij∂yij∗k∗

−∑j

∂(θj + αjXj)

∂yij∗k∗xij

Notice that if k 6= k∗, then∂Y ik∂yij∗k∗

= 0, and if j 6= j∗, then∂xij

∂yij∗k∗

= 0. Thus,

∂Φ(y)

∂yij∗k∗= (µk∗ − βk∗Yk∗)− βk∗Y i

k∗ − cj∗k∗ − (θj∗ + αj∗Xj∗)− αj∗xij∗ .

Observe that Φ(.) is a concave function, thus, we have the following first order condition for an

equilibrium for all provider j∗, purchaser k∗ and intermediary i, who are connected in the network.

αj∗Xj∗ + αj∗xij∗ + βk∗Yk∗ + βk∗Y

ik∗ ≥ µk∗ − θj∗ − cj∗k∗

if strict inequality holds then yij∗k∗ = 0.

Given this equilibrium condition, using a similar argument as in Theorem 4.2, we obtain the

following characterization of equilibria. The proof of this result is provided in Appendix F.

Theorem 7.1. The equilibrium is unique and is the solution of the following convex program

min∑j

αj(Xj)2 +

∑j,i

αj(xij)

2 +∑k

βk(Yk)2 +

∑i,k

βk(Yik )2 (10)

sjt : αjXj + αjxij + βkYk + βkY

ik ≥ µk − θj − cjk ∀j, k, i where i connects j and k. (11)

1The analysis extends for the case that each intermediary i has a different cost cijk.

27

8 Conclusions And Future Work

Sharing economies facilitated by multisided platforms are becoming increasingly popular in a large

number of contexts (Uber, Airbnb being some of the well-known ones). Therefore, understanding

the welfare implications of these platforms is important to guide policy proposals for improving

social welfare. For example, as the sharing economy matures, mergers and acquisitions among

platforms are likely to occur. Yet, there is little guidance from the prior literature to analyze the

welfare implications of such mergers. This is because the majority of the current literature focuses

on monopoly pricing problems. Our paper fills this void by developing a tractable model that

provides insights about the role of network structure in affecting welfare.

Our model considers a two sided market involving both providers and purchasers, where the

market clears based on a Cournot competition. We also assume the providers and purchasers have

linear marginal costs and marginal revenues. In the stylized model, we demonstrate the existence of

a unique equilibrium. The equilibrium can be established as a series of conditions. These conditions

allow us to study the implications of various interesting cases. Using that structure, we show that

social welfare can improve even when two intermediaries merge. This result may appear counter

intuitive given traditional antitrust analyses where decreasing competition always increases the

dead weight loss. We show that the welfare implications vary depending on the resulting network

structure in comparison with the structure that existed before the merger. We also provide some

bounds on the price of anarchy; as defined earlier, the price of anarchy is widely used in the computer

science literature and is simply the ratio between the social welfare obtained under equilibrium and

the maximum social welfare that can be obtained. In that analysis, we define metrics such as the

intermediary capacity of the network that can be used to influence social welfare.

Even though we presented the analysis in a stylized context, the underlying structure is relevant

more broadly. For example, the analysis is also relevant to a physical retail chain context. Visualize

28

the retailers as the platform companies. On one side of the network are manufacturers (e.g.,

Reebok, Under Armour). On the other side of the network are geographic locations where the

retailers compete against one another. The edges between the first side and the platform now

correspond to whether the retailer carries the products from the manufacturers. The edges between

the platform and the second side correspond to whether the retailer has presence in the geographic

location. In such a context, our insights become relevant when studying the impact of mergers

of the retail chains. Again, to the best of our knowledge, supply chain literature has not studied

equivalent problems, and our analysis can provide insights to such different contexts, including some

nontechnological ones. Additionally, the structure is relevant even when considering extension to

the structure. For example, in ad-auction context, a provider can correspond to supply aggregator

firms such ValueClick, AdBrite, Burst Media, Casale Media, Kontera, Chitka, Tribal Fusion, and

Bidvertiser. So, the network may involve an additional layer. Even in this structure, notice that the

analysis on mergers of intermediaries is not affected by extending one side of the graph. A similar

logic may be provided to extend on the other side. In essence, we have considered a parsimonious

structure for analysis which is robust to some generalizations.

In conclusion, we have provided a tractable model of competition among intermediaries. Given

that our main focus was to show how the traditional antitrust analysis will not work in a platform

context, we made many simplifying assumptions. A future extension of this work could be one

which extends the model to include other important features such as more general form of utilities,

uncertainty, asymmetric information, and network formation questions. Also, it would be of interest

to study more deeply in the future the welfare implications when characterizing a multi-partite

graph involving aggregators on the publisher- and/or the provider-side; or when purchasers can

easily substitute among the offerings.

29

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APPENDIX

A Proof of Theorem 4.2

#1 #2s s sX x x= +

2

1 a

b

#2say

#1say

#1sby

#2sx

#1sx

s

#1 #2sa sa saY y y= +

#1sb sbY y=

#2 #2s sax y=

1 #1 #1s sa sbx y y= +

Figure 7: Illustrating example for equilibrium

Proof. We first illustrate the main idea of the proof with an example of the network in Figure 7. The

example will also subsequently be used to demonstrate some seemingly counterintuitive results when

we later study the sensitivity of the results. As for the example, assume a networked structured as

in Figure 7, with two intermediaries (#1 and #2), two purchasers (a and b), and a provider (s).

Readers not interested in the characterization of the proof may skip to the end of the section.

We will use the optimality conditions of individual players’ payoffs to characterize the equilib-

rium. We already know that, at equilibrium, the inventory constraints imply x#1s = y#1

sa + y#1sb

and x#2s = y#2

sa . Further, Intermediary #2 connects seller s and purchaser a and so x#2s = y#2

sa .

34

Therefore, we can write the payoff of Intermediary #2 as functions of y:

Π2(y) = (µsa − βsaYsa)y#2sa − (θs + αsXs)x

#2s

= (µsa − βsaYsa)y#2sa − (θs + αsXs)y

#2sa

= (µsa − θs)y#2sa − βsaYsay#2

sa − αsXsy#2sa .

Notice that Ysa = y#2sa + y#1

sa and Xs = x#2s + x#1

s = y#2sa + y#1

sa + y#1sb , which are functions of y#2

sa .

Thus, the derivative of Π2(y) according to y#2sa is

∂Π2(y)

∂y#2sa

= (µsa − θs)− βsaYsa − βsay#2sa

∂Ysa

∂y#2sa

− αsXs − αsy#2sa

∂Xs

∂y#2sa

= (µsa − θs)− βsaYsa − βsay#2sa − αsXs − αsy#2

sa

Because Π2(y) is concave, we obtain the following necessary condition for Π2(y) to be optimal:

if y#2sa > 0 then

∂Π2(y)

∂y#2sa

= 0 and if∂Π2(y)

∂y#2sa

< 0, then y#2sa = 0. (12)

Similarly, for intermediary #1, with x#1s = y#1

sa + y#1sb , his payoff is:

Π1(y) = (µsa − βsaYsa)y#1sa + (µsb − βsbYsb)y#1

sb − (θs + αsXs)x#1s

= (µsa − θs)y#1sa + (µsb − θs)y#1

sb − βsaYsay#1sa − βsbYsby

#1sb − αsXs(y

#1sa + y#1

sb ).

35

Taking derivative of Π1 according to y#1sa and y#1

sb we have

∂Π1(y)

∂y#1sa

= (µsa − θs)− βsaYsa − βsay#1sa

∂Ysa

∂y#1sa

− αsXs − αs(y#1sa + y#1

sb )∂Xs

∂y#1sa

= (µsa − θs)− βsaYsa − βsay#1sa − αsXs − αs(y#1

sa + y#1sb )

= (µsa − θs)− βsaYsa − βsay#1sa − αsXs − αsx#1

s .

∂Π1(y)

∂y#1sb

= (µsb − θs)− βsbYsb − βsby#1sb

∂Ysb

∂y#1sb

− αsXs − αs(y#1sa + y#1

sb )∂Xs

∂y#1sb

= (µsb − θs)− βsbYsb − βsby#1sb − αsXs − αs(y#1

sa + y#1sb )

= (µsb − θs)− βsbYsb − βsby#1sb − αsXs − αsx#1

s .

The first order conditions for Π1(y) to be optimal are

if y#1sa > 0 then

∂Π1(y)

∂y#1sa

= 0 and if∂Π1(y)

∂y#1sa

< 0, then y#1sa = 0 (13)

if y#1sb > 0 then

∂Π1(y)

∂y#1sb

= 0 and if∂Π1(y)

∂y#1sb

< 0, then y#1sb = 0. (14)

Now, consider the optimal solution of the following quadratic program:

min : Ω = αs2

((Xs)

2 + (x#2s )2 + (x#1

s )2)

+ βsa2

((Ysa)

2 + (y#2sa )2 + (y#1

sa )2)

+ βsb2

((Ysb)

2 + (y#1sb )2

)(15)

s.t : αsXs + αsx#2s + βsaYsa + βsay

#2sa ≥ µsa − θs. (16)

αsXs + αsx#1s + βsaYsa + βsay

#1sa ≥ µsa − θs. (17)

αsXs + αsx#1s + βsbYsb + βsby

#1sb ≥ µsb − θs. (18)

Let z#2sa , z

#1sa , z

#1sb be the dual variables of the constraints (16), (17), and (18), respectively. The

36

Lagrangian relaxation of this convex program is

Ω− z#2sa (αsXs + αsx

#2s + βsaYsa + βsay

#2sa − (µsa − θs))−

z#1sa (αsXs + αsx

#1s + βsaYsa + βsay

#1sa − (µsa − θs))−

z#1sb (αsXs + αsx

#1s + βsbYsb + βsby

#1sb − (µsb − θs)).

(19)

The first order conditions of the Lagrangian relaxation according to Xs give Xs = z#2sa + z#1

sa + z#1sb ;

according to x#2s give x#2

s = z#2sa ; according to x#1

s give x#1s = z#1

sa + z#1sb ; according to Ysa give

Ysa = z#2sa + z#1

sa ; according to Ysb give Ysb = z#1sb ; and according to y#2

sa give y#2sa = z#2

sa ; y#1sa gives

y#1sa = z#1

sa ; and y#1sb give y#1

sb = z#1sb .

Replacing z with y, we obtain a set of conditions that are exactly the equations of the game

captured in Figure 7. We need to show that the first order conditions of the payoff for intermediaries,

(12), (13), and (14) are also satisfied. Consider the first constraint (16). Notice that the KKT

conditions of the optimal solution of the convex program (15)-(17) imply that if the dual variable

z#2sa > 0, then the constraint binds, that is αsXs+αsx

#2s +βsaYsa+βsay

#2sa = µsa−θs. Furthermore,

if the constraint does not bind, i.e, αsXs + αsx#2s + βsaYsa + βsay

#2sa > µsa − θs, then z#2

sa = 0.

Notice that because z#2sa = y#2

sa , this condition is exactly the first order condition of for the payoff

of intermediary #1 in (12). Similarly, we can also have the equivalence between the equilibrium

condition of intermediary #2, (13) and (14), and the constraints (17) and (18). The example above

illustrates the equivalence between the equilibrium condition and the optimality of the quadratic

program (15) to (18). The same idea can be generalized to more general networks. In what follows,

we provide the formal proof.

37

The payoff for intermediary i is

Πi(y) =∑i,j,k

(µjk − θj)yijk −∑jk

βjkYjkyijk −

∑j,k

αjXjyijk

=∑i,j,k

(µjk − θj)yijk −∑jk

βjkYjkyijk −

∑j

αjXj

∑k

yijk.

The derivative of Πi(y) with respect to yijk is

∂Πi(y)

∂yijk= (µjk − θj)− βj,kYjk − βjkyijk − αjXj − αj

∑l∈K

yijl

= (µjk − θj)− βj,kYjk − βjkyijk − αjXj − αjxij

The above equation holds because xij =∑

l∈K yijl. Hence, the first order condition for y to be a

Nash equilibrium is the following:

(µjk − θj)− (βjkYjk + αjXj + αjxij + βjky

ijk) ≤ 0; and equality occurs if yijk > 0. (20)

We will show that (20) has a special property that allows us to characterize its unique solution

by a quadratic convex program. First, consider the unique solution of (6). By the complementarity

slackness condition, x, y,X, Y is the solution if for every i, j, k such that ij and ik are connected in

G, there exists a dual variable zijk satisfying

αj2 2Xj =

∑ik αjz

ijk (21)

αj2 2xij =

∑k αjz

ijk (22)

βjk2 2Yjk =

∑i βjkz

ijk (23)

βjk2 2yijk = (βjk)z

ijk. (24)

38

Furthermore,

if αjXj + αjxij + βjkYjk + βjky

ijk > µjk − θj , then zijk = 0. (25)

Observe that (24) implies that zijk = yijk. Therefore, from (21) , (22), and (23)

Xj =∑ik

yijk;xij =

∑k

yijk and Yjk =∑i

yijk.

Given this, (25) is equivalent to the first order condition in (20).

To see the reverse direction, given a ~z satisfying (20), we introduce zijk := yijk; Xj =∑

ik zijk;x

ij =∑

k zijk and Yjk =

∑i zijk. It is straightforward to see that z, x, y,X, Y satisfy (21-25). Therefore,

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

B Proof of Corollary 5.2 and Numerical examples

Proof of Corollary 5.2

First, if µb ≥ 2µa, then according to the equilibrium characterization, y#2sa = 0 and intermediary 1

will be the only active players. Thus the welfare of the system is independent of k.

It remains to consider the case µa < µb < 2µa. According to the equilibrium characterization,

the amount of goods delivered to a and b are

Ysa = y#1sa + ky#2

sa =k

k + 2(2µa − µb) = 1− 2

(2µa − µb)k + 2

Ysb =(k + 1)µb − kµa

k + 2= (µb − µa) +

2µa − µbk + 2

.

The total number of goods delivered is

Xs = Ysa + Ysb =kµa + µbk + 2

= µa −2µa − µbk + 2

39

Denote t := 2µa−µbk+2 , the welfare is

µaYsa + µbYsb −1

2(Xs)

2 = µa(1− 2t) + µb(µb − µa + t)− 1

2(µa − t)2

= µa + µb(µb − µa)−1

2µ2a + (µb − µa)t−

1

2t2 =: f(t).

When k increases from 1 to ∞, t = 2µa−µbk+2 decreases from t1 := 2µa−µb

3 to t∞ := 0.

Notice that f(t) is a quadratic function and f ′(µb − µa) = 0. Thus, if t1 ≤ µb − µa, then f(t)

decreases if k increases; if t1 > µb − µa, then as k increases f(t) will first increase, then decrease.

Now, t1 = µb − µa implies µb = 5/4µa, which implies our result.

We consider two numerical examples where both α, β are positive.

Example 1: α = 1; β = 1; µa = 3;µb = 9 See Figure 8

Without intermediaries #2, 3, . . . , J+1, intermediary #1 will sell exclusively to purchaser b. y#1sa =

0; y#1sb = 2.25 and the welfare is 15.1875. With intermediary #2, the equilibrium is y#2

sa = 0.2; y#1sa =

0; y#1sb = 2.2 and the welfare is 15.08. As k increases, the social welfare decreases.

14.7

14.75

14.8

14.85

14.9

14.95

15

15.05

15.1

15.15

15.2

15.25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Soci

al W

elfa

re

number of competing intermediaries

Figure 8: Welfare decreases

40

Example 2: µa = 3;µb = 4;α = 1;β = 0.1 See Figure 9

Here, as J increases, the welfare initially increases then decreases below the monopolistic scenario.

5.15

5.2

5.25

5.3

5.35

5.4

5.45

5.5

5.55

5.6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Soci

al W

elfa

re

number of competing intermediaries

Figure 9: welfare increases then decreases

C Proof of Corollary 5.3

Proof. According to the equilibrium computation (7) and (8), as J increases, the total amount of

goods s provides and the amount of goods a receives both increase. This shows that s and b’s payoff

increases. The amount of goods delivered to b, on the other hand decreases. As argued above, this

is a natural consequence of increasing competition for trade between s and a.

The formula for the payoff of intermediary 1 is µay#1sa +µby

#1sb −(y#1

sa +y#1sb )(Ysa+Ysb). Replacing

the solution of (7), we obtain that the payoff of intermediary 1 is

(µb − µa +2µa − µbJ + 2

)2.

Thus, the payoff of intermediary 1 decreases as J increases. The payoff of each intermediary

2, 3, .., J + 1 is µay#2sa − y#2

sa (Ysa + Ysb) = (2µa−µbJ+2 )2, which also decreases as J increases.

41

D Proof of Theorem 6.2

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

SWNash =∑jk

(µjk∑i

yijk −βjk2Y 2jk)−

∑j

(θjXj +αj2X2j ).

Replacing Xj =∑

i,k yijk and Yjk =

∑i yijk, we have

SWNash =∑i,j,k

(µik − θj)ziik −∑jk

βjk2Y 2jk −

∑j

αj2X2j . (26)

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 . (27)

Comparing SWNash and SWOpt, we observe that because of (20), we have

cijk((µjk − θj)− (αjXj + αjx

ij + βjkYjk + βjky

ijk))≤ 0.

Summing over all i, j, k we have

∑i,j,k

(µjk − θj)cijk −∑j

αjAjXj −∑ij

αjxijcij −

∑jk

βjkBjkYjk −∑i,j,k

βjkcijky

ijk ≤ 0. (28)

We use the following inequalities2

BjkYjk ≤1

3B2jk +

3

4Y 2jk;AjXj ≤

1

3A2j +

3

4X2j and

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

4γ2y2.

42

cijxij ≤

1

6(cij)

2 +3

2(xij)

2; cijkyijk ≤

1

6(cijk)

2 +3

2(yijk)

2

to apply to (28) to obtain the following inequality:

∑i,j,k(µjk − θj)cijk −

∑jαj3 A

2j −

∑ijαj6 (cij)

2 −∑

jkβjk3 B

2jk −

∑i,j,k

βjk6 (cijk)

2

≤ 32

(∑jαj2 X

2j +

∑ij αj(x

ij)

2 +∑

jkβjk2 Y

2jk +

∑i,j,k βjk(y

ijk)

2). (29)

To prove the theorem, we will show that the Left hand side of (29) is at least SWOpt and the

Right hand side of (29) is 32SWNash. Focusing on the Left hand side of (29), 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 (29) ≥︸︷︷︸from (27)

∑i,j,k

(µjk − θj)cijk −∑j

αj2A2j −

∑jk

βjk2B2jk = SWOpt. (30)

Focusing on the right hand side of (29), we will show that the right hand side of (29) is equal

to SWNash. To see this, observe that from the Nash equilibrium condition (20), we have

∑i,j,k

yijk ·((µjk − θj)− (αjXj + αjx

ij + βjkYjk + βjky

ijk))

= 0.

This is equivalent to

∑i,j,k

(µjk − θj)yijk =∑j

αjX2j +

∑ij

αj(xij)

2 +∑jk

βjkY2jk +

∑i,j,k

βjk(yijk)

2 (31)

43

Replacing∑

i,j,k(µjk − θj)yijk as in (31) to the formulation of the social welfare at Nash (26),

SWNash =∑

jαj2 X

2j +

∑ij αj(x

ij)

2 +∑

jkβjk2 Y

2jk +

∑i,j,k βjk(y

ijk)

2.

E Proof of Theorem 6.4

Let Y ∗jk be the amount of goods k shown on provider j that gives the optimal social welfare OPT (Eκ)

of the κ-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 κ intermediaries.

DenoteX∗j =∑

k Y∗jk. We introduce the following notations. For any pair j, k that are connected

by at least κ intermediaries, let cjk =Y ∗jkκjk, and let cijk = cjk if i is connected to both j and k; and

0 otherwise.

To compare SWNash and OPT (Eκ), we observe that because of (20), we have

cijk((µjk − θj)− (αjXj + αjx

ij + βjkYjk + βjky

ijk))≤ 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

βjkcijky

ijk ≤ 0. (32)

We use the following inequalities

Y ∗jkYjk ≤κ

2κ+ 1Y ∗jk

2 +2κ+ 1

4κY 2jk;X

∗jXj ≤

κ

2κ+ 1X∗j

2 +2κ+ 1

4κX2i and

cijyij ≤

κ

2(2κ+ 1)(cij)

2 +2κ+ 1

2κ(yij)

2; cijkyijk ≤

κ

2(2κ+ 1)(cijk)

2 +2κ+ 1

2κ(yijk)

2

44

to apply to (32) and obtain the following inequality.

∑i,j,k

(µjk − θj)cijk −∑j

κ

2κ+ 1αjX

∗j2 −

∑ij

κ

2(2κ+ 1)αjc

ij2 −

∑jk

κ

2κ+ 1βjkY

∗jk

2 −∑i,j,k

κ

2(2κ+ 1)βjk(c

ijk)

2

≤ 2κ+ 1

2κ

(∑j

αj2X2j +

∑ij

αj(xij)

2 +∑jk

βjk2Y 2jk +

∑i,j,k

βjk(yijk)

2

)(33)

Similar to the proof of Theorem 6.2, the right hand side of (33) is 2κ+12κ SWNash. In order to

bound the left hand side of (33), we show the following.

κ∑i,j

(cij)2 ≤

∑j

(X∗j )2. (34)

κ∑i,j,k

(cijk)2 ≤

∑jk

(Y ∗jk)2. (35)

Assuming that (34) and (35) are true then the right hand side of (33) is at least

∑i,j,k

(µjk−θj)cijk−∑j

κ

2κ+ 1αjX

∗j2−∑j

1

2(2κ+ 1)αjX

∗j2−∑jk

κ

2κ+ 1βjkY

∗jk

2−∑jk

1

2(2κ+ 1)βjkY

∗jk

2

=∑i,j,k

(µjk − θj)cijk −∑jk

1

2βjkY

∗jk

2 −∑j

1

2αjX

∗j2 = OPT (Eκ).

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

Thus, it remains to show that (34) and (35) are true. To see (35), observe that j, k are connected

by κjk ≥ κ intermediaries, and by definition cijk =Y ∗jkκjk

. Thus,

Y ∗jk2 = κjk ·

∑i,k

(cijk)2 ≥ κ ·

∑i,k

(cijk)2.

45

To see (34), observe that

(X∗j )2 = (∑k

Y ∗jk)2 = (

∑k

κjkY ∗jkκik

)2 = (∑k

κjkcjk)2 =

∑k

κ2jkc2jk + 2

∑k<l

κjkκjlcjkcjl. (36)

On the other hand, ∑i

(cij)2 =

∑i

(∑k

cijk)2. (37)

Notice that cijk = cjk if i is connected to both j and k and 0 otherwise. Thus, to bound∑

i(cij)

2,

we can bound the number of terms c2jk and cjkcjl that appear in (37). There are κjk intermediaries

that connect to both j and k, thus c2jk appears exactly κjk in (37). For k < l; the number of terms

cjkcjl that appear in (37) is twice the number of intermediaries who is connected to all three nodes

j, k and l. Hence the number of terms cjkcjl appearing in (37) is at most 2 minκjk, κjl. So,

∑i

(cij)2 =

∑i

(∑k

cijk)2 ≤

∑k

κjkc2jk + 2

∑k<l

minκjk, κjlcjkcjl.

Comparing this and (36), we have ∑i

(cij)2 ≤ κ(X∗j )2,

which implies (34). With this we finish the proof.

F Proof of 7.1

Consider the convex program

min∑j

αj(Xj)2 +

∑j,i

αj(xij)

2 +∑k

βk(Yk)2 +

∑i,k

βk(Yik )2 (38)

sjt : αjXj + αjxij + βkYk + βkY

ik ≥ µk − θj − cjk ∀j, k, i where i connects j and k. (39)

46

Let ~X, ~Y be its unique optimal solution. Let zijk be the dual variables and set yijk :=zijk2 . We will

show that ~y together with ~X and ~Y satisfies the equilibrium condition. First, observe that the first

order condition of the convex program implies that

Xj =∑i,k

zijk2

=∑i,k

yijk; Yk =∑i,j

zijk2

=∑i,j

yijk;

Xij =

∑k

zijk2

=∑k

yijk; Y ik =

∑j

zijk2

=∑j

yijk.

These equations capture exactly the relationship between the quantities of amount of goods traded.

Second, the complementarity condition for an optimal solution of the convex program implies

that if (39) holds with strict inequality, thenzijk2 = yijk = 0. This condition is exactly the equilibrium

condition to guarantee each intermediary maximizes his own payoff.

To see the reverse direction, let ~X, ~Y , and ~y be the quantities traded at equilibrium. Define

zijk := 2yijk. We need to show that ~X, ~Y is the optimal solution of the convex program and

~z is the dual. First, the relationships among ~X, ~Y , and ~y implies the exactly the first order

condition of the convex program among ~X, ~Y , and ~z . Second, the following equilibrium condition

αjXj + αjxij + βkYk + βkY

ik ≥ µk − θj − cjk if strict inequality holds then yijk = 0 implies the

complementarity and slackness condition between the primal ~X, ~Y and the dual ~z. This shows that

~X, ~Y is the unique optimal solution for the convex program (38-39).

47