Marketmaking Middlemen∗
Pieter Gautier ††‡§Bo Hu‡

Makoto Watanabe§

VU Amsterdam, Tinbergen Institute
September 19, 2019

Abstract

This paper develops a model in which market structure is determined
endogenously by the choice of intermediation mode. There are two representative modes of intermediation that are widely used in real-life markets: one is a middleman mode where an intermediary holds inventories
which he stocks from the wholesale market for the purpose of reselling tobuyers; the other is a market-making mode where an intermediary offersa platform for buyers and sellers to trade with each other. We show that amarketmaking middleman, who adopts a mixture of these two intermediation modes, can emerge in a directed search equilibrium and discuss theimplications of this on the market structure. Our main insight survives
under competing intermediaries.

Keywords: Middlemen, Marketmakers, Platform, Directed Search
JEL Classification Number:D4, G2, L1, L8, R1∗

We thank seminar and conference participants at U Essex, U Bern, U Zurich, Goergetown U, Albany, the Symposium on Jean Tirole 2014, the Search and Matching workshop in Bristol, SaM network
annual conference 2015/2016 in Aix-en-Provence/Amsterdam, Toulouse School of Economics, Tokyo,
Rome, the 2015 IIOC meeting in Boston, the EARIE 2015 in Munich, the 16th CEPR/JIE conference
on Applied Industrial Organization, Workshop of the Economics of Platform in Tinbergen Institute,and the 2016 Summer Workshop on Money, Banking, Payments and Finance in Chicago FED for useful comments.
†
VU Amsterdam, Tinbergen Institute, Address: Department of Economics, VU Amsterdam, De
Boelelaan 1105, NL-1081 HV Amsterdam, The Netherlands, p.a.gautier@vu.nl‡
VU Amsterdam, Tinbergen Institute, Address: Department of Economics, VU Amsterdam, De
Boelelaan 1105, NL-1081 HV Amsterdam, The Netherlands, huboshufe@gmail.com§
VU Amsterdam, Tinbergen Institute, Address: Department of Economics, VU Amsterdam, De
Boelelaan 1105, NL-1081 HV Amsterdam, The Netherlands, makoto.wtnb@gmail.com

1

1

Introduction

This paper develops a framework in which market structure is determined by theintermediation service offered to customers. There are two representative modes ofintermediation that are widely used in real-life markets. In one mode, an intermediary acts as a middleman (or a merchant), who is specialized in buying and selling for
his own account and typically operates with inventory holdings (e.g. supermarkets,
traditional brick and mortar retailers, and dealers in financial and steel markets).
In the other mode, an intermediary acts as a marketmaker, who offers a marketplace for fees, where the participating buyers and sellers can search and trade witheach other and at least one side of the market pays a fee for using the platform (e.g.
auction sites, brokers in goods or financial markets, and many real estate agencies).
The market-making mode became more appropriate since new advanced internet
technology facilitated the use of online platforms in the late 1990s and early 2000s.
In financial markets, an expanded platform sector is adopted in a specialist market, i.e., the New York Stock Exchange (NYSE),1 and even in a typical dealers’ (i.e.,
middlemen’s) market, i.e, the NASDAQ. In goods and service markets, the electronic
retailer Amazon.com and the online hotel/travel reservation agency Expedia.com,
who started as a pure middleman, but now also act as a marketmaker, by allowing
other suppliers to participate on their platform as independent sellers. In housing
markets, some entrepreneurs run a dealer company (developing and owning luxury
apartments and residential towers) and a brokerage company simultaneously in thesame market.
Common to all the above examples is that intermediaries operate both as a middleman and a marketmaker at the same time. This is what we call a marketmaking
1

In the finance literature, the following terminologies are used to classify intermediaries: brokers
refer to intermediaries who do not trade for their own accounts, but act merely as conduits for customer
orders, akin to our marketmakers; dealers refer to intermediaries who do trade for their own accounts,
akin to our middlemen/merchants. The marketmakers (or specialists) in financial markets quote prices
to buy or sell assets as well as take market positions, so they may correspond broadly to our marketmaking middlemen.

2

middleman. Hence, the first puzzle is to explain the emergence of marketmaking
middlemen, i.e., why the middleman or the platform sector has not become the exclusive avenue of trade, despite the recent technological advancements.
We also observe considerable differences in the microstructure of trade in thesemarkets. The NASDAQ is still a more ‘middlemen-based’ market relative to the
NYSE. While some intermediaries in housing markets are marketmaking middlemen, many intermediaries are brokers. Other online intermediaries, such as eBay
and Booking.com, are pure marketmakers, who do not buy and sell on their own
accounts, like Amazon.com and Expedia.com do. They solely concentrate on theirplatform business. So the second puzzle is to explain what determines the positionof an intermediary’s optimal mode in the spectrum spanning from a pure marketmaker mode to a pure middleman mode.
We consider a model in which the intermediated-market structure is determined
endogenously as a result of the strategic choice of a monopolistic intermediary. In
our model, there are two markets open to agents, one is an intermediated market
operated by the intermediary, and the other is a decentralized market where buyers
and sellers search individually. The intermediated market combines two businessmodes: as a middleman, the intermediary is prepared to serve many buyers at atime by holding inventories; as a marketmaker, the intermediary offers a platform
and receives fees. The intermediary can choose how to allocate the attending buyers
among these two business modes.
We formulate the intermediated market as a directed search market in order tofeature the intermediary’s technology of spreading price and capacity information
efficiently – using the search function offered in the NYSE Arca or Expedia/Amazon
website or in the web-based platform for house hunters. For example, one can receive instantly all relevant information such as prices, the terms of trade and stocks
of individual sellers. In this setting, each individual seller is subject to an inventory3capacity of discrete units (normalized to one unit in the model), whereas the middleman is subject to an inventory capacity of a mass K. Naturally, the middleman
is more efficimaent in matching demands with supplies in a directed search equilibrium. The decentralized market represents an individual seller’s outside option that
determines the lower bound of his market utility.
With this set up, we consider two situations, single-market search versus multiplemarket search. Under single market search, agents have to choose which market
to search in advance, either the decentralized market or the intermediated market. This implies that the intermediary needs to subsidize buyers with their expected value in the decentralized market, but once they participate, the intermediated market operates without fear for outside competitive pressure. Given that the
middleman mode is more efficient in realizing transactions, the intermediary uses
the middleman-mode exclusively when agents search in a single market.
When agents are allowed to search in multiple markets, attracting buyers becomes less costly compared to the single-market search case — the intermediary
does not need to subsidize buyers to induce participation. However, the prices/fees
charged in the intermediated market must be acceptable relative to the available option in the decentralized market. Otherwise, buyers and sellers can easily switch tothe outside market. Thus, under multiple-market search, the outside option createscompetitive pressure to the overall intermediated market. In deciding the optimal
intermediation mode, the intermediary takes into account that a higher middlemancapacity induces more buyers to buy from the middleman, and fewer buyers to searchon the platform. This has two opposing effects on its profits. On the one hand, ahigher capacity of the middleman leads to more transactions in the intermediatedmarket, and consequently to larger profits. On the other hand, sellers are less likely
to trade on a smaller-scaled platform and buyers are more likely to trade with alarger scaled middleman, so that more sellers are available when a buyer attempts to4search in the decentralized market. Accordingly, buyers expect a higher value fromthe less tight decentralized market. This causes cross-markets feedback that leads
to competitive pressure on the price/fees that the intermediary can charge, and a
downward pressure on its profits. Hence, the intermediary trade-offs a larger quantity against lower price/fees to operate as a larger-scaled middleman. This tradeoff determines the middleman’s selling capacity and eventually the intermediation
mode.
Single-market search may correspond to the traditional search technology for
local supermarkets or brick and mortar retailers. Over the course of a shopping trip,
consumers usually have to search, buy and even transport the purchased products
during a fixed amount of time. Given the time constraint, they visit a limited numberof shops — typically one supermarket — and appreciate the proximity provided by
its inventory. In contrast, multi-market search is related to the advanced searchtechnologies that are available in the digital economy. It allows the online-customers
to search and compare various options easily. Multiple market search is also relevantin the market for durable goods such as housing or expensive items where customers
are exposed to the market for a sufficiently long time to ponder multiple available
options.
We show that a marketmaking middleman can emerge in a directed search equilibrium. The marketmaking middleman can outperform either extreme intermediation mode. Relative to a pure market-maker, its inventory holdings can reduce theout-of-stock risk, while relative to a pure middleman its platform can better exploit
the surplus of intermediated trade. It is this trade-off that answers the two puzzles
above. Somewhat surprisingly, our result suggests that an improvement in searchtechnologies induces the intermediary to generate inefficiencies to increase profits.
This occurs because platform trade creates more profits but it is at the expense of
more frictional matching.

5We offer various extensions to our baseline model. First, we introduce non-linear
matching functions in the decentralized market, which increases the profitabilityof middleman even with multi-market search. Second, we introduce an aggregate
resource constraint and frictions in the wholesale market, which increases the profitability of using an active platform even with single-market search. Third, we introduce a convex inventory-holding cost function, which reduces the profitability of a
middleman, and sellers’ purchase/production costs that accrue prior to entering theplatform, which reduce the profitability of marketmaker. However, these extensions
do not alter our main insight on the emergence of marketmaking middlemen. Forth,
we introduce competing intermediaries. As is consistent with the monopoly analysis, we show that an active platform of an incumbent intermediary that charges
positive fees can only be profitable when agents search in multi markets and theother intermediary enters with an active platform.
Finally, we provide empirical evidence for our theory. Just as in the last extension
with competing intermediaries, we treat Amazon as the centralized market and eBayas the decentralized market. For our chosen product category, Amazon acts as amarketmaking middleman. Specifically, for 32% of the sample, Amazon acts as a
middleman; for the other 68%, it acts as a platform. Our empirical evidence strongly
supports the model’s prediction that Amazon is more likely to sell the product as a
middleman when the chance of buyers to meet a seller on eBay is low, the buyers’
bargaining power is low, and/or total demand is high.
Our paper is related to the literature on middlemen developed by Rubinstein and
Wolinsky (1987).2 Using a directed search approach, Watanabe(2010, 2018a, 2018b)
2
Rubinstein and Wolinsky (1987) show that an intermediated market can be active under frictions,when it is operated by middlemen who have an advantage in the meeting rate over the original suppliers. Given some exogenous meeting process, two main reasons have been considered for the middlemen’s advantage in the rate of successful trades: a middleman may be able to guarantee the quality ofgoods (Biglaiser 1993, Li 1998), or to satisfy buyers’ demand for a variety of goods (Shevchenko 2004).
While these are clearly sound reasons for the success of middlemen, the buyers’ search is modeled
as an undirected random matching process, implying that the middlemen’s capacity cannot influencebuyers’ search decisions in these models. See also Duffie et al. (2005), Lagos and Rocheteau (2009),
Lagos et al. (2011), Weill (2007), Johri and Leach (2002), Masters (2007), Watanabe (2010), Watanabe

6

provides a model of an intermediated market operated by middlemen with high inventory holdings. The middlemen’s high selling-capacity enables them to serve manybuyers at a time. Because of the lower likelihood of stock-out, it generates a retailpremium of inventories. This mechanism is adopted by the middleman in our model.
Hence, if intermediation fees were not available, then our model would be a simplified version of Watanabe where we added an outside market. It is worth mentioningthat in Watanabe(2010, 2018a, 2018b), the middleman’s inventory is modeled as anindivisible unit, i.e., a positive integer, so that the middlemen face a non-degenerate
distribution of their selling units as other sellers do. In contrast, here we model
the inventory as a mass, assuming more flexible inventory technologies, so that the
middleman faces a degenerate distribution of sales. This simplification allows us
to characterize the middleman’s profit-maximizing choice of inventory holdings — in
Watanabe(2010, 2018b) the inventory level of middlemen is determined by aggregate
demand-supply balancing, and in Watanabe (2018a) it is treated as an exogenous
parameter. More recently, Holzner and Watanabe (2016) study a labor market equilibrium using a directed search approach to model a job-brokering service offered by
Public Employment Agencies, but the choice of intermediation mode is not the scopeof their paper.
Our paper is also related to the two-sided market literature.3 The critical feature of a platform is the presence of a cross-group externality, i.e., the participants’
expected gains from a platform depend positively on the number of participants on
the other side of it. Caillaud and Jullien (2003) show that even when agents have(2018a), Wright and Wong (2014), Geromichalos and Jung (2018), Lagos and Zhang (2016), Awaya et al.
(2019a), Awaya et al. (2019b), Nosal et al. (2015).
3
See, e.g. Rochet and Tirole (2003), Rochet and Tirole (2006), Caillaud and Jullien (2001), Caillaud
and Jullien (2003), Rysman (2009), Armstrong (2006), Hagiu (2006), (Weyl, 2010). Related papers fromother aspects can be found in Baye and Morgan (2001), Rust and Hall (2003), Parker and Van Alstyne(2005), Nocke et al. (2007), Galeotti and Moraga-González (2009), Loertscher and Niedermayer (2008),
Edelman and Wright (2015), Hagiu and Wright (2014), Condorelli et al. (2018), and Rhodes et al.
(2018). Earlier contributions of this strand of literature are, e.g., Stahl (1988), Gehrig (1993), Yavaş
(1994), Yavaş (1996), Spulber (1996), and Fingleton (1997). For platform studies emphasizing matchingheterogeneity, see e.g., Bloch and Ryder (2000), Damiano and Li (2008) and De Fraja and Sákovics(2012).

7

pessimistic beliefs on the intermediated market, the intermediary can make profits
by using “divide-and-conquer” strategies, i.e., subsidizing one group of participantsin order to attract another group and extract the ensuing externality benefit. To
be consistent with this literature, we develop an equilibrium with an intermediary
based on similar pessimistic beliefs. Broadly speaking, if there were no middlemanmode, our model would be a directed search version of Caillaud and Jullien (2003)
in combination with a decentralized market. Further, our result that the intermediary sometimes induces agents to search more than they like is related to the ideaof search diversion in Hagiu and Jullien (2011). They pursue this idea in a modelof an information platform that has superior information about the match between
consumers and stores and that could direct consumers first to their least preferredstore.
Rust and Hall (2003) develop a search model which features the coexistence ofdifferent intermediation markets.4 They consider two types of intermediaries, one isa “middleman” whose market requires costly search and the other is a monopolistic“market maker” who offers a frictionless market. They show that agents segment
into different markets depending on heterogeneous production costs and consumption values, thus these two types of intermediaries can coexist in equilibrium. Their
model is very different from ours in many respects. For instance, selling capability and inventory do not play any role in their formulation of a search rule, but itis the key ingredient in our model. As Rust and Hall (2003) state: “An importantfunction of intermediaries is to hold inventory to provide a buffer stock that offers
their customers liquidity at times when there is an imbalance between supply anddemand. In the securities business, liquidity means being able to buy or sell a reasonable quantity of shares on short notice. In the steel market, liquidity is also
associated with a demand for immediacy so that a customer can be guaranteed ofreceiving shipment of an order within a few days of placement. Lacking inventories
4

See Ju et al. (2010) who extend the Rust and Hall model by considering oligopolistic market makers.

8and stockouts, this model cannot be used to analyze the important role of intermediaries in providing liquidity (page 401; emphasis added).” This is exactly what weemphasize in our model which incorporates Rust and Hall’s observation. We showthat intermediaries can pursue different types of intermediation modes even when
faced with homogeneous agents.
The rest of the paper is organized as follows. Section 2 presents our model ofintermediation, and the benchmark case of single-market search. Section 3 extends
the analysis to allow for multiple-market technologies and presents the key findingof our paper. Section 4 discusses modeling issues. Section 5 discusses some real-life
applications of our theory. Section 6 presents the empirical evidence. Finally, section
7 concludes. Omitted proofs are in the Appendix. Finally, the online appendicescontain our extension to allow for competing intermediaries, unobservable capacityand participation fees, and additional details on the empirical analysis.

2

A basic model with single-market searchThis section studies the choice of intermediation mode under single-market technologies. It serves as a benchmark. We start with the environment in which themonopolistic intermediary operates.

2.1

The framework

Agents. We consider a large economy with two populations, a mass B of identicalbuyers and a mass S of identical sellers. Each buyer has unit demand for a homogeneous good, and each seller is able to sell one unit of that good. The consumption
value for buyers is normalized to 1. Sellers can stock the good from a competitive
wholesale market at a price equal to a constant marginal cost c ∈ [0, 1).

Retail markets.

Buyers and sellers want to trade with each other, but they can

only meet in a retail market. There are two retail markets available — a central9

ized/intermediated market (C market), which is operated by a monopolistic intermediary, and a decentralized market (D market), which serves as the outside option foragents. We consider two different search technologies: single-market search, where
agents can attend only one market, and multi-market search where agents can attend two markets sequentially. This section spells out the details of single-market
search while Section 3 discusses multi-market search.
In general, we let {B C , S C , B D , S D } denote the measures of agents allocatedacross markets, where B C (B D ) ∈ [0, B] is the mass of buyers who join the C (D)market, and S C (S D ) ∈ [0, S] is the mass of sellers who join the C (D) market.
Under single-market search, agents choose either the C or the D market, namely
B D = B − B C and S D = S − S D . We thus denote the measures of participants by
N = {B C , S C } ∈ R2+ in this section. Below, we refer to a buyer’s value as V and aseller’s value as W . We add a superscript when we refer to a market or a supplier
type. These values generally depend on N and on market prices.

Matching and price formation in the decentralized market. In the decentralized market, matching is random and the surplus is split by bilateral bargaining.
It works as follows. Suppose that a measure of B D > 0 buyers and S D > 0 sellers participate in the D market, so the buyer-seller ratio in the D market is xD (B D , S D ) =BD. If all buyers and sellers participate in the D market (B D = B, S D = S), then aSDbuyer meets a seller with probability λb and a seller meets a buyer with probability
λs = λb xD (B, S), satisfying λb , λs ∈ (0, 1). If only a subset of buyers B D ≤ B andD

sellers S D ≤ S participate, then the matching probabilities are given by λb SS andD

λs BB , respectively.5 Matched partners follow an efficient bargaining process, which
5D

The idea behind λb SS is that if a buyer visits a seller but the seller is not available, i.e., he choseD

to offer his product in the C market, then the meeting fails. A similar interpretation applies to λs BB .
By an accounting identity, the number of matched buyers is equal to the number of matched sellers,
DD
B D λb SS = S D λs BB . This matching technology, which is linear in the number of participants on theother side of the market, is a simplified way to formulate the outside option of agents. In Section 4.1,
we show that our main insight is valid with general non-linear matching functions where the meeting
rate (and the expected value) depends on the relative measures of buyers and sellers.

10yields a linear sharing rule of the total surplus, with a share of β ∈ (0, 1] for thebuyer and a share of 1 − β for the seller. In the D market, the expected value for abuyer is given by V D ,V D (N ) = λbSDβ(1 − c),S

(1)and a sellers’ expected value is given by W D ,W D (N ) = λsBD
(1 − β)(1 − c).
B

The intermediary in the centralized market.

(2)

The centralized market is oper-

ated by a monopolistic intermediary whose profit-maximizing mode is the focus of
the model. The intermediary can perform two different intermediation activities. As
a middleman, it purchases a good with mass K ≥ 0 from the wholesale market at a
cost c ∈ [0, 1), and resells it to buyers at a price of pm ∈ [0, 1]. As a market-maker, it
does not buy and sell but instead provides a platform where buyers and sellers caninteract with each other for trade, at a fee. The market-maker mode is also referred
to as the “platform” below. The transaction fees that are charged to buyers and sellers are denoted by f b , f s ∈ [0, 1], respectively. Let f denote the sum of the fees, andassume that the following restriction to the fees applies,6f ≡ f b + f s ∈ [0, 1 − c].

(3)

Trading in the C market is characterized by directed search, which is detailed in thetiming below.

Timing, strategies and equilibrium concept.

The timing of the decisions by

the buyers, the sellers and the intermediary are as follows.
1. Announcement stage. The intermediary announces its intermediation modedenoted by i ∈ {m, p, h}, where m refers to middleman, p to platform (marketmaker) and h to hybrid, and a corresponding plan P i which – depending on the6

Allowing for participation fees/subsidies, which accrue irrespective of transactions in the C market,
will not affect our main result. We offer such an extended model in the online appendix.

11

intermediation mode – may include the platform fees, inventory and price. P iwill be detailed below.
2. Market participation stage. Observing i and P i , buyers and sellers simultaneously decide which market to participate in, the C or the D market. This gives
the measures of participants in the C market, N = {B C , S C }.
3. Trade stage. Trade takes place in each market. Matching in the D market
is random and prices are determined by Nash bargaining. Search in the C
market is directed. Given an announcement i = h, conditional on a positive
mass of buyers and sellers in the C market, the moves in the C market consistof the following stages.
1) Sellers simultaneously post a price ps ∈ [0, 1]. Owing to the advanced
matching technology from the intermediary, the prices and capacities of all
suppliers are publicly observable (individual sellers with 1 unit inventory
post a price ps , the middleman with inventory K posts a price pm ).
2) After observing the prices, buyers simultaneously decide which supplierto visit. Each buyer can visit at most one supplier, either one of the sellers
or the middleman.
3) An individual seller that receives one or more buyers serves one of thesebuyers at random at the announced price. The middleman trades with all
buyers at the announced price as long as the measure of visiting buyers
does not exceed the measure of the inventory. Else, it randomly selects ameasure of K buyers to trade with. If a buyer is served by the middleman
his payoff is 1 − pm , if a buyer is served by an individual seller, his payoff
is 1 − ps − f b , and the seller’s payoff is ps − f s ; if a buyer or a seller does
not trade, his payoff is 0. Finally, the intermediary’s payoff consists of therevenue of platform fees and the profits of inventory sales (see Section 2.4).

12Intermediary
The intermediary announcesi ∈ {m, p, h} and P i

Buyersjoin Cjoin D

Sellersjoin Cjoin D

join D

join C

Sellers
Sellers each post ps

BuyersBuyers choose to visit either the middleman
or one of the sellers on the platform

Figure 1: Timing
The trade under pure intermediary modes is slightly different. With i = p, only
individual sellers post in the C market, and observing their prices buyers simultaneously decide which seller to visit. With i = m, buyers can only visit the middleman.
Figure 1 illustrates the timing and decisions.
The intermediary’s strategy consists of a mode choice i ∈ {m, p, h} and a vector
P i ∈ Pi , where Pi denotes the set of all feasible P i such that K ∈ [0, B], pm , f b , f s ∈[0, 1] and f b + f s ≤ 1 − c. If the intermediary chooses to be a pure middleman, it
announces i = m and P m = (pm , K) ∈ Pm . In this case, the C market is not accessible
to sellers. If the intermediary acts as a pure platform, it announces i = p and P p =(f b , f s ) ∈ Pp . In this case, the middleman sector is inactive. If the intermediary13

adopts a hybrid mode, then i = h and P h = (pm , K, f b , f s ) ∈ Ph is announced. Boththe platform and the middleman sector are available to buyers, and the platform isaccessible to sellers.
The seller’s strategy consists of a participation decision and a posted price. The
participation decision maps the intermediary’s announcement P i ∈ Pi to a seller’s
participation action in the set of {0, 1}, where we refer to 1 as participating and 0 as
not participating in the C market. The price posting decision maps what is known
in the seller’s information set, including whether he has joined the C market, themeasures of participants in the C market N and the intermediary’s announcement
P i to a posted price ps ∈ [0, 1].7
The buyer’s strategy consists of a participation decision (whether or not to participate in the C market) and a decision on which supplier to visit in the C market. The
participation decision maps an announcement P i ∈ Pi to a participation action in{0, 1}. Let µ(ps ) denote the probability measure of sellers’ posted prices. The visiting
decision of a buyer maps what is known in his information set, including whether hehas joined the C market, the probability measure of sellers’ posted prices µ(ps ), themeasures of participants in the C market N , and the announcement of the intermediary P i , to a visiting probability measure denoted by σ(·). We will restrict ourselves
to symmetric anonymous equilibria in the trading stage of the C market (see Section
2.2). In such an equilibrium, µ(ps ) is degenerate, and buyers simply visit the middleman with probability σ m ∈ [0, 1] and visit one of the sellers on the platform withprobability σ s ∈ [0, 1] satisfying σ m + σ s = 1.8
To find an equilibrium of our model, we work backwards and this may hint at
subgame perfection. However, to avoid complications that arise from the fact that in
7
Note that following the histories that the intermediary is a pure middleman, or that the seller
didn’t join the C market, the choice set of ps is “doing nothing” which we can denote by ∅.
8
Let σ(·) denote the probability measure of a buyer’s visiting decision, where σ(m) is the probability
that a buyer visits the middleman, and σ(s, ps ) the probability
that a buyer visits one of the individualR
sellers whose posted price is ps . They satisfy σ(m) + σ(s, ps )dµ(·) = 1, where the second term is theLebesgue integral with respect to µ(ps ). Furthermore, if the buyer didn’t join the C market, the choice
set of which supplier to visit in the C market is ∅.

14the trade stage, both a deviating seller and the expected number of buyers visiting
this deviant have measure zero, we follow the standard market-utility approach (seefor example Wright et al. (2019)). It is based on the notion that the expected numberof buyers per seller is a well-defined object (both on and off the equilibrium path) andthat in a large market, buyers must always receive their market utility, which eachindividual seller treats as given. See section 2.2 for more details.
Section 2.3 examines the participation stage which is a two-sided market problem
and may have multiple equilibria because of indirect network externalities. Those
equilibria depend on off-equilibrium-path beliefs. We follow Caillaud and Jullien(2003) and adopt pessimistic beliefs.9 . Finally, given the pessimistic beliefs and the
directed-search equilibrium, participation decisions are optimal and given the participation decisions, the intermediary decides upon its optimal operation mode (seesection 2.4).

2.2

Trade-in-the-C market stage

In the C market, buyers cannot coordinate which supplier to visit. Hence, there
is a chance that more buyers show up at a given supplier than the supplier canaccommodate, in which case some buyers get rationed. Alternatively, fewer buyers
may show up at a supplier than the supplier can accommodate, in which case thesupplier is rationed.
The coordination frictions are captured by only considering symmetric anonymous equilibria where identical buyers visit identical sellers with equal probabilities.10 Under the visiting probabilities (σ m , σ s ), a measure xm = B C σ m of buyers
visits the middleman sector, and B C σ s of buyers visit the platform, leading to aCsbuyer-seller ratio of xs = BS Cσ on the platform. Since the measure of buyers vis9
Our result on the optimal choice of intermediation mode does not depend on assumptions about
beliefs. For more details we refer to Section 2.4 and footnote 16.
10
Focusing on symmetric anonymous equilibria, rules out asymmetric equilibria where the coordination problem is solved by for example buyer 1 visiting seller 1 with probability 1, buyer 2 visiting
seller 2 with probability 1, etcetera.

15

iting sellers S C xs and the middleman xm should sum up to the total population ofparticipating buyers B C , we have a standard accounting identity
S C xs + xm = B C .

(4)Given P i and N , a directed search equilibrium is a triple (σ s , σ m , ps ) such that:
1. Buyers only visit suppliers (sellers or middleman) that offer them their expected market utility. This determines the visiting probabilities (σ s , σ m ) andthe corresponding expected queues (xs , xm ).
2. Sellers post a price ps to maximize profits subject to the constraint that visiting
buyers must receive their market utility.
3. The expected queues xs and xm satisfy the accounting identity (4).
Since the visiting probabilities are isomorphic to the expected queues, we takethe standard approach and characterize the directed search equilibrium in the tradestage in terms of expected queues. To solve for the equilibrium, we first present the
expected values for a buyer who visits the middleman or a seller.

Buyers’ expected value of visiting the middleman. The middleman sector isopen when the intermediary chooses i = m or h. Suppose that a measure xm > 0of buyers visit the middleman. Since the middleman has capacity K, its expected
revenue is given by min{K, xm }pm . The expected value for a buyer who visits the
middleman isV m (xm ) = η m (xm )(1 − pm ),(5)where η m (xm ) is the matching probability of a buyer at the middleman. For xm > 0,
η m (xm ) ≡ min{ xKm , 1}. For xm = 0, we define η m (·) = 1 if K > 0 and η m (·) = 0 if
K = 0. Obviously, if K ≥ xm > 0, the matching probability η m (xm ) = 1. This is
how the advanced inventory technologies of the intermediary help to improve themmatching efficiency. Under i = m, (5) takes a specific value: V m (B) = K
B (1 − p ).

16

Buyers’ expected value of visiting a seller.

The platform is open when theintermediary chooses i = p or h. Given that we have a continuum of sellers, in asymmetric equilibrium, the probability that an individual buyer visits a particularseller is zero, and the number of buyers visiting a seller, denoted by N , is a random−xs xsnvariable that follows a Poisson distribution, Prob[N = n] = en!

, with an expected

queue xs ≥ 0.11 Therefore, a seller with an expected queue xs > 0 has a probabilitys

1 − e−x (= Prob[N ≥ 1]) of successfully selling, while each buyer has a probability−xsη s (xs ) = 1−exsof successfully buying. Hence, the expected value of a seller on theplatform with a price ps and an expected queue xs is given by W C ,W C = xs η s (xs )(ps − f s − c),and the expected value of a buyer who visits a seller on the platform is given by V s ,V s (xs ) = η s (xs )(1 − ps − f b ).
C

. We now derive the
For i = p, these value functions take specific values with xs = BSCdirected search equilibrium in terms of the equilibrium expected queues and priceps .

Equilibrium queues. Under the announcement P h , sellers’ posted price ps , andB C , S C > 0, we have12




BC
if V m (B C ) ≥ V s (0)



xm =
(0, B C ) if V m (xm ) = V s (xs )




C

 0
if V m (0) ≤ V s ( B
)
SC

(6)

where V j (·) is the equilibrium value of buyers in the C market of visiting a seller if
j = s and the middleman if j = m. Combining (4) and (6) gives the counterpart for
11

Suppose there are b buyers and s sellers, where both b and s are positive integers. If each buyer
visits each seller with equal probability, a seller gets at least one buyer with probability 1 − (1 − 1s )b .
s
Taking the limit as b and s go to infinity and xs = b/s fixed, in a large market, a fraction 1 − e−x ofthe sellers will be matched with a buyer. This process generates an urn-ball matching function. See forexample Butters (1977).
12
Note that (5) describes how V m depends on the posted price.

17

C

xs ∈ [0, B]. If only one sector is open, all participating buyers will join that sector.
SC
That is, under P m , xm = B C , and under P p , xm = 0. Given the equilibrium queues,
buyers’ equilibrium visiting probabilities can be derived accordingly.
Furthermore, a buyer’s market utility V C (N , P i ) is defined as follows. Underi = h, V C (N , P i ) ≡ max{V s (xs ), V m (xm )}, where xs and xm are determined by (4)and (6). Under a pure middleman mode (i = m), V C (N , P m ) = V m (B C ). Under aC). In the following, we willpure market-maker mode (i = p), V C (N , P p ) = V s ( BSC
not write out the explicit dependence of V C (·) on N and P i whenever there is noconfusion.

Sellers’ equilibrium price.

To derive the equilibrium price of sellers on the plat-form, ps , we follow the standard procedure in the directed search literature. Supposethat a potential deviant seller offers a price p0 6= ps that attracts an expected queuex0 6= xs of buyers. Note that given that we have a continuum of sellers, this deviation
has measure zero and does not affect the expected utility of buyers in the C market,
V C.
Since buyers must be indifferent between visiting any seller (including the deviating seller), the market-utility condition holds on and off the equilibrium path andsatisfies
η s x0−x0

where η s (x0 ) ≡ 1−ex0
1 − p0 − f b = V C ,(7)is the probability that a buyer is served by this deviating seller.

Given market utility V C , (7) determines the relationship between x0 and p0 , which we
denote by x0 = x p0 |V C . This yields a downward sloping demand curve faced by theseller: when the seller raises his price p0 , the expected queue of buyers x0 becomesshorter and this corresponds to a lower trading probability for the seller, and viceversa. The seller’s optimal price must satisfy−x(p0 |V C )ps V C = arg max
1−ep0 − f s − c0p18Substituting out p0 using (7), the sellers’ objective function can be written as

1 − e−x0(1 − f − c) − x0 V C ,where f ≡ f b +f s and x0 = x p0 |V C satisfies (7). Since choosing a price is isomorphic
to choosing an expected queue, the first order condition is0e−x (1 − f − c) − V C = 0.
The second order condition is also satisfied. Arranging the first order condition using(7) and evaluating it at xs = x ps |V C , we can back out the equilibrium price ps =ps V C which is equal tos 
xs e−x(1 − f − c) .
p =f +c+ 1−
1 − e−xsss(8)Therefore, inserting (8) into the buyer’s value V s (·) yieldss

V s (xs ) = e−x (1 − f − c),(9)and the seller’s expected value W C can be written asss

W C (P i , N ) = (1 − e−x − xs e−x )(1 − f − c),(10)

for i = p, h. We now turn to the buyers’ and sellers’ participation decisions.

2.3

Participation stage

The equilibrium in the participation stage is derived taken the directed search equilibrium in the next stage (that we derived above) as given. Following the intermediary’s announcement, each infinitesimal agent has expectations about how all agentswill participate in the C market, and in equilibrium those expectations are correct.
Our definition of the participation equilibrium is therefore a rational expectation
equilibrium which is consistent with the literature, e.g., Caillaud and Jullien (2003)
and Hagiu (2006).

19Definition 1 (Participation equilibrium) A participation equilibrium given P i isa pair N = (B C , S C ) such that
B C = B · I V C (P i , N ) ≥ V D (N )and

S C = S · I W C (P i , N ) ≥ W D (N ) ,where I {·} is an indicator function which equals 1 if the condition in brackets is
satisfied, and otherwise equals 0. V j (W j ) is the buyer’s (seller’s) value function inmarket j, j = C, D, that has been described in previous sections. A participation
allocation is a mapping N (·) that maps each intermediary announcement P i into aparticipation equilibrium N (P i ).
In the above definition, we make it explicit that a buyer’s and a seller’s value
in the C market ultimately depends on the intermediary announcement P i and themeasures of participants N . We also make the tie-breaking assumption that agentschoose to participate in the C market if they are indifferent between visiting the C
and the D market.
As in the two-sided market literature, there exist cross-group externalities in the
C market, namely a participant’s gain from the C market positively depends on thenumber of participants on the other side of the market. Therefore, for each P i there
may exist multiple participation equilibria.
Although our conclusion about the optimal intermediary mode does not dependon the selection of beliefs, we stick to the pessimistic beliefs assumption in the main
analysis following Caillaud and Jullien (2003).13 Under pessimistic beliefs, buyers
and sellers coordinate on a participation distribution so that the C market is empty
whenever possible. In our set-up, despite the pessimistic beliefs, the intermediary
13
Alternatively, under the optimistic beliefs, N (·) = (B, S C ) whenever possible, where S C = S if
i = h, p and S C = 0 if i = m. We provide an analysis of the optimistic beliefs in footnote 16.

20

can convince buyers that if they do join the C market, they have access to the middleman inventory and their expected value is higher than if they would visit the Dmarket. That is, the intermediary can “divide” buyers using middleman inventory.
And knowing buyers will join the intermediary, the sellers will also join the intermediary whenever possible (namely, when the platform is open under i ∈ {h, p}).14
To make the “divide” step most challenging for the intermediary, we assume thatagents coordinate on N̂ = (B, 0) whenever possible. Namely, (a) no sellers participate(S C = 0), and only the middleman supplies the good; and (b) all buyers participate(B C = B), so each buyer has the minimum chance to be matched due to a congestion
effect in the matching technology. Indeed, if the intermediary can guarantee buyers
a higher expected utility than in the D market under N̂ , then buyers can be securely“divided” under any alternative beliefs. More specifically, we define a pessimistic
allocation as follows.
Definition 2 A pessimistic allocation N (·) is given by (B, S C ) if V C (N̂ , P i ) ≥ λb β(1 − c);iN (P ) = (0, 0)otherwise;
where i ∈ {m, h}, N̂ = (B, 0) and S C ∈ {0, S}.
In words, assuming the measures of participants N̂ = (B, 0), the only scenario
where buyers are willing to join the C market is the one where the middleman holds
enough inventory and charges a price that guarantees more than the expected value
in the nonempty D market. Since S C = 0, V s (·) = 0, a positive measure of participating buyers requires
V C (N̂ , P i ) =K(1 − pm ) ≥ λb β(1 − c).
B

14

(11)

While Caillaud and Jullien (2003) focus on the divide-and-conquer strategy which employs a negative entry fee (namely a participation subsidy), we restrict our analysis to a zero participation fee anddefer the analysis of divide-and-conquer strategies to the online appendix. There we show that ourconclusion on the optimal intermediary mode continues to hold with divide-and-conquer strategies.

21

There are two issues that warrant further discussion. First, the exact value of S C
follows from a seller’s optimal decision that whenever B C = B and the platform isopen, we have S C = S, and when the platform is not open, we have S C = 0. Second,since dividing buyers is based on middleman inventory, a pure market-maker i = p
will not be able to attract any agent under the pessimistic beliefs. We, therefore,
focus on i ∈ {m, h} for the optimal intermediation mode in the next subsection.15

2.4

Announcement stage

Given the participation allocation under pessimistic beliefs and the corresponding
directed search equilibrium, the intermediary chooses i and P i to maximize its profits. The intermediation mode defined above by i can be succinctly stated in terms of
the equilibrium allocation, xm .
Definition 3 (Intermediation Mode) Suppose B C ∈ (0, B] buyers and S C ∈ [0, S]
sellers participate in the C market. Then we say that the intermediary acts as:• a pure middleman if xm = B C ;• a market-making middleman if xm ∈ (0, B C );• a pure market-maker if xm = 0.

The pure middleman.

Under i = m, xm = B C . The profit maximizing problem ofa pure middleman is
max
m
m

P ∈P

n

o
min{B, K}pm − Kc

subject to (11), which implies that all buyers visit the C market (xm = B C = B).
The maximum profit is achieved at P m∗ = 1 − λb β(1 − c), B . That is, the pure15

Our results about the optimal intermediation mode does not rely on the in-feasibility of i = p underpessimistic beliefs. Under optimistic beliefs or with a subsidy to divide buyers/sellers to participate,a pure platform mode i = p is feasible. However, the profit-maximizing intermediary mode remains
the same. See footnote 16 for a discussion on optimistic beliefs. We refer to the online appendix for ananalysis of participation subsidies.

22

middleman serves all buyers with enough inventory (xm = K = B). This leads to a
profit ofΠm ≡ B(1 − λb β)(1 − c).

(12)

Suppose a hybrid mode is announced (i = h so that xm < B C

The active platform.

is allowed), the intermediary chooses a P h ∈ Ph to maximize its profitssΠh (Ph ) = S(1 − e−x )(f b + f s ) + min{K, xm }pm − Kc
subject to the accounting identity (4), the directed search equilibrium condition (6)and the participation constraint (11). The intermediary’s expected profits consistsof the revenue of platform fees, S(1 − e−x )(f b + f s ), plus the revenue of inventory
sales minus inventory cost, min{xm , K}pm − Kc. If condition (11) is satisfied, then allbuyers join the C market (B C = B). As a result, all sellers find it profitable to join
the C market (S C = S) according to Definition 1.
The following proposition delivers the main message of our single-market searchanalysis, where we show that creating an active platform gives a lower profit thanΠm .16
Proposition 1 (Pure middleman) Under single-market search technologies, the intermediary will not open the platform and will act as a pure middleman with xm =K = B, serving all buyers for sure.
16

Our conclusion that the pure middleman is optimal under single-market search does not rely on
pessimistic beliefs. Under optimistic beliefs, the measures of participants are given by N = (B, S)
whenever possible. Hence, a pure middleman obtains a profit of B(1 − c) by announcing P m = (1, B).
This profit is higher than the profits under an active platform:Πh (·)s=S(1 − e−x )f + min{K, xm }pm − Kc≤

Sxs f + min{K, xm }(pm − c) + (min{K, xm } − K)c≤(Sxs + xm ) max{f, pm − c} < B(1 − c),for xm < B. The last inequality follows from f ≤ 1 − c and pm ≤ 1. Therefore, using the middleman
inventory to break the pessimistic beliefs is not the reason that makes the pure middleman mode
optimal.

23

Proof. Observe that for xm < B,
V m (xm ) ≥ V m (B) =K(1 − pm ),B(13)and by (11) and K ≤ B, we have pm − c ≤ (1 − λb β)(1 − c). Similarly, with xs > 0 and(11), combining (5), (13) and V s (xs ) ≥ V m (xm ) (which holds for xm ∈ (0, B)) yieldsse−x (1 − f − c) ≥ V m (xm ) ≥K(1 − pm ),Bindicating that 1 − f − c ≥ λb β(1 − c) or f < (1 − λb β)(1 − c). Therefore, we have thatfor xm < B,sΠh (·) = S(1 − e−x )f + min{K, xm }pm − Kc
≤ Sxs f + min{K, xm }(pm − c) + (min{K, xm } − K)c≤ (Sxs + xm ) max{f, pm − c}
< B(1 − λb β)(1 − c) = Πm .
Hence, opening the platform is not profitable under single-market search.
The intuition behind the occurrence of a pure middleman mode is as follows.
Given the frictions on the platform, a larger middleman sector creates more transactions. To achieve the highest possible number of transactions, the intermediary
shuts down the platform. The middleman’s capacity is the most efficient way to
distribute the good and, if agents search within a single market, the intermediaryis guaranteed the highest possible surplus by choosing this mode. The allocation
characterized here serves as a benchmark for the rest of our analysis.17

3

Multi-market search

In this section, we extend our analysis to multi-market search technologies where
agents can search in both the C and the D market. To facilitate the presentation
17
According to equilibrium condition (6), the choice of pm is isomorphic to the choice of xm for xm < B.
We can define the profits of a hybrid intermediary as a function of its middleman sector scale xm .
Complement to Proposition 1, the intermediary’s profits are monotonically increasing in xm . The proof
is available upon request.

24

of our key idea, we make the assumption that the C market opens prior to the Dmarket.18 Apart from the fact that this appears to be the most natural setup in our
economy, it can be motivated by the first mover advantage of the intermediary: its
expected profit is higher if the C market opens before the D market. Hence, thissequence arises endogenously if the intermediary is allowed to select the timing of
the market sequence.19 Below we maintain the set-up of matching and price formation in both markets, and we show that under multi-market search the intermediary
adopts a hybrid mode, where it acts both as a marketmaker and a middleman.
The timing in a multi-market search environment is illustrated in figure 2 andformally stated as follows.
1. The intermediary announces i ∈ {m, p, h} and P i .
2. After observing P i , buyers and sellers simultaneously decide whether to participate in the C market or not. Then the C market opens, where participating
sellers simultaneously post a price ps , and then participating buyers simultaneously decide which supplier to visit (one of the individual sellers or the
middleman if possible). Agents who have traded successfully leave the retailmarkets.
3. Remaining agents simultaneously decide whether or not to participate in the Dmarket. Then the D market opens. In the D market, matching is random andsurplus is split by Nash bargaining.
The multi-market search set-up brings several changes to the analysis. First,
joining the C market does not rule out the possibility of trading in the D market.
18
If the two markets opened at the same time, we would have to deal with the agents’ beliefs about
what other agents would choose when they turn out to be matched in both markets. This would giverise to multiple equilibria which complicates the analysis significantly. Our sequential setup avoids
this issue. In an infinite horizon model, one can construct a stationary equilibrium relatively easily
where the order of the markets does not matter (see Watanabe 2018a).
19
In a recent study without intermediation, Armstrong and Zhou (2015) show that a seller often
makes it harder or more expensive to buy its product later than at the first opportunity.

25

Intermediary
The intermediary announcesi ∈ {m, p, h} and P i

Buyersjoin C

Sellers

not join C

not join C

join Cjoin Cnot join C

Sellers
Sellers each post ps

BuyersBuyers choose to visit either the middleman
or one of the sellers on the platform......

......

......

Unmatched buyers
join D

not join D

Unmatched sellers
join D

not join D

join D

not join D

Followed by random searchand Nash bargaining

Figure 2: Timing under Multi-market search
We thus denote the measures of participants across markets by a quadruple Ñ ={B C , S C , B D , S D }.20 .
Second, since an agent can always refuse to trade if it yields a negative value, thevalue of joining the C and the D market sequentially is always larger or equal to thatof only joining the D market. This holds for any configuration of P i and expected Ñ .
Therefore, independent of the agents’ beliefs, the only participation equilibrium thatis consistent with Definition 1 is the one where all agents first visit the C market
and then the D market whenever possible. That is, B C = B for all announcements
20

We need to extend N to Ñ (adding B D and S D ) because agents may participate in multiple markets,
i.e., B C + B D ≥ B and S C + S D ≥ S.

26

of the intermediary, and S C = S whenever the platform in the C market is open.21
Third, while inducing participation in the C market is easier, convincing participants to trade in the C market becomes more difficult. The intermediary needs toensure that the offers from the C market must be weakly better than the participants’ expected utility in the D market. This is imposed by the incentive constraints
which are derived below (see Section 3.2).
Forth, the expected utility of buyers and sellers in the D market is affected by
the terms of trade that the intermediary commits to in the C market. Hence, there
exists cross-market feedback. The intermediary takes this into account in choosing
the optimal mode. The feedback from the D market makes it optimal to adopt a
hybrid mode which is the key message of Section 3.3. We work backwards and startwith the equilibrium value in the D market.

3.1

Trade-in-the-D market stage

Suppose that in equilibrium the offers from the C market make it weakly better for
buyers and sellers to go there than to go directly to the D market.22 Then, the agents
who ultimately join the D market are those who failed to trade in the C market.
Denote the expected queue at the middleman by xm , and the expected queue at anindividual seller by xs . Both satisfy the accounting identity (4). The population ofsmatched sellers in the C market is Sxs η s (xs ) = S(1 − e−x ). Hence, sellers who are
not matched in the C market automatically join the D market,ss

S D (xs ) = S − S(1 − e−x ) = Se−x .
The population of matched buyers in the C market consists of two groups, the buyers
matched with the middleman, min{K, xm }, and the buyers matched with one of the21

We thus will skip the analysis of participation equilibrium below. Note further that irrespectiveof agents’ beliefs, an empty D market cannot occur in equilibrium. This is because even when buyers
are extremely pessimistic about the D market so that sellers are indifferent between entering and notentering, there will always be sellers who fail to sell in the C market and they will be automaticallypresent in the D market.
22
Conditions for this to hold are incentive constraints (15), (16) and (17), which are derived below.

27ssellers on the platform, (B − xm )η s (xs ) = S(1 − e−x ). Hence, the measure of buyers
joining the D market is given bys

B D (xm , K) = B − min{K, xm } − S(1 − e−x ).
Note that the pure modes are special cases. With the pure middleman announcement
i = m, S D = S D (0) = S and B D = B D (B, K) = B − K. With the pure platform
−B
S ,announcement i = p, there is no middleman inventory. Hence, S D = S D B
S = Se
B
B D = B D (0, 0) = B − S 1 − e− S .23
Inserting S D into (1) gives the equilibrium value for buyers in the D market,s

V D (xm ) = λb e−x β (1 − c) .
s

e−x is the probability that a seller fails to trade in the C market. β(1 − c) is thebuyer’s payoff when matched with a seller in the D market, which happens withs

probability λb e−x . Hence, the larger the platform size xs , the higher the chancethat a seller trades in the C market, and consequentially the lower the chance thata buyer can trade successfully in the D market and the lower his expected outside
payoff V D is.24
Inserting B D into (2) gives the equilibrium value for sellers in the D market,
W D (xm , K) = λs ξ (xm , K) (1 − β) (1 − c) ,where ξ (xm , K) is the probability that a buyer fails to trade in the C market and it
is given byξ (xm , K) ≡ 1 −



B−xm
1 
.
min {K, xm } + S 1 − e− S
B

(14)

m

A buyer visits the middleman sector with probability xB and is served with probability min xKm , 1 (if xm > 0). Alternatively he visits the platform with probability
23

In the following, we stick to that there is no middleman inventory (K = 0) when i = p is announced.
As under the single-market search, the buyers’ expected value in the D market generally depends
on the measures of participants Ñ . We aim to be more precise in this section and emphasize that V D
depends on the intermediary mode in the C market represented by xm . We follow the same notation
for W C (·), W D (·) and V C (·) below.
24

28

Sxs
1−e−x
s s
B and is served with probability η (x ) =xss. Hence, the second term of (14)represents the probability that the buyer trades in the C market. W D (·) equals theseller’s payoff conditional on having a buyer, (1 − β)(1 − c), multiplied by the probability that a seller meets a buyer in the D market, λs ξ(·). Note that both V D (xm ) andW D (xm , K) take specific values with xm = B under i = m and xm = K = 0 underi = p.

3.2

Trade-in-the-C market stage

In this section, we derive the directed search equilibrium for the C market given thevalues in the D market and the participation decisions (B C = B, S C = S). Relative tosingle-market search, what is new here is that agents always expect a non-negativevalue of visiting the D market when deciding whether or not to accept an offer in the
C market. Therefore, the prices/fees in the C market must be low enough to inducebuyers/sellers to visit and trade.

Incentive constraints to trade in the C market. Whenever the platform is active, it must satisfy the following incentive constraints:
1 − ps − f b ≥ V D (xm ),(15)

ps − f s − c ≥ W D (xm , K).

(16)

Condition (15) states that the offered price/fee on the platform is acceptable for abuyer only if the offered payoff, 1 − ps − f b , weakly exceeds the expected value thatbuyers can obtain in the D market. Since V D (xm ) is increasing in xm , the intermediary has the incentive to lower xm in order to lower the buyers’ expected outside
payoff. Condition (16) is the incentive constraint for sellers to trade in the C market,
which states that the payoff in the C market ps − f s − c should be no less than the
expected payoff in the D market. Both conditions apply to i ∈ {p, h}.
A similar incentive constraint must be satisfied for buyers to trade in the middle-29

man sector:
1 − pm ≥ V D (xm ).

(17)

It states that the middleman’s price must be acceptable for buyers relative to the
expected payoff in the D market under i ∈ {h, m}. Specifically, for i = m, (17) becomes
1 − pm ≥ λb β(1 − c).

(18)

Under conditions (15) to (17), agents are weakly better off trading in the C market. Therefore, under our tie-breaking assumption, indifferent buyers trade in the C
market whenever they are matched there.

Values. Suppose the incentive constraints hold, then under i = h, the value ofbuyers in the C market equals V C (xm ) = max{V s (xs ), V m (xm )}, where
V s (xs ) = η s (xs ) 1 − ps − f b + (1 − η s (xs )) V D (xm )(19)

for an active platform xs > 0 andV m (xm ) = η m (xm ) (1 − pm ) + (1 − η m (xm )) V D (xm )(20)

for an active middleman sector xm > 0, where xs and xm satisfy the accounting
identity (4). Intuitively, if a buyer visits a seller (or a middleman), then he gets
served with probability η s (xs ) (or η m (xm )) and his payoff is 1 − ps − f b (or 1 − pm ).
If not served in the C market, he enters the D market and finds an available sellerswith probability λb e−x , and obtains a payoff of β(1 − c). It follows naturally that
V C (B) = V m (B) under i = m, and V C (0) = V s ( BS ) under i = p. Similarly, the valueof participating sellers on the platform is given by
W C (xs , K) = xs η s (xs ) (ps − f s − c) + (1 − xs η s (xs )) W D (xm , K),(21)

under i ∈ {p, h}. A seller trades successfully in the C market platform with probability xs η s (xs ) and if this occurs, he receives ps − f s − c. If not successful in the C30

market, the seller can meet a buyer in the D market with probability λs ξ (xm , K) andobtains a payoff of (1 − β) (1 − c).
In the directed search equilibrium of the trade stage, buyers search optimallyand only visit a supplier who offers their market utility V C (·) and sellers set ps thatmaximizes profits. Condition (6) in the previous section continues to characterize theequilibrium expected queues. To derive the equilibrium price ps , we again follow thestandard procedure in the directed search literature.

For simplicity, we drop the explicit dependence of

Sellers’ equilibrium price.

value functions on xm and K in this derivation. Consider a seller who deviates toa price p0 6= ps and attracts an expected queue x0 6= xs of buyers, subject to themarket-utility condition (which holds on and off the equilibrium path):V−x0

where η s (x0 ) ≡ 1−ex0

C

=η

s

x

0



0

1−p −fb+ 1 − η s x0



V D,

(22)

is the probability that a buyer is served by this deviating seller.

If not served, which occurs with probability 1 − η s (x0 ), the buyer receives V D . For agiven market utility V C , (22) determines the relationship between x0 and p0 , which
we denote by x0 = x(p0 |V C ). This yields a downward sloping demand curve: when
the seller raises his price p0 , the expected queue of buyers x0 becomes smaller, andvice versa.
Given the search behavior of buyers described above and the market utility V C ,the seller’s optimal price must satisfy
o
n



−x(p0 |V C )−x(p0 |V C )
0
s
D
1
−
e
p
−
f
−
c
+
e
W
.
ps V C = arg max
0
p

(23)

The deviating seller trades successfully in the C market platform with probability0C
1 − e−x(p |V ) and in that case he receives p0 − f s − c. Otherwise, the seller has achance to meet a buyer in the D market and he obtains an expected value of W D .
Substituting out p0 in (23) using (22), we can rewrite the sellers’ objective function31

as follows,

1 − e−x0(v (xm , K) − f ) − x0 V C − V D + W D ,where v (xm , K) is the intermediated trade surplus, i.e., the total surplus in the C
market net of the outside options, and it is defined byv (xm , K) ≡ 1 − c − V D (xm ) − W D (xm , K).
Since choosing a price p0 is isomorphic to choosing an expected queue x0 , the firstorder condition is0e−x (v (xm , K) − f ) − V C − V D = 0.
The second order condition is also satisfied. Arranging the first order condition using(22) and evaluating it at xs = x ps |V C , we obtain the equilibrium price ps = ps V C
which can be written ass 
xs e−x(v(xm , K) − f ) + W D (xm , K).
p −f −c= 1−
1 − e−xs
s

s

(24)

Equation (24) states that the optimal price ps net of fee f and cost c guarantees the
s −xs

x e
seller a profit that equals the seller’s outside value W D plus a share 1 − 1−e−xs of theintermediated trade surplus that the intermediary is willing to give to buyers andsellers, v(xm , K) − f .
For the platform to be active, the price and fees must satisfy the incentive constraints (15) and (16). Substituting in (24) yields
f ≤ v(xm , K),(25)

which states that for the platform to be active (xs > 0), the total transaction fee
f should not be greater than the intermediated trade surplus, v (xm , K). (25) also
applies to i = p where K = 0 and (25) then becomesihBf ≤ v(0, 0) = 1 − λb e− S β − λs ξ (0, 0) (1 − β) (1 − c) ,

32

(26)

where ξ (0, 0) = 1 − η s ( BS ) according to (14). Whenever (15) and (16) are satisfied, (25)must hold, and vice versa. Hence, (25) is a sufficient condition for an active platform.
Observe that K > xm cannot be profitable for the intermediary since it only
increases the capacity costs. For K ≤ xm , the intermediated trade surplus v (xm , K)
can be rewritten as
"

m
− B−x
S

v (xm , K) = 1 − λb eβ − λsK + S(1 − e−
1−
B

B−xm
S

)

!

#
(1 − β) (1 − c) ,which is decreasing in xm . This occurs because a larger sized platform (i.e., a lowerxm ) crowds out the D market transactions and lowers the outside option of the buyers.

3.3

Announcement stage

Given the participation allocation under multi-market search and the corresponding
directed search equilibrium, we now examine the optimal mode of the intermediary.
We start with writing down the payoffs for each possible intermediation mode.

Pure middleman.

If the intermediary acts as a pure middleman (i = m, xm = B),then all sellers are active in the D market. Hence, the intermediary’s problem is tochoose P m ∈ Pm to maximize its profits min{B, K}pm − Kc subject to (18). As under
single-market search, the middleman selects P m∗ = 1 − λb β(1 − c), B , serves all
buyers for sure and makes a profit which we refer to as Πm :
Πm ≡ B(1 − λb β)(1 − c).

(27)

Pure market-maker. If the intermediary acts as a pure market-marker (i = s, xm =0), then xs = B
S . The intermediary’s problem becomes



−B
b
s
S
max
S
1
−
e
f
+
f
, s.t. (26).
p
p

P ∈P

33

At the optimality, the constraint (26) is binding and this yields f ∗ = v (0, 0) . The
maximum profit for the pure market-maker referred to as Πp :B

Πp ≡ S(1 − e− S )v(0, 0).

(28)

If the intermediary is a market-making middleman, then xm ∈ (0, B) and xs ∈ 0, B
S , satisfying the condition that buyers must
Market-making middleman.

be indifferent between visiting the middleman or the platform V m (xm ) = V s (xs ).
Using the equilibrium values in (19), (20), and (24), this indifference condition generates the following expression for the price pm = pm (xm )sspm = 1 − λb e−x β (1 − c) −xm e−x(v (xm , K) − f ) .
min {K, xm }(29)

Together with (4), this equation defines the relationship between pm and xm . Applying this expression, we can see that condition (17) is eventually reduced to (25). The
profit for the hybrid mode isΠh ≡ max

P h ∈Phnos
S(1 − e−x )f + min {K, xm } pm − Kc , s.t. (25) and (29).

As a first pass to solve the problem, the following conditions imply that the intermediary’s capacity should satisfy all the forthcoming demands, and the intermediation
fee should be set to extract the full intermediation surplus.
Lemma 1 The market-making middleman sets: K = xm and f = v(xm , K).
Proof. See Appendix A.1.

Profit-maximizing intermediation mode.

First, note that relative to the puremiddleman mode, an active platform together with multiple-market search can undermine the D market by lowering the available supply. This influences the middleman’s price in the following way. With v(·) = f , the incentive constraint (17) isbinding, and the middleman’s equilibrium price is given bypm = 1 − λb e−

B−xm
S

34

β(1 − c)for any xm ∈ (0, B) (see (29)). This shows that pm decreases in xm . The outside
option of buyers depends positively on the size of the middleman sector, since a larger
scale of the middleman crowds out the platform and increases the chance that abuyer can find an active seller in the D market (who was not matched on the C
market platform). Hence, in order to extend the size of the middleman sector, theintermediary must lower the price pm . In other words, a larger platform allows for a
price increase by reducing agents’ alternative trade opportunities.

Proposition 2 (Market-making middleman/Pure Market-maker) Given multimarket search technologies, there exists a unique equilibrium with active intermediation. The intermediary will open a platform and act as• a market-making middleman if λb β ≤ 12 or if λb β > 12 and B
S ≥ x̄,• a pure market-maker if λb β > 12 and B
S < x̄,where x̄ > 0 is uniquely defined by Θ(x̄) = 0 withhiΘ(x) ≡ −e−x 1 − λb e−x β − λb x − 1 + e−x (1 − β) + 1 − λb β + λb (1 − e−x )2 .
Proof. See Appendix A.2.
With multiple-market search technologies, there is cross-market feedback fromthe D market to the C market, which makes using the platform as part or all of
its intermediation activities profitable. Additionally, the intermediary must decide
whether or not to operate as a pure market maker. Our results show that the equilibrium mode of the intermediary depends on parameter values. If λb β ≤ 12 then thebuyers’ outside option value is low. In this case, the middleman sector generates highenough profits for the market-making middleman mode to be adopted for any value1bof B
S . If instead λ β > 2 then the buyers’ outside option value is high, and attracting

buyers to the middleman sector is costly. In this case, the intermediary will only act

35

as a market-making middleman if B
S is high, since the D market is tight for buyers
and they expect a low value from it. The intermediary acts as a a pure market maker
if B
S is low, since buyers expect a value from the D market that is so high that it iscostly for the intermediary to attract buyers to the middleman. Indeed, the same
logic applies to the following comparative statics result.

Corollary 1 (Comparative statics) Consider a parameter space in which the marketmaking middleman mode is profit-maximizing. Then, an increase in the buyer’s bargaining power β or meeting rate λb in the D market, or a decrease in the buyer-sellermsratio, B
S , leads to a smaller middleman sector x and a larger platform x .

Proof. See Appendix A.3.

4

Extensions

This section considers extensions to the model. As we show below, our main insight,that the benefits of using a platform (as part of) the intermediation business is relatively large when agents can search in multiple markets rather than in a single
market only, is robust to these extensions.25

4.1

Matching functions

So far, we assumed a linear matching function in the D market. In this section,
we allow for a more general matching function. As is standard in the literature,
we assume that the matching function M (B D , S D ) is homogeneous of degree one in
D

D

D

D

D

B D and S D , M (1, x1D ) = M (BB D,S ) and M (xD , 1) = M (BS D,S ) , where xD = B
is theSDbuyer-seller ratio in the D market. Then, individual match probabilities depend on
the buyer-seller ratio.
λb (xD ) = M (1,1) and λs (xD ) = M (xD , 1) = xD λb (xD )
xD

25

(30)

For expositional simplicity, we let c = 0 and make the tie-breaking assumption that when the
middleman is indifferent between K = xm and K > xm we set K = xm .

36

where λb (xD ) is strictly concave and decreasing in xD .
For single-market search technologies, the result will not be affected by thisextension (for instance, under the pessimistic beliefs of Definition 2, the matchingprobability in the D market is simply replaced by another constant λi (xD ), i = b, s,with xD = B
S ). Therefore, we only consider multi-market search technologies. With
this modification, the buyers’ probability to meet an available seller changes froms

λb e−x to λb (xD ), and the sellers’ probability to meet an available buyer changes
from λs ξ(xm , K) to λs (xD ) = xD λb (xD ).
In what follows, we derive a condition for a pure middleman mode to be selected under multi-market search technologies. This is the case when, for example,0

λb (xD ) = 0, i.e., when there is no feedback from the D-market to the intermediary’s
decision in the C market. We proceed with the following steps. First, note that, as
before, there is no gain from having excess capacity K > xm . In addition, a pure middleman wants to avoid stockouts (K < xm ) if the first order derivative of its profitsΠm (K, pm ) = Kpm = K 1 − λb (xD )β with respect to K satisfies
K 0dΠm (K, pm )= 1 − λb (xD )β + λb (xD )β > 0,dKS
for any xD = B−K≥ 0, which states that the elasticity of the middleman’s priceS
pm = 1 − λb (xD )β should satisfy0z(K) ≡ −Kλb (xD )β∂pm /∂K
=
−
≤ 1.
pm /K
S(1 − λb (xD )β)This condition guarantees that a pure middleman should satisfy all forthcoming demand K = xm .
Second, when all buyers are served by the middleman xm = K = B, the marginal
gain of allocating buyers to the platform, measured by the intermediation fee,
f = 1 − λb (xD )β − xD λb (xD )(1 − β),can not exceed the marginal opportunity cost, measured by the lost revenue in the37

middleman sector,0

1 − λb (0)β − Kλb (0)βdxD (K, xs (K))|xs (K)=0 ,dK

where xs (K) = B−KandSs−S + (B − K − S)
dxD (K, xs (K))
d B − K − S(1 − e−x (K) )|xs (K)=0 =|K=B = 0.
|xs (K)=0 =
s (K)
−x
dK
dK
Se
S 2 e−xs (K)Hence, the intermediary can be a pure middleman even with multiple-market searchtechnologies.

Proposition 3 With a non-linear matching function in the D market outlined above,a pure middleman mode can be profitable even with multi-market search technologies
if the middleman’s price is inelastic at full capacity xm = K = B. Otherwise, theintermediary should be a marketmaking middleman or a pure market maker.
Proof. See Appendix A.4.
m

Figure 2 plots the size of the middleman sector xB and the elasticity of the middleman’s price with respect to capacity, evaluated at xm = K = B. It shows thatmwhen a pure middleman mode is selected, xB = 1, the price is inelastic: z(B) < 1,
whereas when an active platform is used, the price is elastic: z(B) > 1. This confirms that given the appropriate restriction on the meeting rate λb (xD ), our main
conclusion in the baseline model remains valid. When the middleman’s price is elastic, there is sufficient negative feedback from the D market on the price to make theexclusive use of the middleman mode not profitable.

4.2

Endowment economy

In our baseline model, we simplified the middleman’s inventory stocking by assuming that the good is supplied by a competitive frictionless wholesale market. In thissection, we study the implication of wholesale-market frictions in an endowment
38m(a) The optimal size of the middleman sector ( xB ) for different values of B and β(b) Price elasticity z(B)Figure 3: The optimal size of the middleman sector and price elasticity at B under anonlinear matching function
Note: The upper figure plots the optimal size of the middleman sector, xm /B, using colors to informits values, against the mass of buyers B on the vertical axis and the buyer’s bargaining power β on
the horizontal axis. The lower figure is a contour plot on the price elasticity of pm with respect to K
at K = B, z(B), against B on the vertical axis and β on the horizontal axis. All values are calculated−xD

based on S = 1 and λb (xD ) = 1−exD.

39economy. Suppose that each seller owns one unit of endowment. In total, a massof S commodities are available. In the wholesale market, the middleman can access a fraction α of sellers, where α ∈ [λs , 1] is exogenous.26 Then, the middleman’s
inventory should satisfy the aggregate resource constraint,K ≤ αS.

(31)

In an economy with unlimited production capacity, sellers are willing to supply
as long as the wholesale price, denoted by pw , is enough to compensate them for themarginal cost; whereas in an endowment economy, sellers are only willing to supply if pw is high enough to compensate them for the foregone trading opportunities
elsewhere. Once contacted by the middleman, sellers choose among selling the endowment to the middleman, or joining the C market platform and/or joining the Dmarket. To simplify the analysis, we abstract from the influence of what sellers can
expect from the D market on the determination of the wholesale price, and assume
that sellers in the D market receive a zero trade share, β = 1. Our main conclusion does not depend on this simplification. Then, the middleman’s offer to buy fromsellers is accepted if and only if
pw ≥ W C (xs ),(32)

where W C (xs ) is the expected value of sellers to operate in the C market platform.

Single-market search. The determination of the intermediation mode depends
on the available resources. If B ≤ αS, then the middleman can stock the full inventory (K ∗ = B) to cover the entire population of buyers. In this case, by closing theplatform S C = 0, the middleman makes the highest possible profit, Π = B(1 − λb ),with the wholesale price pw = 0, just like in the baseline model. If B > αS, then the26
We require α ≥ λs for single-market search. Under single-market search, the pessimistic beliefs
require that the intermediary “divides” buyers using middleman inventory. Specifically, the offers fromthe middleman must satisfy K(1−pm ) ≥ λb . To ensure that the middleman price pm ≥ 0, the minimumBbinventory is Bλ . Thus, we need αS ≥ Bλb to have an active intermediary under single-market search.
Our analysis of multi-market search does not depend on α ≥ λs .

40

middleman’s inventory will not be enough to cover all buyers, and so the intermediary may wish to use a platform even under single-market search technologies.
With the wholesale price pw determined by the binding constraint (32), the fee fand the price pm determined by the binding participation constraint of buyers, theintermediary’s problem can be written as the choice of inventory K and allocation
xm that maximizes the profitss(S − K)(1 − e−x )f + min {K, xm } pm − Kpwmwhere xs = B−x
S−K , subject to the resource constraint (31).
As expected, the solution is characterized by the binding resource constraint (31)and an active platform xs > 0 when B > αS. Although deactivating the platform
would lead to the lowest wholesale price for the middleman pw = 0, this is not profitable. The benefit of fee revenue from the active platform outweighs the cost savingsof the middleman. Hence, even under single-market search, the aggregate resource
constraint can be one reason for the intermediary to open the platform sector in theendowment economy.

Proposition 4 Consider the endowment economy outlined above with single-market
search technologies, and the zero trade share of sellers in the D market (β = 1). Theintermediary chooses to be:• a pure middleman if B ≤ αS;• a market-making middleman with K = αS ≤ xm if B > αS.
Proof. See Appendix A.5.
The result xm ≥ K occurs because, in line with the previous setup, an excess inventory means extra costs in the middleman sector and lost revenue on the platform.
Figure 4 demonstrates that when B > αS, it is possible that the intermediary attracts an excessive number of buyers to the middleman sector (xm > K) in order to41

lower the wholesale price paid by the middleman. This results in stockouts. When
this occurs, the resource constraint is tight and the outside value of sellers is high sothat economizing on stocking costs is relatively important.

Multi-market search. With multi-market search technologies, the participation
constraint of agents is not the issue but the intermediation fee and the middleman’s
price should be acceptable relative to the outside value. Hence, the intermediary
faces incentive constraints (15) – (17) (see details in the proof of Proposition 5). As
before, these conditions are reduced to f ≤ v(xm , K). To be consistent, we maintainthe assumption of a zero trade share of the sellers in the D market (β = 1). This
assumption now implies that sellers are fully exploited in the C market, thus pw =W C (xs ) = 0 for any xs ≥ 0.
Under multi-market search, the buyers’ outside option depends positively on thenumber of sellers available in the D market. This has the following consequences.
First, just as in the baseline setup, a pure middleman mode can never be profit
maximizing. Second, in our endowment economy, the intermediary may wish tostock more inventories than the number of buyers visiting the middleman sector.
This is because a larger K will crowd out the supply available in the D market, which
will eventually lower the outside value of buyers and increase the profit. Therefore,
unlike in all the previous setups, the solution here allows for an excess inventory inthe middleman sector.
Proposition 5 Consider the endowment economy outlined above under multi-market
search and a zero trade share of sellers in the D market. The intermediary chooses tobe a market-making middleman or a pure market-maker with xm ≤ K = αS.
Proof. See Appendix A.6.
Figure 5 shows the occurrence of excess inventory holdings in the middleman
sector with high values of λb and α. This confirms our intuition that the crowding42

out effect of excess inventory is stronger when the buyer’s outside value in the Dmarket is higher.
Comparing Proposition 4 and 5, we can summarize the implications of search
frictions in wholesale markets represented by α and the agents’ search technologies
in retail markets on the choice of intermediation mode in our endowment economy
as follows.
• For αS ≥ B, the middleman can stock the full inventory that satisfies all thebuyers’ demand. As in the benchmark setup, the intermediary chooses to be apure middleman under single-market search, while it also opens an active platform under multi-market search. Unlike in the previous setup, the middlemanholds an excessive amount of inventory.
• For αS < B, holding a full inventory is not possible due to an aggregate resource constraint. The intermediary uses a platform irrespective of whether
agents search in a single or in multiple markets. Our main insight is stillvalid. Namely, the intermediation mode is further away from the pure middleman mode when agents search in multiple markets, rather than in a single
market. The size of the middleman sector, measured by xm , is smaller undermulti-market search than under single-market search technologies.

43

0.35
0.9
0.30

0.8
0.7

0.25

0.6
0.20
0.5
0.15

0.4
0.3

0.10
0.2
0.05

0.1
0.1

0.2

0.3

0.4

0.5
b

0.6

0.7

0.8

0.9

0.00

Figure 4: Stockouts of the middleman (xm − K) under single-market search in endowment economy
Note: The figure plots the level of stockouts, represented by the value of xm − K, against α on thevertical axis and λb on the horizontal axis. The figure is drawn with B = 0.8, S = 1, and α ≥ λs .

44

0.9

0.00

0.8

0.02

0.7

0.04

0.6

0.06

0.5
0.08
0.4
0.10
0.3
0.12
0.2
0.14

0.1

0.1

0.2

0.3

0.4

0.5

0.6

b

0.7

0.8

0.9

1.0

1.1

1.2

0.16

Figure 5: Excessive inventory holdings (xm − K) with multi-market search in endowment economy
Note: The figure plots the level of stockouts, represented by the value of xm − K, against α on thevertical axis and λb on the horizontal axis. The figure is drawn with B = 0.8 and S = 1.

45

4.3

Cost functions

Inventory Costs.

In the baseline model, we assume zero inventory costs of the

middleman. In this section, we consider a convex inventory-cost function C(K) thatsatisfies C 0 (K) ≥ 0, C 00 (K) ≥ 0, C(0) = 0, and C 0 (K) < 1 − λb . The last condition
guarantees that C(B) < B(1 − λb ). We assume β = 1 for simplicity. With positive
inventory costs, it may be profitable to activate a platform even under single-market
search. Still, we show that our main insight is valid.
As in the baseline model, profit maximizing requires K = xm . Under singlemarket search, the problem of the intermediary can be described as maximizingsΠs (xm ) = S(1 − e−x )f (xm ) + xm pm (xm ) − C(xm ),(33)

by choosing an xm ∈ [xm , B] with xm ≡ max{B −S log(1/λb ), Bλb } subject to pm (xm ) =
B b
1
m m
1 − xBm λb , f (xm ) = 1 − e−xs m λ , and (4). p (x ) is derived from condition (11), andxsf (xm ) is given by the buyers’ indifference condition, V m (·) = V s (·) = e−x (1 − f ).
The platform fee f = f (·) is strictly increasing in xm . Intuitively, the tighter
the platform, the lower the fee that the intermediary can charge in order to makebuyers indifferent between the platform and the middleman sector. The negative
dependence of the platform fee on the platform size favors the middleman mode.
The first order condition becomesB b 1 − e−x∂Πs (xm )
−xs
=
1
−
e
+
λ +
∂xm
xm
e−xs

s



S
B b
1+ m
λ − C 0 (xm ) ≡ ΘSf oc (xm ) ≥ 0.(34)xxm

Observe that in (34): ΘSf oc (B) = λb − C 0 (B) and the second order condition is satisfied.27 Therefore, the pure middleman mode is profit-maximizing if λb ≥ C 0 (B).
Otherwise, the optimal intermediation mode xm∗ ∈ (xm , B) satisfies ΘSf oc (xm∗ ) = 0.
27

The second order condition follows thatse−x
Θ0Sf oc (xm ) = −−S

B b B bλ − mλxm2xs

1
1 − e−x+xm e−xs
Se−xs

46

!

S
1+ m
x



s

1 − e−x S+
e−xs xm2!
−C 00 (xm ) < 0.

The intuition is as follows. Although, with inventory cost, the intermediary may
adopt a hybrid mode under single-market search, the pure middleman is still preferred if λb is relatively high. This is because with a higher λb , expanding the midm

m

)
b
= (xB
dleman scale allows for a higher price increase. That is, ∂p∂x(xmm )2 λ increasesin λb .
Under multiple-market search, the objective function is the same as in (33) exceptthat the inventory is not required to conquer the pessimistic beliefs. As in the baseline model, the intermediary optimally holds just enough inventory, that is K = xm .
The profit maximization problem is to maximize the profitssΠm (xm ) = S(1 − e−x )f (xm ) + xm pm (xm ) − C(xm ),by choosing xm and f , subject to pm (xm ) = f (xm ) = 1−λb e−

B−xmS(by (29) and v(·) = f

as in Lemma 1). As before, the positive dependence of the middleman’s price and theplatform fee on the platform size favors the market-maker mode. The first orderderivative iss

λb xm e−x∂Πm (xm )
−xs
b −xs
(1
−
e
)(1
−
2λ
e
)
−
− C 0 (xm ) ≡ ΘM f oc (xm ).
∂xm
S

(35)

b

The second order condition is satisfied.28 Observe that ΘM f oc (B) = − λSB −C 0 (B) < 0.
Therefore, a market-making middleman is the profit-maximizing mode if
BBΘM f oc (0) = (1 − e− S )(1 − 2λb e− S ) − C 0 (0) > 0.

(36)

Otherwise, a pure market-maker mode is selected.
Comparing the optimal modes under two search technologies, we find that under single-market search, the middleman is always active. The platform might be
activated, but a pure platform is never optimal. In contrast, under multi-market
28

The second order condition follows that thesi λb xm e−xs
s
s
e−x h
Θ0M f oc (xm ) = −1 − λb e−x + 3λb (1 − e−x ) −− C 00 (xm ) < 0.
S
S2

47

search, the platform is always active and a pure middleman is never optimal. For
the interior case, we haveΘM f oc (xm ) = ΘSf oc (xm ) −B b
1 − e−xss
λ − 2λb e−x (1 − e−x ) −
m
x
e−xs

s



s
B b λb xm e−x
S
1+ m
λ
−
x
xm
S

< ΘSf oc (xm ),implying that the marginal profit of increasing the size of the middleman sector
is smaller under multi-market search than under single-market search. The logic
behind this is essentially the same as in the baseline model.
Proposition 6 Consider the convex inventory costs of a middleman defined above.
Then, a platform can be activated even under single-market search. Still, the size ofthe platform under multi-market search is larger than or equal to that under singlemarket search.

Prior production/purchase before joining the platform.

In real-life markets,sellers sometimes need to prepare (produce or purchase) their product for sale prior
to market entry. For example, online sellers find it important to display their product’s image and keep it ready for delivery before actual transactions occur. A similar
issue arises when asset holders are required to commit to their portfolio before trading with their brokers. In these situations, because sellers incur costs irrespective of
their success on the platform, attracting sellers to the platform is costly and so therelative profitability of the market-maker mode is reduced. We show, however, that
our insight remains valid in such a setting. Interestingly, we also find that a platform can be activated even when the net profit obtained from the platform businessis negative.
The only modification that is required now is to introduce a participation constraint for sellers to operate on a platform. Under single-market search, this is irrel-

48

evant because the pure middleman mode remains profit maximizing. With multiplemarket search, the participation constraint is given by
W C (xs ) − fp ≥ cE ,(37)

where cE ≥ 0 are the entry costs of sellers to the platform, fp ≥ 0 (or fp ≤ 0) is
a platform participation fee (or subsidy) to be paid by each individual seller, andW C (xs ) = η(xs )(pm − f ) is the equilibrium value of sellers who participate on theplatform. With β = 1, i.e., zero payoff in the D market for sellers, the intermediaryssets f = pm = 1 − λb e−x , satisfying the incentive constraint (25) (note that theparticipation in the D market does not require prior production/purchase as before),and fp = −cE . That is, the intermediary should subsidize the entry cost and fully
extract the trade surplus in the platform. The profit of a market-making middlemanissΠ̃(xm ) = S (1 − e−x )f − cE + xm pm ,while the profit of a middleman isΠ̃(B) = B(1 − λb ).
Comparing these profits, one can find a value of xm < B (e.g. imagine a neighborhoodof xm = B − Sxs ≈ B) and cE > 0 for which the platform profit is negative butΠ̃(xm ) > Π̃(B). This leads to the following result.
Proposition 7 Suppose sellers incur production/purchasing costs prior to platform
entry. Then, an active platform can be profit maximizing even when the platform
entry cost is higher than the platform fee revenue.
One benefit of having an active platform in the C market for the intermediary is
to reduce competition so that it can set a higher price in the middleman sector. This
benefit can be the major source of profits for market-making middlemen even when
the platform-entry costs are so high that the net profit from the platform business isnegative.
49

4.4

Competing intermediaries

Our framework can be extended to study competing intermediaries. We consider
two intermediaries who make a simultaneous choice of platform fees and/or priceof their good. In particular, we are interested in whether an active platform withpositive fees of an incumbent intermediary, referred to as I, can be profitable when
the other intermediary, referred to as E, enters with adopting a pure middlemanmode or a pure market-maker mode. To simplify the analysis we abstract away fromdecentralized market trade, and assume zero marginal costs and zero entry costs.

Single-market search. With single market search, irrespective of the intermediation mode of E (and beliefs of agents on which intermediary to be favorable), I has
no strict incentive to activate a platform with positive fees. To see this, let V E be thebuyer’s value of visiting E. If I chooses to be a pure middleman then its profit is Bpm
with price pm = 1 − V E . If I activates a platform, then, just like in our benchmark
setup, the fee should satisfy f ≤ 1 − V E and so its maximum attainable profit withpositive fees is strictly less than B(1 − V E ). The intuition remains the same as before — with single market search, the middlemen mode achieves the highest trade
surplus.

Multiple-market search with a pure middleman E.

To be consistent with theprevious analysis, we assume that agents visit I prior to E by default. The idea
behind this assumption is that I is a well-established intermediary in the market,
whereas E is a newcomer which has no regular customers.
When E is a pure middleman with price pE ∈ [0, 1], I’s price/fee (pm , f ) should
satisfy the incentive constraints,pm ≤ pE and f ≤ pE ,respectively. The major difference from the benchmark is that, as a pure middleman,50

E would undercut any positive price/fee of I and so an active platform with positive
fees can never survive.

Multiple-market search with a pure market-maker E.

When E is a puremarket-maker with a fee f E , the incentive constraints becomepm ≤ 1 − V E (f E ) and f ≤ 1 − V E (f E ).
E could either act as a “second source” for intermediation service, or undercut I and
be the “sole source”. Given the strategic choice of E, a pure middleman I can notexist in equilibrium because it is only profit maximizing when buyers’ outside option
is zero. However, E has an incentive to undercut I, leading to a positive buyer’svalue at E. A pure market-making I can neither exist in equilibrium. This follows
from the fact that E has an incentive to undercut I as long as the transaction fee
is positive; at a fee of zero, E would rather increase the fee to the highest level andextract the full surplus, and I’s best response is a pure middleman.
In the online appendix, we show that there exists a pure-strategy equilibrium
when undercutting is costly for E. In equilibrium, E operates as the second source,
and it adjusts f E in a way that takes into account the responses of the participating
buyers and sellers. Surprisingly, the transaction fee of E affects the intermediation
mode of I. To see this, note that a lower f E improves the outside option for buyers.
The buyer now finds it more attractive to visit an individual seller on I’s platform
rather than I’s middleman sector, since even if he is not matched at I, the outside
option to trade on the platform of E (with a lower f E ) has a favorable prospect. As
such, with a lower f E , more buyers switch from I’s frictionless middleman to I’s frictional platform, leaving more unmatched buyers and less unmatched sellers joining
E. Ultimately, this trade-off of E between more participating buyers (by decreasingf E ) and more participating sellers (by increasing f E ) leads to an equilibrium withf E < 1 and a positive expected value for buyers. Given buyers’ positive outside value,51

an active platform can better exploit I’s intermediated surplus as we presented in
the benchmark model. Hence, using an active platform with positive fees can bea profitable business mode for an intermediary when the other intermediary also
activates a platform.

Proposition 8 Consider two competing intermediaries, one is an incumbent (I), justlike our original intermediary, and the other is an entrant (E) that replaces the Dmarket. Then, I activates a platform with positive fees only when agents search inmultiple markets and E also adopts an active platform.
Proof. See details in the online appendix.

5

ExamplesOur analysis shows that a marketmaking middleman is more likely to emerge undermulti-market search technologies than under single-market technologies. In thissection, we offer some real market examples.

Online retailers.

The electronic commerce company Amazon.com is traditionallyan online retailer, who mainly aims at selling its inventories to customers. In the late
1990s, Amazon was facing fierce competition from local brick and mortar rivals, aswell as chain stores such as Walmart, Sears, etc., and especially from eBay. According to the book, The Everything Store: Jeff Bezos and the Age of Amazon, Jeff Bezos
worried that eBay may become the leading online retailer who attracts the majorityof customers. In the summer of 1998, he invited eBay’s management team and suggested the possibility of a joint venture or even of buying out their business. Thisis perhaps Amazon’s first attempt to set up an online marketplace. In the end, thistrial failed. After several more trials and errors, however, Amazon finally launchedtheir own marketplace in the early 2000s. The entry version of our model in Section
4.4 captures well Amazon’s reaction to the entry by eBay.
52

Amazon’s launch of the platform business influenced significantly the book industry. On the one hand, Amazon attracts many of its competitors to join their platform.
Indeed, Amazon drove physical book and record stores out of business, and many
bookstore owners re-launched their business on the Amazon-website platform. On
the other hand, Amazon lowers the chance of buyers to trade outside. As local bookstores disappeared, it became the habit for most book buyers to start their everyday
online-shopping using Amazon as the prime site (De los Santos et al. 2012). Overall,these observed phenomena are in line with our theory.2930 Not surprisingly, Amazon
promoted this shopping pattern to customers in other product categories.
The general picture of the online travel agency industry is similar. Before the riseof Internet, most intermediaries in this industry acted as a pure middleman. In the
middleman mode, hotels sell rooms to a middleman in bulk at discounted prices. The
middleman then sells them to customers at a markup price. With the online reservation system, a market-making mode became popular, wherein hotels pay a market
maker (e.g. Booking.com) commission fees upon successful reservations. The hotels
post their services and prices on the platform. Expedia used to be a pure middleman
but is nowadays a representative market-making middleman who employs both ofthese intermediation modes.
29
Nowadays, most buyers and sellers use Amazon as the main website (the first one to visit). On
the seller side, according to a survey on Amazon sellers conducted in 2016, more than three-quartersof participants sell through multiple channels, online marketplaces, webstores and bricks-and-mortar
stores. The second most popular channel, after Amazon, is eBay, with 73% selling through this marketplace. On the buyer side, according to a recent Reuters/Ipsos poll, 51 percent of consumers plan to
do most of their shopping on the Amazon.com.
30
An alternative (or complementary) to our theory would be a product selection story where Amazon
uses the platform for third-party sellers to add new products with the demands too small for Amazon
to offer. Once a product is “tested” to be popular enough, Amazon starts to also offer it through the
middleman sector. This would be certainly a valid explanation but by far not the exclusive one. First,
if this explanation were correct, we should eventually observe that most popular products are listedby Amazon, and most not-so-popular products are listed by independent sellers. In reality, however,many high-demand products are listed by both Amazon and third-party sellers at the same time, andimportantly, they are competing with each other. This competition goes against the proposed explanation, but is more in line with our theory. In fact, Amazon could avoid fierce competition with strong
competitors operating in the Amazon marketplace, such as GreenCupboards or independent sellers
who own ‘Buy Boxes’, by giving up dealing with such a product in the middleman sector, which shouldin turn increase their fee revenue.

53Specialist markets.

The New York Stock Exchange (NYSE) is a specialist market,

which is defined as a hybrid market that includes an auction component (e.g., afloor auction or a limit order book) together with one or more specialists (also called
designated market makers). The specialists have some responsibility for the market:
as brokers, they pair executable customer orders; and as dealers, they post quotes
with reasonable depth (Conroy and Winkler 1986).
As for their role as dealers in the exchanges, our model suggests that, at least forless active securities (represented by a lower outside option), the specialists’ market
can provide predictable immediacy and increase the trading volume and liquidity.
This is consistent with the trend to adopt hybrid markets in derivative exchanges
and stock markets around the world, especially for thinly-traded securities. For example, several European stock exchanges implemented a program which gives lessactive stocks an option of accompanying a designated dealer in the auction market.
These initiatives were effective not only in enhancing the creation of hybrid specialist markets, but also in increasing trade volumes and reducing liquidity risks(Nimalendran and Petrella 2003, Anand et al. 2009, Menkveld and Wang 2013, andVenkataraman and Waisburd 2007.)Another prediction from our analysis is related to the changing competitive environment faced by securities exchanges. As a broader implication, our result thatthe increased outside pressure goes hand in hand with more decentralized trades,
captures the background trend in general: the market for NYSE-listed stocks was
highly centralized in 2007 with the NYSE executing 79% of volume in its listings;
in 2009, this share dropped to 25% (Securities and Exchange Commission 2010); today, the order-flow in NYSE-listed stocks is divided among many trading venues –11 exchanges, more than 40 alternative trading systems, and more than 250 brokerdealers in the U.S. (Tuttle 2014). As a more specific implication, we show that theincreased pressure from outside markets will scale up the platform component. This54

is indeed the case. Starting from 2006, the NYSE adopted the new hybrid trading
system featuring an expanded platform sector “NYSE Arca”, which allows investors
to choose whether to trade electronically or by using traditional floor brokers andspecialists. The new system is further supplemented by several dark pools, akin toplatforms, owned by the NYSE. These strategies are also adopted by NASDAQ which
has been thought of as a typical dealers’ market. In addition, the use of fees is widelyadopted, as is consistent with our theory. For instance, in 2014, the NYSE offeredbanks a discount of trading costs by more than 80% conditional on their agreement
to stay away from the outside dark pools and other off-exchange venues.31

Real estate agencies. While intermediaries in housing markets are mostly thoughtof as brokers, i.e., platforms, the business mode employed by the Trump family is amarketmaking middleman. The Trump Organization holds several hundred thousand square feet of prime Manhattan real estate in New York City (NYC) and somemore in other big cities. Besides developing and owning residential real estate, theTrump family operates a brokerage company that deals with luxury apartments,
the Trump International Realty. Both of these companies target the same market
in NYC. Indeed, the Trump’s business mode is a marketmaking middleman – bothowning his own residential towers, and offering broker services. According to Forbes,
the latter portion of Trump’s empire becomes by far his largest business with a valuation of 562 million in 2006. Another example is Thor Equities, a large-scale realestate company, which owns and redevelops retail properties in Soho, Madison Avenue, and Fifth Avenue, and also runs brokerage agencies, Thor Retail Advisors andTown Residential.
In the endowment economy version of our model, we show that the marketmaking middleman over-invests in inventory with multi-market search, up to the point
31
See a report “NYSE Plan Would Revamp Trading” in the Wall Street Journal, 2014.
http://www.wsj.com/articles/intercontinental-exchange-proposing-major-stock-market-overhaul1418844900.

55

where the resource constraint is binding. Perhaps, the real estate market in NYC
is an appropriate example of this since it is well known to be competitive and tightfor house/apartment hunters. In addition, most new developments in big cities arerenovations of old houses, and so we can roughly regard the total supply as fixed.
Notably, top real estate firms in NYC attempt to expand their business by being engaged in many new joint projects with developers. Mapped into our model, these
efforts are aimed at relaxing their resource constraint and increasing their inventory. For example, Nest Seekers, a real estate brokerage and marketing firm in NYC,
works tightly with constructors on new developments. They work together from thevery early stage of layout design and fund raising (in some cases Nest Seekers offers
their own capital) to the later marketing stage. Nest Seekers provides qualified sales
and administrative staff to the sales office, prepares pricing schedules, manages all
contracts with the brokerage community, and is eventually in charge of the entire
marketing process. This co-development business is one step beyond the middlemanmode formulated in our theory, but is considered as an alternative way to secure theirinventory.32 This business mode is adopted in many other big real-estate companies
in NYC, such as Douglas Elliman, Stribling, and Corcoran.
Finally note that some intermediaries do not only help to promote new developments, but also manage apartment complexes, which constitutes another source of“inventory”. For example, Brown Harris Stevens provides residential management
service for its customers since cooperative apartments were first introduced to NYC.
These cooperative apartments usually contain hundreds of units in one building, andBrown Harris Stevens is then in charge of listing these properties when they are forrent or on sale.
32

Strictly speaking, Nest Seekers does not own properties, but becomes the exclusive
agent of projects.
So far, they have co-developed/marketed more than 30 projects.
See
https://www.nestseekers.com/NewDevelopments. A report titled “Inside the fight for Manhattans mostvaluable new development exclusives” by The Real Deal introduces more detailed information on how
brokers cooperate with developers, which is available in http://therealdeal.com/2016/03/15/inside-thefight-for-manhattans-most-valuable-new-development-exclusives/ (visited on July 15, 2016).

56

6

Empirical evidence

The model’s predictions on the choice of intermediation mode can be empiricallytested. As in the last extension of competing intermediaries, we take Amazon as
the centralized market and eBay as the decentralized market. Our model predicts
that Amazon is more likely to sell the product as a middleman when the chance ofbuyers to meet a seller in the decentralized market, λb , is low, the buyers’ bargaining
power β is low, and the total demand B is high. That is,Pr(Amazon’s middlemen mode is active) = f (λb , β, B ),− − +where − (+) indicates a negative (positive) correlation.
We collected data from www.amazon.com and www.ebay.com, focusing on one
product category, namely pans. We think this product choice is appropriate not only
because there are many observations for both eBay and Amazon but also because
pans require some minimum consideration and search before a purchase decision ismade. In addition, we analyze the theoretically most relevant case where Amazon
acts as a marketmaking middleman: For 32% of the sample, Amazon acts as a middleman; for the other 68%, Amazon acts as a pure market maker. Our data matches
each product on sale at Amazon to a list of the offers at eBay. Using information on
individual prices and sellers, we construct proxies for key parameters in the model:
λb , β and B as explained below (see the Online Appendix for more details). Since ourdata are not experimental, our evidence should be interpreted as suggestive rather
than causal.

Table 1 summarizes various cross-sectional regressions of Amazon’s intermediation mode, represented by sellByAmazon, which is a dummy variable that takes avalue of 1 if the product is sold by Amazon. It is an indicator that Amazon is anactive middleman for that product. Other variables we use for the linear regressionin Column 1 are discussed below. sellersEbayRelative is a proxy for λb and is defined57

Table 1: Regressions for Amazon’s intermediation mode(1)
LinearsellersEbayRelative(2)
Linear

-0.00630∗∗∗(0.000765)(3)
Probit(4)
Probit-0.00778∗∗∗(0.00119)sellersEbayRefined-0.00159∗∗(0.000595)-0.00156∗(0.000698)

sellersAmazon-0.000552(0.000917)-0.000669(0.000996)

log(rank)-0.103∗∗∗(0.00442)-0.107∗∗∗(0.00463)-0.105∗∗∗(0.00500)-0.110∗∗∗(0.00517)

priceDiff

0.105∗∗∗(0.00820)

0.111∗∗∗(0.00830)

0.124∗∗∗(0.0107)0.133∗∗∗(0.0111)log(price)

0.0390∗∗∗(0.00608)

0.0406∗∗∗(0.00613)

0.0484∗∗∗(0.00680)

0.0510∗∗∗(0.00688)

listedDays

0.0602∗∗∗(0.00480)

0.0660∗∗∗(0.00486)

0.0717∗∗∗(0.00599)

0.0784∗∗∗(0.00604)

Observations
Adjusted R2

6457
0.136

6457
0.130

6457

6457

Note: Columns (1) and (2) use linear probability model, and columns (3) and (4)use probit model. For probit models, the marginal effects evaluated at the sample
mean are reported. sellersEbayRefined is the number of sellers on a refined listof sellers on eBay by matching the title of offers with the Amazon product title
and restricting the price of offers between 0.5 and 1.5 times the Amazon price.
sellersAmazon is the number of third-party sellers on Amazon. Robust standard
errors are reported in parentheses. Other variables are explained in the main text.
∗
denotes p < 0.05, ∗∗ denotes p < 0.01, ∗∗∗ denotes p < 0.001.

58

as the number of sellers on eBay divided by the number of third-party sellers on
Amazon. rank is a proxy for total demand B and is defined as the sales rank within
the product category. Rankings are negatively correlated with sales (e.g., a productwith a rank value of 100 is associated with more sales than a product with a rank
value 200). priceDiff is a proxy for β and is defined as the log of the median price ofeBay offers minus the log of the Amazon price. listedDays controls for the numberof days since the product was first listed on Amazon. Our theoretical model predicts that sellByAmazon is negatively correlated to sellersEbayRelative, negatively
correlated to log(rank) and positively correlated to priceDiff. As shown in Table 1,
all explanatory variables have the expected signs and are statistically significant inall the specifications (including those where we use alternative proxies and probit
regressions).33 To quantify the effect of available options on eBay, we find that the
chance that Amazon acts as a middleman decreases by 3.7 percent for a one-standard
deviation increase in sellersEbayRelative (λb ), and increases by 0.1 percent for a one
percent increase in the median eBay price relative to the Amazon price (proxied bypriceDiff, β). In the Online Appendix, we give more detailed information on the data
and we experiment with a number of different specifications, but none of them alters
our main results.

7

Conclusion

This paper developed a model in which market structure is determined endogenously
by the choice of intermediation mode. We considered two representative businessmodes of intermediation that are widely used in real-life markets: a market-making
33
Recent work by Zhu and Liu (2018) also examines empirically the product choice by Amazon. While
their approach is very different from ours, it is interesting to note some common evidence. For instance,
we find that the number of sellers on Amazon is negatively associated with the likelihood of Amazon
to act as a middleman. This may reflect a crowding out effect of Amazon on third-party sellers. Similarly, Zhu and Liu (2016) find that Amazon’s entry could discourage third-party sellers and eventually
force them to leave the platform. Also, our evidence suggests that Amazon is more likely to sell moreestablished products of higher prices. This is also consistent with Zhu and Liu (2018)’s findings that
Amazon may be targeting successful products to exploit the surplus from third-party sellers.

59mode and a middleman mode. We derived conditions for a mixture of the two modes,
a marketmaking middleman to emerge.
One implication of our theory is that intermediaries can use a platform to reduce
competition with sellers in the decentralized market. However, this is done by inducing consumers to search excessively and so generates inefficiencies. For future
research, it would be interesting to examine this from the viewpoint of a regulator.

60AppendicesA

Omitted proofs

A.1

Proof of Lemma 1

Using K ≤ xm and (29), the intermediary’s problem becomesmax Π (xm , f, K) ,xm ,f,KwhereΠ (xm , f, K)= S(1 − e−B−xmS)f + K(1 − λb e−

B−xmSβ)(1 − c) − xm e−

B−xmS(v(xm , K) − f ) ,subject to (25) and
0 < K ≤ xm < B.
Observe that: limxm →B Π (xm , f, K) = Πm and limxm →0 Π (xm , f, K) = Πp (P p ), whereΠ = B(1 − λb β)(1 − c) is the profit for the pure middleman mode (27) and Πp (P p ) =B
S(1 − e− S )f is the profit for the pure market-maker mode. Hence, we can compactify theconstraint set and set up a general problem to pin down a profit-maximizing intermediation
mode using the following Lagrangian:m

L = Π (xm , f, K) + µk (xm − K) + µb (B − xm ) + µv (v(xm , K) − f ) + µ0 K,
where the µ’s ≥ 0 are the lagrange multiplier of each constraint. In the proof of Proposition
2, we show that the following first order conditions are necessary and sufficient:
∂L
∂xm
∂L
∂f
∂L
∂K

=
=
=

∂Π (xm , f, K)∂v (xm , K)
+ µk − µb + µv
= 0,
m
∂x
∂xm
∂Π (xm , f, K)− µv = 0,∂f∂Π (xm , f, K)∂v (xm , K)
− µk + µ0 + µv
= 0.
∂K
∂K

(38)
(39)
(40)

The solution is characterized by these and the complementary slackness conditions of thefour constraints.
We now prove the claims in the lemma. First, (39) implies that we must havess

µv = S(1 − e−x ) + xm e−x > 0,
which implies the binding constraint (25),




m
m
K
S
s
− B−x
m
b − B−xSS− (1 − ef = v (x , K) = 1 − λ e
β−λ 1−) (1 − β) (1 − c).
B
B
Second, applying µv from (39) into (40) gives
h


i
B−xm
B−xm
µk = 1 − λb e− S β + λb 1 − e− S
(1 − β) (1 − c) + µ0 > 0,
which implies that K = xm . This completes the proof of Lemma 1. 

61

A.2

Proof of Proposition 2

Active platform. First of all, we show that the platform will always be active (i.e., xm < B)
in equilibrium. Substituting µk , µv into (38),




B−xm
B−xm
B−xm
B
1 − λb e− S β − λb(1 − c)−1 (µb − µ0 ) = −e− S− (1 − e− S ) (1 − β) (41)
S
m
B−xm
B−xm
x
−λb
e− S + 1 − λb β + λb (1 − e− S )2S
≡ φ(xm | B, S, β, λb ).
JbSuppose that the solution is xm = B. Then, (41) yields φ(B | ·) = (1 − c)−1 µb = − BS λ β < 0,mwhich contradicts µb ≥ 0. Hence, the solution must satisfy x < B (which implies µb = 0).

J

Market-making middleman or pure market-maker. Second, we derive the condition for apure market-maker xm = 0 or a market-making middleman xm > 0. Since φ(B | ·) < 0,
if φ(0 | ·) > 0, there exists xm ∈ (0, B) that satisfies φ(xm | ·) = 0, i.e., a market-making
middleman. Further,
m
m
m i
B−xm
∂φ(xm | ·)1hλb − B−xm
b
b
− B−x
2
b
− B−x
− B−x
S
S
S
S
|
=
−
1
−
λ
β
+
λ
(1
−
e
)
)e
−
e
(1 − e− S ) < 0.
+
2λ
(1
−
e
φ=0
m
∂x
S
S

This implies that the allocation of the middleman sector xm ∈ (0, B) is unique (if it exists),
and that if φ(0 | ·) < 0 then φ(xm | ·) < 0 for all xm ∈ [0, B] and the solution must be a puremarket maker, xm = 0.
Now, we need to investigate the sign of it:
φ(0 | B, S, β, λb ) = −e−x 1 − λb e−x β − λb x − 1 + e−x (1 − β) + 1 − λb β + λb (1 − e−x )2≡ Θ(x),
where x ≡ B
S . Observe that:Θ(0) = 0 < 1 − λb β + λb = Θ(∞),
and


∂Θ(x)
= e−x 1 − λb x + λb β(x − 2) + 4λb (1 − e−x ) .
∂xbThis derivative has the following properties: ∂Θ(x)∂x |x=0 = 1 − 2λ β;∂Θ(x)|Θ(x)=0 = 1 − λb β(1 + e−x ) + λb (1 − e−x )(1 + 2e−x ) ≡ Υ(x).
∂xThere are two cases.
∂Θ(x)• When λb β ≤ 12 , we have ∂Θ(x)∂x |x=0 ≥ 0 and ∂x |Θ(x)=0 > 0, implying that no x ∈ (0, ∞)exists such that Θ(x) = 0. Hence, Θ(x) = φ(0 | ·) > 0 for all x ∈ (0, ∞).

• When λb β > 21 , we have ∂Θ(x)∂x |x=0 < 0. Hence, there exists at least one x̄ ∈ (0, ∞) suchthat Θ(x) < 0 for x < x̄ and Θ(x) ≥ 0 for x ≥ x̄. Below we show that such a value has
to be unique. For this purpose, observe that:Υ(0) = 1 − 2λb β < 0 < 1 + λb (1 − β) = Υ(∞),∂Υ(x)= λb e−x (4e−x − 1 + β),
∂x

∂Υ(x)
∂ 2 Υ(x)|x=0 = λb (3 + β) > 0,| ∂Υ(x) =0 = −4e−x λb e−x < 0.
∂x
∂x
∂x2
These properties imply that there exists an x0 ∈ (0, ∞) such that Υ(x) < 0 for all x < x0and Υ(x) ≥ 0 for all x ≥ x0 . This implies that x̄ is unique.

62

To summarize, we have shown that if λb β ≤ 21 then the solution is a market-making
1bmiddleman xm ∈ (0, B) for all x = B
S ∈ (0, ∞). If λ β > 2 then there exists a unique criticalvalue x̄ ∈ (0, ∞) such that the solution is a market-making middleman for x ≥ x̄ and is apure market-maker xm = 0 for x < x̄.
Second order condition. Finally, we verify the second order condition. Define X ≡ [xm , f, K]
and write the binding constraints asJ

h1 (X) = v (xm , K) − f, h2 (X) = xm − K.
The solution characterized above is a maximum if the Hessian of L with respect to X at thesolution denoted by (X∗ , µ∗ ) is negative definite on the constraint set {w : Dh (X∗ ) w = 0}
with h ≡ [h1 (X), h2 (X)]. This can be verified by using the bordered Hessian matrix, denoted
by H.

H

"

0
T
Dh (X∗ )



0
0

≡

0
 0

 ∂h1
= 
 ∂xm
 ∂h1
 ∂f
∂h1
∂K



∂h2
∂xm
∂h2
∂f
∂h2
∂K

Dh (X∗ )2DX
L (X∗ , µ∗ )
∂h1
∂xm
∂h2
∂xm
2
∂ L(X∗ ,µ∗ )∂xm2∂ 2 L(X∗ ,µ∗ )∂xm ∂f∂ 2 L(X∗ ,µ∗ )∂xm ∂K

0
0



b
s

=  − λS e−x (1 − c)


−1

λs
B (1 − β) (1 − c)

0
0
1
0
−1

#

∂h1
∂f
∂h2
∂f
∂ 2 L(X∗ ,µ∗ )∂f ∂xm∂ 2 L(X∗ ,µ∗ )∂f 2∂ 2 L(X∗ ,µ∗ )
∂f ∂K
b

s


∂h1
∂K
∂h2


∂K
2
∗
∗ 
∂ L(X ,µ )

m
∂K∂x

∂ 2 L(X∗ ,µ∗ ) ∂K∂f∂ 2 L(X∗ ,µ∗ )∂K 2− λS e−x (1 − c)1
∂ 2 L(X∗ ,µ∗ )∂xm2
xm −xsS e∂ 2 L(X∗ ,µ∗ )∂xm ∂Kλs
B (1 − β) (1 − c)



−1
0

−1

x
−x
S e

∂ 2 L(X∗ ,µ∗ )∂xm ∂K

0

0

0

0

m

s








with
∂ 2 L (X∗ , µ∗ )∂xm2
∂ 2 L (X∗ , µ∗ )∂xm ∂K






b
1 −xs
1 xm b −xsxmλb −xs λ −xs
−xs
−xs
= − e
v+ −
λ e
β+2 1+ee−(1 − c) ,
1−e
e
S
S S
S
S
S
 b



s
s λ
λ −xs
xm
= −
e
β+ 1+
(1 − β) (1 − c) .
e−x
S
S
B

The determinant is given by




1 −xs
xm b −xs
m
∗
b −xs
−xs
|H| = −ev (x , K ) +λ e
β (1 − c) + 3λ e1−e(1 − c) < 0.
S
S
Thus, the sufficient condition is satisfied. This completes the proof of Proposition 2. 

63

A.3

Proof of Corollary 1

In (41), we have:∂φ (xm | ., ., β, .)|(φ(xm |·)=0)∂β∂φ (xm | B, ., ., .)|(φ(xm |·)=0)∂B∂φ (xm | ., S, ., .)|(φ(xm |·)=0)∂S
∂φ xm | ., ., ., λb|(φ(xm |·)=0)
∂λb

=
=
=

s

s

−λb (1 − e−2x ) − λb e−xsB− 1 + e−x
S


< 0,

i
s
s
s
1h
1 + λb (1 − β) − λb e−2x + λb e−x (1 − e−x ) > 0
S 

xs
b
b −2xsb −xs B−xs−
1 + λ (1 − β) − λ e+λ e( s −e
) <0
S
x
s

=

−

1 − e−x
< 0.
λb

m

m

|·)
Hence, since ∂φ(x|(φ(xm |·)=0) < 0 (see the proof of Proposition 2), it follows that: ∂x∂xm∂β < 0;
∂xm
∂xm
∂xm
<
0.
This
completes
the
proof
of
Corollary
1.

<
0;
>
0;
∂B
∂S
∂λb

A.4

Proof of Proposition 3

The proof takes steps that are very similar to the ones we made in the proof of Proposition
1. With the non-linear matching function, the intermediary’s profit function is modified toΠ (xm , f, K)

s=

S(1 − e−x )f + min {K, xm } pm=

S(1 − e−

B−xm
S

m

where xD = max{B−min{xSe,K}−S(1−e−xs)f + K(1 − λb (xD )β) − xm e−

−xs

),0}

B−xm
S

(v(xm , K) − f ) ,, and the surplus function tov (xm , K) = 1 − λb (xD )β − λs (xD )(1 − β).
With these profit and surplus functions, the constraints and the Lagrangian remain unchanged, and the first orders are given by (38) – (40) (the second order conditions are presented below). As before, (39) implies that we must havess

µv = S(1 − e−x ) + xm e−x > 0,
and the binding constraint (25). Further, substituting µv from (39) into (40) givesµk = µ0 + 1 − λb (xD )β +S
1 − e−x  b 0 D
K
0
b0 D
λ
(x
)β
+
λ (x )β + (λb (xD ) + xD λb (xD ))(1 − β) . (42)
s
s
−x
−x
Se
e

Substituting µk , µv into (38) gives,
µb

=


B−K K
b0 D
1 − λb (xD )β − λb (xD )xD (1 − β) + 1 − λb (xD )β +s λ (x )β
−x
S Se
S

B − K 1 − e−x  b 0 D0+
λ (x )β + (λb (xD ) + xD λb (xD ))(1 − β) .
(43)
S
e−xs

µ0 − e−xs

Suppose now that xm = B and K > 0. Then, µk > 0 in (42) if and only if
1 − λb (xD )β +K b0 Dλ (x )β > 0,S

and µb ≥ 0 in (43) if and only if
B−K
K b0 Db D(1 − β)λ (x ) + λ (x )β ≥ 0,S
S64with xD = B−K
S . Both of these conditions are satisfied only when K = B (which implies
xD = 0, satisfying the latter condition) and1 − λb (0)β +B b0
λ (0)β > 0S(44)(satisfying the former condition with xD = 0). Under this condition, the solution is unique,
K = B = xm , xs = 0 and f = v(B, B). Hence, we have shown that the solution can be apure middleman xs = 0 only if (44) holds and otherwise the solution must be xs > 0 (either amarketmaking middleman or a pure marketmaker).
Finally, we verify the second order condition. With the modified profit and surplus functions, as before, the bordered Hessian matrix is given by"#0
Dh (X∗ )H ≡T
2
Dh (X∗ )DX
L (X∗ , µ∗ )


∂h1
∂h1
∂h1
0
0
∂xm
∂f
∂K
∂h2
∂h2
∂h2
 0

0


∂xm
∂f
∂K
2
∗
∗
2
∗
∗
2
∗
∗ 
 ∂h1
∂
L(X
,µ
)
∂
L(X
,µ
)
∂
L(X
,µ
)
∂h
2


=  ∂xm ∂xm∂xm2∂f ∂xm∂K∂xm  ∂h1
∂ 2 L(X∗ ,µ∗ )∂ 2 L(X∗ ,µ∗ )∂ 2 L(X∗ ,µ∗ ) 
∂h2
 ∂f

∂f
∂xm ∂f
∂f 2
∂K∂f
∂h1
∂K

∂h2
∂K

0
0

0
0





∗

)
=  ∂v(X
m
∂x

 −1

∂v(X∗ )∂K

1
0
−1

∂ 2 L(X∗ ,µ∗ )∂xm ∂K∂ 2 L(X∗ ,µ∗ )∂f ∂K∂v(X∗ )
∂xm

−1
0

1
∂ 2 L(X∗ ,µ∗ )
∂xm2
B
S
2
∂ L(X∗ ,µ∗ )∂xm ∂K∂ 2 L(X∗ ,µ∗ )∂K 2∂v(X∗ )
∂K

−1





0

∂ 2 L(X∗ ,µ∗ ) ∂xm ∂K 

0

0

B
S

0



with
∂ 2 L (X∗ , µ∗ )∂xm2
2
∂ L (X∗ , µ∗ )∂xm ∂K1∂ 2 λbB ∂v(X∗ )= − f − B m 2 (X∗ )β − (2 + )
,
S
∂x
S ∂xm
∂λb
∂ 2 λbB ∂v(X∗ )= − mβ − B m
β−
.
∂x
∂x ∂K
S ∂K
0

The determinant is |H| = − S1 (1 − λb (0)β) − SB2 λb (0)β < 0. This completes the proof of
Proposition 3. 

A.5

Proof of Proposition 4

As stated in the main text, for αS ≥ B the intermediary can achieve the highest possible
profit by choosing to be a pure middleman. What remains to be proven here is the partfor αS < B. Under αS < B, if the intermediary chooses i = m, then the profits are Kpm .
Note that the inventory cost is pw = 0 as the platform in the C market is shut down. Thensthe optimal announcement of the pure middleman strategy is P m∗ = 1 − λα , αS , which
achieves a profit of αS − Bλb .
Now we turn to i = h. Applying the analysis in the previous section, gives the seller’sssvalue W C (xs ) = (1 − e−x − xs e−x )(1 − f ). The indifferent condition for buyers becomess
V m (xm ) = V s (xs ) with V m (xm ) = η m (xm )(1 − pm ) and V s (xs ) = e−x (1 − f ). The binding
B b
participation constraint for buyers implies that pm = 1 − K
λ andf (xm , K) = 1 −1 m m B b
η (x ) λ .
e−xs
K

65

(45)

s

s

The binding constraint (32) implies that pw = (1 − e−x − xs e−x )(1 − f ).
To guarantee f ≥ 0, we require xm ≥ xm ≡ max B − (1 − α)S log(1/λb ), 0 . To guaranteepm ≥ 0, we require K ≥ Bλb . The following analysis assumes B > αS ≥ Bλb which follows
from α ≥ λs . Inserting the expression for pw and the indifference condition V m (xm ) = V s (xs )yieldssΠ(xm , K) = (S − K)(1 − e−x )f (xm , K) + min{K, xm }(1 −ssB bλ ) − K(1 − e−x − xs e−x ) (1 − f (xm , K)) ,Kmm
where xs = B−x
S−K and f (x , K) satisfies (45). Differentiation yields

B b∂ min{K, xm }K∂f (xm , K)∂Π(xm , K)
−xs
s −xs
=
−
e
(1
−
,
f
(·)
+
λ
)
+
x
e
(1
−
f
(·))
+
A
∂xm
∂xm
K
S−K
∂xm
(46)
where
!

1 min{K, xm } ∂ min xKm , 1
−
,
S−K
xm
∂xm

∂f (xm , K)
Bλb
=
∂xm
e−xs Ksss

A ≡ (S − K)(1 − e−x ) + K(1 − e−x − xs e−x ).
m,K)
If xm < K, we have ∂f (x> 0. Also note that according to V s (xs ) = V m (xm ), we have∂xm
∂Π(xm ,K)B bmf (x , K) < 1 − K λ . Hence,> 0 for xm < K. The optimal solution of the problem∂xm
must satisfy xm ≥ K.
Observe that: limxm →B Π(xm , K) = K − Bλb where the right-hand-side is the profit function for the pure middleman mode with an inventory K. Hence, as before, we can compacify
the domain of xm and find a profit-maximizing intermediation mode using the following Lagrangian:L = Π(xm , K) + µk (xm − K) + µxm (xm − xm ) + µb (B − xm ) + µk (K − Bλb ) + µs (αS − K).
The first order conditions are∂L∂Π(xm , K)=
+ µk + µxm − µb = 0,
(47)
∂xm
∂xm
m
∂L
∂Π(x , K)=− µk + µk − µs = 0,
(48)
∂K
∂K
where


1
∂Π(xm , K)
K
1
1 B b
−xs
s −xs
=
−e
f
+
x
e
(1
−
f
)
+
A
+
λ ,
∂xm
S−K
S−K
xm e−xs xm


s
s
s
∂Π(xm , K)∂Π(xm , K)
1 1 B b= −(1 − e−x )f + 1 − (1 − e−x − xs e−x )(1 − f ) − xs
−
A
λ
.
∂K
∂xm
xm e−xs xm
Suppose xm = B, then we have µxm = 0 and µk = 0 (since B > αS ≥ K) and so (47) implies
m
m
,K)
,K)
we must have ∂Π(x|(xm =B) = µb ≥ 0. However, ∂Π(x|(xm =B) = −(1 − λb ) < 0, so we
∂xm
∂xm
m
get a contradiction. Hence, the solution must satisfy x < B (and µb = 0), i.e., the platform
is optimally activated.
Summing up the two first order conditions with µb = 0,µs − µk==∂Π(xm , K) ∂Π(xm , K)+
+ µ xm
∂K
∂xm


K
1
1
1 B b
s −xs
s −xs
(1 + x )e(1 − f ) +x e(1 − f ) + A
+ m
λ
−x
S−K
S−K
x
e s xm∂Π(xm , K)
1 1 B
−xs
− A m −xs m λb + µxm > 0,
m
∂x
x e
x

66

m

,K)
where the last inequality follows from (47) and µb = 0 that implies ∂Π(x= −µk −µxm ≤ 0.
∂xm
This implies µs > 0. That is, resource constraint (31) is binding, K = αS. Finally, note that
Π(B, αS) = αS − Bλb ≥ 0 given α ≥ λs . However, Π(B, αS) is suboptimal, which implies
that the optimal profit is even larger (hence must be positive). This completes the proof of
Proposition 4. 

A.6

Proof of Proposition 5

In our endowment economy, the middleman’s inventory purchase influences market tightness not only in the C market platform, but also in the D market. Given all sellers arein the D market, the probability that a buyer meets a seller available for trade in the Ds−xsmarket changes from λb e−x to λb S−K. With this change and using the analysis of
S emulti-market search shown in the previous section, the value of sellers becomes W C (xs ) =ss−xs(1 − e−x − xs e−x )(v(xm , K) − f ), and the middleman’s price becomes pm = 1 − λb S−K−
S es

xm e−xmmb S−K −xs.
min{xm ,K} (v(x , K) − f ), where v(x , K) = 1 − λS e

Substituting these expressions
into the profit function, it becomes immediate that the profit is strictly increasing in the fee
f . Hence, the incentive constraints are binding, f = v(xm , K). Using this result, we canwrite the profit function assΠ(xm , K) = (S − K)(1 − e−x )(1 − λb

S − K −xs
S − K −xse
) + min{K, xm }(1 − λb
e
),
S
S

m

where xs = B−x
S−K . Differentiation yields∂Π(xm , K)
∂xm

=



min{K, xm } b −xs
b S − K −xs
bS − K
−xs
−e
1−λ
e
−λ
(1 − e
) −
λ e
S
S
S


∂ min{K, xm }S − K −xs+e,1 − λb∂xm
S−xs

which is negative if min{K, xm } = K. Hence, the solution has to satisfy xm ≤ K.
Suppose xm = B. Then,∂Π(xm , K)
B
|xm =B = − λb < 0.
∂xm
SHence, the solution has to be xm < B, i.e., an active platform.
The Lagrangian then becomes
L = Π(xm , K) + µ0 xm + µk (K − xm ) + µs (αS − K).
The first order conditions are
∂L
∂xm
∂L
∂K

=
=

∂Π(xm , K)
+ µ0 − µk = 0,∂xm
∂Π(xm , K)
+ µk − µs = 0,
∂K

(49)
(50)

where
∂Π(xm , K)
∂K

s −xs



s bS − K−xsxs xm b −xs
) +
λ e
S

e
+x λ
(1 − e
S
S




s
S − K −xsxm b −xs+ 1 − e−x1 − 2λb
+
e
λ e
.
S
S

= x e

1−λ

b S − K −xs

67Combining (49) and (50) yieldss∂Π(xm , K) ∂Π(xm , K)+
= xs e−x
∂xm
∂K




S−K
xm b
b S − K −xs−xsbe(1 − eλ = µs − µ0 ,
1−λ
+
)λ +
S
S
S

which implies µs > 0 and K = αS. This completes the proof of Proposition 5. 

68

B

Web Appendix: Competing Intermediaries

B.1

Set-ups

This appendix is an extension on competing intermediaries that explains the emergence of marketmaking middlemen in a duopoly. We maintain our assumptions onbuyers, sellers and the intermediary that operates the C market, and replace the Dmarket by another strategic intermediary. We call the original C market intermediary the incumbent I and the new intermediary the entrant E, and we assume thatbuyers and sellers can only meet via one of the intermediaries.
Intermediary I.

This intermediary is the same as the C market intermediary inthe benchmark model. Intermediary I can combine the middleman mode and themarket-maker mode. We maintain the inventory advantage of the middleman mode,i.e., the middleman is able to hold a continuum of inventory to lower the out-of-stock
risk. In this appendix, we model this advantage by assuming that the middleman
has continuous access to the production technology which has a constant marginal
cost normalized to zero. Hence, the middleman can deliver any order from a buyer.
Effectively, the out-of-stock probability is zero. Specifically, intermediary I chooses
a mode i ∈ {m, p, h} and associated prices/fees (m for the pure middleman, p for thepure platform, and h for a hybrid mode). For i = m, I announces an inventory pricepm ∈ [0, 1]; for i = p, I announces a transaction fee f I ∈ [0, 1],34 and for i = h, both pm
and f I are announced.
Intermediary E.

We consider two scenarios. In one scenario, E is a pure middle-man, who has continuous access to the production technology and sets an inventory
price pE ∈ [0, 1]. In another scenario, E operates as a pure marketmaker who owns
the random matching technology, just as the D market in the benchmark model. In
this scenario, E makes profits by extracting a transaction fee f E ∈ [0, 1] imposed onsellers. A matched buyer and seller engage in an efficient bargaining process. We
focus on the case where the buyer gets the full trading surplus (β = 1). Throughout
this appendix, we take the intermediation mode of E to be exogenous. Our focus ison the optimal intermediation mode of I when it competes with either a pure middleman or a pure market maker E.
Timing. The timing is as follows. (1) Two intermediaries set fees/prices. Intermediary I announces i ∈ {m, p, h} and associated fees/prices. It charges a fee f I ∈ [0, 1]
34

Without loss of generality, we assume the transaction fee is imposed on the seller.

69

to a seller if the platform is open, and an inventory price pm ∈ [0, 1] to a buyer if the
middleman sector is open. Intermediary E announces a transaction fee f E ∈ [0, 1]
to a seller if it is a pure market-maker or an inventory price pE ∈ [0, 1] if it is apure middleman. The announced prices/fees from both intermediaries together arereferred to as P.35 (2) Observing i and P, buyers and sellers simultaneously decide
whether to participate in one or both of the intermediaries, yielding an allocationof buyers/sellers across intermediaries. (3) At intermediary I, search is directed as
specified in the main text. At intermediary E, agents search randomly and follow theefficient sharing rule for the trade surplus if E is a pure platform, or buyers trade
directly with E at the announced price pE if E is a pure middleman.
Equilibrium concept.

Let N = {B I , B E , S I , S E } denote the measures of buyers

and sellers across intermediaries, where B I (B E ) is the mass of buyers visiting I(E), S I (S E ) is the mass of sellers visiting I (E). Let V i (·) (or W i (·)), i ∈ {I, E} be
the expected value of a buyer (or a seller) who visits intermediary i. This value,
generally depends on P and N and will be specified below. Following Definition 1,
we say N is an equilibrium for given P if
B I = B · I{V I (P, N ) ≥ V E (P, N )}andS I = S · I{W I (P, N ) ≥ W E (P, N )}.
We define a market allocation as a mapping N (·) that assigns equilibrium measuresof buyers/sellers N (P) to I and E’s announcements P. Hence, each N (·) generates a
reduced-form price-setting game between intermediaries with a corresponding Nashequilibrium.
Definition 4 Given a market allocation N (·), an equilibrium of the game between I
and E is a price/fee vector P and the corresponding measures of buyers and sellers
N (P) where neither I nor E has an incentive to deviate from N (·).

B.2

Single-market Search

In this section, we show that, under single-market search, I is a pure middleman inequilibrium. Consistent with the main analysis, we assume that buyers and sellers
hold pessimistic beliefs against I. However, our conclusion does not depend on theselection of beliefs. The pessimistic beliefs mean that buyers and sellers all visit
35

Depending on the modes of both intermediaries, P varies to reflect the announcements. For example, if I announces i = p, and E is exogenously set as a pure platform, then P = {f I , f E }.

70

intermediary E whenever possible. Under pessimistic beliefs, if E is a middleman,then V E = 1 − pE where pE is the inventory price. If E is a market-maker with fee
B

f E , then V E = e− S (1 − f E ). We have dropped the dependence of V E on P and N for
simplicity.
Under pessimistic beliefs, a pure market-maker I won’t be active since buyers
expect an empty platform. I needs to convince buyers using its inventory that if
buyers do join the I market, by visiting the middleman sector, their expected value
at I is higher than V E . This requirespm ≤ 1 − V E .
Therefore, the maximum profit of a pure middleman I is
Π̃(B) = B(1 − V E ).
If I activates its platform (i = h), then the prices/fees must satisfyη s (xs )(1 − ps ) = 1 − pm ≥ V E ,(51)

ps − f I ≥ 0.

(52)(51) states that buyers must be indifferent between the two modes of I when bothare active. (52) states that sellers must be better off by trading on I’s platform than
any alternative options given that all buyers have joined I. (51) and (52) imply thatsf I ≤ 1 − V E . The resulting profits of a hybrid I satisfy S(1 − e−x )f I + xm pm <(Sxs + xm ) max{f I , pm } ≤ B(1 − V E ) = Π̃(B). Therefore, I optimally works as a puremiddleman in equilibrium.

B.3

Multi-market search: E is a pure middleman.

Under multi-market search, we assume that I opens prior to E. As has been shownin the main analysis, independent of agents’ beliefs, the only participation equilibrium that is consistent with Definition 1 is the one where all agents first visit I andthen E whenever possible. When the offer that an agent gets from I is at least as
good as his expected value at E, the agent will trade at I. We call I a first source, andE a second source. Otherwise, the agent chooses to forgo the trading opportunity at
I and only trade at E which then makes E the sole source.
We first consider the case that E is a pure middleman. To activate the platform,
I is subject to the following incentive constraints:
1 − ps ≥ 1 − pE ,ps − f I≥ 0.
71

The first constraint says that the buyer’s payoff if he is matched on the platform of
I must be higher than the outside option value of trading with middleman E. The
second constraint says that the payoff for a seller if he is matched on the platformof I must be positive. These constraints imply: f I ≤ pE . Similarly, to activate themiddleman, I is subject topm ≤ pE .
If max{pm , f I } ≤ pE , then trade can occur in either one of the sectors, and so I is a
first source.
Given pE , the maximum profit of I under i = m is Π̃(B) = BpE , and under i = p
B−S)pE < Π̃(B). Under i = h, the profit function of I becomes
it is Π̃( BS ) = S(1 − essΠ̃(xm ) = S(1 − e−x )f I + xm pm ≤ S(1 − e−x ) + xm max{f I , pm }≤ B max{f I , pm } ≤ BpE ≡ Π̃(B)m

for all xm < B with xs = B−x
S . Therefore, when facing a pure middleman competitor,an active platform is not profitable for I. Since the two intermediaries compete byprice, any equilibrium must be subject to Bertrand undercutting, leading to pm =pE = 0 and zero profits.

B.4

Multi-market Search: E is a pure market-maker

When E is a pure market-maker, the expected value of buyers when visiting E iss

V E = λb e−x (1 − f E ).

(53)

This value is the same as in the main analysis except that f E is subtracted fromtotal surplus. With this modification, as long as
max{pm , f I } ≤ 1 − V E ,(54)

trade can occur at one or both of the modes of I and I then becomes the first source.
B.4.1

The best response of intermediary I

We start with a characterization of the best responses of I.
Proposition 9 Under multi-market search technologies, given f E ∈ [0, 1], I’s best
responses are as follows:• If f E = 1, then I adopts a pure middleman mode (i = m) with inventory pricepm = 1;72

• If λb e−B/S ≥ 12 and f E ≤ 1 − 2λb e1−B/S , then I adopts a pure market-maker modeB(i = p) with transaction fee f I = 1 − λb e−x S (1 − f E ) ;• If λb e−B/S < 21 , or λb e−B/S ≥ 12 and f E > 1 − 2λb e1−B/S , I adopts a hybrid modesm(i = h) with the optimal price/fee pm = f I = 1 − λb e−x (1 − f E ) and xs = B−x
S ,where xm is characterized bysfE

S(1 − e−x ).
= b (x ) ≡ 1 −(2S(1 − e−xs ) + xm )λb e−xs
I

m

(55)

Proof. See B.5.1.
If f E = 1, then we are in a similar scenario as under single-market search —a buyer’s outside option is zero and a pure middleman with pm = 1 is optimal. If
f E < 1, there is cross-market feedback from E to I, which makes using the platform
as part or all of I’s intermediation activities profitable. Furthermore, whether I
operates as a pure market-maker or not depends on buyers’ outside option values. If
λb e−B/S < 12 , then the buyers’ outside option value is low. In this case, the middleman
sector generates high enough profits for the market-making middleman mode to be
adopted for any value of f E . If instead λb e−B/S ≥ 12 , then the buyers’ outside option
value is high, and attracting buyers to the middleman sector is costly. In this case,the intermediary will act as a market-making middleman if f E > 1 − 2λb e1−B/S , sobuyers expect a low value from E market, and as a pure market maker if f E ≤1 − 2λb e1−B/S , so buyers expect a high value from the E market.
Under i = h, (55) characterizes the optimal intermediation structure that I is
willing to pursue. bI (xm ) is monotonically increasing in xm , implying that as f E
decreases, I’s optimal mode moves towards a pure platform. Eventually (in the casethat λb e−B/S ≥ 21 ), as f E approaches 1 − 2λb e1−B/S , I’s optimal mode becomes the puremarket-maker mode.36
Armed with Proposition 9, we can rule out any equilibrium where I either acts asa pure middleman or as a pure market-maker (i = m or p). A pure middleman I does
not arise in equilibrium because, according to Proposition 9, I only adopts a puremiddleman mode with pm = 1 when f E = 1. Since all transactions are implemented
by I, E makes zero profit. But facing pm = 1, E would rather set f E slightly lower
than 1 to become the sole source and makes a profit of Bλb f E > 0.373611This is so because limxm →0 bI (xm ) = 1 − 2λb e−B/S, limxm →B bI (xm ) = 1. And if 1 − 2λb e−B/S< 0,then limf E →0 xm > 0.
37
When I is a pure middleman, E can only maketransactions if 1 − pm < λb (1 − f E ). That is, to1−pm
Eundercut I, E sets f slightly lower than 1 − λb as long as f E ≥ 0. In this way, I becomes inactiveand E makes a profit of Bλb f E .

73Turn to the case when I is a pure market-maker. According to the incentive
constraints, when V E = λb e−B/S (1 − f E ) ≤ 1 − f I , E is the second source; otherwise, I becomes inactive and E is the sole active source. Consider an equilibrium
candidate where E sets a fee f E ∈ [0, 1] and a pure market-maker I sets a fee
f I ≡ 1 − λb e−B/S (1 − f E ).38 The following discussion depends on the value of f E .
If f E > 0, it is profitable for E to undercut I by setting f E0 = f E − ε for some small
but positive ε. Then E becomes the sole source and makes a profit of Bλb f E0 . If
f E = 0, then E would rather work as a second source and take the full surplus ofeach transaction by deviating to f E0 = 1 and make a profit of Bλb e−B/S > 0. We
summarize these observations in the following corollary.
Corollary 2 There does not exist a pure strategy equilibrium where intermediary I
operates as a pure (middleman or market maker) mode.
B.4.2

The best response of intermediary E

We now turn to the optimal strategy of E. The analysis in the last section showsthat a pure mode I can not exist in equilibrium. Since the optimal hybrid mode I
implies pm = f I , below we refer to the level of price/fee by ψ and we discuss the best
responses of E under ψ = pm = f I . According to Proposition 9, the best response of Isfeatures ψ = 1 − λb e−x (1 − f E ) ≥ 1 − λb which holds with equality only when f E = 0
and xm = B. But according to Corollary 2, xm = B cannot be an equilibrium. Hence,
we further restrict ψ > 1 − λb for the best response analysis of E.
We start with a condition that determines whether E is a sole source or a secondsource. With the price/fee level of ψ > 1 − λb , the incentive constraint (54) becomesfE ≥ 1 −1(1 − ψ),λb e−xs

for some xs ∈ [0, B/S]. The right-hand side takes the minimum value at xs = B/S.
Therefore, as long asfE ≥ 1 −1
λb e−B/S(1 − ψ),(56)

E is the second source for some xm ∈ [0, B]. Otherwise, E is the sole source.
Suppose that E adopts the sole source strategy, then E undercuts ψ and sets f E
slightly lower than the right hand side of (56), which yields a profit of

b
ΠE
(ψ)
=
Bλ
1−
sole

1


(1
−
ψ)
.
λb e−B/S(57)

38
According to Proposition 9, f I is required to satisfy the best response of I. Any other fee level would
lead to a deviation of I.

74

Observe that when ψ is low, i.e., ψ < 1 − λb e−B/S , E will suffer a loss by undercutting
I (ΠEsole < 0).
Suppose that E adopts the second source strategy, then E chooses an f E thatsatisfies (56), and its profit maximization problem is


s
m
−xs
B
−
x
−
S(1
−
e
)
λb e−x f E ,
ΠE
(ψ)
=
max
2nd
f E ∈[1−1(1−ψ),1]λb e−B/S(58)

subject to (4) and (6).
The first step to solve the problem is to note that it is in E’s interest to havebuyers be indifferent between I’s two sectors. This is formally stated in the following
lemma.
Lemma 2 Given pm = f I = ψ, the optimal solution for problem (58) implies that thebuyers’ value of visiting the middleman of I equals that of visiting the platform of I:
V m (ψ) = V s (xm , ψ, f E ).

(59)

Proof.
Let {fˆE , x̂m } denote the optimal solution for problem (58). Suppose at the optimum, V m (ψ) > V s (x̂m , ψ, fˆE ), then x̂m = B. It follows thatV s (x̂m , ψ, fˆE ) = 1 − f I = 1 − ψ = 1 − pm = V m (ψ),
which contradicts with the assumption that V m (ψ) > V s (x̂m , ψ, fˆE ). Suppose at the
optimum, V m (ψ) < V s (x̂m , ψ, fˆE ). That is, x̂m = 0 andV s (·) = e−B/S (1 − ψ) + (1 − e−B/S )λb e−B/S (1 − fˆE ) > 1 − ψ.
This implies that fˆE < 1 − λb1−ψ. But then E gains a higher profit by deviating
e−B/S
1−ψ
E
E
to f˜ = 1 − b −B/S . At f˜ , E maintains the same trading volume while extracting
λ ehigher fees from each transaction, thus gains a higher profit according to (58).
The intuition is as follows. On the one hand, V m (·) > V s (·) leads to a pure middleman incumbent, leaving zero market share to E. On the other hand, V m (·) < V s (·)
means E needs to set f E unnecessarily low to attract buyers to I’s platform — E
could increase f E without changing the allocation of buyers/sellers yet obtain higher
profits.
Inserting the expression of V i (·), i = m, s, into (59), we haves

λb e−x (1 − f E ) = 1 − ψ.

(60)

Increasing f E leads to less favorable outside values for buyers on I’s platform, hence
more buyers visit I’s middleman sector (xm increases), and there are more unmatched75

s−x increases). Lemma 2 and (60) indicate that we can further resellers left for E (e1−ψ
1−ψ
.
B , 1 − λb
λb e− S
Substituting for f E from (60) and inserting into (58) yields

strict f E to 1 −ΠE
2nd (ψ) = maxxm ∈[0,B]

By choosing an f E


∈ 1−

s

B − xm − S(1 − e−x )1−ψ
B , 1 − λb
λb e− S
1−ψ






s
λb e−x − (1 − ψ) .

(61)

, E essentially chooses an xm ∈ [0, B].

While xm is the scale of the middleman of I, it also determines the measures ofparticipating buyers and sellers to E. A higher xm implies that less buyers join Esafter trading at I, since the population of participating buyers B − xm − S(1 − e−x )decreases in xm . At the same time, a higher xm implies that more sellers join E
because sellers are less likely to trade on the platform of I. This is reflected by thesmatching probability λb e−x which increases in xm .
The optimal xm depends on ψ. When ψ is high, it is profitable to have more buyers
at I’s platform by lowering xm and ultimately increase participating buyers on E. E
can achieve this by decreasing f E . When ψ is low, E finds it less profitable to havemore participating buyers, and the optimal f E should be higher. Given that I setsf I = pm = ψ ∈ (1 − λb , 1], the following proposition characterizes the best responsesof E using xm . The corresponding f E can be derived from (60).
Proposition 10 Under multi-market search technologies, given f I = pm = ψ ∈ (1 −λb , 1], E’s optimal strategy can be characterized as follows:
E• For ψ ∈ (1 − λb , 1 − λb e−B/S ], we have ΠE
2nd (ψ) > 0 ≥ Πsole (ψ), and E acts as thesecond source with xm ∈ (0, B) satisfyingss

1 − ψ = λb e−x

1−

B − xm − S(1 − e−x ) ;S(1 − e−xs )(62)• For ψ ∈ (1 − λb e−B/S , 1], both the second and the sole source generate positive
profits, and E chooses the one that is more profitable. If E operates as the second
1source, then xm ∈ [0, B) satisfies (62). A special case occurs when B
S ≤ − log 
4,

−B/S
B−S(1−e
)
b
b
b
−B/S
then for ψ ∈ (1 − φ(B, S, λ ), 1], where φ(B, S, λ ) ≡ λ e
1 − S(1−e−B/S ) ,E
we have ΠEsole (ψ) > Π2nd (ψ) > 0, and E optimally chooses to be the sole source.

Proof. See B.5.2.
The intuition is as follows. When ψ is low, further undercutting I is not profitable
for E despite that it generates more demand. E optimally chooses to be the secondsource. When ψ is high, the sole-source strategy may become profitable for E. In
76

particular, when the market is relatively tight ( B
S is low), the residual demand for asecond source E is small and the sole source strategy becomes optimal.
Equation (62) is the key element of the best response of E operating as the secondsource. It describes the optimal market structure (represented by xm ) that E would
like to choose. Consider a range of [xm , B] where xm ≥ 0 ensures that the right handside of (62) is non-negative. It then follows that
s
s
s
1
∂ψ
λb e−x (1 − e−x − xs e−x ) =− 1−ψ+< 0,∂xm
S(1 − e−xs )2
for xm ∈ [xm , B]. This means that as ψ increases, E finds it profitable to competewith I. To do so, E lowers f E to make the middleman sector of I less favorable
to buyers. Correspondingly, the middleman sector of I shrinks (xm decreases), andmore unmatched buyers join E.
B.4.3

Equilibrium analysis

We start with a lemma on the non-existence of pure strategy equilibrium where E is
the sole source.
Lemma 3 There does not exist a pure strategy equilibrium where E is the only active
intermediary.
To understand the lemma, suppose that E is the sole source in equilibrium and I
is inactive with zero profit. From Proposition 10, we know that in this equilibrium,
the price/fee of I must satisfy ψ > 1 − λb e−B/S . It is then a profitable deviation for I
to set ψ smaller than 1 − λb e−B/S but bigger than 1 − λb so that I can make a positive
profit as the first source.
Consider the candidate equilibrium where E acts as the second source. The equilibrium should jointly solve the optimal responses of two intermediaries, namely (55)and (62), together with the equilibrium condition (4). Inserting (60) into (62) gives
an alternative expression for the best response of E:sf E = bE (xm ) ≡B − xm − S(1 − e−x ).
S(1 − e−xs )(63)Hence, the equilibrium xm∗ ∈ (0, B) should solve for bI (xm∗ ) = bE (xm∗ ). Proposition
11 gives sufficient conditions for the existence and uniqueness of the equilibrium.
−x

1−e
b
Proposition 11 Define x ≡ B
S > 0, and assume that either x < x̄ and λ < 2e−x (2(1−e−x )−x) ,or x ≥ x̄, where x̄ > 0 is uniquely given by 2(1 − e−x̄ ) − x̄ = 0. Then there exists a
unique pure strategy equilibrium if
1 − λb e−B/S ≥ ψ ∗ > 1 − λb ,
77

(64)

where ψ ∗ = 1 − λb e−x

s∗




s∗ −(1−e−xs∗ )m∗
1 − x (1−e, xs∗ = B−x, and xm∗ ∈ (0, B) solves−xs∗ )SbI (xm∗ ) = bE (xm∗ ).

(65)

The equilibrium features a market-making middleman I as the first source and amarket-maker E as the second source who both make positive profits. We can char∗, and the measures ofacterize the equilibrium by f I = pm = ψ ∗ , f E = 1 − λb1−ψ
e−xs∗
s∗

s∗

participants N = {B, B − xm∗ − S(1 − e−x ), S, Se−x , xm∗ }.
Proof. See B.5.3.
Figure 6 illustrates the equilibrium by two variables, the price/fee level represented by f E , and the market structure represented by xm . It plots bI (xm ) andbE (xm ) and we have marked the function values as xm approaches 0 and B.39 The
interaction of the two best responses gives the equilibrium value f E∗ and xm∗ . The
equilibrium allocation of participants and prices/fees can be derived accordingly, as
stated in the proposition.
For comparative statics, consider an exogenous change in the buyer’s meeting
rate λb and the population of buyers B. First, as λb increases to λb0 , bI (xm ) moves up
while bE (xm ) does not shift, leading to a smaller xm∗ . This is illustrated in Figure 7.
E

I

m

m

(x )
(x )
> 0 and ∂b ∂B<0Second, consider an exogenous change in B. Note that ∂b ∂B

for xm ∈ (0, B). That is, as the population of buyers B increases, bI (xm ) moves down
while bE (xm ) moves up, leading to a higher xm∗ . This is illustrated in Figure 8 (themass of buyers increases from B to B 0 ). Similar comparative statics can be done forthe population of sellers S. We summarize these observations in Corollary 3.
Corollary 3 (Comparative statics) Consider a parameter space in which a purestrategy equilibrium exists. Then, an increase in the buyer’s meeting rate λb at intermediary E, or a decrease in the buyer-seller ratio B/S, leads to a smaller middleman
sector xm and a larger platform xs of intermediary I in equilibrium.
Proof. See B.5.4.
39

We make use of the following observations:1,
lim bI (xm ) = 1,xm →B
2λb e−B/S
B − S(1 − e−B/S )lim bE (xm ) =,lim bE (xm ) = 0.
xm →0
xm →B
S(1 − e−B/S )lim bI (xm ) = 1 −

xm →0

−B/S

)
1
We have plotted one particular scenario that 1 − 2λb e−B/S> 0 and B−S(1−e< 1. But theseS(1−e−B/S )restrictions are not required for the existence of an equilibrium.

78

fE
1
B−S(1−e−B/S )S(1−e−B/S )bI (xm )f E∗

1 − 2λb e1−B/SbE (xm )

0
xm∗

B

xm

Figure 6: Equilibrium under multi-market searchfE
1bIλb0 (xm )B−S(1−e−B/S )S(1−e−B/S )bIλb (xm )f E∗0
1 − 2λb0 e1−B/S
1 − 2λb e1−B/SbE (xm )

0
xm∗0

xm∗

B

xm

Figure 7: Comparative Statics w.r.t. λbfE0

B 0 −S(1−e−B /S )S(1−e−B 0 /S )1bIB (xm )B−S(1−e−B/S )S(1−e−B/S )bIB 0 (xm )

f E∗
f

0

E∗

1 − 2λb e1−B/SmbE
B 0 (x )1
1 − 2λb e−B
0 /S

m
bE
B (x )

0
xm∗

0

xm∗

Figure 8: Comparative Statics w.r.t. B

79

B

xm

Numerically, we can verify that sufficient condition (64) is satisfied at least forsome reasonable parameter values. For example, taking B = S = 1, and setting agrid of λb with two decimals from 0.01 to 0.99, shows that (64) holds for all λb grid
points smaller than 0.95. For λb grid points between 0.95 and 0.98, despite that thesufficient condition (64) is violated, the second source is still more profitable than
the sole source for E, so a pure strategy equilibrium continues to exist. For the grid
point λb = 0.99, E finds it more profitable to undercut I and become the sole source.
Then there does not exist a pure strategy equilibrium. By applying Theorem 5 ofDasgupta and Maskin (1986), a mixed strategy equilibrium exists.
Proposition 12 There exists a mixed strategy equilibrium under multi-market search.
Proof. See B.5.5.
Given that in the mixed strategy equilibrium f E < 1 happens with positive probability, according to Proposition 9, I activates its platform with positive probability
in equilibrium.
Corollary 4 In the mixed strategies equilibrium, I’s platform is activated with positive probability.

80

B.5

Omitted proofs

B.5.1

Proof of Proposition 9

Under i = h, inserting the binding constraint (54) on pm and f I , the intermediary’s problem
can be written as choosing an xm to maximize the profits

 


B−xm
Π (xm ) =S 1 − e− S
+ xm 1 − V E xm , f E ,where 0 < xm < B and V E (·) is buyers’ value of visiting E defined in (53), and we have madeit explicit that V E depends on xm and f E .
We can further compactify the constraint set of xm to [0, B] sincelim Π (xm ) = B 1 − λb (1 − f E ) = Π̃(B),
m
x →B




m
−B/S
lim
Π
(x
1 − V E (0, f E ) = Π̃(0),
)
=
S
1
−
e
m

x →0

where Π̃(B) is the profit for the pure middleman mode and Π̃(0) is the profit for the puremarket-maker mode. Then we pin down a profit-maximizing intermediation mode using thefollowing Lagrangian:L = Π (xm ) + µ0 xm + µb (B − xm ),where the µ’s ≥ 0 are the Lagrange multiplier of each constraint. Since Π(xm ) is concave in
xm , the solution is characterized by the following first order condition and the complementary slackness conditions,∂L∂xm∂Π (xm )+ µ0 − µb = 0.
∂xm

=

(66)
s

A special case is that if f E = 1, then V E (xm , f E = 1) = λb e−x (1 − f E ) = 0. Thesintermediary’s profits become Π (xm ) = S(1 − e−x ) + xm , where Π (xm ) is concave in xm . Thesfirst order condition with respect to xm is 1 − e−x = 0. Therefore, at the optimal xm = B.
In general, if f E < 1, we show that I’s platform is active, i.e., xm < B at the optimality.
(66) can be rewritten as follows,µb − µ0

=

s

s

s

(1 − e−x )(1 − λb e−x (1 − f E )) − (xm + S(1 − e−x ))≡ φ(xm | B, S, λb , f E ).

λb −xse(1 − f E )
S
(67)

b
E
At xm = B, (67) yields φ(B | ·) = µb = − B
S λ (1 − f ) < 0, which contradicts to µb ≥ 0.
m
Hence, the solution must satisfy x < B (which implies µb = 0). At xm = 0, (67) yields
φ(0 | ·) = −µ0 = (1 − e−B/S )(1 − 2λb e−B/S (1 − f E )), which requires f E ≤ 1 − 2λb e1−B/S . This
leads to the conditions in the proposition. Set µb = µ0 = 0, we have φ(xm | B, S, λb , f E ) = 0
according to (67). This gives condition (55) for xm ∈ (0, B).

B.5.2

Proof for proposition 10b
First of all, from (57) and (61), it is straightforward to see that ΠE
2nd ≤ 0 if ψ ≤ 1 − λ ; andEb −B/SΠsole ≤ 0 if ψ ≤ 1 − λ e. These observations give the signs of the profits in all cases in
the proposition.
Second, we then discuss the optimal xm when E acts as the second source. Define theprofit function as



s
m
m
−xs
ΠE
) λb e−x − (1 − ψ) .
(68)
2nd (x |ψ) ≡ B − x − S(1 − e81The first order condition condition issmss
B − xm − S(1 − x−x ) b −xs∂ΠE
2nd (x |ψ)= −(1 − e−x )(λb e−x − (1 − ψ)) +
λ e
= 0.
m
∂x
S

(69)

m
m
Since ΠE=
2nd (x |ψ) is continuously differentiable on [0, B], the maximum point is either x∂ΠE (xm |ψ)0, xm = B or some x̂m such that 2nd|xm =x̂m = 0. With ψ > 1 − λb , we know that E can∂xmmobtain a positive profit as a second source. However, ΠE
2nd (B|ψ) = 0, thus x = B does
 not
−B/S

∂ΠE (xm |ψ))give the maximum. When 2nd|xm =0 > 0, that is 1 − λb e−B/S 1 − B−S(1−e> ψ,∂xm
S(1−e−B/S )mm
x = 0 does not satisfy the necessary condition. For an x̂ that satisfies the first ordercondition, we rearrange (69) to get (62). These observations give the first bullet point of the
proposition.


−B/S
)
.
The following discussion depends on the sign of φ(B, S, λb ) ≡ λb e−B/S 1 − B−S(1−e
−B/S
S(1−e
)

E
If φ(B, S, λb ) < 0, then for ψ ∈ [1−λb e−B/S , 1], E compares ΠEsole (·) and Π2nd (·) to decide whichbb −B/Sis more profitable. If φ(B, S, λ ) ≥ 0, then again for ψ ∈ (1 − λ e, 1 − φ(B, S, λb )), boththe sole source and the second source can be profitable and E chooses the more profitableone. For ψ ∈ [1 − φ(B, S, λb ), 1], we provide a sufficient condition that as a second source E
optimally chooses xm = 0. Yet in this case, the second source profit is strictly dominated by
the sole source profit:−B/SΠE)(λb e−B/S − (1 − ψ)) < B(λb e−B/S − (1 − ψ)) = ΠE
2nd (ψ) = (B − S(1 − esole (ψ).

The following elaborates the condition that xm = 0 is optimal in this case. Rearrange (69) tomake it in terms of xs , and refer to it as ω(xs ):
s

s

s

s

ω(xs ) ≡ −(1 − e−x )(λb e−x − (1 − ψ)) + (xs − (1 − e−x ))λb e−x .

(70)Under ψ > 1 − φ(B, S, λb ), we have ω(B/S) < 0. It is straightforward to verify that ω(0) = 0.
Furthermore, given that
s

s

s

s

s

ω 0 (xs ) = −e−x (λb e−x − (1 − ψ)) + 3(1 − e−x )λb e−x − xs λb e−x ,we have ω 0 (0) = 1 − λb − ψ < 0 under ψ > 1 − λb . Hence, there exists an ε > 0 that ω(ε) < 0.
∂ΠE (xm |ψ)2nd
To show xm = 0 is optimal, it is enough to show that
< 0 for xm ∈ (0, B), or
∂xm
s
s
s
equivalently ω(x ) < 0 for x ∈ (0, B/S). Suppose there exists an x1 ∈ (0, B/S) such thatω(xs1 ) = 0. Because ω(B/S) < 0 and ω(ε) < 0, then either there exists another xs2 ∈ (xs1 , B/S)
such that ω(xs2 ) = 0, or ω 0 (xs1 ) = 0. The latter can be easily excluded since the xs ’s that allows
ω(xs ) = ω 0 (xs ) does not give ω(xs ) = ω 0 (xs ) = 0. To derive a condition for the contradiction in
the former case, notice
Further, let’s definesssΩ(xs ) ≡ ω 0 (xs )|ω(xs )=0 = −(ψ − (1 − λb e−x )) + 2(1 − e−x )λb e−x .
If both xs1 and xs2 exist, then we shall have Ω(xs1 ) > 0 > Ω(xs2 ). And this is impossible ifssΩ0 (xs ) = λb e−x (4e−x − 1) ≥ 0.
1
B
1bThis condition is guaranteed by B
S ≤ − log( 4 ). Indeed, for S ∈ (0, − log( 4 )], we have φ(B, S, λ ) >0. We therefore proved that xm = 0 is optimal for ψ ∈ (1 − φ(B, S, λb ), 1]. 

82

B.5.3

Proof for proposition 11

Defineg(xm ) ≡ bI (xm ) − bE (xm ).
First, notice g(xm ) is continuous differentiable with respect to xm . Taking the limits of xm tob
0 and B, and define x ≡ B
S . Under the stated assumption about x and λ in the proposition,
we havelim g(xm ) = m
lim bI (xm ) − mlim bE (xm )

xm →0

x →0

x →0

1
x − (1 − e−x )= 1 − b −x −2λ e
(1 − e−x )b −x−x
2λ e (2(1 − e ) − x) − (1 − e−x )=< 0.
2λb e−x (1 − e−x )andlim g(xm ) = mlim bI (xm ) − mlim bE (xm )

xm →B

x →B

x →B

= 1 − 0 > 0.
According to the Intermediate Value Theorem, there exists an xm∗ ∈ (0, B) such that bI (xm∗ ) =bE (xm∗ ).
Second, the equilibrium xm∗ is unique. This is because g is monotone increasing in xm
on (0, B). Taking the first order derivate with respect to xm , we have00

g 0 (xm ) = bI (xm ) − bE (xm ) > 0,
where
s

0

bI (xm ) =s(1 + 2(1 − e−x ))S(1 − e−x ) + xm
> 0,
(2S(1 − e−xs ) + xm )2 λb e−xsss

1 − e−x − xs e−x
b (x ) = −< 0.
S(1 − e−xs )2E0m

Thirdly, if ψ ∈ (1 − λb , 1 − λb e−B/S ], then according to Proposition 10, E has no incentive
to deviate to the sole source strategy since ΠEsole ≤ 0.
Finally, according to Proposition 9 and Proposition 10, both intermediaries make positive
profits. 

B.5.4

Proof for corollary 3

Let’s consider a marginal increase in λb , B and S in turns.
Consider an increase in λb : λb0 = λb + ε with ε > 0. We show the equilibrium market structure under λb0 follows xm∗0 < xm∗ , where xm∗ is the equilibrium expected queue length underλb and xm∗0 is the equilibrium expected queue length under λb0 . We denote the best response function under λb by biλb (xm ) and that under λb0 (xm ) by biλb0 (xm ), i = I, E. SinceI∂bE (xm )
(xm )
m
E
m
I
m
I
m
= 0, ∂b∂λ> 0, for xm ∈ (0, B), we have bEbλb0 (x ) = bλb (x ), and bλb0 (x ) > bλb (x ).
∂λb
m∗0
m∗

Suppose x=x, thenm∗m∗
m∗0bIλb0 (xm∗0 ) > bIλb (xm∗0 ) = bIλb (xm∗ ) = bE
) = bE) = bE).
λb (xλb0 (xλb0 (xm∗0
But bIλb0 (xm∗0 ) > bE
) implies that xm∗0 = xm∗ is not in equilibrium.
λb0 (x83

Suppose xm∗0 > xm∗ , then
m∗
m∗
m∗0
bIλb0 (xm∗0 ) > bIλb (xm∗0 ) > bIλb (xm∗ ) = bE
) = bE) > bE).
λb (xλb0 (xλb0 (xm∗0
Again, bIλb0 (xm∗0 ) > bE
) implies that xm∗0 > xm∗ can not be in equilibrium.
λb0 (xConsider an increase in B: B 0 = B + ε with ε > 0. We show the equilibrium market structure under B 0 follows xm∗0 > xm∗ . Sincess∂bE (xm )1 − e−x − xs e−x
> 0,
=
∂B
S(1 − e−xs )2s∂bI (xm )2S(1 − e−x )2 + xm
< 0,
=−
∂B
(2S(1 − e−xs ) + xm )2 λb e−xs
m
E
m
I
m
I
m
for xm ∈ (0, B), we have bE
B 0 (x ) > bB (x ), and bB 0 (x ) < bB (x ).
m∗0m∗
Suppose x= x , then
m∗0
m∗
m∗
).
) = bEbIB 0 (xm∗0 ) < bIB (xm∗0 ) = bIB (xm∗ ) = bE
) < bEB 0 (xB (xB 0 (xm∗0
But bIB 0 (xm∗0 ) < bE
) implies that xm∗0 = xm∗ can not be in equilibrium.
B 0 (xSuppose xm∗0 < xm∗ , then
m∗
m∗
m∗0
bIB 0 (xm∗0 ) < bIB (xm∗0 ) < bIB (xm∗ ) = bE
) < bE) < bE).
B (xB 0 (xB 0 (xm∗0) implies that xm∗0 < xm∗ can not be in equilibrium.
Again, bIB 0 (xm∗0 ) < bE
B 0 (xConsider an increase in S: S 0 = S + ε with ε > 0. We show the equilibrium market structure under S 0 follows xm∗0 < xm∗ . Sincess∂bE (xm )(1 − e−x − xs e−x )xs
=−
< 0,
∂S
S(1 − e−xs )2
i
h
−xs
m
s
xs 2(1 − e−x )2 + xS (1 − 1−exs )∂bI (xm )=> 0,∂SS(2(1 − e−xs ) + xm /S)2 λb e−xs
m
E m
I
m
I
m
for xm ∈ (0, B), we have bE
S 0 (x ) < bS (x ), and bS 0 (x ) > bS (x ).
m∗0m∗
Suppose x= x , then
m∗
m∗
m∗0
bIS 0 (xm∗0 ) > bIS (xm∗0 ) = bIS (xm∗ ) = bE
) > bE) = bE).
S (x
S 0 (xS 0 (xm∗0
Sut bIS 0 (xm∗0 ) > bE
) implies that xm∗0 = xm∗ can not be in equilibrium.
S 0 (xSuppose xm∗0 > xm∗ , then
m∗
m∗
m∗0
bIS 0 (xm∗0 ) > bIS (xm∗0 ) > bIS (xm∗ ) = bE
) > bE) > bE).
S (x
S 0 (xS 0 (xm∗0
Again, bIS 0 (xm∗0 ) > bE
) implies that xm∗0 > xm∗ can not be in equilibrium. This comS 0 (xpletes the proof of corollary 3.

B.5.5

Proof for proposition 12

Consider a game between I, who selects (f, pm ) ∈ [0, f¯] × [0, p̄] with a payoff ΠI = ΠI (f, pm |f E ), and E, who selects f E ∈ [0, 1] with a payoff ΠE = ΠE (f E | f, pm ). Here, we set f¯, p̄ > 1
and f > 1 (p̄ > 1) leads to an inactive platform (middleman sector). We apply Theorem 5 ofDasgupta and Maskin (1986) to show there exists a mixed strategy equilibrium.
Given Theorem 5 of Dasgupta and Maskin (1986), it is sufficient to show that ΠI (ΠE )is bounded and weakly lower semi-continuous in f and pm (in f E ), and ΠI + ΠE is uppersemi-continuous. Clearly, ΠI (ΠE ) is bounded in (f, pm ) ∈ [0, f¯] × [0, p̄] (in f E ∈ [0, 1]).

84

Both of the profit functions are continuous except atmin{f, pm } = 1 − V E (f E ),(71)

where V E (f E ) is evaluated at xm = 0. So we shall pay attention to this discontinuity point.
First, we show that ΠI (f, pm | f E ) is weakly lower semi-continuous in (f, pm ). Give thediscontinuous point in (71), we have(sS(1 − e−x )f + xm pm , if min{f, pm } ≤ 1 − V E (f E )ImEΠ (f, p | f ) =0otherwise,
where in the second situation, the price/fee of I is not competitive to the fee of E, hence agentswill trade via E, rather than I, and so I will become inactive. Consider some f E ∈ [0, 1],
and some f, pm > 0 such that min{f, pm } = 1 − V E (f E ). For any sequence {(f (j) , pm (j) )}converging to (f, pm ) such that no two f (j) ’s, and no two pm (j) ’s are the same, and f (j) ≤f, pm (j) ≤ pm , we must have min{f (j) , pm (j) } ≤ 1 − V E (f E ). Hence,lim ΠI (f (j) , pm (j) | f E ) = ΠI (f, pm | f E ),j→∞satisfying the definition of weakly lower semi-continuity (see Definition 6 in page 13 of Dasgupta and Maskin, 1986, or condition (9) in page 384 of Maskin, 1986).
Second, we shall show that ΠE (f E | f, pm ) is lower semi-continuous in f E . Consider apotential discontinuity point f0 ∈ (0, 1) satisfying (71) such that(Bλb f E ,if f E < f0EEm
Π (f | f, p ) =s(B − xm − S(1 − e−x ))λb f E if f E ≥ f0 .
Clearly, this function is lower semi-continuous, since for every  > 0 there exists a neighborhood U of f0 such that ΠE (f E | ·) ≥ ΠE (f0 | ·) −  for all f E ∈ U .
Finally, we prove the upper semi-continuity of ΠI + ΠE . For this purpose, consider all
sequences of {f (j) , pm(j) , f E(j) } that converges to {fˆ, p̂m , fˆE } that satisfies min{fˆ, p̂m } = 1 −V E (fˆE ).
Consider first an extreme in which case min{f (j) , pm (j) } ≤ 1 − V E (f E(j) ) for all j. As theequilibrium is that I is visited prior to E, we must havelim ΠI (f (j) , pm(j) | f E(j) ) + ΠE (f E(j) | f (j) , pm(j) ) = ΠI (fˆ, p̂m | fˆE ) + ΠE (fˆE | fˆ, p̂m ).

j→∞Consider next the other extreme in which min{f (j) , pm (j) } > 1 − V E (f E(j) ) for all j. Then, inthe equilibrium only E is active and we must havelim ΠI (f (j) , pm(j) | f E(j) ) + ΠE (f E(j) | f (j) , pm(j) ) = Bλb fˆE .

j→∞

If fˆ ≥ pˆm , thenΠI (fˆ, p̂m | fˆE ) + ΠE (fˆE | fˆ, p̂m ) = B p̂m = B(1 − λb (1 − fˆE )) > Bλb fˆE .
If fˆ < pˆm , thenΠI (fˆ, p̂m | fˆE ) + ΠE (fˆE | fˆ, p̂m )BB= B(1 − e− S )fˆ + Bλb e− S fˆE
B
B> B[(1 − e− S ) + λb e− S ]fˆE > Bλb fˆE .

Thus,lim ΠI (f (j) , pm(j) | f E(j) ) + ΠE (f E(j) | f (j) , pm(j) ) < ΠI (fˆ, p̂m | fˆE ) + ΠE (fˆE | fˆ, p̂m ).

j→∞

85

As these two extreme cases give the upper and lower bounds, respectively, all other sequences
give some limits in between. Therefore,lim ΠI (f (j) , pm(j) | f E(j) ) + ΠE (f E(j) | f (j) , pm(j) ) ≤ ΠI (fˆ, p̂m | fˆE ) + ΠE (fˆE | fˆ, p̂m ),j→∞

for any of the sequences converging to {fˆ, p̂m , fˆE }, and so ΠI + ΠE is upper semi-continuous.
This completes the proof of Proposition 12. 

86

C

Web Appendix: Participation fees

In this Additional Appendix, we show that our main result does not change in a version of our model where the middleman’s supply is not observable in the participation
stage, but instead the intermediary can use participation fees or subsidies. Supposethat in the first stage the intermediary announces a set of fees F ≡ f b , f s , g b , g s forthe platform, where f b , f s ∈ [0, 1] is a transaction fee charged to a buyer or a seller,
respectively, and g b , g s ∈ [−1, 1] is a registration fee charged to a buyer or a seller,
respectively.
As is consistent with the main analysis, we follow the literature of two-sided markets and assume that agents hold pessimistic beliefs on the participation decision of
agents on the other side of the market (Caillaud and Jullien, 2003). Agents believethat the intermediary would never supply anything at all unless the C market attracts some buyers. This is the worst situation for the intermediary. A pessimistic
belief of sellers means that sellers believe the number of buyers participating in the
C market is zero wheneverλb β > −g b ,where λb β is the expected payoff of buyers in the D market and −g b is the payoffbuyers receive in the C market (it is a participation subsidy when g b < 0).
Single-market search: To induce the participation of agents under those beliefs,
the best the intermediary can do is to use a divide-and-conquer strategy, denoted byh. To divide buyers and conquer sellers, referred to as h = Db Cs , it is required that
Db : −g b ≥ λb β,(72)

Cs : W − g s ≥ 0.

(73)

The divide-condition Db tells us that the intermediary should subsidize the participating buyers so that they receive at least what they would get in the D market,
even if the C market is empty. This makes sure buyers will participate in the C market whatever happens to the other side of the market. The conquer-condition Cs
guarantees the participation of sellers, by giving them a nonnegative payoff – theparticipation fee g s ≥ 0 should be no greater than the expected value of sellers in
the C market, W = W (xs ). Observing that the intermediary offers buyers enoughto participate, sellers understand that all buyers are in the C market, the D market
is empty, and so the expected payoff from the D market is zero. Here, the expectedvalue of sellers in the C market W is defined under the sellers’ belief that the intermediary will select the capacity level optimally given the full participation of buyers.
87Similarly, a strategy to divide sellers and conquer buyers, referred to as h = Ds Cb ,requires thatDs : −g s ≥ λs (1 − β),(74)

Cb : V − g b ≥ 0.

(75)

where V = max{V s (xs ), V m (xm )} is the expected value of buyers in the C market.
Given the participation decision of agents described above, the intermediary’s
problem of determining the intermediation fees F = {f b , f s , g b , g s } for h = {Db Cs , Ds Cb }is described asΠ ≡ max{Bg b + Sg s + max
Π(pm , f, K)},
m
F,h

p ,K

subject to (72) and (73) if h = Db Cs , or (74) and (75) if h = Ds Cb . Here, Bg b and Sg s
are participation fees from buyers and sellers, respectively, and Π(·) is the expected
profit in the C market described above. Under either of the divide-and-conquer
strategies, the choice of participation fees g i , i = b, s, does not influence anyone’s
behavior in the C market. The choice of transaction fees affects the expected valueof agents and consequently also the participation fees and intermediary’s profits.
However, it does not alter the original solution, a pure middleman, remains optimal.

Proposition 13 With unobservable capacity and with participation fees, the intermediary sets f > 1, pm = 1 and K = B. All the buyers buy from the middleman,xm = B, and the platform is inactive, xs = 0. The intermediary makes profits,Π = B − min{Bλb β, Sλs (1 − β)},guaranteeing the participation of agents by h = Db Cs if β < 21 and h = Ds Cb if β > 12 .
Proof. Consider first h = Db Cs . Then, by (72) and (73), g b = −λb β and g s = W .
For f > 1, no buyers go to the platform xs = 0 and all buyers are in the middleman sector xm = B, yielding g s = W = 0. By selecting K = B and pm = 1, theintermediary makes profits,Π = −Bλb β + Π(pm , 1, B) = (−λb β + 1)B.
To show that this is indeed the maximum profit, we have to check two possible cases.
ms
Suppose f = f b + f s ≤ 1 and K = 0. Then, xs = B
S and x = 0, and g = W (B/S) ≥ 0,88

if there is a non-negative surplus in the platform for buyers, f b + ps ≤ 1, and forsellers, f s ≤ ps . The resulting profit satisfies
B

B

Bg b + Sg s + Π(pm , f, 0) = −Bλb β + S(1 − e− S )(ps − f s ) + S(1 − e− S )fB

= −Bλb β + S(1 − e− S )(f b + ps )< −Bλb β + B = Π
for all f b + ps ≤ 1. Hence, this is not profitable.
Suppose f = f b + f s ≤ 1 and K ∈ (0, B], and both sectors have a non-negative
surplus to buyers, i.e., pm ≤ 1 and f b + ps ≤ 1. This leads to xm ∈ (0, B) andxs ∈ (0, B
S ) that satisfy the add-up requirement (4) and the indifferent condition (6).
Then, g s = W (xs ) ≥ 0, and the resulting profit isBg b + Sg s + Π(pm , f, K)ss= −Bλb β + S(1 − e−x )(ps − f s ) + S(1 − e−x )f + min{K, xm }pm< −Bλb β + Sxs (f b + ps ) + xm pm≤ −Bλb β + (Sxs + xm ) max{f b + ps , pm }≤ −Bλb β + B = Π
for all f b + ps ≤ 1 and pm ≤ 1. Hence, this is not profitable either. All in all, nodeviation is profitable for h = Db Cs .
Consider next h = Ds Cb . Then, by (74) and (75), g s = −λs (1−β) and g b = V . Whenf > 1, no one go to the platform xs = 0 and all buyers are in the middleman sector
xm = B as long as pm ≤ 1. This yields g b = V = V m (B) ≥ 0 and Π(pm , f, B) = Bpm
with K = B. The profits areΠ = −Sλs (1 − β) + B(1 − pm ) + Π(pm , f, K) = −Sλs (1 − β) + B.
To show that this is indeed the maximum profit, we have to check two possible cases.
m = 0, and g b = V =Suppose f = f b + f s ≤ 1 and K = 0. Then, xs = B
S and xV s (B/S) ≥ 0, if there is a non-negative surplus in the platform for buyers, f b +ps ≤ 1,
and for sellers, f s ≤ ps . This leads to
B

s

b

m

s

Sg + Bg + Π(p , f, 0) = −Sλ (1 − β) + B

1 − e− SB
SB= −Sλs (1 − β) + S(1 − e− S )(1 − ps + f s )< −Sλs (1 − β) + B = Π
for all f s ≤ ps . Hence, this is not profitable.
89B(1 − ps − f b ) + S(1 − e− S )f

Suppose f = f b + f s ≤ 1 and K ∈ (0, B], and both sectors have a non-negative
surplus to buyers, i.e., pm ≤ 1 and f b +ps ≤ 1. This leads to xm ∈ (0, B) and xs ∈ (0, B
S)
that satisfy the add-up constraint (4), Sxs + xm = B, and the indifferent condition(6), V s (xs ) = V m (xm ). Then, g b = V = V s (xs ), and the resulting profit isSg s + Bg b + Π(pm , f, K)s
1 − e−xss(1 − ps − f b ) + S(1 − e−x )f + min{K, xm }pm .
= −Sλ (1 − β) + B
xs
There are two cases. Suppose K ≥ xm . Then, the indifferent condition (6) implies
thatspm = 1 −1 − e−x(1 − ps − f b ).
xs

Applying this expression to the profits, we get
Sg s + Bg b + Π(pm , f, K)ss
1 − e−x1 − e−x
s
b
−xs
m
s
b
s
(1 − p − f ) + S(1 − e )f + x
1−(1 − p − f )= −Sλ (1 − β) + Bxsxss
1 − e−xs= −Sλs (1 − β) + (B − xm )(1 − ps − f b ) + S(1 − e−x )f + xmxss
= −Sλs (1 − β) + S(1 − e−x )(1 − ps + f s ) + xm
< −Sλs (1 − β) + B
for all f s ≤ ps . Suppose K < xm . Then, the indifferent condition implies thatspm

xm 1 − e−x(1 − ps − f b ).
=1−
K
xs

Applying this expression to the profits, we get
Sg s + Bg b + Π(pm , f, K)ss
1 − e−xxm 1 − e−x
s
s
b
−xs
s
b
= −Sλ (1 − β) + B(1 − p − f ) + S(1 − e )f + K 1 −(1 − p − f )
xs
K
xs
−xs
s
s
m 1−e
(1 − ps − f b ) + S(1 − e−x )f + K= −Sλ (1 − β) + (B − x )sxs= −Sλs (1 − β) + S(1 − e−x )(1 − ps + f s ) + K< −Sλs (1 − β) + B
for all f s ≤ ps . Hence, any deviation is not profitable for h = Ds Cb .
Finally, since the intermediary makes the maximum revenue B for either h,which side should be subsidized is determined by the required costs: noting Bλb =Sλs , we have Bλb β R Sλs (1 − β) ⇐⇒ β R 21 . This completes the proof of Proposition
13. 

90

Multi-market search: Under multiple-market search, any non-positive registration fee ensures that agents are in the C market, since participation to the C market
is not exclusive. Hence, attracting one side of the market becomes less costly. By contrast, conquering the other side becomes more costly, since the conquered side stillholds the trading opportunity in the D market. The Ds Cb condition with multiplemarket search is
Ds : − g s ≥ 0,s

Cb : max {V s (xs ), V m (xm )} − g b ≥ λb e−x β (1 − c) .
The divide-condition Ds tells that now a non-positive fee is sufficient to convince
one side to participate. The conquer-condition Cb now needs to compensate for theoutside option in the D market. Similarly, the Db Cs condition becomes
Db : − g b ≥ 0,
Cs : W (xs ) − g s ≥ λs ξ (xs , xm ) (1 − β) (1 − c) .
Participation fees are designed to induce buyers and sellers’ participation. Once
agents join the C market, the participation fees become sunk costs, and will not
influence their trading decision.
The intermediary’s problem of choosing F = b s b s
f ,f ,g ,gtogether with h ={Db Cs , Ds Cb } and pm , K ∈ [0, B] are described asΠ ≡ max

b

s

m



Bg + Sg + max
Π(p , f, K) ,
m
p

F,h,K

(76)

s

where Π(pm , f, K) = S(1 − e−x )f + min {K, xm } pm − Kc. Besides the divide-andconquer constraints, this maximization problem is also subject to the incentive constraints as described in the main text.

Proposition 14 In the extended problem described in (76) with unobservable capacity, participation fees and multiple-market search, the determination of the profitmaximizing intermediation mode is identical to the one described in Proposition 2,
with g i = 0, i = s, b.
Proof. It suffices to prove that the solution is g i = 0, i = s, b for each intermediation mode, since then (76) becomes identical to the problem we have already solved in
the main text. For a pure middleman mode (xm = B), the intermediary sets g b = 0 to
divide buyers, with pm = 1−λb β(1−c) satisfying (17). For a pure market-maker mode91

(xs = 0), either with Db Cs or Ds Cb , the intermediary sets the transaction fee to satisfy

the binding incentive constraint (25), f = v(0, 0) = 1 − λb e−B/S − λs ξ(0, 0) (1 − c),and g b = g s = 0.
For a hybrid mode, the intermediary’s problem is subject to the incentive constraint (25), and pm satisfying (29) such that buyers are indifferent between the twomodes. We can rewrite the maximization problem (76) as a two-stage problem overa vector X ≡ (xm , f, K) ∈ X, where X ≡ [0, B] × [0, 1] × [0, K]:noStage 1: max Bg b (X) + Sg s (X) + Π(xm (f, K), f, K)

(A)

(f,K)

s.t. 0 ≤ f ≤ v(xm (f, K), K), 0 ≤ K ≤ B.
Π(xm , f, K)
Stage 2: maxmxs.t. f ≤ v(xm , K), 0 ≤ xm ≤ B,
where g b (X) and g s (X) are given by the binding divide-and-conquer conditions,sg b (X) = 0, g s (X) = 1 − e−x − xs e−xs(v (xm , K) − f ) ,if h = Db Cs , orsg s (X) = 0, g b (X) = e−x (v (xm , K) − f ) .
if h = Ds Cb . As our objective is to prove g i (X) = 0, i = s, b, all that remains to beshown here is that f = v(xm , K) at the candidate equilibrium solution. This follows
however from the fact that the objective function in (A) is strictly increasing in f
and any change in f (< v(xm , K)) does not influence the other constraints. Hence,
as in the original problem, we must have f = v(xm , K). This completes the proof of
Proposition 14. 

92

DWeb Appendix: Empirical Appendix

Data and Variables. From www.amazon.com and www.ebay.com, we retrieve all
paginated results listed in the category of Amazon called: “All Pans”, which is asubcategory of “Home & Kitchen: Kitchen & Dining: Cookware”. This subcategory
includes 400 pages of more than 9000 products as of August 2018.40 For each pan,we obtain the price (price), the sales rank in the category “Home & Kitchen” (rank),
the listing days since the first listed date on Amazon by either Amazon or some othersellers on Amazon’s market-maker platform (listedDays), the number of third-party
sellers that sell this product on Amazon (sellersAmazon), whether the product is soldby Amazon itself (sellByAmazon) and the title of the product.
Sellers could offer various prices for a product on Amazon. We obtain price information from the default page Amazon displays when users search for a product.
This gives us the price at which the majority of transactions are processed. Amazon
does not publish sales data but does provide a sales ranking for each product. Since
ranking information is provided at different levels of categories, in order to makethe sales ranking as comparable as possible, we adopt the ranking at the highest
possible level “Home & Kitchen”. This gives us the variable rank.
The title of the product is used to link each product on Amazon to the outside
option available at eBay as the theory develops. For each product collected on Amazon, we search its “Amazon product name” on eBay to obtain all related offers. As aproxy for the buyers’ matching probability in the decentralized market λb , we countthe number of all the offers shown in eBay’s raw search result. We call this variable
sellersEbayAll. Admittedly, this is a very noisy measure. EBay tends to provide along list with offers that are only loosely related to the product. For example, insome cases a pan offered on eBay only matches with some key features of a panoffered on Amazon such as size but it does not match other features such as materials. In this case, we compare the similarity between the eBay product title andthe Amazon product title. In some other cases, the titles are similar but the products turn out to be different. For example, searching a pan on eBay only yields anoffer of the lid of the same pan on Amazon. To solve this issue, we use the following
two-step procedure. We first select offers with product names similar to the Amazon
product name.41 We then refine the list by restricting the offer price between 0.5
40

The URL of the list of all pans is https://www.amazon.com/pans/b?node=3737221 (visited on
August 24, 2018).
41
Here, we use the Fuzzy String Matching Library in Python which computes a score between 0 and
100, with 100 indicating the exact matching. The function fuzz.token set ratio() computes thescore and only selects offers with a score higher than 80. We also tried other criteria scores such as 60
and 90. The results are robust.

93and 1.5 times the Amazon price. The rationale for this procedure is that if the offer
price is far away from the Amazon offer, the product is likely to vary in quality or
could even be a distinct product. Counting the number of sellers in this refined list
leads to another proxy for λb , sellersEbayRefined. This is a more precise measure forthe relevant number of sellers on eBay. We will use sellersEbayRefined in our main
regressions, and use sellersEbayAll as a robustness check.
As an alternative proxy for λb , we could use the number of sellers on eBay relative
to that of Amazon,42
sellersEbayRelative =sellersEbayRefined.
sellersAmazon

This measure proxies the relative success probability of meeting a seller on eBay versus Amazon. It is constructed based on a typical buyer’s online shopping experience.
When a buyer discovers dozens of sellers on Amazon, it is relatively less likely that
he can find even better offers outside Amazon, so the perceived outside value of going
to eBay is low. In contrast, for a buyer who observes only few sellers on Amazon, theexpected payoff of searching on the outside market is high. sellersEbayRelative istherefore likely to be positively correlated with λb .
As a proxy for buyers’ bargaining power in the outside market, β, we compute theprice difference between eBay and Amazon. For each product, we find the median
price in the refined eBay offer list and compute the log of this price minus the logof Amazon price. This defines the variable priceDiff. We expect this variable to be
negatively correlated with β (recall that a higher β implies a larger share of thesurplus for the buyer so a lower price in the decentralized market).
Descriptive Analysis. We collected information for 9066 products on Amazon andfound matched eBay offers for 7944 of them. Variables may have missing values
leading to a smaller sample size. For example, ranking information might be provided not in the aggregate category “Home & Kitchen” but in some subcategories
with incomplete ranking. We did try different (sub)categories to extract ranking information, and it turned out that “Home & Kitchen” gave us the largest valid sample
with 7942 observations. In the regressions below, we exclude products without any
matched eBay offers to avoid missing priceDiff, and exclude products without any
third-party sellers on Amazon to avoid missing sellersEbayRelative. Finally, we only
collect offers for brand new products.
42

We use the number of third-party sellers (that is excluding Amazon if Amazon sells) in the denominator.

94

Table 2: Summary Statistics
Variables

Obs.

Mean

Std.

Min.

Max.

sellByAmazon
listedDays
price
rank
sellersAmazon
sellersEbayAll
sellersEbayRefinedsellersEbayRelative
priceDiff
sellersEbayAll 60
sellersEbayRefined 60
sellersEbayRelative 60
priceDiff 60

9066
8168
8856
7942
9066
9066
9066
8487
7944
9066
9066
8487
8349

0.32
1759.5464.03
440711
3.70
14.69
6.53
3.30
0.07
20.88
10.54
5.59
0.07

0.46
1458.12107.00314288
5.36
14.13
8.05
5.35
0.74
15.87
10.53
7.69
0.72

0.00
8.00
0.01
28
0.00
0.00
0.00
0.00
-5.08
0.00
0.00
0.00
-5.08

1.00
6864.00
2118.83
2581111
77.00
60.00
43.00
43.00
6.86
62.00
48.00
44.00
6.66

Note: The table reports summary sample statistics for the merged scraped data fromwww.amazon.com and www.ebay.com. The last four variables sellersEbayAll 60, sellersEbayRefined 60, sellersEbayRelative 60, priceDiff 60 are defined on a dataset constructed by searching only the first 60 characters of Amazon product title in eBay’s
search engine. They are used in robustness checks. Finally, we calculate the statistics of each variable with all valid observations in the dataset.

Table 2 presents summary statistics for our main variables of interest. For 32% of
the products in our sample, Amazon acts as a middleman; for the other 68% products,
Amazon acts as a pure platform. On average, the products have been on sale atAmazon for almost 5 years, although this varies across products from several days
to 18 years. There is a large variation in the price and ranking. The maximum priceis as high as $2118.83. The mean price is $64, the 25th percentile is $18.9, the 75th
percentile is $72.9. The number of third-party sellers for a product ranges from 0 to 77
with a mean of 3.7 sellers. The number of sellers on eBay is much larger with a meanof 14.69 for sellersEbayAll and a mean of 6.53 for sellersEbayRefined. On average, thenumber of sellers on eBay is more than three times as high as the number of sellers
on Amazon. Finally, variables with suffix 60 come from another dataset constructed
for robustness checks and will be discussed later. In table 3, the linear correlations
among proxies are very weak. Correlations among proxies for different parameters
are around 0.1.
Robustness Checks. We shall pursue a number of robustness checks. A first concern is that our result could be driven by the way that we count the number of eBay
offers. To address this issue, instead of refining the list of eBay offers, we use theraw list of eBay offers to calculate the number of sellers, sellersEbayAll, and replace95

Table 3: Correlations among proxy variables
logRank

priceDiff

sellerEbay
RefinedsellerEbay
All

logRank

1.0000

priceDiff-0.1359

1.0000sellersEbay
RefinedsellersEbay
AllsellersEbay
Relative-0.1462-0.0901

1.0000-0.0595-0.0797

0.70631.00000.06839-0.1203

0.67900.4866sellerEbay
Relative

1.0000sellersEbayRelative by sellersEbayAll/sellersAmazon. Our results are robust to thischange as shown in Table 4: Although the coefficients of sellersEbayAll and sellersEbayRelative become smaller, they remain negative. The coefficient of sellersEbayAll
now becomes non-significant, while sellersEbayRelative is still statistically significant. Relative to the result summarized in Table 1, the coefficients of the othervariables remain almost the same.
A second concern is a bias caused by using the eBay search engine. We find thatthe number of offers provided by the eBay search engine is negatively correlatedwith the length of search text. In general, the longer the search text is, the fewer
results that the eBay search engine can provide. Hence, the longer the product nameis, the less likely that eBay can find good matches in its database. This implies that
we may ignore good matches if we provide a very long product name with too muchinformation. For example, the same product may have different product titles bydifferent sellers emphasizing different product features, such as size and color of thepan. In some cases, eBay can not give any offer when searching the whole Amazon product title, but does give the right offers when searching part of the Amazon
product title. More importantly, there exists anecdotal evidence showing that the
product title on Amazon is longer if it is registered by Amazon itself rather than by
third-party sellers. If this is true, we may have spurious correlations. To solve thisissue, we construct a second dataset by searching all product names using only the
first 60 characters on eBay.43
The variables sellersEbayAll 60, sellersEbayRefined 60, sellersEbayRelative 60
and priceDiff 60 are constructed in this new dataset. As shown in the last four rows
43
In our data, the median length of product title is 65, the minimum is 9, and the maximum is 245.
We also tried to cut the first 50 or 80 characters. The results are similar.

96

Table 4: Regressions for Amazon’s intermediation mode using the raw eBaysearch results(1)LinearsellersEbayRelative(2)
Linear

-0.00354∗∗∗(0.000428)(3)
Probit(4)
Probit-0.00478∗∗∗(0.000613)

sellersEbayAll-0.000226(0.000382)-0.000439(0.000426)

sellersAmazon-0.000691(0.000915)-0.000765(0.000996)

log(rank)-0.101∗∗∗(0.00444)-0.106∗∗∗(0.00461)-0.102∗∗∗(0.00500)-0.108∗∗∗(0.00513)

priceDiff

0.100∗∗∗(0.00844)

0.112∗∗∗(0.00847)

0.122∗∗∗(0.0111)0.136∗∗∗(0.0112)log(price)

0.0346∗∗∗(0.00614)

0.0402∗∗∗(0.00622)

0.0440∗∗∗(0.00689)

0.0503∗∗∗(0.00698)

listedDays

0.0606∗∗∗(0.00479)

0.0664∗∗∗(0.00485)0.0719∗∗∗(0.00596)

0.0788∗∗∗(0.00604)

Observations
Adjusted R2

6457
0.135

6457
0.129

6457

6457

Note: This table reports the robustness check using the raw eBay search results
reflected in sellersEbayAll and sellersEbayRelative. Except for this change, thespecification is the same as before.

97

in the summary statistics Table 2, the average number of sellers for each product becomes larger. For example, in terms of the length of the raw search list, the averageincreases from 14.69 to 20.88. However, the relative prices between eBay and Amazon do not change much. The results using this new dataset are reported in Table 5
and yield similar relationships as our main ones. There are more observations in the
regressions because some Amazon product titles which had no eBay offer before can
now be matched. As in our previous regressions, the coefficients of other variables
remain almost unchanged.
Table 5: Regressions for Amazon’s intermediation mode using first 60 characters to search eBay offers(1)
LinearsellersEbayRelative(2)
Linear

-0.00450∗∗∗(0.000595)(3)
Probit(4)
Probit-0.00488∗∗∗(0.000816)

sellersEbayRefined-0.000352(0.000499)-0.000170(0.000556)

sellersAmazon-0.00587∗∗∗(0.000847)-0.00708∗∗∗(0.00112)log(rank)-0.101∗∗∗(0.00423)-0.0950∗∗∗(0.00460)-0.102∗∗∗(0.00468)-0.0984∗∗∗(0.00515)

priceDiff

0.114∗∗∗(0.00819)

0.126∗∗∗(0.00849)

0.134∗∗∗(0.0105)

0.146∗∗∗(0.0106)log(price)

0.0431∗∗∗(0.00602)

0.0503∗∗∗(0.00614)

0.0529∗∗∗(0.00658)

0.0586∗∗∗(0.00667)

listedDays

0.0611∗∗∗(0.00449)

0.0646∗∗∗(0.00480)

0.0741∗∗∗(0.00566)

0.0747∗∗∗(0.00576)

Observations
Adjusted R2

6822
0.138

6822
0.100

6822

6822

Note: This table reports the robustness check based on eBay search results using only the first 60 characters of the Amazon product title. Except for using new
variable reflecting this change, sellersEbayAll 60, sellersEbayRefined 60, sellersEbayRelative 60 and priceDiff 60, the specification remains the same as before.

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