sourcing through intermediaries

Sourcing Through Intermediaries

_______________

Elena BELAVINA Karan GIROTRA 2010/82/TOM

Sourcing Through Intermediaries

Elena Belavina*

Karan Girotra**

October 2010

* PhD Candidate in Technology and Operations Management at INSEAD, Boulevard de Constance 77305 Fontainebleau Cedex Ph: 33 (0)1 60 72 92 23 Email: [email protected]

** Assistant Professor of Technology and Operations management at INSEAD, Boulevard de

Constance 77305 Fontainebleau Cedex Ph: 33 (0)1 60 72 91 19 Email: [email protected]

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SOURCING THROUGH INTERMEDIARIES

Abstract. We study firms that repeatedly source products or services from external suppliers.

Due to changes in exchange rates, tariffs, transportation costs, etc., the lowest-cost supplier changes

frequently. Further, sourcing creates opportunities for self-interested behavior that reduces supply

chain profits– this “decentralization inefficiency” could arise due to non-contractible actions such as

investments in inventory or capacity, following environmental or social norms, leakage of proprietary

information, unobservable efforts, etc. Inspired by the phenomenal rise of sourcing intermediaries

such as Li & Fung Ltd., we compare two sourcing structures in this context: 1) Direct Sourcing,

where firms deal with the suppliers directly and, 2) Mediated Sourcing, where an intermediary

consolidates demand from multiple firms and sources for each of these firms. We model each

structure as an infinitely repeated dynamic game with random states. Our analysis illustrates that

sourcing through an intermediary reduces the supplier-side incentives for self-interested, inefficient

behavior by limiting the immediate gains from such behavior and by providing more long-term

sourcing business in its absence. Each of these advantages arises out of the intermediary’s cost-

minimizing allocation of business among suppliers that is better distributed or more egalitarian than

that of any individual buyer. This implies that it is more lucrative for an intermediary rather than a

direct buyer to provide sufficient levels of business to more suppliers over the long-run and to limit

exposure in any sourcing period to a suppliers’ self-interested behavior. Our analysis illustrates

that procurement through intermediaries is most beneficial when there is an intermediate level of

inefficiency and the suppliers are, on average, equally preferred or when the inefficiency is high and

one supplier has an advantage over the other supplier.

1. Introduction

Over the last two decades, reduction in trade barriers, improved communication technologies, dif-

ferences in input costs between geographies, and geographical skill specialization have led firms

to employ increasing levels of sourcing of products and services (Boston Consulting, 2009). This

magnifies two sourcing challenges. First, distributed supply chains expose firms to uncertainty in

currency exchange rates, tariffs on trade, transportation, telecommunication costs, input prices, sup-

ply disruptions and other risks from changes in the business environment of the firm (cf. Kouvelis

Date: October 2010. This is the first version of this paper.Key words and phrases. Global Sourcing, Intermediaries, Relational Contracts, Supply Chain Relationships, Flexi-bility, Contracting, Li and Fung, Supply Chain Management.

1

2 SOURCING THROUGH INTERMEDIARIES

et al., 2004). In particular, the lowest-cost supplier may change frequently and firms must remain

responsive in the face of these changes (Goyal and Netessine, 2010). Second, outsourcing leads to a

reduced ability to monitor, contract and control the actions of suppliers. This exposes the firm to

self-interested or opportunistic behavior by their suppliers. For instance, non-verifiable supplier ac-

tions lead to non-conformance with environmental and social norms, inadequate upstream capacity

investments, insufficient inventory levels, poor quality, leakage of proprietary information and higher

supply chain costs of capital. These lead to a "decentralization inefficiency" from outsourcing.

In the face of these risks, firms offering sourcing services, sourcing intermediaries, have grown

rapidly. These intermediaries partner with multiple firms, often competitors from the same industry,

to manage their sourcing processes. They often completely take over the sourcing function; they

select, verify and approve suppliers. They allocate business between different suppliers and manage

the relationship, including providing incentives for good performance, provisioning credit, etc.

A notable sourcing intermediary is Li & Fung Ltd., which provides sourcing services to major

brands and retailers worldwide, including Walmart, Target, Zara, Marks & Spencer Plc, Levis

and Philip Morris (McFarlan et al., 2007; Magretta, 1998; Cheng, 2010). Li & Fung Limited has

grown at a compounded annual rate of 23 percent for the last 14 years. As of 2009, the company’s

annual turnover exceeded US$ 14 billion. The group operates over 80 offices covering over 40

economies across North America, Europe and Asia, with an ever-growing global sourcing network

of nearly 12,000 international suppliers and over 1,000 buyers worldwide. Yet, Li & Fung does not

own any means of production, nor is it in the business of directly retailing the vast majority of

products it sources. Essentially, it provides only an interface between multiple buyers and suppliers.

Olam International is another fast-growing sourcing intermediary that focuses on the sourcing of

agricultural products (Bell and Shelman, 2009).

Li & Fung’s customers explain their decision to work with the company as resulting from its superior

abilities to lock up ample supplier capacity for orders, to ensure quality and timely delivery of

orders, and to ensure compliance with environmental and social norms (Loveman and O’Connell,

1995). Many of Li & Fung’s customers operate in highly competitive industries with low margins

and must source products at the lowest prices to stay competitive. These sourcing prices often

depend on a multitude of ever-changing trade restrictions that regulate imports from different

supplying countries, exchange rates, transportation costs, etc. This makes it imperative for these

firms to retain flexibility in their choice of suppliers. William Fung, Li & Fung’s managing director,

SOURCING THROUGH INTERMEDIARIES 3

explains: “These customers must switch suppliers if the price variation is 10% to 15%. [...] You

need to be constantly roaming the market and looking for better prices” (Loveman and O’Connell,

1995). Li & Fung’s customers claim that the size of Li & Fung’s supplier network allows the firm to

switch production sites quickly when the economics of purchase transactions change. They further

state that working with Li and Fung provides them with a competitive advantage (Loveman and

O’Connell, 1995) in sourcing from lower-cost suppliers more often.

This paper develops a stylized model to understand the advantages and disadvantages of sourcing

through intermediaries such as Li & Fung Ltd. Specifically, we model two buyers who repeatedly

source products from two suppliers. Due to changes in the business environment, the cheapest

supplier for any buyer may change frequently. Further, there are decentralization inefficiencies

associated with the outsourcing exercise. Put differently, the sourcing relationship is most valuable

to buyers and suppliers if they partner and work to the joint interest of the supply chain, rather

than behaving opportunistically in their self-interest. For instance, supply chain costs may be lower

if the buyer pays the supplier in advance and the supplier then adequately meets the buyer’s needs,

including non-contractible dimensions. However, knowing that the supplier’s performance incentives

will be diminished with advance payment, the buyer does not do so, and the supplier often must

use expensive financing. Part of these financing costs are then passed on to the buyer, making both

the buyer and supplier worse-off.

In this setup we compare two supply chain structures: 1) Direct Sourcing, where the firm selects

and manages the suppliers directly, in line with traditional relationship-based sourcing practices at

firms like Toyota (Nishiguchi and Beaudet, 1998), IKEA (Baraldi, 2008), etc. 2) Mediated Sourcing,

where the firm delegates choosing the supplier and managing the relationship to a third party, a

sourcing intermediary like Li & Fung Ltd., that consolidates demand from multiple buyers. We

model each structure as an infinitely repeated dynamic game with random states. Our analysis

illustrates that the intermediary’s consolidation of demand from multiple buyers leads to three

key phenomena that differentiate direct and mediated sourcing. First, the consolidated demand

from multiple buyers leads to preferences over suppliers such that mediated sourcing profits are

maximized at a more egalitarian allocation of business between suppliers than those from direct

sourcing. In other words, an intermediary is better off than a direct buyer with a more distributed

allocation of business. This increases the number of suppliers that receive substantial business

over the long-run from this sourcing exercise. Consequently, there is a larger group of suppliers


that can be incentivized to desist from self-interested behavior using inter-temporal incentives,

thereby reducing the decentralization inefficiency while still allowing for sourcing from the lower-

cost supplier. Second, the larger consolidated demand volume that flows through the intermediary

allows it to better distribute the demand in any one sourcing period across multiple suppliers, which

limits its exposure to self-interested behavior by any individual supplier. Due to the larger demand

volume, the intermediary can do this while still sourcing more often from the cheaper suppliers.

Again, this implies that mediated sourcing prevents self-interested behavior from a larger group of

suppliers, while simultaneously allowing for more flexibility in sourcing from the cheapest supplier.

Finally, direct sourcing has an advantage– direct sourcing buyers with very strong preferences over

suppliers can focus on incentivizing the favored supplier, while the intermediary’s consolidated

preferences attenuate any individual buyer’s preferences and thus it can be more difficult for it1 to

incentivize an individual supplier.

Taken together, our analysis suggests that in procurement situations where both the losses from

the decentralization inefficiency and the impact of cost changes are significant and comparable,

mediated sourcing improves on direct sourcing. The benefits are most substantial when there is an

intermediate level of inefficiency and the suppliers are equally preferred, or when the inefficiency

is high and one supplier has a competitive advantage over the other supplier. We provide this

analysis in a very general setup, where we allow for any kind of buyer-supplier interactions that

can be captured by a finite extensive-form game. Further, beyond the possibility of inefficient,

self-interested behavior, we make no assumptions about the form or structure of the buyer actions,

the supplier actions, the profit functions, the distribution of cost advantages, etc.

Our comparison of direct and mediated sourcing makes three key contributions: First, we identify

and illustrate the advantages and disadvantages of sourcing through intermediaries. We use this

understanding to develop actionable managerial prescriptions on appropriate sourcing strategies

for buyers. Second, to the best of our knowledge, this is the first paper to systematically model

and study the business practices of a fast-growing and important segment of the logistics industry,

sourcing intermediaries. Our setup and analysis provide a foundation to investigate this increasingly

important segment. Third, our analysis contributes to the sourcing and procurement literature by

bringing together the largely parallel literatures on operational flexibility (cf. Goyal and Netessine,

2010) and relational contracting (cf. Taylor and Plambeck, 2007a). The operational flexibility

1We use masculine pronouns for buyers, feminine pronouns for suppliers and impersonal pronouns for the intermediary.


literature studies flexible sourcing strategies that allow for a free change of suppliers in response

to changes in the business environment (cf. Li and Debo, 2009 and, Lu and Van Mieghem, 2009).

On the other hand, to limit self-interested behavior by partners, an opposite strategy of promising

future business to a limited and fixed set of preferred suppliers, is recommended in the relational

contracting literature (cf. Taylor and Plambeck, 2007b). Tunca and Zenios (2006) study the trade-

off between these two. Our analysis captures the key elements of both setups and we identify

how mediated sourcing breaks the trade-off between the competing prescriptions of flexibility and

commitment from these two literatures. Finally, our results also challenge the conventional wisdom

from existing theory pertaining to the disadvantages of adding additional tiers in a supply chain

(Cachon and Lariviere, 2005).

The rest of the paper is organized as follows. In Section 2, we provide a brief overview of the relevant

literature. In Section 3, we set up the mathematical model that captures this operating environment,

and we formally define and analyze the two sourcing structures– direct and mediated sourcing. In

Section 4, we compare mediated sourcing and direct sourcing, we isolate the key differences between

direct and mediated sourcing, and we identify conditions where mediated sourcing outperforms

direct sourcing and vice versa. We provide managerial prescriptions and discuss the robustness of

our results in Section 5.

2. Literature Review

Strategies for sourcing have been a central focus of recent research in supply chain management.

An established stream of literature has identified maintaining a large flexible network of suppliers

as a strategy to manage changing preferences over suppliers. A largely distinct stream of literature

has dealt with managing the decentralization inefficiencies in outsourcing that arise out of non-

contractible actions. Long-term or relational contracts with a restricted set of suppliers are suggested

to eliminate these inefficiencies. Finally, more recent work studies the trade-off between these two

competing recommendations. For each of the three categories, all existing work has studied these

issues exclusively in the context of direct sourcing. To the best of our knowledge, this is the first

paper to study mediated sourcing in these situations. Next, we provide a brief summary of the

literature on flexible sourcing, long-term contracting and the trade-off between the two.


Studies on Flexible Sourcing. Sourcing when preferences over suppliers change due to changes in

the business environment is an active area of research. The main suggested strategy is to develop

and appropriately source from multiple distributed suppliers. Kouvelis et al. (2004) demonstrate

the pervasive exposure of global sourcing firms to risks arising out of subsidized financing, tariffs,

regional trade rules and taxation. Kouvelis et al. (2001) and Ding et al. (2007) develop strategies

for a firm that faces uncertainty in currency exchange rates. Lu and Van Mieghem (2009) study the

choice between sole and dual sourcing strategies and consider the influence of changing logistics costs

and foreign trade barriers on configuring global networks that best combat multi-market demand

uncertainty. Flexible sourcing strategies are also suggested for supply chain resilience and managing

supply disruption risk (cf. Wang et al. 2010a and Yang et al. 2009). Bakshi and Kleindorfer (2009)

propose contracting solutions for settings where cooperative investments are necessary for preventing

supply disruptions. Wang et al. (2010b) study alternate sourcing strategies in the face of changing

trade regulations. Finally, Tomlin (2006) and Chod et al. (2010) examine the value of these flexible

sourcing strategies under different contingencies.

Studies on Relational Contracting. This literature addresses inefficiencies that arise due to profit-

relevant non-contractible actions of sourcing partners. There is a growing body of operations lit-

erature that highlights the use of informal agreements (relational contracts) as a remedy to these

inefficiencies. Taylor and Plambeck (2007a,b) study settings where price and capacity are non-

contractible and are the primary sources of inefficiency. Debo and Sun (2004) study a setup where

inventory levels are non-contractible and where building long-term relationships can mitigate the

traditional double marginalization losses. Plambeck and Taylor (2006) study joint production with

unobservable utility-relevant actions and a non-contractible output yield. Ren et al. (2010) consider

forecast sharing by a buyer in a setup where he has an incentive to inflate the forecasts. In all these

settings, the buyers and suppliers have non-contractible actions that can significantly impact the

profit of the other party. Further, firms can derive immediate benefits from opportunistic behavior.

This leads to a decentralization inefficiency. In each of these papers, building long-term relationships

is presented as a mechanism for providing inter-temporal incentives that mitigate myopic oppor-

tunistic behavior. In line with this literature, the engagement step game of our model (introduced

in Section 3.1) captures these non-contractible aspects of sourcing interactions. In fact, rather than

model any of the specific non-contractible actions studied in this literature, we consider a generic

game which captures key elements of each of the above settings.


Trade-off between Flexible Sourcing and Relational Contracting. Flexible sourcing and relational

contracting are competing strategies. Since many firms are simultaneously exposed to changing

preferences over suppliers and decentralization/outsourcing inefficiencies, the trade-off between the

two sources of inefficiencies and the consequent optimal strategy is of significant interest. Tunca and

Zenios (2006) consider the trade-off between relational contracts and flexible procurement auctions

in a setting with multiple buyers and sellers. Relational contracts allow the firm to overcome

opportunistic behavior and to guarantee the supply of quality parts, while auctions allow for an

increase in supply chain efficiency by choosing the least costly supplier. Tunca and Zenios (2006)

demonstrate that depending on the economics of the product in question, the relational contract

or the procurement auction may be the optimal sourcing strategy. Swinney and Netessine (2009)

look at the same trade-off when there is a possibility of supplier bankruptcy or default. Li and

Debo (2005, 2009) illustrate the long-term shortcomings and benefits of committing to source from

a single supplier when future sourcing options may change.

Our study continues in the tradition of examining the trade-off between relational contracts and

flexible sourcing, and demonstrates the utility of mediated sourcing in relieving this trade-off. While

sourcing intermediaries have never before been studied in the context of this trade-off, there is

an active debate between mediation and disintermediation in the economics and financial market

literature. Wu (2004) provides a summary of the functions that an intermediary may provide and

synthesizes the debate from a supply chain perspective. Finally, while most of the literature cited

above pertains to supply chains for physical goods, the sourcing issues described above arise equally

in the outsourcing of services (Ren and Zhang, 2009; Roels et al., 2010).

To summarize, this paper contributes to and extends the literature that has studied the trade-off

between flexible sourcing and relational contracts. In particular, to the best of our knowledge, this

is the first paper that systematically studies the role of mediated sourcing in relieving this trade-off.

3. Model Setup, Direct and Mediated Sourcing

3.1. Model Preliminaries. Consider two buyers, b1 and b2, that repeatedly source products or

services available from two potential suppliers, s1 and s2. Each supplier has ample capacity to

supply one or both buyers. Buyers and suppliers discount future profits with a discount factor δ,

which captures the time value of money and the probability of exit of the buyers and suppliers from

the market.


SUPPLIER SELECTION INFORMATION GATHERING

Relative Cost Advantage is revealed ex. Exchange rates, transportation costs, tariffs, input prices, etc. 𝑋𝑖~𝐹 𝑥 is drawn

ENGAGEMENT

Supplier Actions, 𝑎𝑠 ∈ 𝐴𝑠 (ex. Quality, confidentiality, social norms)

Buyer Actions, 𝑎𝑏 ∈ 𝐴𝑏 (ex. Timely payments, information sharing, rewards, etc.)

𝑡 𝑡 + 1

STAGE GAME

Figure 3.1. The Stage Game

We model the repeated trade between these buyers and suppliers as an infinitely repeated game– in

each stage game of this repeated game, both buyers source the product. In particular, the sourcing

proceeds in three steps (Figure 3.1). First is the Information Gathering step, where the differences

in the costs of sourcing from the two suppliers are revealed. Second is the Supplier Selection step,

where a supplier is selected for each buyer. Finally, the product or service is actually sourced in

the Engagement step. These three steps are repeated in every period t ∈ 0, 1, .... We describe the

three steps in detail below.

The Information Gathering Step. In this step, the current lower-cost supplier for each buyer is

revealed. The cost advantage of a supplier in supplying a buyer, could be different for the two buyers

and it changes from one sourcing period to another due to changes in exchange rates, transportation

or telecommunication costs, cross-border tariffs, pass-through input costs, etc. So, for any buyer

i, at times, supplier i is the lower-cost supplier, while at times it is the other supplier, denoted as

i. In particular, the profits of buyer i, i ∈ 1, 2 if he sources from supplier i, include an additive

component, Xti , drawn from a probability distribution with cdf, F (x) that has support on the

interval [−x, x]; x, x > 0. In every sourcing period, t, Xti is independently and publicly drawn for

the two buyers. For buyer i, Xti , captures the relative cost advantage of one or the other supplier–

if the current realization of Xti is negative, supplier i is the lower-cost supplier for buyer i, else it is

supplier i.

The Supplier Selection Step. One supplier is selected for each buyer. We denote buyer bi’s

selected supplier as zi, zi ∈ 1, 2.

The Engagement Step. The actual sourcing of the product or service takes place in this step.

Each buyer and the corresponding selected supplier undertake actions that influence the profits of

their sourcing partner. On the supplier side, these could include efforts in ensuring quality, ensuring


timely delivery, conforming to technical and labor standards, following environmental and social

norms, maintaining confidentiality of proprietary information, providing prompt after-sales service

and support, etc. On the buyer’s side, these could include timely payments, access to new business

opportunities, cross-investments, access to capital, training, technology transfer, accurate sharing

of demand information, recommendations, rewards, sanctions, etc.

We model these interactions between the buyer and the selected supplier in the engagement step as

a completely general finite extensive-form two-player game, Γ, that can capture each of the above

mentioned actions. In game Γ, the buyer and supplier’s feasible actions are denoted as Ab, As⊂ Rk.

The set of feasible action profiles is then given by A ≡ As×Ab. Each element of set A, a, describes

the actions undertaken by both the buyer and supplier in this game. On completion of game Γ, the

action profile a is perfectly and publicly observable. Buyer and supplier profits are given by general

profit functions ub, us: A → R. We denote the Nash equilibrium of game Γ as aN ∈ A, associated

with actions corresponding to “opportunistic behavior ”, and we assume that it is unique and the

payoff associated with it is inefficient. In particular, there exists an efficient outcome aC ∈ A,

associated with “cooperative behavior ”, that makes each player better off.

The above setup allows any number of sequential or simultaneous buyer or supplier actions, and the

profits can be any function of these actions. We consider situations where self-interested behavior

and the consequent Nash equilibrium outcome are inefficient. For example, consider a supply

chain where the buyer has lower costs of financing and the product includes an element of supplier

performance derived quality that is difficult to contract, verify, observe or enforce– such as supplier

performance in providing responsive and timely support and installation services. Both buyers

and suppliers would be better off if the buyer raises capital, pays the supplier in advance and the

supplier performs well (an efficient, cooperative outcome). However, knowing that the supplier’s

performance incentives will be diminished with advance payment, the buyer does not do so, and the

supplier then uses more expensive financing. Part of these financing costs are then passed on to the

buyer, making both the buyer and supplier worse off (an opportunistic, Nash outcome). Tunca and

Zenios (2006) provide a detailed example of another such situation with unobservable quality.

The cooperative outcome may also be interpreted as the outcome if the buyer and supplier ac-

tions were undertaken by one centralized entity. Our assumption on the existence of a cooperative

outcome that is preferred to the opportunistic outcome implies that there exist decentralization


inefficiencies in the supply chain. With this interpretation, our setup also covers many classic sup-

ply chain issues such as inefficiency due to double marginalization (Cachon and Lariviere, 2005),

inefficiencies due to limited information sharing (Ren et al., 2010), poor performance on unobserv-

able quality dimensions and the accompanying low prices (Tunca and Zenios, 2006), insufficient

investments in capacity (cf. Taylor and Plambeck, 2007b for a general version), etc. Additionally,

it also captures some key issues from the service outsourcing literature related to service quality,

capacity building, utilization, etc. (Ren and Zhang, 2009; Roels et al., 2010). The classic prisoner’s

dilemma type game (Mailath and Samuelson, 2006) also conforms to this setup.

Alternate Supply Chain Structures. We model two alternate sourcing structures:

(1) Direct Sourcing : Each buyer sources directly from the suppliers. The buyers select sup-

pliers, and each buyer acts in the engagement step. This structure captures traditional

sourcing practices as exemplified by long-term sourcing relationships at Toyota (Nishiguchi

and Beaudet, 1998), IKEA (Baraldi, 2008), etc.

(2) Mediated Sourcing: Both buyers source through a third party, the Intermediary. The inter-

mediary selects the supplier for each buyer and the intermediary acts for both buyers in the

engagement step. This structure captures firms that source through an intermediary like

Li & Fung Ltd. (Loveman and O’Connell, 1995) or Olam International (Bell and Shelman,

2009).

In the next sections, we describe the game in each of the two sourcing setups, and we compare the

equilibrium outcomes in Section 4. A formal, technical definition of the games and the equilibria is

provided in the Appendix.

3.2. Direct Sourcing. In direct sourcing, the buyers act independently, and their choices can be

analyzed in two identical but distinct games.

The Stage Game. For buyer i, the stage game involves the suppliers, sj , j ∈ i, i, and the buyer,

bi. The game is illustrated in Figure 3.2.

First, the random component, Xi, is drawn. Next, the buyer chooses supplier zi. Finally, buyer i

and supplier zi play the engagement step game Γ, which we denote as Γzi . The payoffs of buyer bi


SUPPLIER SELECTION INFORMATION GATHERING ENGAGEMENT

Γ𝑖

𝑁 𝑋𝑖 Γ𝑖

𝑏𝑖 𝑠𝑖

𝑠𝑖

Nature draws 𝑋𝑖~𝐹 𝑥 Buyer 𝑖 Chooses Supplier, 𝑧𝑖 Actions 𝑎 and Payoffs

𝑢𝑏 𝑎 𝑢𝑠 𝑎 0

𝑢𝑏 𝑎 + 𝑋𝑖 0 𝑢𝑠 𝑎

𝐵𝑢𝑦𝑒𝑟𝑖 𝑆𝑢𝑝𝑝𝑙𝑖𝑒𝑟𝑖 𝑆𝑢𝑝𝑝𝑙𝑖𝑒𝑟𝑖

Figure 3.2. The Direct Sourcing Stage Game for Buyer i.

and suppliers sj are given as

ubi(Xi, zi, a) =

ub(a), if zi = i;

ub(a) +Xi, otherwise;

usj (zi, a) =

us(a), if zi = j;

0, otherwise;

where Xi ∼ F (x). The action profile α∗d, that prescribes zi = i, if Xi ≤ 0; zi = i, otherwise;

followed by actions aN in each of the games Γi and Γi, is a subgame-perfect equilibrium of the

direct sourcing stage game (formally shown in Lemma 3 in the Appendix).

The Repeated Game. In the repeated game, the above stage game is played in each period

t ∈ 0, 1, .... Players discount future payoffs with a common discount factor δ. The present value

of the expected normalized profits of player n, n ∈ s1, s2, bi, are given by

(3.1) Un (σ) = (1− δ)Eσ ∞∑t=0

δtun(Xti , z

ti (σ) , at (σ)

),

where the strategy profiles, σ, or player actions in the repeated game, prescribe the choice of the

supplier and the actions in the engagement step game, as a function of the realization of Xti and

the history of these actions in previous periods.

Potential Equilibrium Strategies. We show that to analyze buyer and supplier profits it is

sufficient to consider stationary threshold policies for supplier selection (Lemma 4 in the Appendix).

Specifically, in any period t, buyer i chooses supplier i if Xti ≤ xθ, and supplier i, otherwise. The

long-term share of the sourcing business allocated to supplier i is θ, θ = Pr(Xti < xθ

).

Based on the selection step choices, the buyer chooses between two engagement step games that

are identical in all respects except for the supplier involved. In each of these two games the players


may play the cooperative or the Nash actions. Specifically, three kinds of behavior may arise in

equilibrium: 1) the buyer and both suppliers always play the Nash actions, or 2) the buyer and one

supplier (i or i) play the cooperative actions in the engagement games that involve them and the

Nash actions in the engagement step game that involves the other supplier; or 3) the buyer and

both suppliers always play the cooperative action. We call these the direct transactional (dt), the

direct sole (ds) and the direct dual (dd) sourcing strategies, respectively. Note that in all three of

these strategies, the buyer can source from both suppliers and allocate different levels of long-term

business to the suppliers (θ); the difference lies in the nature of the relationship when sourcing from

different suppliers: is it characterized by cooperative or opportunistic behavior?

Formally, ∀k ∈ dt, dsi, dsi, dd, strategy σk (θ) prescribes the following play: if in all past play, the

outcomes of the selection and engagement step actions prescribed below were observed, continue to

play the corresponding selection and engagement step actions; else play action α∗d (the stage game

equilibrium) forever.2

Selection Step Actions: ∀k ∈ dt, dsi, dsi, dd, the choice of supplier, zi is given as zi = i, if

Xti ≤ xθ, and zi = i otherwise.

Engagement Step Actions: The prescribed actions are(aN , aN

)for strategy dt;

(aC , aN

)for strategy

dsi;(aN , aC

)for strategy dsi and

(aC , aC

)for strategy dd. The first action denotes the actions

after selection of supplier i, and the second after i.

The expected normalized profits from playing these strategies are computed using equation 3.1 and

are provided in Table 2 (Appendix, page 37). The ability to sustain the above strategy profiles as

subgame-perfect equilibria of the repeated game depends on the incentives for the buyers and the

suppliers to deviate from the strategy. We summarize the necessary and sufficient conditions for

this in the next Lemma.

Lemma 1. Equilibrium Outcomes of the Direct Sourcing Game.

(1) A strategy profile σdt (F (0)) is the only transactional subgame-perfect equilibrium of the

repeated game.

(2) Strategy profile σk (θ), k ∈ dsi, dsi, dd, is a subgame-perfect equilibrium of the repeated

game if and only if the following two conditions hold:

2These and all the other strategies proposed in this paper are Nash reversion trigger strategies, that is, on observationof a deviation from the equilibrium path the Nash outcome is played in all future periods. The Nash outcome isthe most severe credible punishment for cheating; therefore one may, without loss of generality, restrict attention totrigger strategies (Taylor and Plambeck, 2007b).


(a) The difference in the buyer’s expected normalized utility from strategies σk (θ) and

σdt (F (0)) is greater than 1−δδ max

Gb,

∣∣xθ∣∣− ηb.(b) The difference in the supplier’s(s’) expected normalized utility from strategies σk (θ) and

σdt (F (0)) is greater than 1−δδ Gs.

Gs (Gb) denotes the gain from the most profitable deviation of the supplier (buyer) in the

subgame Γ. This is the difference between the profit of the best-response action to the coop-

erative actions of the other player in the game Γ, and the profit of the cooperative action.

ηb ≡ ub(aC)− ub

(aN), ηs ≡ us

(aC)− us

(aN)are the buyer’s and supplier’s gain from

cooperation.

Proof. The formal proof is provided in the Appendix (Page 37), and the intuition follows. In direct

transactional sourcing, player actions do not influence subsequent stages of the game and the stage

game equilibrium, repeated in every period, is the subgame-perfect equilibrium of repeated direct

sourcing game. Sustaining the latter three relational strategy profiles as equilibrium outcomes

requires an inter-temporal trade-off between the immediate benefits of any unilateral deviation and

the future benefits from continuing cooperation on the efficient outcome. In particular, the loss

in continuation benefits following deviation from the strategy must be greater than the immediate

benefits of the most profitable deviation. The conditions in Part 2 of Lemma 1 capture this trade-off

for the buyer and the suppliers. The conditions in Part 2 of Lemma 1 are monotone in δ, and as

such, they define a threshold δ, above which the strategy is an equilibrium. We denote the threshold

discount factors required to sustain strategy k as δk.

There are three key observations from the above analysis:

(1) The profits for buyers are highest with dual sourcing followed by sole and transactional

sourcing, on account of more cooperative behavior. Dual sourcing strategies are harder to

sustain as an equilibrium (they require higher discount factors) than sole sourcing strategies,

which in turn are harder to sustain than transactional strategies.

(2) For each strategy k there is a profit maximizing long-term allocation of business between

suppliers, θk, that derives out of choosing the lower-cost supplier for each realization of the

cost advantage.3 More egalitarian allocation of business between suppliers (θ → 1/2) makes

it easier to sustain dual sourcing strategies in equilibrium, whereas more unequal allocation

3Supplier i gets θk units of average per-period business. θdd,dt = F (0), θdsi = F (ηb) , θdsi = F (−ηb). This is also

her "market share".


of business between suppliers (θ → 1, 0) makes it easier to sustain the sole sourcing strategy

with the favored supplier as an equilibrium.

(3) Increases in the benefits of cooperation (ηb ↑ or ηs ↑) and decreases in the benefits of devia-

tion (Gb ↓ or Gs ↓) make it easier to sustain higher profit strategies (sole and dual strategies)

as equilibrium.

In our setup, there are benefits to having more cooperative behavior. To realize these benefits in

equilibrium, the firms must be sufficiently patient and the suppliers should get enough sourcing

business over the long run to discourage them from opportunistic behavior. However, ensuring

enough long-term business for each supplier may require the buyer to source from a higher-cost

supplier, thus increasing the costs of sourcing. This trade-off between enforcing more cooperative

behavior as an equilibrium and choosing the lowest-cost supplier is a central facet of our model.

In effect, this is a trade-off between the benefits of retaining flexibility in the choice of suppliers

and the benefits of committing to a restricted set of suppliers. Tunca and Zenios (2006) study this

trade-off in detail. Our study focuses on how mediated sourcing systematically alters this trade-off.

3.3. Mediated Sourcing. With mediated sourcing, both buyers delegate their supplier selection

and their engagement step actions to a third party, the intermediary. The intermediary chooses

the supplier for each buyer and acts on behalf of the buyers in the engagement step. In lieu of the

sourcing services provided by the intermediary, the buyer pays a share of his profit from the sourcing

exercise to the intermediary. Specifically, the intermediary gets a fraction, β, of each buyer’s profit.4

As with our analysis of direct sourcing, we first describe the stage game, followed by the repeated

game. We then describe possible equilibrium outcomes and the conditions required to sustain them.

The Stage Game. The mediated sourcing stage game is illustrated in Figure 3.3. First, in the

information gathering step, the differences in the costs of sourcing from different suppliers are

revealed, i.e. X1 and X2, are independently drawn from F (x). Then, the intermediary chooses the

supplier for each buyer’s orders. We denote this decision by the vector z, z = (z1, z2). Finally, actual

sourcing takes place in the engagement step, and the intermediary and its chosen supplier(s) play

4We assume here that the buyer and the intermediary engage in a revenue sharing contract. While alternate contractsmay be possible, the analysis of contracting forms is not the focus of this study, and therefore we choose a simplecontract that is known to align incentives and does not introduce any superfluous effects in our model. Further,commission contracts are most widely used by existing intermediaries with typical commissions that range between6-15% (Loveman and O’Connell (1995)). Note that this choice completely aligns the incentives of the buyers and theintermediary, resulting in the same preference over sourcing strategies for the buyers and the intermediary.


𝑧 = (1,1) 𝑢𝑏 𝑎1 + 𝑢𝑏 𝑎2 + 𝑋2 𝑢𝑠 𝑎1 + 𝑢𝑠 𝑎2 0

𝑧 = (1,2) 𝑢𝑏 𝑎1 + 𝑢𝑏 𝑎2 𝑢𝑠 𝑎1 𝑢𝑠 𝑎2

𝑧 = (2,1) 𝑢𝑏 𝑎1 + 𝑢𝑏 𝑎2 + 𝑋1 + 𝑋2 𝑢𝑠 𝑎2 𝑢𝑠 𝑎1

𝑧 = (2,2) 𝑢𝑏 𝑎1 + 𝑢𝑏 𝑎2 + 𝑋1 0 𝑢𝑠 𝑎1 + 𝑢𝑠 𝑎2

𝐼𝑛𝑡𝑒𝑟𝑚𝑒𝑑𝑖𝑎𝑟𝑦 𝑆𝑢𝑝𝑝𝑙𝑖𝑒𝑟1 𝑆𝑢𝑝𝑝𝑙𝑖𝑒𝑟2 (Profits are divided by 𝛽)

SUPPLIER SELECTION INFORMATION GATHERING ENGAGEMENT

𝑁 𝑋1, 𝑋2 𝐼

Nature draws 𝑋1, 𝑋2 Intermediary Chooses Supplier(s) 𝑧 = 𝑧1, 𝑧2

Actions 𝑎𝑧1 ,𝑎𝑧2 and Payoffs

Figure 3.3. The Mediated Sourcing Stage Game

the engagement game Γ for each buyer’s order. This game is identical to the one that buyers played

in direct sourcing, except that the buyer is replaced by the intermediary. Finally, the suppliers

and buyer earn their profits. The action profile α∗m, that prescribes zi = i, if Xi ≤ 0, and zi = i,

otherwise; and actions aN in both engagement step games is a subgame-perfect equilibrium of the

mediated sourcing stage game (Lemma 5 in the Appendix).

The Repeated Game. In the repeated game, the stage game is played in each period t ∈ 0, 1, ....

Players discount future payoffs with a common discount factor δ. The present value of the normalized

expected payoff of player n ∈ I, s1, s2 from a strategy profile σ is given by

(3.2) Un (σ) = (1− δ)Eσ ∞∑t=0

δtun(Xt

1, Xt2, z

t (σ) , atz1 (σ) , atz2 (σ))

.

The strategy profile σ prescribes the choice of the suppliers and the actions in the engagement step

game as a function of the realizations, Xt1, X

t2, and the history of all actions taken in previous

periods.

Potential Equilibrium Strategies. As with direct sourcing, we again show that considering sta-

tionary supplier selection strategies is sufficient (Lemma 6 in the Appendix). Supplier selection is a

function of the realization of the relative cost advantages, Xt1 and Xt

2. With respect to engagement

step actions, the choices follow along the same lines as those in direct sourcing. Specifically, the

intermediary and the chosen supplier(s) may play Nash actions in all games, or the intermediary

and one (preferred) supplier may play the cooperative actions in engagement games that involve

them and the Nash actions in the engagement step games that involve the other supplier, or the

intermediary and each supplier may always play the cooperative action. We call these mediated

transactional, mediated sole and mediated dual sourcing strategies, respectively.


There is an additional strategy available to the intermediary that is not available in direct sourcing.

In each of the strategies where the intermediary plays the cooperative action, it might choose to

constrain the maximum amount of business that is allocated to the cooperating supplier in any

given period. When a supplier gets two orders in the same period, she can deviate on both and

can gain the most from this deviation. By restricting the maximum business allocated to any

cooperative supplier at one unit, the suppliers’ maximum gain from deviation is limited over all

sample paths, thus reducing the exposure of the intermediary to opportunism. In dual sourcing

strategies, where there are two preferred or cooperating suppliers, this implies that in each period

each supplier always receives one unit of business. Such a strategy gets as much cooperation as a

dual sourcing strategy and limits supplier incentives for opportunistic behavior. But, this comes at

the cost of restraining free supplier selection. In sole sourcing strategies, where there is only one

cooperating supplier, such a restrictive strategy is dominated by other strategies (Lemma 7 in the

Appendix). In addition to the types of strategies considered in direct sourcing, we now also consider

this restrictive dual sourcing strategy, where the intermediary cooperates with both suppliers but

restricts the maximum business ever given to any supplier. We will call these restrictive strategies

mediated limited exposure sourcing strategies, denoted by ml. In practice, sourcing intermediaries

are known to employ sourcing rules such as “30-70 sourcing”5 which are similar in spirit to the

limited exposure strategy. Note that in direct sourcing the maximum business allocated by any

individual buyer to a supplier on any sample path was only 1 unit; thus, there were no effective

restrictive strategies, or the restrictive strategies collapsed to the supplier selection step actions.

Formally, ∀k ∈ mt,ms,md, ml, strategy σk (θ) prescribes the following play: if in all past play

only outcomes of actions prescribed below were observed, continue to play the corresponding selec-

tion and engagement step actions, else play action α∗m (the stage game equilibrium) in all subsequent

stage games.

Selection Step Actions: Choose suppliers z, if(Xt

1, Xt2

)∈ Ωz (θ).6 Supplier 1 supplies on average θ

orders, and supplier 2, 2 − θ orders, where θ = 2 Pr(Xt

1, Xt2 ∈ Ω1,1 (θ)

)+ Pr

(Xt

1, Xt2 ∈ Ω1,2 (θ)

)+

Pr(Xt

1, Xt2 ∈ Ω2,1 (θ)

). θ/2 is the market share of supplier 1.

5At Li & Fung Ltd., no supplier is ever given more than 70% of the sourcing business, and never less than 30% (Funget al., 2008).6∪zΩz (θ) = [−x, x] × [−x, x], and the intersection of any two different Ωz (θ) is empty. For k = ml, Ω1,1 (θ) andΩ2,2 (θ) are constrained to be empty.


Engagement Step Actions: The prescribed actions are(aN , aN

)for strategy mt;

(aC , aN

)for strat-

egy ms;7 and(aC , aC

)for strategies ml and md. The first action denotes the actions in the game

with supplier z1 and the second with supplier z2.

Like their counterparts in direct sourcing, strategies mt, ms and md allow for a free choice of the

supplier based on the realization of the cost advantage. In strategy ml, on the other hand, supplier

selection is constrained. As before, the strategies prescribe different contingencies for cooperative

behavior in the engagement step actions. Note that both strategy ml and md always prescribe

cooperative behavior. The players’ profits from each of the strategies are computed using equation

3.2 and are provided in Table 4 (Appendix, page 42). Next we provide the necessary and sufficient

conditions to sustain a strategy profile σk (θ) as an equilibrium.

Lemma 2. Equilibrium Outcomes of the Mediated Sourcing Game

(1) Strategy profile σmt (1) is the only transactional subgame-perfect equilibrium of the mediated

sourcing game.

(2) Strategy profile σk (θ), k ∈ ms,md is a subgame-perfect equilibrium of the mediated sourc-

ing game if and only if the following two conditions hold:

(a) The difference in the supplier’s(s’) expected normalized utility from strategies σk (θ) and

σmt (1) is greater than 1−δδ 2Gs.

(b) The difference in the intermediary’s expected normalized utility from strategies σk (θ)

and σmt (1) is greater than 1−δδ βmax

2Gb, x

Z − 2ηb.

(3) Strategy profile σml (1) is a subgame-perfect equilibrium of the mediated sourcing game if and

only if the following two conditions hold:

(a) The difference in the supplier’s(s’) expected normalized utility from strategies σml (1)

and σmt (1) is greater than 1−δδ Gs.

(b) The difference in the intermediary’s expected normalized utility from strategies σml (1)

and σmt (1) is greater than 1−δδ βmax 2Gb,max x, x − 2ηb .

xZ represents the intermediary’s maximum selection step deviation gain.

Proof. The equilibrium conditions for strategies mt, ms and md follow along the same lines as with

direct sourcing. For strategy ml the benefits of the deviations available to the supplier are halved as

7In direct sourcing, we considered two types of sole sourcing strategies depending on the supplier (i or i) that thebuyer chose to cooperate with. In mediated sourcing, this is no longer necessary as both suppliers are ex-ante identicalfrom the intermediary’s point of view. Thus, we only consider the strategy where the intermediary cooperates withsupplier 1.


compared to ms and md strategies. The intermediary’s benefits from deviation and from continuing

on the path prescribed by strategy ml are also changed by the restrictions in supplier selection. A

formal proof is provided in the Appendix (Page 42).

The equilibrium conditions and profits in mediated sourcing share the three characteristics previ-

ously highlighted for direct sourcing. Again, more cooperative outcomes are more profitable but

harder to sustain; there is a profit maximizing allocation of business,8 and more business to a sup-

plier makes it easier to sustain the cooperative equilibrium with this supplier. Consequently, as

before, there is a trade-off between choosing the lower-cost supplier and providing enough business

to the supplier(s) for sustaining cooperative behavior. However, with respect to this trade-off there

is one key difference– the profit maximizing market shares are different from the ones in direct

sourcing.

In particular, in dual sourcing strategies, the market shares for the suppliers implied by a direct

buyer’s cost minimization are (F (0) , 1− F (0)), whereas for the intermediary they are (1/2, 1/2).9

Similarly, in sole sourcing strategies the shares are (F (ηb) , 1− F (ηb)) in direct sourcing (supplier i),

and (1/2 (1 + F (ηb)− F (−ηb)) , 1/2 (1− F (ηb) + F (−ηb))) in mediated sourcing. In both dual and

sole sourcing, the market shares that accrue to suppliers aremore egalitarian with mediated sourcing

than with direct sourcing. From the direct buyer, the average per-period business that comes any

supplier’s way is the probability that this supplier is the higher-profit supplier. On the other hand,

the business that comes any supplier’s way from the intermediary includes the incidence when this

supplier is the high-profit supplier for buyer 1 and also the incidence when this supplier is the high-

profit supplier for buyer 2. Business in the mediated relationship arises from two buyers, and even

if a supplier is expensive for one buyer, it is possible that she may be cheap for the second buyer.

There is no such opportunity in direct sourcing. Put differently, if one supplier is, on average, much

superior to the other supplier, her average per-period business will be higher with direct sourcing

than with mediated sourcing. The consolidation of buyer preferences by the intermediary that leads

to the long-term business share attenuates any supplier’s idiosyncratic advantage and makes the

intermediary’s allocation more egalitarian. Taken together, while the idiosyncratic preferences of

direct buyers over suppliers lead to a more unequal allocation of business between suppliers, the

8Supplier 1 gets θk units of average per-period business. θmd,mt = 1, θms = 1 +F (ηb)−F (−ηb). The correspondingmarket shares are θ/2.9The first component is supplier i’s (1’s) share of buyer i’s (intermediary’s) sourcing business.


intermediary averages the preferences out, leading to a more Egalitarian Allocation of business.10

In the next section, we examine how these more egalitarian allocations play into the choice between

direct and mediated sourcing.

4. The “Benefits” of Intermediation

We compare the total equilibrium buyer-side surplus or the “sourcing profits”, π, earned in direct

and mediated sourcing strategies. In the case of direct sourcing, this is the sum of the two buyers’

profits. In the case of mediated sourcing, it is the sum of the buyers’ and the intermediary’s profits.

For any given discount factor, only a subset of the strategies discussed above are sustainable as

an equilibrium.11 Of these equilibrium strategies, the one that earns the highest sourcing profit

determines the preferred supply chain structure. If the highest sourcing profit strategy is a mediated

sourcing strategy, then there exists a revenue sharing contract, β′, that ensures that both buyers are

better off employing mediated sourcing strategy, and the intermediary can earn a positive profit.

On the other hand, if the highest sourcing profit strategy is a direct sourcing strategy, then there

is no benefit to employing an intermediary, and direct sourcing is preferred. The highest sourcing

profit strategy for any discount factor derives from a comparison of the sourcing profits and the

threshold discount factors that determine which strategies can be sustained as an equilibrium.

The equilibrium conditions and sourcing profits share the three characteristics highlighted previously

for the direct buyer’s and intermediary’s profits. Thus, the difference between the equilibrium

sourcing profits from direct and mediated sourcing strategies also derives from the trade-off between

providing incentives to suppliers and sourcing from the lower-cost supplier most often. To further

develop some insights into these systematic differences between direct and mediated sourcing, we

first compare sourcing profits and threshold discount factors for strategies with θ chosen such that

the sourcing profit of this strategy is maximized, θk = θk (defined in footnotes 3 and 8). This analysis

allows us to compare the upper bound on the profits that can be achieved through a strategy class

k and a range of discount factors where this upper-bound sourcing profit can be sustained as an

equilibrium.

10The paper develops the results for the case where the two cost differences follow the same distribution and areindependent. However, the spirit of the egalitarian argument in this paragraph and in the subsequent discussionpersists with different distributions and correlations of the cost difference variable.11As is typical in repeated games, we express our equilibrium conditions in terms of the discount factor. However, theseconditions can equally be interpreted as conditions on all exogenous parameters- the distribution, F (x), the generalprofit functions, ub and us, the gains from deviation G·, and the benefits from cooperation η· = u·

(aC

)− u·

(aN

).


4.1. Upper Bound Sourcing Profits.

Proposition 1. Comparison of Upper Bound Sourcing Profits

(1) Dual Sourcing: The upper bound on the sourcing profits is the same in direct and mediated

sourcing, πdd = πmd.

(2) Sole Sourcing: The upper bound on the sourcing profits follow the relationship: πdsr ≤ πms ≤

πdsr , where πdsr = maxπdsi , πdsi

; πdsr = min

πdsi , πdsi

.

Proof. For this and all subsequent propositions, formal proofs are provided in the Appendix. The

intuition typically follows the proposition.

In both mediated and direct dual sourcing, the cost minimizing supplier is always chosen and the

cooperative outcome always ensues. Thus, the sourcing profits are the same.

In direct sole sourcing, the buyer must choose supplier i or i as the cooperative supplier (depending

on πdsi and πdsi). In general, this is a different supplier for each buyer. In the mediated sole

sourcing strategy, the intermediary must pick one cooperative supplier for both buyers’ orders.

This supplier is not the best supplier for both buyers’ orders. Thus, effectively the independent

decisions of the two buyers in direct sourcing gives the buyers more Selectivity in choosing the

preferred supplier, whereas the intermediary being limited to choose one supplier for the two buyers

has lower selectivity, thus, πdsr ≤ πms ≤ πdsr .

Proposition 2. Supplier Incentives to follow Sourcing Profit Maximizing Strategies

(1) Dual Sourcing: The intermediary can better incentivize suppliers to follow dual sourcing

strategies, δmds ≤ δdds .12

(2) Sole Sourcing: The intermediary can incentivize one supplier better than direct buyers and

the incentives for the other are worsened, minδdsis , δ

dsis

≤ δmss ≤ max

δdsis , δ

dsis

.

Suppliers’ incentives to follow any strategy are related to, first, the long-run average business that

is allocated to them that determines their continuation benefits, and second, the maximum business

allocated to them on any sample path that determines the maximum gain from any deviation.

Dual Sourcing: In dual sourcing strategies, two suppliers must be incentivized to cooperate and the

supplier with less business defines the binding constraint and the threshold discount factor for the

12We use δk· to denote the threshold discount factors for the profit maximizing strategy. Note that δk· ≥ δk· .


dual sourcing strategy. Recall that the sourcing profit maximizing allocation in mediated sourcing

is more egalitarian.13 This implies that the critical supplier’s market share is lower with direct

sourcing, and thus direct dual sourcing is harder to enforce than mediated dual sourcing. Effectively

for one supplier, cooperating with the intermediary who pools the idiosyncratic preferences of two

buyers brings a higher long-term market share than cooperating with the direct buyer that is less

likely to source from her. Thus, due to a more egalitarian allocation with mediation, dual sourcing

strategies are easier to sustain with mediation.

Sole Sourcing: The effect of more egalitarian allocation in mediated sourcing again plays out,

but this time with different consequences. In sole sourcing strategies, only one supplier must be

incentivized to cooperate. The supplier who gets more business is easier to incentivize with direct

sourcing, whereas the supplier with less business is easier to incentivize with mediated sourcing.

Part 2 of Proposition 2 follows.

Proposition 3. The Limited Exposure Strategy

(1) Dual Sourcing: Suppliers are easier to incentivize with limited exposure, δmls < δmds ≤ δdds .

However, the profits from strategy ml are lower than those from dual sourcing strategies.

πml < πmd = πdd

(2) Sole Sourcing: Profits from strategy ml are higher than the direct sole sourcing strategy,

πml > πds, iff

2ηb + E[X1 +X2|X1 +X2 ≥ 0] > 2 max ηbF (ηb) + E[Xi|Xi ≥ ηb], ηbF (−ηb) + E[Xi|Xi ≥ −ηb] .

Dual Sourcing: Recall that strategy ml is a restricted version of the dual sourcing strategies that

restricts the maximum amount of sourcing business that can be allocated to any supplier in one

period. This restriction limits the maximum benefit any supplier can get from deviating; thus, the

limited exposure strategy is easier to enforce than the dual sourcing strategy. However, this also

limits the selection of the cost minimizing supplier and this ensures that the sourcing profits are

slightly lower in comparison with dual sourcing strategies.

Sole Sourcing: The limited exposure strategy may earn higher sourcing profits than direct sole

sourcing. The restriction of the limited exposure strategy appears to be severe; if the intermediary

13Specifically, in the profit-maximizing mediated dual sourcing strategy, any supplier’s long-run market share is theaverage of the probabilities that she is the lower-cost supplier for the two buyers, whereas with direct sourcing themarket share is the probability that the supplier is the lower-cost supplier for the buyer in question.


Table 1. Differences between Direct and Mediated Sourcing

Key Differencebetween Direct andMediated Sourcing

Potential Consequences forSole Sourcing (Cooperation

with one supplier)

Potential Consequences forDual Sourcing (Cooperation

with two suppliers)

Selectivity: Direct sourcingallows buyer to pick suppliersbetter matched to their needs.

Direct Sole Sourcing leads tohigher sourcing profits.

-

Egalitarian Allocation: Inmediated sourcing, costminimization leads to a moreegalitarian long-termallocation of business betweensuppliers.

Direct Sole sourcing providesbetter supplier incentives forcooperation to one of the twosuppliers and worse incentivesto the other.

Mediated Dual sourcingprovides better incentives forthe critical supplier.

Limited Exposure: In anysourcing period, mediatedsourcing can limit exposure toopportunistic behavior.

Mediated Limited Exposureretains a substantial amountof the flexibility of dualsourcing due to the potentialrerouting of flows.

Mediated Limited Exposurestrategies are easier to enforcethan dual sourcing strategies.

can source a maximum of one unit from any supplier, he must effectively source from each supplier

in each period; thus appearing to lose all supplier flexibility. However, in reality this restriction is

much less restrictive. Even though the intermediary must always source from each supplier, it has

the ability to flexibly use any supplier for any buyer. In particular, the intermediary can reroute

flows from any supplier to any buyer. Effectively, there is a substantial flexibility that remains even

while limiting exposure to any supplier in any period. Thus, while the sourcing profits are lower

in strategy ml than the unrestricted dual sourcing strategies; but this impact may be much less

significant than if a direct buyer tried to limit exposure to a supplier. So, as a result of maintaining

a high level of cooperation (the intermediary always cooperates with two suppliers) while retaining

a substantial amount of flexibility, the limited exposure strategy can outperform the direct sole

sourcing strategy.

Taken together, the results of Propositions 1, 2 and 3 suggest that the effect of mediation is dif-

ferent for sole sourcing and dual sourcing strategies. Further, there are three key phenomena that

differentiate direct and mediate sourcing; higher Selectivity in direct sourcing, a more Egalitarian

Allocation in mediated sourcing, and the benefits of Limited Exposure in mediated sourcing. We

summarize these three key differences and their impact on the profits and sustainability of sourcing

strategies in Table 1.


4.2. The Preferred Supply Chain Structure. In the previous section, we set θ=θk, where θk

was the allocation that maximized sourcing profits. This gave us upper bounds on the sourcing

profits that can be achieved. We then identified the range of discount factors where this upper

bound could be sustained as an equilibrium outcome(δ < δk

). In this section, we will now consider

all choices of θ, which will expand the range where the strategies can be sustained as equilibrium

but at the cost of some sourcing profits, giving rise to a trade-off in the choice of θ between max-

imizing sourcing profits and incentivizing suppliers. However, even after decreasing some of the

sourcing profits, the strategy may still have higher sourcing profits than the next-best equilibrium

strategy. This effect of choosing θ along with the effects in the previous section drive the following

main proposition, which succinctly identifies the conditions where direct and mediated sourcing are

preferred. We define a few additional building blocks before we describe our main proposition.

Definition 1. ∀δ, define πk (δ) = πk (θ∗k (δ)), where θ∗k (δ) = argmaxθπk (θ), such that strategy k

is an equilibrium for δ.

Essentially, for any given δ, πk (δ) gives us the best sourcing profit that can be obtained in equilib-

rium from strategy k while considering all possible allocations of business between suppliers. Unlike

θ, θ∗ takes into account both the desire to maximize sourcing profits and incentivizing suppliers to

sustain that equilibrium. As there are two possible direct sole sourcing strategies, we further define

πds (δ) = maxπdsi (θ∗dsi (δ)) , πdsi

(θ∗dsi

(δ))

.

Definition 2. Define δ′1 as the highest δ such that πds (δ) ≤ πms (δ), δ′2 as the highest δ < δ′1 such

that πds (δ) ≥ πms (δ), and δ′3 as the highest δ < δ′2, such that πds (δ) ≤ πms (δ), and so on. Denote

the lowest such δ as δ′L, hence L is the number of such changes and it can take values from 0 to ∞.

We set δ′0 = δmd. Further, define δl = minδ′l , δ

md.

In principle, at these discount factors the higher-profit strategy changes from ds to ms or vice versa.

We show that δ′1 involves a change from the domination of ds to ms.

Definition 3. Define δ′ as the highest δ such that πds (δ) ≤ πml (δ).

Proposition 4. Mediated or Direct Sourcing:

(1) On account of a more Egalitarian Allocation of business, mediated sourcing outperforms

direct sourcing for sourcing environments with: (a) δ ∈(δmd, δdd

], (b) δ ∈ (δl+1, δl], l ∈

0, ..., L− 1, and (c) δ ∈ (δms, δL] (if L is odd).


(2) On account of Limited Exposure, mediated sourcing outperforms direct sourcing for sourcing

environments with: δ ∈(δml,min

δ′, δmd

]≡ ∆.

(3) On account of Selectivity, direct sourcing outperforms mediated sourcing for sourcing envi-

ronments with: (a) δ ∈ (δl+1, δl] \∆, if l is even, l ∈ 0, ..., L− 1 and, (b) δ ∈(δds, δL

]\∆

(if L is even).

(4) Direct and Mediated sourcing achieve the same sourcing profits for all other sourcing envi-

ronments.

If L is infinite, the regions in parts 1b, 1c and 3a, 3b do not exist. If δ′ does not exist, the region

defined in Part 2 also does not exist.

Figure 4.1 illustrates the comparison between direct and mediated sourcing as described in Propo-

sition 4. The figure is drawn for F ∼ U [−40, 20], ub(aN)

= 40, ub(aC)

= 70, Gb = 10, and

us(aN)

= 10, us(aC)

= 20, Gs = 20. This leads to L = 2, i.e. l ∈ 0, 1. Since L is even, region 1c

does not exist. The other regions in the figure are labeled to correspond to the parts of Proposition

4. The dotted line shows the best profit from a direct sourcing strategy, the solid line the best profit

from a mediated sourcing strategy, and the mixed dash-dot line shows an overlap of the two.

Part 1 of Proposition 4: From Proposition 2 Part 1, we know that dual sourcing is easier to sustain

with mediation due to a more egalitarian business allocation. This effect plays out in sourcing

environments that fall in region (1a). Here, the profit maximizing mediated dual sourcing strategy

can be sustained as an equilibrium, but to sustain the direct dual sourcing strategy, the direct buyers

must now allocate business in a fashion that compromises on sourcing profits. Thus, mediated

sourcing outperforms direct sourcing. From Proposition 2, Part 2, we know that sole sourcing is

easier to sustain with mediation for one supplier and is harder for the other. When it is easier for

the higher-profit supplier to sustain the equilibrium, mediated sourcing outperforms direct sourcing.

We see this in region (1b).

Part 2 of Proposition 4: From Proposition 3 Part 2, we know that the limited exposure strategy

ml is easier to enforce than the direct dual sourcing strategy because of the limited benefits from

supplier deviations. Further, it can earn higher profits than either of the sole sourcing strategies

due to the higher level of cooperation and substantial flexibility in supplier selection despite the

restrictions on supplier choice. For sourcing environments that fall in region 2, this advantage of


12

0

12

5

13

0

13

5

14

0

14

5

(4)

75 0

.50

0.5

50

.60

0.6

50

.70

0.7

50

.80

0.8

50

.90

Sourcing Profit, 𝜋𝑘

Dis

cou

nt

Fa

cto

r, 𝛿

𝜋𝑚𝑙>𝜋𝑑𝑠

𝜋𝑚𝑠>𝜋𝑑𝑠

𝜋𝑚𝑑>𝜋𝑑𝑑

𝜋𝑚𝑑=𝜋𝑑𝑑

𝜋𝑑𝑠>𝜋𝑚𝑠

𝜋𝑑𝑠>𝜋𝑚𝑡

Ga

in fr

om

med

iati

on

Lo

ss f

rom

med

iati

on

Med

iate

d s

ou

rcin

g

Dir

ect

sou

rcin

g

(3𝑏)

(3𝑎)

(2)

(4)

(1𝑎)

(1𝑏)

Eg

ali

tari

an

All

oca

tio

n

Lim

ited

Exp

osu

re

Sele

ctiv

ity

𝜋𝑚𝑡=𝜋𝑑𝑡

𝛿𝑑𝑠

𝛿 𝑑𝑑

𝛿𝑑𝑑=𝛿𝑚𝑑

𝛿 1

𝛿𝑚𝑠

𝛿𝑚𝑙

Figure 4.1. Performance Comparison: Mediated vs. Direct Sourcing


the limited exposure strategy, only available to intermediaries, allows mediated structure to earn

higher profits than direct buyers can.

Part 3 of Proposition 4: From Proposition 1 Part 2, and Proposition 2 Part 2, we know that

direct buyers can better choose their sole supplier. This increases their profits and also allows them

to incentivize one supplier better as compared to mediated sourcing. This selectivity drives the

superior performance of direct sourcing in sourcing environments that fall in regions (3a) and (3b).

Alternately, this may also be interpreted as the effect of Egalitarian Allocation favoring direct sole

sourcing with one supplier.

5. Implications and Robustness

5.1. Prescribed Sourcing Strategy. In the above analysis we identified three mechanisms through

which direct sourcing differs from mediated sourcing. Depending on the characteristics of the pro-

curement exercise, one of these effects is dominant and drives the choice between Toyota-like direct

sourcing or sourcing through an intermediary like Li & Fung. Managerially, these characteristics

can be thought of along two dimensions. The first is the extent of the decentralization inefficiency,

which is driven by benefits that can be realized by coordinating on cooperative outcomes as opposed

to opportunistic behavior. Decentralization inefficiencies are high in sourcing situations where the

product specifications are unclear, unobservable or costly to verify, it is hard to ensure conformance

of the production process with labor, environmental and social norms, where there is significant

proprietary information, where accurate forecasting is not possible, where significant relationship-

specific investments must be made by buyers or suppliers, etc. In our model, these correspond to a

higher difference between u·(aC)and u·

(aN)(or η·).

The second key factor is related to the competitive advantage of one supplier over the other. In

particular, it is the difference between the likelihoods of suppliers being lower-cost. If neither

supplier has a significant advantage over the other, both suppliers are equally likely to be lower-cost

and the suppliers are comparable. If, on the other hand, one supplier has a distinct advantage over

the other supplier, she is more likely to be lower-cost than the other. We expect this competitive

advantage to be low when both suppliers are equally well positioned in terms of their geographical

location, costs of production, capabilities, etc. In the above model, when F (0) ≈ 1/2, the suppliers

are comparable, and when F (0)→ 1 or 0, one supplier has a significant competitive advantage.


Decentralization Inefficiency Low Medium High

Co

mp

etit

ive

Ad

van

tag

e

One Supplier has a

distinct Advantage

Mediated or Direct (Depends on the

suppliers’ characteristics)

Mediated Direct

Mediated Direct (More egalitarian allocations in dual

sourcing)

Suppliers are comparable

Mediated Direct

Mediated Direct (Limited exposure allows

for cooperation and flexibility)

Mediated Direct

, > , and indicate small, medium and, large differences in sourcing profits, respectively.

Figure 5.1. Prescribed Sourcing Strategy

The issues studied in this paper are relevant in sourcing situations where there is a significant de-

centralization inefficiency and where no supplier has an exceptionally high advantage over the other.

For these sourcing situations, the prescribed sourcing strategy depends on the level of decentraliza-

tion inefficiency and the competitive advantage of one supplier over the other. Figure 5.1 illustrates

the preferred strategy for different values of these two key factors in the sourcing situation.

For sourcing situations where decentralization inefficiency is high, the benefits of cooperative be-

havior are high. In the context of our model, these situations are ripe for dual sourcing. More

generally, a larger network of preferred suppliers can be developed. In these situations, the egal-

itarian allocation benefit of mediated sourcing is at work. However, the egalitarian allocation is

performance improving if and only if the suppliers are different i.e. one supplier has a distinct

competitive advantage (upper right corner of Figure 5.1). If the suppliers are comparable, mediated

sourcing offers less advantage over direct sourcing.

For sourcing situations with medium levels of decentralization inefficiency, the benefits of cooperative

behavior are not high enough to sustain dual sourcing, but with the limited exposure strategy offered

by mediated sourcing it may still be possible to source in a cooperative fashion from both suppliers

while compromising minimally in the free choice of suppliers on account of potential rerouting of

orders. With higher supplier differentiation, the opportunities for this rerouting become less and less

likely. Thus for medium levels of decentralization inefficiency, mediated sourcing most outperforms

direct sourcing when suppliers are comparable, and the benefits diminish when one supplier is

advantaged (as in the middle column of Figure 5.1)


Finally, when decentralization inefficiency is even lower, only sole sourcing may be sustained. In

these cases, the egalitarian allocation of mediated sourcing and the selectivity of direct sourcing

are at work. Both these effects are operative only when one supplier has a competitive advantage.

Thus with comparable suppliers, the difference between direct and mediated sourcing are smaller,

whereas when the suppliers are different, the net effect of selectivity and egalitarian allocation drives

the preferred sourcing strategy. If one supplier is much preferred over the other, direct sourcing

may dominate mediated sourcing. This is illustrated in the left column of Figure 5.1.

To summarize, our analysis suggests that the benefits of mediation are most substantial when

there is an intermediate level of inefficiency and the suppliers are equally advantaged, or when the

inefficiency is high and one supplier has a competitive advantage over the other supplier.

5.2. Robustness of our Analysis. The trade-off between committing to source from a restricted

set of suppliers in order to get benefits of long term relationships (cooperation) and flexibly sourcing

from whichever supplier is the cheapest is the central focus of this paper. Our stylized model above

captures these trade-offs succinctly, and abstracts from other relevant issues in sourcing such as

the effect of additional sources of heterogeneity between buyers and suppliers such as distinct and

correlated profit shock distributions, different discount factors, asynchronous demand, the effect

of changing the number of buyers and suppliers, and the effect of imperfect public and private

monitoring of engagement step actions. Next, we discuss the implications of considering some of

these effects in our model.

While the buyers and suppliers in our model are not homogeneous (they differ in their average pref-

erence over suppliers), they share common features. Specifically, we assume that the distributions

of random shocks to buyers’ profits are stationary, identical, independent. While this makes the

exposition easier, but we don’t think this drives the effects in this paper. We expect our central

insights to continue to hold even if we assume any general bi-variate, time-varying distributions for

these shocks. We also assume that all players have the same discount factor δ. We expect that

different discount factors will enhance the differences between suppliers and that the effects arising

out of mediation’s egalitarian allocation will become more salient.

We assume that both buyers are interested in the same product and both suppliers can produce it.

Again, we conjecture that this assumption is not necessary for our results. It is sufficient that both

suppliers have the capability of producing the products that each of the buyers might be interested


in. A good example of such capabilities can be found in the garment industry, where different buyers

are interested in different collections at any given period, but suppliers have the labor force and

generic equipment to produce a wide variety of clothes. Our model considers only two buyers and

two suppliers. Our analysis largely derives from the intermediary’s consolidation of demand and

attenuation of idiosyncratic supplier preferences; thus we expect that with an increase in the size of

the intermediary’s network, the effects highlighted above will again become more salient.

We assume that all the actions of the players can be observed at the end of each sourcing pe-

riod. In many practical situations the actions of the suppliers, buyers, or intermediaries, may not

be fully observed, but only some noisy signals of these actions may be observed. Such imperfect

observability adversely affects the set of payoffs achievable in equilibrium; in particular, the achiev-

able payoffs might be bounded away from the Pareto frontier that we exploit in our current model

(Mailath and Samuelson (2006), p. 284). The role of intermediation in such settings would require

a separate study, however, and based on the insights of our current model we conjecture that as

mediation improves observability in the system through more frequent interactions with suppliers,

the intermediary may be able to better address the imperfect observability issue and the benefits

of intermediation may be even more significant. Belavina and Girotra (2010) study these effects in

detail.

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Appendix A. Formalization of Section 3.1 (Model Preliminaries)

A.1. Notation for The Engagement Game Γ. Let Ξ be the collection of initial nodes of the

subgames of the extensive-form game Γ, with ξ0 being the initial node. Denote the subgame of Γ

with initial node ξ ∈ Ξ by Γξ, hence Γξ0 = Γ. It is partially ordered by precedence relation, where

ξ < ξ′ , if ξ′ is a node in Γξ. Denote Y , a set of terminal nodes, with typical element y. An action

for player s (b) specifies a move for player s (b) at each information set owned by that player. At

the end of the period, the players observe terminal node y reached as a result of play. A unique

terminal node is reached under a path of play implied by a. Given a node ξ ∈ Ξ, us(a|ξ) (ub(a|ξ))

is player s’s (b’s) payoff from Γξ, given the moves in Γξ implied by a. The terminal node reached

by a conditional on ξ is denoted by y(a|ξ).

Appendix B. Formalization and Proofs for Section 3.2 (Direct Sourcing)

B.1. Notation for The Direct Sourcing Stage Game Gi. Collection of initial nodes of the

subgames of game Gi is ΞGi ≡ ξzi ∪ ΞΓ1 ∪ ΞΓ2 where ξzi is the supplier selection node, and

ΞΓj is the set of initial nodes of the subgames of game Γ played between buyer i and supplier j. To

distinguish between nodes in game Γ with suppliers 1 and 2, we will add superscript j to all nodes,

so the initial node of game Γ, ξ0 will become ξoj ; with precedence relation ξzi < ξ, ∀ξ ∈ ΞΓ1∪ΞΓ2,

and ∀ξ ∈ ΞΓj , ξ < ξ′ , if ξ′ is a node in Γjξ. Denote the subgame of Gi with initial node ξ ∈ ΞG

i by

Giξ, hence Giξ0j = Γj . The set of terminal nodes is Y Gi = Y Γ1 ∪ Y Γ2 .

B.2. Notation for The Repeated Direct Sourcing Game Gi∞. The set of period t, t ≥ 0,

ex-ante histories is given byH t = (X ×A )t, X ≡ [−x, x], A ≡ 1, 2 × A, identifying the state(Xti

)and action profile (A ) in each previous period. The set of period t, t ≥ 0, ex-post histories

is given by H t = (X ×A )t ×X , identifying the state and action profile in each previous period

and identifying the current state. Let H = ∪∞t=0Ht, H = ∪∞t=0H

t, we set H 0 = ∅, hence

H 0 = X . Pure strategies for player n is a mapping σn : H → An, associating an action with each

ex-post history.

B.3. Additional Lemmas.

Lemma 3. Direct Sourcing: The Stage Game Equilibrium

α∗d is a subgame-perfect Nash equilibrium of Gi.


Proof. By definition, an action profile α∗d is a subgame-perfect equilibrium if for every node ξ ∈ ΞGi ,

the profile α∗d|ξ is a Nash equilibrium of subgames Giξ. Stage game Gi starts with choice of supplier–

initial node ξzi , i.e. Gi = Giξzi , and depending on the supplier chosen, it is followed by subgames

starting from ξ01 or ξ02. Giξ0j = Γj = Γ. We know that aN is a subgame-perfect equilibrium of

Γ, i.e. for every node ξ ∈ Ξ, the profile aN |ξ is a Nash equilibrium of Γξ. For every node ξ ∈ ΞΓj

α∗d|ξ = aN |ξ, and so α∗d|ξ0j is a subgame-perfect equilibrium of Giξ0j = Γj . Hence, we only need

to show that α∗d|ξzi is a Nash equilibrium of Gi. Choosing supplier i, the buyer will end up with

utility ub(aN ), and choosing supplier i he receives ub(aN ) +Xi. Hence, the buyer will maximize his

utility by choosing supplier i if Xi ≤ 0 and choosing supplier i if Xi > 0.

Lemma 4. Optimality of the Stationary Threshold Policy in Direct Sourcing

Proof. For a given expected normalized profit of supplier(s) that depends on the long-run business

split determined by θ, we will be interested in the best profit that the buyer can achieve in equi-

librium. The proof proceeds in three steps: first, we will derive conditions for any policy to be an

equilibrium; second, we will show that in any period the buyer will be at least weakly better off

using threshold policy; and third, we will show that the stationary threshold policy is optimal.

1. Along the same lines as the Proof of Lemma 1 Part 2 (which appears in the next sub-section), we

have: for all histories that featured no deviation, and ξ that belong to the equilibrium path, (h, ξ),

it should hold:

(B.1) δ(Ub

(σ|(h,y(αk|ξ))

)− Ub

(σdt))≥ (1− δ) max

Gb, |xθh | − ηb

,

(B.2) δ(Uj

(σ|(h,y(αk|ξ))

)− Uj

(σdt))≥ (1− δ)Gs.

Where j ∈ J , J is the set of cooperative suppliers and |xθh | is the “highest” realization of the random

component such that the buyer would want to deviate.

2. In any period t, based on the realization of the random component, Xti , and the previous history,

h, buyer i can employ any policy. We are interested in choosing a policy such that the buyer’s

expected profit is maximized and the constraints above are satisfied. Note that only the expected

profit from any given period, t, enters the supplier-side constraint (B.2), or in other words, the

supplier only cares about her long-run expected share of business determined by θt. The buyer


can achieve it in many different ways. Next, we will show that the threshold policy allows him to

simultaneously achieve the highest expected profit in any period t and the lowest deviation gain

in the supplier selection step, |xθt | − ηb. We will do so for the case xθt > 0; the opposite case

follows along exactly the same lines. For the threshold policy, we have θt =∫ xθt−x f(x)dx. Next,

we will consider a policy that prescribes to source from supplier i for an arbitrary realization of

the random components, denoted by Ω, and still achieves the same expected share θt, i.e. θt =∫Ω f(x)dx. Unless Ω =

[−x, xθt

], it must contain an interval [a, b] such that b > xθt , i.e. Ω =

Ω ∪ [a, b]. We will show that employing a policy that prescribes to source from supplier i for

realizations in Ω′

= Ω∪ [a− εa, b− εb], where εa (εb) are chosen so as to satisfy θt =∫

Ω′ f(x)dx, will

improve the buyer’s expected profit. The difference in the buyer’s profit from these strategies is:∫[−x,x]\Ω′ xf(x)dx −

∫[−x,x]\Ω xf(x)dx, which equals:

∫ bφ xf(x)dx −

∫ φa−εa xf(x)dx ≥ φ

∫ bφ f(x)dx −

φ∫ φa−εa f(x)dx ≥ 0, where φ = max a, b− εb and φ = min a, b− εb, and last inequality follows

from θt =∫

Ω′ f(x)dx =∫

Ω f(x)dx. In other words, the buyer can improve any policy by shifting

all ranges where he is sourcing from supplier i as much to the left as possible. By doing so, the

maximum deviation in the supplier selection step is restricted to xθ, and the maximal deviation

gain is moved as much to the left as possible.

3. Next, we will show that the buyer would always prefer to follow a stationary threshold pol-

icy. To do so, we will compare an arbitrary threshold policy θt and the stationary thresh-

old policy θ. As before, the buyer will seek to maximize his profit, so that constraints B.1,

B.2 are satisfied. When none of the constraints bind, unconstrained profit maximization by the

buyer will lead to a stationary threshold policy where he employs fully flexible sourcing. If the

buyer-side constraint, B.1, binds for the highest profit achievable by the buyer (unconstrained

optimization), no other strategies can be enforced. Next, we will consider the situation when

only the supplier-side constraint, B.2, binds. Below we will present the proof for sole sourc-

ing from supplier i; other cases follow along exactly the same lines. The expected continuation

profit, for any period t, is∑∞

t=0 δt(ub(aN)

+∫ xxθt xf(x)dx+ θtηb

). Constraint B.2 can be rewrit-

ten: for all t,∑∞

y=t δy−t(∫ xθy−x f(x)dx

)≥ (1−δ)Gs+δUsi(σ

dt)δus(aC)

≡ ϕ. We are considering the case

where the buyer’s myopically optimal (unconstrained optimization) stationary threshold policy

does not satisfy constraint B.2. Hence, to maximize his profit, the buyer would ideally want to

satisfy the supplier’s constraint without the slack. Unless θt is stationary, the buyer will have

to make sure that for some t, the constraint is satisfied with equality; denote such a t as t, but


Table 2. Expected Normalized Profits

Strategy Buyer i Supplier i Supplier i

σdt (θ) ub(aN)

+ µxθ θus(aN)

(1− θ)us(aN)

σdsi (θ) ub(aN)

+ µxθ + θηb θus(aC)

(1− θ)us(aN)

σdsi (θ) ub(aN)

+ µxθ + (1− θ) ηb θus(aN)

(1− θ)us(aC)

σdd (θ) ub(aN)

+ µxθ + ηb θus(aC)

(1− θ)us(aC)

where, µx = E[Xi|Xi ≥ x], ηb = ub(aC

)− ub

(aN

).

for some other t, it must be satisfied with slack, hence reducing his profit. Next, we will com-

pare profits from stationary and non-stationary policies.∑∞

t=0 δt(ub(aN)

+∫ xxθt xf(x)dx+ θtηb

)≤∑∞

y=t δy−t(ub(aN)

+∫ xxθy xf(x)dx+ θyηb

)which in turn, taking into account that the inequality

B.2 binds, equals 11−δub

(aN)

+ ϕ+∑∞

y=t δy−t ∫ x

xθy xf(x)dx. Similarly the profit for the stationary

policy equals 11−δub

(aN)

+ ϕ+∑∞

y=t δy−t ∫ x

xθ xf(x)dx, where t is actually any t. Subtracting from

the latter, the former we have:∑∞

y=t δy−t(∫ xθy

xθ xf(x)dx)≥∑∞

y=t δy−t(xθ∫ xθyxθ f(x)dx

)= 0. The

last equality follows from the binding supplier-side constraint:∑∞

y=t δy−t(∫ xθy−x f(x)dx

)= ϕ =∑∞

y=t δy−t(∫ xθ−x f(x)dx

)⇒∫ xθyxθ f(x)dx = 0.

B.4. Expected Normalized Profits from Direct Sourcing Strategies. The expected normal-

ized profits are computed using equation 3.1 and are provided in Table 2 .

B.5. Proof of Results in Main Paper.

Proof of Lemma 1 (Section 3.2, Page 12).

Proof. Part 1. We showed in Lemma 3 that α∗d with xθ = 0, i.e. θ = F (0), is a subgame-perfect

equilibrium of Gi and hence it is a subgame-perfect equilibrium of Gi∞. No other xθ can be

maintained as, in any period, the buyer could deviate to xθ = 0 and improve his profit.

Part 2. In order to establish this, we need to show that for all histories (h, ξ), (1− δ)un(αk|ξ

)+

δUn

(σk|(h,y(αk|ξ))

)≥ (1− δ)un

(α′n, α

k−n|ξ

)+δUn

(σk|(h,y(α′n, αk−n|ξ))

)for all α′n and for all players

n, where αk is an action profile prescribed by σk following the ex-post history h.

We will start with histories that included a deviation. Following such a history, the strategy σk

is prescribing the stage game equilibrium to be played for ever after, hence Un(σk|(h,y(α∗d|ξ))

)=

Un

(σk|(h,y(α′n, α∗−n|ξ))

). Now, we only need to show that ∀ ξ: un (α∗d|ξ) ≥ un

(α′n, α

∗−n|ξ

), which is

shown in the proof of Part 1 of Lemma 1.


For histories that do not include a deviation, the strategy is prescribing αk to be played if no

deviations were observed, and the stage game equilibrium following any deviation. Next, we will

divide all initial nodes of subgames, ξ ∈ ΞGi , into two classes: the ones that are on and the ones

that are off the equilibrium path.

For all ξ that are off the equilibrium path, it holds Un(σk|(h,y(αk|ξ))

)= Un

(σk|(h,y(α′n, αk−n|ξ))

),

as the strategy is prescribing the stage game equilibrium to be played for ever after. Then we only

need to show un(αk|ξ

)≥ un

(α′n, α

k−n|ξ

), αk|ξ = aN |ξ and α′n|ξ = a′|ξ, i.e. we need to show

un(aN |ξ

)≥ un

(a′n, a

N−n|ξ

), which holds as aN is a Nash equilibrium of Γ.

For all ξ that belong to the equilibrium path, Un(σk|(h,y(α′n, αk−n|ξ))

)= Un

(σdt), Un

(σk|(h,y(αk|ξ))

)=

Un(σk)for all cooperating players. For the non-cooperating supplier, denoted by sn, (in strategy

dsi it is supplier i, and in strategy dsi it is i) Usn(σk|(h,y(α′sn , αk−sn |ξ))

)= Usn

(σk). In other

words, even if the supplier that the buyer does not cooperate with deviates, this would not influence

the continuation of cooperation among cooperating players. We need to ensure that for cooperat-

ing players the following holds: (1− δ)un(αk|ξ

)+ δUn

(σk)≥ (1− δ)un

(α′n, α

k−n|ξ

)+ δUn

(σdt).

For the non-cooperative supplier: (1− δ)usn(αk|ξ

)≥ (1− δ)usn

(α′sn , α

k−sn |ξ

). The latter holds

as αk|ξ = aN |ξ, α′n|ξ = a′|ξ and due to aN being a subgame-perfect equilibrium of Γ. For

cooperative suppliers, suppliers j only have deviations inside Γj , we need to show: for all a′s,

ξ, δ(Uj(σk)− Uj

(σdt))≥ (1− δ)

(us(a′s, a

Cb |ξ)− us

(aC |ξ

)). Hence, δ

(Uj(σk)− Uj

(σdt))≥

(1− δ)Gs, where we denoted Gs = maxa′s,ξ(us(a′s, a

Cb |ξ)− us

(aC |ξ

)), which will ensure that the

above holds for all a′s, ξ.

For the buyer also, ∀ξ the following should hold:

δ(Ub(σk)− Ub

(σdt))≥ (1− δ)

(ub(α′b, α

k−b|ξ

)− ub

(αk|ξ

)).

The buyer can deviate at the initial node ξzi , which is immediately detectable:

∀ xi ≤ xθ : δ(Ub(σk)− Ub

(σdt))≥ (1− δ)

(ub(aN)

+ xi − ub(aC))

= (1− δ) (xi − ηb);

∀ xi > xθ : δ(Ub(σk)− Ub

(σdt))≥ (1− δ)

(ub(aN)− ub

(aC)− xi

)= (1− δ) (−ηb − xi).

As this should hold for all xi, it should hold for most extreme xi = xθ: δ(Ub(σk)− Ub

(σdt))≥

(1− δ)(|xθ| − ηb

). If there were no deviations in the selection step: δ

(Ub(σk)− Ub

(σdt))≥

(1− δ)(ub(a′b, a

Cs |ξ)− ub

(aC |ξ

)), for all a′b, ξ. As Gb = maxa′b,ξ

(ub(a′b, a

Cs |ξ)− ub

(aC |ξ

)), we

need δ(Ub(σk)− Ub

(σdt))≥ (1− δ)Gb. This establishes all inequalities of Lemma 1 Part 2.


Appendix C. Formalization and Proofs for Section 3.3 (Mediated Sourcing)

C.1. Notation for The Mediated Sourcing, The Stage Game, GI . Denote the collection of

initial nodes of the subgames of GI as ΞGI

= ξz∪ΞΓz , where ξz is the supplier selection node and

ΞΓz is the set of the initial nodes of the subgames of engagement step games Γz played between the

intermediary and suppliers z1 and z2. Each Γz is a merge of two engagement step games, Γz1 and Γz2 ,

as the intermediary now needs to source for two buyers. In Γz compared to the supplier that took

the actions in game Γ, both suppliers z1 and z2 can now take the very same actions, and whenever

the buyer was supposed to act, the intermediary is now taking two such actions. The set of terminal

nodes is Y GI = Y G1×Y G2 . Further, 1βuI (az1 , az2) = ub (az1)+ub (az2)+χ2 (z1)X1 +χ1 (z2)X2 and

uj (z, az1 , az2) = χj (z1)us (az1) + χj (z2)us (az2), j ∈ 1, 2 and χy (x) =

1, if x = y;

0, if x 6= y.

In other

words, the supplier only receives profits from her own actions and the actions of the intermediary

in the game(s) with her.

C.2. Notation for The Repeated Mediated Sourcing Game GI∞. The set of period t ≥ 0

ex-ante histories is given byH t =(X 2 ×A

)t, A ≡ (1, 1) , (1, 2) , (2, 1) , (2, 2)×A×A, identifying

the state and action profile in each previous period. The set of period t ≥ 0 ex-post histories is

given by H t =(X 2 ×A

)t ×X 2, identifying the state and action profile in each previous period

and identifying the current state. Let H = ∪∞t=0Ht, H = ∪∞t=0H

t, we set H 0 = ∅, hence

H 0 = X 2. Pure strategies for player n are mappings σn : H → An, associating an action with

each ex-post history.

C.3. Additional Lemmas.

Lemma 5. Mediated Sourcing: The Stage Game Equilibrium

α∗m is a subgame-perfect Nash equilibrium of GI .

Proof. As in the proof of Lemma 3, we again will use the definition of the subgame-perfect Nash

equilibrium in extensive-form games, the fact that aN is a subgame-perfect Nash equilibrium of

sourcing game Γ and additive separability of utility functions of all players in GI with respect to

actions az1 and az2 .

By definition, an action profile α∗m is a subgame-perfect equilibrium if for every node ξ ∈ ΞGI,

the profile α∗m|ξ is a Nash equilibrium of GIξ . Stage game GI starts with the choice of supplier -


initial node ξz, i.e. GI = GIξz and depending on the suppliers chosen it is followed by subgames

starting from ξ0z, where ξ0z is the initial node of Γz. Γz is a merge of two sourcing games Γ, so

it is easy to see the correspondence between the nodes in Γz and Γ. Initial node ξ0 of Γ indicates

that no actions relevant to the sourcing exercise itself have been taken, specifies players that have

to move (supplier, buyer or both), and for both players s and b, specifies an action space A0s, A

0b

of all action available at this node (if a player does not get to move his or her action space is

empty). Hence, in Γz, ξ0z indicates that no actions relevant to the sourcing exercise itself have

been taken, specifies players that have to move (whenever the supplier was supposed to move in Γ,

both supplier z1 and z2 will move simultaneously; when the buyer was supposed to move now the

intermediary will take two such actions) and action space A0z1,z2 = A0

s if z1 6= z2 and if z1 = z2:

A0z1 = A0

s × A0s and it is empty for a not-chosen supplier; A0

I = A0b × A0

b . And so forth for the

consequent nodes. From this structure, additive separability of the utilities of the intermediary and

the suppliers with respect to actions az1 and az2 in the two merged games, Γz1 and Γz2 , respectively,

and that aN is a subgame-perfect equilibrium of Γ, i.e. for every node ξ ∈ Ξ, the profile aN |ξ is

a Nash equilibrium of Γξ, it follows that α∗m|ξ0z is a subgame-perfect equilibrium of GIξ0z = Γz.

Thus, we only need to show that the supplier selection choice is optimal for the intermediary, or

ub(aN)

+ ub(aN)

+ χ2 (z1)X1 + χ1 (z2)X2 ≥ ub(aN)

+ ub(aN)

+ χ2

(z′1

)X1 + χ1

(z′2

)X2. The

prescribed choice satisfies this inequality. So, α∗m is a subgame-perfect equilibrium of GI .

Lemma 6. Optimality of the Stationary Policy in Mediated Sourcing

Proof. The proof follows along the same lines as the proof of Lemma 4 Part 3.

Lemma 7. Sole Restricted Strategy of the Mediated Sourcing is Dominated

Proof. We will show that mediated sole restricted sourcing strategy, msr, is dominated either by ml

or mt. We will start by deriving the best outcome the intermediary can achieve by following this

strategy. The intermediary can place no more than one unit of business to a cooperative supplier

(we will consider cooperation with supplier 1; analysis for supplier 2 is almost identical).

Place order of buyer(s) Intermediary’s Profit (divided by β) Preferred in

1 with s1 2ub(aN)

+ ηb X1 +X2 < 0, X1 < ηb

2 with s1 2ub(aN)

+ ηb +X1 +X2 X1 +X2 > 0, X2 > −ηb

1&2 with s2 2ub(aN)

+ 0ηb +X1 X1 > ηb, X2 < −ηb


Table 3. Comparison of mediated limited exposure, transactional and restrictedsole strategies.

Region πml† πmsr† πmt† πml−msr πmsr−mt

1 X1 > 0, X2 ≥ 0 2ηb +X1 +X2 ηb +X1 +X2 X1 +X2 ηb ηb

2 X1 +X2 ≥ 0, 0 ≥ X2 ≥ −ηb 2ηb +X1 +X2 ηb +X1 +X2 X1 ηb ηb +X2(≤ ηb)3 X1 +X2 ≥ 0, −ηb ≥ X2 2ηb +X1 +X2 X1 X1 2ηb +X2 0

4 X1 +X2 < 0, X1 ≥ ηb 2ηb X1 X1 2ηb −X1 0

5 X1 +X2 < 0, 0 ≤ X1 ≤ ηb 2ηb ηb X1 ηb ηb −X1(≤ ηb)6 X1 ≤ 0, X2 < 0 2ηb ηb 0 ηb ηb7 X1 +X2 < 0, 0 ≤ X2 ≤ ηb 2ηb ηb X2 ηb ηb −X2(≤ ηb)8 X1 +X2 < 0, X2 ≥ ηb 2ηb ηb X2 ηb ηb −X2

9 X1 +X2 ≥ 0, X1 ≤ −ηb 2ηb +X1 +X2 ηb +X1 +X2 X2 ηb ηb +X1

10 X1 +X2 ≥ 0, 0 > X1 > −ηb 2ηb +X1 +X2 ηb +X1 +X2 X2 ηb ηb +X1(≤ ηb)† 2ub

(aN)is subtracted.

Following along the same lines as the proof of Lemma 3, we can derive the necessary and suffi-

cient conditions to sustain strategy msr as equilibrium: the supplier’s continuation benefit of the

strategy should be greater than 1−δδ Gs. As in all other strategies, the intermediary in every pe-

riod has two orders on hand and hence can deviate on both in the engagement step; in addition

it can deviate in the supplier selection step, the most profitable deviation will bring max x, x,

but it loses the cooperation benefit ηb, hence, the intermediary’s continuation benefits of the strat-

egy should also be greater than 1−δδ max 2Gb,max x, x − ηb . Comparing this to the conditions

for strategy ml ( Lemma 2), we can see that the supplier-side constraint is the same and the

buyer-side is weakly less strict in ml. So we are left with a comparison of continuation benefits

of the two. In ml, the supplier always gets one unit of business, while in msr, she gets one unit

at most. Hence, msr performs worse on incentive provision to suppliers. Intermediary contin-

uation benefits are simply π· − πmt, so we only need to compare πml and πmsr. We will show

that if πml < πmsr then πmt > πmsr. In other words strategy msr is always dominated by ml

or mt. From Table 3, as∫∫

3 (ηb + x2) f(x1)f(x2)dx1dx2 =∫∫

9 (ηb + x1) f(x1)f(x2)dx1dx2 and∫∫8 (ηb − x2) f(x1)f(x2)dx1dx2 =

∫∫4 (ηb − x1) f(x1)f(x2)dx1dx2, taking expectations, we obtain

πml − πmsr > πmsr − πmt, QED.

C.4. Expected Normalized Profits from Mediated Sourcing. The players’ profits from me-

diated sourcing strategies are computed using equation 3.2 and are provided in Table 4.

C.5. Proof of Lemma 2 in the Main Paper.


Table 4. Expected Normalized Profits

Strategy Intermediary (divided by β) Supplier 1 Supplier 2

σmt (θ) 2ub(aN)

+ 0ηb + µZ θus(aN)

(2− θ)us(aN)

σms (θ) 2ub(aN)

+ θηb + µZ θus(aC)

(2− θ)us(aN)

σml (1) 2ub(aN)

+ 2ηb + µZR us(aC)

us(aC)

σmd (θ) 2ub(aN)

+ 2ηb + µZ θus(aC)

(2− θ)us(aC)

where, µZ =∑

(z1,z2)∈Z

∫∫Ω(z1,z2)

(χ2 (z1) x1 + χ1 (z2) x2) f (x1) f (x2) dx1dx2, χy (x) =

1, if x = y;

0, if x 6= y.

Proof of Lemma 2 (Section 3.3, Page 17).

Proof. Following along the same lines as the proof of Lemma 3, we can derive the necessary and

sufficient conditions to sustain respective strategies in equilibrium. The only difference is that, in

strategies ms and md, suppliers at times have two orders on hand and can deviate from both. In

strategy ml they have only one order, so the supplier-side constraints now follow. The intermediary

always deals with two orders, hence can deviate on both, and its deviation gain in the engagement

step is 2Gb, while in the supplier selection in strategy ml, the most profitable deviation will bring

max x, x and in ms and md xZ .

Appendix D. Proofs for Section 4 (The Benefits of Intermediation)

Proof of Proposition 114 (Section 4, Page 20).

Proof. Part 1. From Table 2: πdd = 2ηb + 2µ0. From Table 4: πmd = 2ηb + 2µ0, which is exactly

πdd.

Part 2. πdsi = 2µ−ηb + 2 (1− F (−ηb)) ηb, πdsi = 2µηb + 2F (ηb) ηb and πms = F (ηb) ηb +

(1− F (−ηb)) ηb+µηb+µ−ηb . It is easy to see that πms = 1/2(πdsi + πdsi

), and hence, min

πdsi , πdsi

= πdsr ≤ πms ≤ πdsr= max

πdsi , πdsi

.

Proof of Proposition 2 (Section 4, Page 20).

Proof. From Lemmas 1, 2 we have the necessary and sufficient conditions for sustaining respective

strategies in equilibrium. From proof of Proposition 1 we have values of θs that bring the highest

profit to the buyer/intermediary in any given class of strategies: dd: θ = F (0); md: θ = 1;

142ub(aN

)is subtracted from all π·.


dsi: θ = F (ηb); dsi: θ = F (−ηb); ms: θ = F (ηb) + 1 − F (−ηb). Plugging these values into

Lemmas 1, 2 we obtain the thresholds. The relationships between the thresholds follow, noting that

2Gs(1−δmss )

δmss=

Gs(

1−δdsis

)δdsis

+Gs

(1−δ

dsis

)δdsis

.

Proof of Proposition 315 (Section 4, Page 21)

Proof. Part 1. Suppliers in both strategies ml and md get the very same expected profit: Umls1 =

Umls2 = Umds1 = Umds2 ≡ us(aC). Punishment path utility is again the same for all strategies and for

both suppliers: Umts1 = Umts2 = us(aN). However, in the limited exposure strategy, the intermediary

never places more than one order with each supplier, while in dual sourcing there are states of the

world where it is sourcing from both buyers’ orders from the same supplier. Hence, the maximal

possible deviation in ml is only Gs, while in md it is 2Gs. So, we have δmls = GsGs+us(aC)−us(aN )

<

δmds = 2Gs2Gs+us(aC)−us(aN )

. From Table 4: πml = 2ηb + µZR , πmd = πdd = 2ηb + 2µ0, as µZR < 2µ0,

the inequality follows.

Part 2. The condition follows directly from expressions for upper-bound profits.

Proof of Proposition 4 (Section 4, Page 23)

Proof. As we have seen in Proposition 1, the first-best utility that both sourcing structures can

achieve is πdd = πmd. Hence, if δ is such that both structures can maintain this profit, they

will perform equally well. Next, we will compare the ranges of δ where these levels of profits

can be maintained. We start by comparing δmd and δdd. δk = maxδkb , δs

k. From Proposition

2 and Lemmas 1 and 2 as well as from profit expressions in Tables 2 and 4, we have δmd =

max

GsGs+

12ηs, GbGb+ηb

≤ δdd = max

Gs

Gs+minF (0),1−F (0)ηs ,Gb

Gb+ηb

. Hence if this inequality is strict,

mediated sourcing will perform better in some range of δ, as the intermediary will be able to maintain

the first-best profit while the buyer will have to provide more incentives for suppliers to maintain

cooperation with both, which comes at a cost for him: reduced profits. Further, the intermediary

will be able to maintain this level of profit up to δmd = δmd = max

GsGs+

12ηs, GbGb+ηb

. Hence, in range(

δmd, δdd), mediated sourcing will perform better. This range would be non-empty if δmd < δdd,

this is only possible if F (0) 6= 1/2 and Gsηb > Gb min F (0) , 1− F (0) ηs. This is stated in Part

1a of the Proposition.

152ub(aN

)is subtracted from all π·.


Next, we will show that strategy md may be an equilibrium in potentially a wider range of δ than dd.

This means that the next-best strategy available in mediated sourcing, once md can’t be sustained

in equilibrium, is only to be compared to the next-best strategy after dd that is available in direct

sourcing. δmd = max

GsGs+

12ηs, GbGb+ηb

≤ δdd = max

δdds , δ

ddb

, where δdds = Gs

Gs+12ηs. In providing

incentives for cooperation, the buyer is making sure that the supplier’s constraint is satisfied with

equality and he can do so until the maximum possible incentives are provided, δddb = max a, b.

Where, a is a point where the buyer prefers/has to switch to the sole sourcing strategy and b is a δ

that solves δ (ηb + µxθ − µ0) = (1− δ) maxGb,

∣∣xθ∣∣− ηb, withxθ =

F−1

(1δ

Gsus(aC)

− Gsus(aC)

+ F (0)us(aN)us(aC)

), p ≤ 1

2 ;

F−1

(1− 1

δGs

us(aC)+ Gs

us(aC)− (1− F (0))

us(aN)us(aC)

), p > 1

2 .

Point b represents a point where the deviation gains for the buyer outweigh the benefits of coop-

eration. Further, δddb > GbGb+ηb

as the latter is a solution to δηb = (1− δ)Gb and the former is a

solution to δ (ηb + µxθ − µ0) = (1− δ) maxGb,

∣∣xθ∣∣− ηb, with LHS smaller than ηb and, the RHS

potentially higher.

After md can’t be sustained in equilibrium, the next-best choice for the intermediary is either ms

or ml, while direct buyers have to resort to the best sole sourcing strategy. We will first consider

the performance of the mediated structure without taking into account the possibility of using ml,

and after that we will formulate how ml alters the situation.

πds (δ) = maxπdsi (δ) , πdsi (δ)

. For δ > δdsk , πds (δ) = πdsk , while the maximum value πms (δ)

can take is πms < πdsk . Hence, until πds (δ) becomes less than πms (δ) for the first time (going from

right to left), δ′1, direct sourcing will perform better. Until they “cross” the next time, δ′2 - mediated

sourcing will perform better, and so forth until one of the sole sourcing strategies will cease to be

an equilibrium. This is described in Parts 1b, c and 3a, b or the proposition.

Next, as the intermediary also has strategy ml at its disposal, we need to exclude regions where

ml performs better than ds from the area where direct sourcing performs better (Part 3 of the

Proposition) and add such areas to the dominance regions of mediated sourcing (Part 2 of the

Proposition). So, if πds (δ) ≤ πml (δ) = πml, or in other words, there exists a δ′ , the mediated

structure will perform better in the region(δml, δ

′]. Though, as we know that in region

(δmd, δdd

]the intermediary will always choose md, we exclude this region.

sourcing through intermediaries

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