Working Paper Series Innovation.pdfsourcing process should incentivize innovation and also allow for...

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Sourcing Innovation: Integrated System or Individual Components?

Zhi Chen

INSEAD, [email protected]

Jürgen Mihm INSEAD, [email protected]

Jochen Schlapp

Frankfurt School of Finance & Management, [email protected]

Scholars have developed a sound theory of how to design procurement mechanisms for sourcing a simple innovative product. In practice, however, many (if not most) procurement efforts involve complex products consisting of multiple interacting components. Yet theory offers few guidelines regarding the design of procurement efforts for such complex innovations-despite an abundance of pertinent questions. Of these, perhaps the most crucial is whether a firm procuring a complex innovation should purchase (a) an integral product from a single supplier or rather (b) the product's constituent components from (possibly) different suppliers. We use a game-theoretic model to identify the two most critical factors for answering this question: the extent of the required innovation and the size of the supplier base. We also characterize favorable designs of procurement contests that aim to source (respectively) a full product or individual components. Keywords: Innovation Contest; Procurement; Tournament; Product Component; Modular System

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1. Introduction

Innovative products are a key source of competitive advantage in many industries (Hall and Rosen-

berg 2010, Pisano 2015). Firms have traditionally tried to innovate by undertaking internal research

and development (R&D). More recently, however, the focus has shifted to an alternative source

of innovation: the firm’s supplier base (Cabral et al. 2006). When thus procuring innovation from

the outside, companies inevitably face the challenge of how best to structure their procurement

efforts toward the end of fully exploiting and even further stimulating their suppliers’ innovative

potential. The design of effective procurement mechanisms for innovative products—and, in partic-

ular, the design of innovation contests—has accordingly received growing attention in the academic

literature. Yet this research has focused almost exclusively on the procurement of a single product

(e.g., Taylor 1995, Terwiesch and Xu 2008). In reality, though, most custom-engineered innovations

consist of multiple interacting components that together form the final product. More technically,

in the case of complex innovations there are different modules that jointly determine the perfor-

mance of an integrated system. This paper considers a firm that intends to source such a complex

innovation. We address in particular the critical question of when that firm should purchase the

full product as a whole, and when it should rather procure the product’s components individually

from (possibly) different suppliers.

The automotive industry exemplifies the effective use of suppliers as a source of complex inno-

vations. Many such innovations, especially electronic and mechatronic advancements, have indeed

been driven mainly by suppliers. Consider, for instance, recent advances in automotive lighting

systems—with which one of this paper’s authors has experience as regards sourcing practices. Over

the course of the last century, the initial tungsten filament technology (introduced to vehicle design

in 1898) was replaced first by halogen lighting in 1962 at the urging of a consortium of European

manufacturers of bulbs and headlamps. This development was in turn followed by the introduction

of xenon lamps in 1991, driven by the supplier Hella, and LED headlamps in 2006 and 2007, which

was driven by the suppliers Koito and Magneti Marelli. It is worth noting that LED headlamps

may lead to improved traffic safety because they allow for selectively lighting, and thereby drawing

attention to, objects in the road or near the side of a street (e.g., street signs, children running

toward the street).

This lighting system example displays some interesting features that are common to the pro-

curement of complex innovations. First, it is typically the supplier that has a deeper grasp of the

underlying light technology and hence has a clearer vision of what can and cannot be done in

the next major or minor iteration of the lighting system. Second, the car manufacturer’s main

goal—with respect to high-end innovative lighting systems—is to incentivize and exploit innova-

tion on the part of its supplier base (i.e., innovation that will distinguish the automaker from its

1

2 Article Short Title

competitors) and not to instill a mechanism that ensures the lowest supply cost. Third, advanced

lighting systems consist of at least two major components: a light module, which is part of the car’s

exterior design and is designed to emit light; and an electronic control module, which regulates

the light module’s functionality. For the “selective” lighting technologies alluded to previously, the

control module will need to feature a considerable amount of new functionality and so is a key

success factor. Fourth, the light module and the electronic control module must clearly function

together and yet their technological interface is relatively straightforward; since their integration

is practically trivial, these modules can be sourced separately.

The inferior technological understanding of automakers, when combined with their goal of incen-

tivizing innovation, raises the question of how a car manufacturer can best create a sourcing process

that is tailored to foster innovation (rather than simply to reduce cost) in its supplier base. That

sourcing process should incentivize innovation and also allow for enough flexibility that the best

design can be selected after completion of the most risky steps in the development process. Com-

plex systems typically contain multiple components that can often be sourced separately, which

raises the question of whether—when attempting complex innovation—a firm should buy the full

system from a single supplier or, instead, source the individual components from different suppliers.

With respect to sourcing, the academic literature has promoted a mechanism that is information-

ally parsimonious, induces high innovation efforts, and allows for an ex post selection of the best

product alternative: the contest (Rogerson 1989, Taylor 1995, Terwiesch and Xu 2008). Under this

mechanism, the contest holder announces a challenge and promises a prize for the best submission.

Contestants then compete, at their own expense, to identify and present the best solution and

thereby win the prize. The contest holder selects the winner ex post after reviewing all contestants’

solutions. Automotive firms roughly follow a contest structure when sourcing innovative products,

as confirmed by our own experience and well documented in an extensive practitioner-oriented case

literature (e.g., Liker and Meier 2006, McIvor et al. 2006, Aoki and Wilhelm 2017). In organizing

a procurement contest, the car manufacturer announces only “inspirational” specifications for an

innovative but not yet developed product (e.g., an LED headlamp) because it lacks the knowl-

edge that would be required for giving a full specification (Bonaccorsi and Lipparini 1994). The

promised award—which typically is a valuable supply contract for the winning design (Cooper and

Slagmulder 1999, p. 108)—incentivizes the suppliers to engage in development activities at their

own cost in which they consider both technological and aesthetic aspects of the product (McIvor

2001). In what may end up being an iterative process, the manufacturer evaluates the submitted

designs and then selects the best one ex post (Langner and Seidel 2009).

The literature on contests has developed a thorough understanding of how contest-based procure-

ment processes for innovative goods should optimally be structured. That research has provided

Article Short Title 3

insight on, inter alia, the optimal number of participants (Taylor 1995, Fullerton and McAfee 1999,

Che and Gale 2003) and the optimal award structure (Moldovanu and Sela 2001, Terwiesch and

Xu 2008, Ales et al. 2017b). However, scholars have not systematically addressed the question of

how best to source a complex innovation that comprises multiple components. In such cases, we

ask whether the firm should hold separate component contests (i.e., for the product’s constituent

elements) or rather a single system contest that requires suppliers to propose complete products.

This question of contest format arises regularly in practice. Consider again our automotive head-

lamp example: Whereas the light source and the electronic control module of the standard xenon

headlamps in the Volkswagen Golf VI were both sourced from supplier Hella, the high-end LED

lamps and the corresponding control modules in the Volkswagen Golf VII were being purchased

from suppliers Hella and Keboda, respectively.

In this paper we develop a game-theoretic model that enables us to determine the circumstances

under which a buyer should prefer a system contest over a component contest (and vice versa).

We identify different contingencies—regarding the product and also the supplier base—that affect

a buyer’s optimal choice. In terms of product characteristics, we focus on the extent of innovation

required to develop each component and on the technological relationship between components.

For instance, do both components require only incremental innovation or must one (or both) of

them be radically new? Also, are the components best described as technological substitutes or

complements? In terms of supplier characteristics, we examine the role played by the size of the

supplier base as well as the fact that different supplier characteristics may lead different suppliers

to develop systems whose components exhibit different levels of performance correlation. Thus we

ask how many suppliers should be asked to participate in a procurement contest, and how the

optimal contest design is affected by the likelihood that a supplier that is good (resp. bad) at

developing one component is also good (resp. bad) at developing other components. In short, we

identify a contingency plan for how to structure the sourcing of complex innovation.

In detail, the contributions of this paper to the literature are twofold. First, we show that the

firm should use a system contest if all product components are merely incremental innovations

and the firm’s supplier base is relatively small. However, if at least one component requires radical

innovation or if the firm has a relatively large supplier base, then it is preferable to hold a separate

procurement contest for each individual component. As an interesting side result we find that these

general guidelines are independent not only of the technological relationship between components

but also of any performance correlation between those components.

Second, once the decision has been made about what type of contest (i.e., system or component)

to hold, we are interested in discovering the best way to organize that contest. More specifically,

for each contest format we identify which suppliers should be invited to participate. Our results


give rise to the following managerial recommendations. When holding component contests, the firm

benefits from inviting the same set of suppliers—rather than different sets of suppliers—to each

individual contest; these benefits manifest themselves particularly when components are technolog-

ical complements. When holding a system contest, the firm’s optimal choice of suppliers depends

on the degree of innovation in the desired product. In the case of incremental innovations, it is best

to invite suppliers that exhibit relatively low levels of performance correlation across components

(e.g., suppliers with a divisional organizational structure); yet when radical innovation is required,

the firm should seek higher performance correlation (e.g., from suppliers with an integrated devel-

opment process).

In addition to characterizing optimal structures for procuring complex innovations, this paper

makes a more subtle and purely conceptual point. Previous academic work on contests has most

often been motivated by the observation of online crowdsourcing platforms (e.g., Terwiesch and Xu

2008, Bimpikis et al. 2018), and the research questions typically asked reflect that setting. However,

it is often the case that also industrial procurement efforts for innovative products can best be

characterized as contests. Therefore, reframing some industrial sourcing processes as contests can

open up a set of novel research directions—only one of which is explored in this paper.

2. Related Literature

The design of effective procurement mechanisms has been a long-standing concern in the academic

literature (Vickrey 1961, Rob 1986, Laffont and Tirole 1993, Elmaghraby 2000, Beil 2010). There is

in particular a vast body of work, originating in the economics literature, that focuses on sourcing

a single component at the lowest cost (Rob 1986, Dasgupta and Spulber 1990, Che 1993); however,

the operations management (OM) literature has since accounted for a number of factors other than

cost. These factors include product quality (Beil and Wein 2003), delivery lead time (Cachon and

Zhang 2006), transportation cost (Chen et al. 2005, Kostamis et al. 2009), supplier qualifications

(Wan and Beil 2009, Wan et al. 2012), supplier reliability (Chaturvedi and Martınez-de-Albeniz

2011, Yang et al. 2012), the integration of quantity and price decisions (Chen 2007, Duenyas

et al. 2013), purchase pooling (Gur et al. 2017), future changes in competitive structures (Li and

Debo 2009), long-run stability of the supplier base (Chaturvedi et al. 2014), the deployment of

a test auction to learn suppliers’ cost (Beil et al. 2018) and suppliers’ motivation to invest in

cost reduction efforts (Li and Wan 2016). Research in the OM field has even addressed issues of

sourcing multiple items simultaneously (see, e.g., Elmaghraby and Keskinocak 2004, Chu 2009,

Olivares et al. 2012). Although these papers widely differ in what aspect of procurement they focus

on, they all agree in proclaiming the auction as their preferred procurement mechanism. In this

sense, much of the procurement literature can be viewed as a subset of the literature on auction


theory (initiated by Vickrey 1961). However, there is an explicit assumption in all models based on

auction mechanisms—namely, that the product to be auctioned can be fully characterized ex ante

by verifiable metrics. In other words, the good being auctioned is fully specified and so the main

focus of the procurement process is to reduce information asymmetries (e.g., regarding production

cost) between the buyer and suppliers. Such assumptions capture well the procurement of standard

goods (e.g., tires in the automotive sector), and hence much of the extant procurement literature

concentrates on devising optimal procurement mechanisms for those standard products (see, e.g.,

Cachon and Zhang 2006, Chen 2007, Wan and Beil 2009, and the references therein).

In the procurement of an innovative product, however, neither the buyer nor the supplier can

fully envision or specify—at the start of the procurement process—the product’s final performance

or its final feature set. It is therefore impossible for a supplier to credibly and verifiably stipulate

a product’s future characteristics. Furthermore, the buyer cannot identify ex ante (i.e., prior to

development of the product) which supplier should be preferred. In this situation, the procurement

mechanism’s main function is not so much reducing asymmetric information about a known entity

but rather to discover and incentivize innovation.

Both theory and practice have demonstrated that holding a contest is the best way to overcome

the auction’s limitations and is hence the most effective means for sourcing innovative products

(Lichtenberg 1988, Rogerson 1989, Elmaghraby 2000, Scotchmer 2004, Deng and Elmaghraby 2005,

Cabral et al. 2006). In a procurement contest for an innovative good, the buyer gives only a rough

description of its vision for the product, announces a prize, sets a deadline at the start of the contest,

and ultimately awards the prize to the contestant who has submitted the best product (Taylor

1995). Thus there is no need for the performance of the submitted product(s) to be verifiable,

because the buyer has no reason to manipulate the contest outcome—that is, because the award

must be paid out to one of the contestants in any case. As a result, contests are both easy to

implement and “informatively parsimonious” (Cabral et al. 2006). Yet, the competition for the

contest prize induces effort in the supplier base and the ex post selection of the winner allows the

buyer to choose among many different solutions.

Inspired by these benefits, a wealth of research has evolved from the question of how best to

design and manage effective innovation contests. Building on the pioneering work of Lazear and

Rosen (1981), Green and Stokey (1983), and Nalebuff and Stiglitz (1983), this literature offers

rich insights into such fundamental issues of contest design as the structure of optimal award

schemes (Moldovanu and Sela 2001, Siegel 2009, 2010, Ales et al. 2017b), open versus closed entry

to the contest (Taylor 1995, Fullerton and McAfee 1999, Terwiesch and Xu 2008, Korpeoglu and

Cho 2017), bid caps (Gavious et al. 2002) and auctions as efficient ways of limiting access to

contests (Fullerton and McAfee 1999), motivating internal employees (Nittala and Krishnan 2016)


and motivating participation more generally (Stouras et al. 2017), helpful problem specifications

and design spaces for open-ended design contests (Erat and Krishnan 2012), using intermediate

feedback to affect the contest as it unfolds (Aoyagi 2010, Jiang et al. 2016, Mihm and Schlapp 2017,

Bimpikis et al. 2018), employing dynamic formats such as elimination and round-robin contests

(Yucesan 2013), and holding repeated contests among the same contestants (Konrad and Kovenock

2009). Taken together, the papers cited here present a coherent theory of how a contest can be

used to source a single innovative product.

However, we are more interested in the design of a procurement contest for a multi-module

product: How should a buyer simultaneously procure multiple heterogeneous—but technologically

dependent—product components? This question augurs a new subfield of contest research that has

not, with the exception of Hu and Wang (2017), received adequate attention. The focus on multi-

object contests entails that our paper and that of Hu and Wang share some technical primitives.

Yet those authors focus on the temporal aspects of contest design—that is, on whether two contests

should be held sequentially or rather in parallel—and so we end up posing research questions and

we thus formulate model setups that differ substantially from theirs.

In particular, we make the following contributions to the theory of procurement contests. First,

we show which contest format is the best to use for sourcing complex innovative products that

consist of multiple components; for that purpose, we determine which product characteristics and

which supplier base properties govern the optimal choice. By considering the many different factors

that affect this choice, we cover a large variety of common products found in practice.

Our second main contribution is to shed light on the ideal structure for each type of contest.

Thus we show how different supplier characteristics that influence performance correlation across

different component development projects call for subtle variations in the ideal structure of an

associated contest’s format. In particular, we focus on the question of what type of supplier to

invite to which contest format.

3. The Model

Consider a firm (hereafter “the buyer”) that wishes to organize a procurement contest among

n≥ 2 suppliers toward the end of acquiring an innovative product that comprises two components

j ∈ {1,2}. To source the product, the buyer can hold either a system contest or a component

contest. In a system contest, the buyer asks the suppliers to submit solutions for the full product

(i.e., the integrated system) and promises an award A> 0 to the supplier that delivers the product

with the best performance. In a component contest, the buyer holds a separate contest for each

of the two components and offers an award A1 = pA (resp., A2 = (1− p)A) with p ∈ [0,1] to the

supplier that delivers the best-performing component j = 1 (resp., j = 2).


Table 1 Performance of the Full Product

System contest Component contest

Technological substitutes Ssyssub = maxi{vi1 + vi2} Scposub = maxi{vi1}+ maxi{vi2}

Technological complements Ssyscml = maxi{min{vi1, vi2}} Scpocml = min{maxi{vi1},maxi{vi2}}

The buyer’s goal is to maximize the performance S of the product to be procured. Because the

full product consists of two components, its performance depends on two attributes as follow: (i) the

individual performance vj ∈R of each component j, and (ii) the technological relationship t : R2→

R between the two components. With regard to the second attribute, we follow the literature and

suppose that—depending on the context—components can be either substitutes or complements

(see, e.g., Roels 2014, Gurvich and Van Mieghem 2015, Bhaskaran and Krishnan 2009, and the

references therein). If the two components are technological substitutes then the buyer can trade

off their respective performance, in which case the integrated system’s performance is simply the

sum of the performance of its individual components. If the two components are technological

complements, then both are essential to the integrated system’s performance; in this case, then,

overall performance is the minimum of the components’ respective performance.

Both the contest format and the components’ technological relationship determine the perfor-

mance S of the integrated system. To see this, let vij be the performance of component j ∈ {1,2}

as developed by supplier i∈ {1, . . . , n}. In a system contest, each supplier i submits an integrated

system with performance Si = t(vi1, vi2) = vi1 + vi2 if the components are substitutes or with Si =

t(vi1, vi2) = min{vi1, vi2} if they are complements. Then the buyer chooses the product with the

highest performance; that is, Ssys = maxi{Si}. In a component contest, however, the buyer procures

the two best components v1 = maxi{vi1} and v2 = maxi{vi2} individually and so the performance

of the full product is Scpo = t(v1, v2) = v1 + v2 if the components are substitutes or Scpo = t(v1, v2) =

min{v1, v2} if they are complements. Table 1 summarizes the system’s performance as a function

of (a) the contest format and (b) the technological relationship between components.

3.1. Sequence of Events

The procurement process begins when the buyer publicly announces the contest format (i.e., a

system or a component contest), the total award A, and—in the case of a component contest—the

prize split p. In a procurement setting, A is most usefully viewed as the value of the supply contract

to the suppliers. (For now we assume symmetrical suppliers in the sense that A is the same for all

of them. In Section 6 we extend our base model to account for heterogeneity in suppliers’ award

valuations, in which case each supplier i obtains a different utility αiA, αi ∈ (0,1], from winning

an award of size A.) As is common in the contest literature, A is exogenous and so the buyer’s

decisions are to choose the contest format and the prize split p (if the component contest is picked).


Figure 1 Sequence of Events

Once the buyer has publicly announced the contest, each supplier i∈ {1, . . . , n} decides whether

to participate in any of the offered contests. If participating, supplier i invests an unobservable

solution effort eij ≥ 0 to develop component j ∈ {1,2} at a private cost c(eij), where c is a twice con-

tinuously differentiable, increasing, and strictly convex function with c(0) = 0 and c′(0) = 0. Each

supplier then receives a stochastic performance shock ζij. The final performance of component j

as developed by supplier i is hence given by vij = r(eij) + ζij, where r(·) captures the deterministic

relationship between effort and performance. We assume that r is a twice continuously differen-

tiable, strictly increasing, and concave function with r(0) = 0 and limx→∞ c′(x)/r′(x) =∞; note

that the realization of ζij is supplier i’s private information. For each supplier i, ζi = (ζi1, ζi2) follows

a bivariate Normal distribution with marginal distributions ζi1 ∼ N(0, σ2) and ζi2 ∼ N(0, k2σ2)

and correlation ρ ∈ [0,1), where σ > 0 reflects the components’ inherent innovativeness and k ≥ 1

captures the asymmetry in innovativeness between components. The term ζi is assumed to be

independent across suppliers.

Finally, after receiving the suppliers’ submissions, the buyer evaluates their respective perfor-

mance and awards the pre-announced prize to the supplier with the best performance. In a system

contest, supplier i wins prize A if t(vi1, vi2)> t(vl1, vl2) for all l 6= i. In component contest j, sup-

plier i analogously wins prize Aj if vij > vlj for all l 6= i. In all cases, ties can be broken by invoking

an arbitrary rule. All primitives of the model are common knowledge. Figure 1 summarizes this

sequence of events.


3.2. Model Implications

In presenting this model, we have deliberately ignored integration costs. This approach is equivalent

to assuming either that integration costs are negligible or that they are much the same regardless of

whether integration is performed by the supplier (in the case of a system contest) or by the buyer

(in the case of component contests). Either assumption seems to be appropriate for a wide range of

real-world products—for instance, many automotive systems consisting of a mechanical component

and an electronic control module (as in our previous headlamp example). Such products have

a straightforward interface between components, which simplifies the integration task and thus

emphasizes component development. However, there are many other products for which integrating

components is itself a major task. For such products, this model is not a good approximation of

their procurement.1

Note that the buyer’s goal is to maximize the expected performance of the best integrated system

that can be built based on the suppliers’ submissions: Π =E[S], where S is as described in Table 1.

Unlike the buyer, the suppliers are not concerned with the absolute performance of their respective

submissions; instead, suppliers are interested only in their relative performance vis-a-vis that of

their competitors. More precisely, each supplier’s primary interest is to win any of the contests

in which it participates. In a system contest, supplier i gains utility Ui = A− c(ei1)− c(ei2) if it

wins the contest or Ui = −c(ei1) − c(ei2) if it loses; in a component contest, supplier i’s utility

from winning component contest j is similarly Uij =Aj − c(eij) whereas losing leads to a utility of

Uij =−c(eij). We thus normalize the utility of each supplier’s outside option to zero, which implies

that all suppliers find it worthwhile to participate in any of the offered contests.

We are interested in symmetric, pure-strategy, perfect Bayesian Nash equilibria of the different

contest formats. To ensure that such equilibria exist, we must invoke two additional technical

assumptions.

Assumption 1. For components that are technological substitutes (resp. complements), we

assume that σ > σsub (resp. σ > σcml); here σsub (resp. σcml) is the smallest σ such that, for all

σ > σsub (resp. σ > σcml), supplier i’s expected utility function has a unique maximum for any

contest format provided that all other suppliers exert effort xj ≥ 0 for all j ∈ {1,2}.

Assumption 1 is commonly invoked in the contest literature (see, e.g., Nalebuff and Stiglitz 1983,

Aoyagi 2010, Ales et al. 2017a, Mihm and Schlapp 2017). In a nutshell, it requires that the per-

formance uncertainty involved in any of the contest formats be sufficiently large—a condition that

1 However, our model can easily accomodate some extensions. Suppose, for example, that the buyer and suppliers dopay different but fixed integration costs Ib and Is (respectively); then our analysis still goes through but with theresults containing one additive term ∆IC = Ib − Is. The intuition for ∆IC is that, compared with our main results,it extends the region for which system contests dominate component contests (if it is positive) or shrinks that region(if it is negative).


is typically true for innovative products. To appreciate the intuition underlying this assumption,

suppose to the contrary that performance uncertainty is negligible. Then any pure-strategy equilib-

rium must be symmetric, for otherwise the losing player(s) could profitably deviate by exerting no

effort at all; yet under symmetric pure strategies, each supplier could profitably deviate by raising

effort by even an infinitesimal amount and thus could almost surely win the contest. In sum, no

pure-strategy equilibrium could exist. It is also easy to show that the thresholds σsub and σcml

always exist because each supplier’s utility function becomes strictly concave for large enough σ.2

Assumption 2. The following inequality holds: 3r′′(x)r′(x) <

c′′′(x)r′(x)−c′(x)r′′′(x)c′′(x)r′(x)−c′(x)r′′(x) for all x≥ 0.

Although Assumption 2 looks unwieldy, it is but mildly restrictive and is satisfied by many

standard functional relationships for r and c that appear in the contest literature (e.g., polynomial

or logarithmic r, and polynomial or exponential c). Furthermore, this assumption has an impor-

tant managerial interpretation: it ensures that the buyer is actually interested in procuring both

components. Were this condition not satisfied, the buyer could find it optimal to buy only one

component, and—at the end of the procurement process—be left with an incomplete product. We

omit this case from further consideration because it does not reflect any practical circumstances of

interest.

4. Optimal Contest Format

In this section, we characterize the buyer’s optimal choice of contest format and detail how that

choice is affected by the following contextual parameters: (i) the extent of innovation involved

in developing each component, as measured by σ, and the two components’ relative extent of

innovativeness, as measured by k; (ii) the technological relationship between components; (iii) the

size n of the supplier base; and (iv) the correlation ρ between suppliers’ performance. We shall start

with the simplest case and identify the optimal contest format for a product whose components

are technological substitutes (Section 4.1), after which we characterize the optimal contest format

for a product with complementary components (Section 4.2). To streamline the presentation and

focus on the managerial implications of our analysis, we do not derive any mathematical results in

the main text. Readers who are technically inclined may refer to the Appendix for proofs.

4.1. Technological Substitutes

We best answer the question of which contest format is superior for the buyer by building an under-

standing of (a) the main benefits of each contest format and of (b) how those benefits are affected

2 In fact, when deriving more explicit sufficient conditions for the existence of symmetric pure-strategy equilibria,papers in the contest literature typically focus on verifying the concavity of a supplier’s utility function (Aoyagi 2010,Ales et al. 2017a).


by our parameters. Recall that vij, the performance of component j as developed by supplier i,

depends on the supplier’s development effort eij and also on a technological shock ζij. Hence the

optimal contest format is the one that best allows the buyer to incentivize significant development

efforts and also to hedge against (resp., benefit from) negative (resp., positive) technological shocks.

A system contest is well suited to incentivizing effort because its monetary stakes are higher than

those in a component contest. That said, a component contest allows the buyer to choose (and

combine) the two best components ex post; thus it provides an excellent hedge against—and a good

opportunity to exploit—the high technological uncertainty typical of many innovation projects.

Before weighing the benefits of the two contest formats, we analyze the effect that those formats

have on suppliers’ innovation efforts.

Lemma 1. Suppose that components are technological substitutes.

(i) For k= 1, the unique symmetric perfect Bayesian Nash equilibrium for both system and com-

ponent contests satisfies the following properties: (a) the buyer’s optimal award split in a component

contest is p∗ = 1/2; and (b) esys1 = esys2 > ecpo1 = ecpo2 > 0.

(ii) For k→∞, the unique symmetric perfect Bayesian Nash equilibrium for both system and

component contests satisfies the following properties: (a) the buyer’s optimal award split in a com-

ponent contest is p∗ = 1; and (b) esys1 = esys2 = 0 whereas ecpo1 > ecpo2 = 0.

Lemma 1(i) addresses the case where the two components require an equal level of innovation

(i.e., k= 1); thus both components bear the same level of risk and opportunity during development.

It follows that, ex ante, each component has an equally strong effect on the integrated system’s

performance; this is why both the buyer (in terms of the award split) and the suppliers (in terms

of effort) “spread their bets” equally across both components.

To understand the intuition behind this result in greater detail, consider the buyer’s and suppli-

ers’ decisions in a component contest. The buyer seeks to maximize the performance of submitted

components, which necessitates maximizing the suppliers’ development efforts in each of the com-

ponent contests. Since suppliers invest more effort in contests that feature a higher promised award,

it follows that the buyer must carefully balance the distribution of awards between the two com-

ponent contests. But why is it optimal for the buyer to split the total award equally (i.e., for

p∗ = 1/2)? Recall from Section 3.1 that for each supplier, the marginal return decreases in effort

but the marginal cost increases in effort. It is therefore most efficient for the buyer if each supplier

invests equal effort in each of the two component contests, and the buyer can induce such a bal-

anced distribution of effort only by offering the same award in each contest. As a result, equilibrium

efforts are also identical across component contests: ecpo1 = ecpo2 .


A similar reasoning applies to system contests. Yet in these contests it is not the buyer that is

most interested in an efficient distribution of efforts across components (since the buyer cares only

about the integrated system’s final performance); it is rather the suppliers that try to maximize

the efficiency of their effort provision. As before, each supplier’s marginal return (resp. costs)

decreases (resp. increase) in effort; hence each supplier splits its efforts evenly between components:

esys1 = esys2 .

Finally, the most intriguing result in Lemma 1(i) is the relation between the equilibrium efforts in

a system contest versus a component contest: if components require the same extent of innovation

(i.e., k= 1), then suppliers unequivocally invest higher efforts in a system contest. This monotonic-

ity is worth mentioning because the change from a system contest to a component contest results

in two diametrically opposed effects. On the one hand, the award for which the suppliers compete

is twice as high in a system contest as in each of the individual component contests (A vs. A/2); of

course, higher financial stakes induce greater effort. On the other hand, each supplier encounters

more uncertainty in a system contest than in two separate component contests, and greater risk

will tend to deter suppliers from investing in development effort. Lemma 1(i) states that the former

effect, which favors an increase in effort, always dominates the latter effect for components with

k= 1 (i.e., for components requiring an equivalent level of innovation).

It is interesting that Lemma 1(ii) reverses this finding when one component requires a signifi-

cantly higher extent of innovation than the other (i.e., as k→∞). More specifically, if components

have a sizeable difference in their levels of required innovation then the system contest’s primary

benefit, which is to induce greater development efforts, is entirely absent. The reason is that, for

large k, performance of the full product is determined mainly by the performance of the more

innovative (and thus riskier) component—and for such a component it is largely technological

uncertainty, and not effort, that ultimately determines performance. In other words: since develop-

ment efforts play a subordinate role in determining the full product’s final performance, suppliers

refrain from exerting high levels of effort. In a component contest, however, effort still matters for

the less innovative component. For this component the suppliers’ (and also the buyer’s) return on

effort expended is substantially larger than for the more innovative component; hence the suppliers

dedicate their development efforts to the less innovative component. This dynamic explains why,

as k→∞, equilibrium development efforts are greater in a component contest than in a system

contest.

With σsub as in Equation (22) in the Appendix (its precise definition is not material to the

interpretation of results), Proposition 1 allows us to characterize the buyer’s optimal choice of

contest format.


Figure 2 Optimal Contest Format

1

1.2

1.4

1.6

0.15 0.25 0.35 0.451

1.4

1.8

2.2

0.12 0.18 0.24 0.31

1.2

1.4

1.6

0.15 0.25 0.35 0.451

1.4

1.8

2.2

0.12 0.18 0.24 0.3

Note. The graphs show a buyer’s optimal choice of contest format for technological substitutes (left panel) and

technological complements (right panel). The buyer prefers a component contest in the white area (region C) but

prefers a system contest in the light gray area (region S); no pure-strategy equilibrium exists in the dark gray area

(region N). The parameters used here are r(x) = x, c(x) = x2, n= 2, A= 1, and ρ= 0.

Proposition 1. Suppose components are technological substitutes. Then the following profit

relations hold in equilibrium.

(i) If σ ∈ (σsub, σsub), then there exists a threshold ksub > 1 such that Πsyssub >Πcpo

sub if k ∈ [1, ksub).

There is also an n such that, for n>n, (σsub, σsub) = ∅.

(ii) For any fixed σ > σsub, there exists a threshold ksub <∞ such that Πcposub >Πsys

sub if k ∈ (ksub,∞).

(iii) For any fixed k≥ 1, there exists a threshold σsub <∞ such that Πcposub >Πsys

sub if σ ∈ (σsub,∞).

This proposition establishes the buyer’s optimal contest format choice for a product with substi-

tute components; it shows that this choice depends both on the level of innovation required for the

two components and on the size of the supplier base—but not on performance correlation. The left

panel of Figure 2 illustrates Proposition 1 for a numerical example. Two key managerial insights

are evident.

First of all, the buyer should use a system contest to procure the full product only if two

conditions are simultaneously satisfied: (i) both components require only incremental innovation

(i.e., σ < σsub and k < ksub, as in Proposition 1(i)); and (ii) the supplier base is relatively small

(small n, also as in Proposition 1(i)). If either of these two conditions is violated—that is, if

at least one component requires radical innovation (high k or high σ, as in part (ii) or (iii),

respectively, of the proposition) or if the buyer’s supplier base is large—then the buyer should

opt for a component contest and procure components individually. The intuition for this result

is instructive. For incrementally innovative components and a small supplier base, the supplier’s

development efforts have a sizeable effect on the final product’s performance. Hence the buyer


chooses the contest format offering the highest effort incentives, which is a system contest. By

Lemma 1, however, the effect of these effort incentives decreases quickly when at least one of the

components requires more radical innovation. In such a setting (i.e., when the effort–performance

link becomes more tenuous) the possibility of combining the best components ex post, which is

the main advantage of a component contest, becomes more valuable; therefore, the buyer prefers

a component contest.

The second managerial insight concerns how the size of the supplier base affects the buyer’s choice

of contest format. Two mutually reinforcing effects are at work here. From a supplier’s perspective,

the presence of more competitors reduces the odds of winning the contest and thereby leads each

supplier to invest less effort in development; a component contest thus becomes more attractive.

From the buyer’s perspective, the likelihood of a single supplier developing an exceptional overall

system increases—but slowly—with the number of suppliers; in contrast, the likelihood of a single

supplier developing an exceptional design in either of the component contests increases much more

rapidly with the number of suppliers. This dynamic, too, makes the component contest more

attractive. We therefore conclude that, if n is sufficiently large, then the buyer should hold a

component contest irrespective of the components’ required extent of innovation.

From a practical standpoint, Proposition 1 can be summarized by the following decision rule.

If the buyer has access to a large supplier base, which is typically the case in crowdsourcing

environments, then a separate contest should be held for each component. Yet if there are only

a handful of suppliers—which often occurs for complex technological products (e.g., automotive

lighting systems)—then it is the level of innovation required of the components that ultimately

determines whether a system or component contest should be held.

4.2. Technological Complements

In the previous section we identified the optimal contest format when the buyer’s intention is to

procure a product whose components are technological substitutes. At this point, it remains an

open question whether the technological relationship between the components affects our findings

and thus our managerial recommendations. The main purpose of this section is to answer that

question.

As a starting point for our discussion, suppose the buyer hosts a component contest (i.e., a sepa-

rate contest for each component) and consider a supplier that participates in each of the component

contests. Because it cares only about winning each of the contests, the supplier focuses on develop-

ing the best component for each one and is not concerned about how those components contribute

to the full product’s overall performance. In other words, a component contest induces suppliers

to consider each component in isolation and so their effort choices are unaffected by whether the


components are technological substitutes or complements. In a system contest, however, the sup-

plier may well consider the technological relationship between components for its effort choices.

Lemma 2 is the analogue of Lemma 1: both compare suppliers’ equilibrium efforts in a system

versus a component contest, but Lemma 2 addresses the case of technological complements.

Lemma 2. Suppose that components are technological complements.

(i) For k = 1, there exists a symmetric perfect Bayesian Nash equilibrium for both system and

component contests that satisfies the following properties: (a) the buyer’s optimal award split in a

component contest is p∗ = 1/2; and (b) esys1 = esys2 > ecpo1 = ecpo2 > 0.

(ii) For k→∞, the unique symmetric perfect Bayesian Nash equilibrium for both system and

component contests satisfies the following properties: (a) the buyer’s optimal award split in a com-

ponent contest is p∗ = 1; and (b) ecpo1 > esys1 > 0 whereas ecpo2 = esys2 = 0.

Lemma 2 shows that our most important observations for technological substitutes immediately

transfer to the case of technological complements. As of part (i) of the lemma, for products that

require both components to exhibit the same level of innovation (i.e., k = 1), it remains true that

buyer and suppliers split their investments equally across both components and that suppliers

always invest more development effort in a system contest than in a component contest. As of

part (ii) of the lemma, for systems in which one of the components requires a significantly greater

extent of innovation than the other (i.e., as k→∞), component contests induce more effort than

do system contests—as was also true for substitutes. Note that these findings do not depend on

the size of the supplier base or on each supplier’s performance correlation.

Based on this observation, it is no longer surprising that structurally our results regarding the

buyer’s optimal contest format choice carry over from the case of technological substitutes to the

case of technological complements. With σcml as in Equation (27) in the Appendix (again, its

precise definition is not material to the interpretation of results), we have the following result.

Proposition 2. Suppose that components are technological complements. Then the following

profit relations hold in equilibrium.

(i) If σ ∈ (σcml, σcml) then there exists a threshold kcml > 1 such that Πsyscml >Πcpo

cml if k ∈ [1, kcml).

There is also an n such that, for n>n, (σcml, σcml) = ∅.

(ii) For any fixed σ > σcml, there exists a threshold kcml < ∞ such that Πcpocml > Πsys

cml if k ∈

(kcml,∞).

(iii) For any fixed k≥ 1, there exists a threshold σcml <∞ such that Πcpocml >Πsys

cml if σ ∈ (σcml,∞).

As in the case of technological substitutes, the buyer uses a system contest when both components

require only incremental innovation and the supplier base is limited in size (i.e., when σ < σcml,


k < kcml, and n is small as in Proposition 2(i)). In contrast, with a product that requires more

radical innovation for at least one of its components (i.e., high k or high σ, as in parts (ii) and (iii)

of the proposition, respectively) or when there are many suppliers (large n), the buyer should

generally seek to implement a component contest. The reason is again that, in contests with a

large number of participants or with at least one highly innovative component, the buyer is willing

to forgo some of the suppliers’ solution efforts in favor of having the option to select, ex post, the

best components from a large set of alternatives.

From a managerial perspective, we can now distill our findings into a set of parsimonious guide-

lines: The buyer should prefer a system over a component contest only for systems with incremen-

tally innovative components and a small supplier base. This rule applies regardless of whether the

full product consists of technological substitutes or complements.

5. A Second Take on Performance Correlation

So far, we have assumed that there were no structural differences between different contest formats

in that our main modeling contingencies—the level of component innovation (σ and k), the nature

of technological interactions between components (as substitutes vs. complements), the size n

of the supply base, and performance correlation ρ—were not affected by the choice of contest

design. This assumption not only ensures comparability but is often a good approximation of

reality. For instance, it would seem contrived to suppose that the contest format affects either the

buyer’s requirements for level of innovation or the technological interaction between components.

Furthermore, there is no reason to assume that differences in the size of the supplier base should

have anything but trivial effects. Yet systematic differences in performance correlation between

different contest formats, and even within a single contest format, are relevent both theoretically

and practically.

Differences in performance correlation may arise for two reasons. On the one hand, variation

in ρ may be a distinct possibility that the decision maker must take into account when designing

a procurement contest. On the other hand—and of perhaps greater interest—is that the contest

holder may be able to vary ρ deliberately; that is, the buyer could design a contest in which

performance correlation can be systematically influenced.

It is certainly realistic to suppose that holding a system contest entails more performance cor-

relation between components than does holding two individual component contests. In a system

contest, each supplier brings its established development culture to bear on both components: it

will apply similar development methodologies and processes; it will deploy similar technical capa-

bilities and technical talent; it may even choose the same project manager. In a component contest,

however, different suppliers will likely have different development cultures. Hence one naturally

expects more correlation in the former scenario than in the latter.


For these reasons, neither buyer nor supplier should view performance correlation as an unal-

terable factor; indeed, each party can make many decisions that affect the level of performance

correlation in a contest. Consider, for example, a component contest. The buyer could decide to

allow each supplier to participate in only one contest, thereby precluding any correlation between

the two component contests. At the other extreme, the buyer could encourage as many suppliers

as possible to participate in both component contests, thus introducing a higher level of corre-

lation between the two. In a system contest, the buyer could similarly choose to invite suppliers

with specific characteristics: they could have a divisional organizational structure or an integrated

structure; they could have an extremely diverse or a more focused technology base. In this way, the

buyer can directly influence the level of performance correlation in its procurement mechanism.

In this section we systematically explore the role of correlation differentials. We begin by looking

at how differences in correlation between a system and a component contest affect the buyer’s

optimal choice of contest format (Section 5.1). We next investigate whether the buyer that holds

a component contest should prefer more or instead less correlation (Section 5.2)—and then do

likewise for the case of a system contest (Section 5.3). By adopting each of the last two perspectives

in turn, we deepen our understanding of contest design beyond the question of whether to hold a

component or a system contest.

5.1. Heterogeneity in Performance Correlation

We have argued that it is natural to expect that in a system contest—where each supplier develops

the full product rather than an isolated component—performance correlation may be at least as

high as in a component contest. To incorporate this possibility into our model setup, we let ρsys

and ρcpo denote the performance correlation in (respectively) a system contest and a component

contest and then examine how the buyer’s choice of contest format changes when ∆ρ= ρsys−ρcpo ≥

0. (In our base model, we assumed that ∆ρ= 0.) The following proposition summarizes our findings.

Proposition 3. (i) Suppose that components are technological substitutes. Then the same pref-

erence ordering between contest formats as given in Proposition 1 applies for any ∆ρ≥ 0.

(ii) Suppose that components are technological complements. Then: (a) for small k and small

σ, we have Πsyscml >Πcpo

cml for any ∆ρ≥ 0; and (b) there exists a threshold σρ > 0 such that, for all

σ ∈ (σρ,∞), there exist thresholds ∆ρ and ∆ρ with 0 < ∆ρ < ∆ρ < 1; here Πcpocml > Πsys

cml for any

∆ρ<∆ρ and Πsyscml >Πcpo

cml for any ∆ρ>∆ρ.

The results presented in Proposition 3 emphasize that, generally speaking, our initial results

on the buyer’s optimal contest format choice (as presented in Propositions 1 and 2) remain valid

even when different contest formats exhibit varying levels of performance correlation. In particular:


whereas a system contest is the buyer’s optimal choice whenever both components require incre-

mental innovation, the component contest is preferred for more innovative components. However,

part (b) of Proposition 3(ii) shows that there is one notable exception to this general rule when

the components are technological complements: if a system contest exhibits much higher levels of

performance correlation than does a component contest (i.e., if ∆ρ>∆ρ), then the buyer prefers a

system contest even when both components call for radical innovation (i.e., when σ is large). Why

is this the case? Recall that, for radical innovation, supplier efforts have a relatively small effect

on product performance; instead it is technological uncertainty that mainly determines the final

product’s performance. Yet as ∆ρ increases, more correlation—and thus, in effect, more overall

uncertainty with regard to the final product—is introduced into the system contest (note that this

effect applies only in the case of technological complements, since the full product’s performance

depends primarily on its weakest component). In contrast, the component contest exhibits less

correlation (and thus less effective overall uncertainty) with increasing ∆ρ. A threshold value of ∆ρ

is eventually reached, beyond which the system contest format is the buyer’s optimal choice when

both components require radical innovation.

5.2. Performance Correlation in a Component Contest

It is clear that the buyer, by altering the composition of the contestant pool, can systematically vary

the level of performance correlation in a component contest and hence the success of the entire pro-

curement process. A salient question is whether the same or different suppliers should be invited to

participate in each of the component contests—a decision that, respectively, increases or decreases

performance correlation across the two component contests. To address this issue of contest design

(and possibly other issues that affect performance correlation), it is imperative to understand when

the buyer should prefer more (or less) performance correlation. Our next proposition establishes

that the buyer’s preferences in this regard depend mainly on the technological relationship between

components (i.e., on whether the components are substitutes or complements).

Proposition 4. In equilibrium, the buyer’s expected profit from holding a component contest is

a function of the following dynamics.

(i) If the components are technological substitutes, then Πcposub is invariant with respect to ρ.

(ii) If the components are technological complements, then Πcpocml increases in ρ.

Part (i) of this proposition states that performance correlation has no effect on the buyer’s profits

in a component contest with substitutable components. To see why, recall that the suppliers in such

a setup consider each of the two contests in isolation; hence suppliers’ development efforts are not

affected by performance correlation. Also, since the two components are substitutes, it follows that

the buyer cannot exploit correlation to influence the effective level of technological uncertainty. So


given that neither the suppliers’ efforts nor the effective overall level of technological uncertainty is

affected by performance correlation when the components are technological substitutes, the buyer’s

profits are invariant with respect to ρ.

As shown by Proposition 4(ii), this observation does not hold for technological complements.

Even though supplier development efforts remain unaffected by performance correlation for techno-

logical complements (as they did for technological substitutes), in this case the buyer can exploit the

effect of correlation on the level of technological uncertainty. Recall that the higher the performance

correlation between individual components, the more similar they are in terms of performance. It

follows that, since the full product is only as good as its weakest component, the buyer prefers

components exhibiting similar performances to components whose performance is relatively more

heterogeneous. In short, the buyer always benefits from higher levels of performance correlation.

Proposition 4 has immediate implications for practice. A buyer always (weakly) benefits from

performance correlation in a component contest, and this finding holds regardless of whether the

components require incremental or radical innovation and irrespective of the size of the supplier

base. Moreover, the buyer can in practice take simple yet effective measures that will increase

performance correlation in its procurement contests. For instance, the buyer could induce suppliers

to participate in both contests and not limit access to only one contest. The buyer could also invite

only those suppliers that are known to develop components exhibiting well-balanced performance—

that is, rather than inviting overly specialized suppliers.

5.3. Performance Correlation in a System Contest

In section 5.2 we established that, for a component contest, the effect of performance correlation

on the buyer’s profits is driven by the technological relationship between components. Somewhat

surprisingly, in a system contest it is not the technological relationship that determines the sensi-

tivity of buyer profits to correlation; the decisive roles are played rather by the size of the supplier

base and the components’ required levels of innovation.

Proposition 5. Let µ(n) denote the expected value of the maximum of n independent and iden-

tically distributed standard Normal random variables, and define η(·) = (r ◦ (c′/r′)−1)(·).

(i) Suppose that the focal components are technological substitutes. Then Πsyssub increases in ρ if

and only if nσ2(1 + k2 + 2kρ)≥ 2Aη′(Aµ(n)/(nσ

√1 + k2 + 2kρ)

).

(ii) Suppose that the components are technological complements. Then Πsyscml decreases (resp.

increases) in ρ if k and σ are sufficiently small (resp. large).

The main finding of Proposition 5 is as follows. For products that are made of two incrementally

innovative components (i.e., when σ and k are both low) and for which the supplier base is narrow

(small n), the buyer prefers contests with a low level of performance correlation; in any other


situation (i.e., if at least one component requires radical innovation or if the supplier base is

wide), the buyer prefers contests with a high level of performance correlation. We can better

understand this result from the buyer’s perspective by recalling that, of the two factors that

determine a contest’s success—namely, the suppliers’ development efforts and the management

of performance uncertainty—it is supplier effort that drives the final product’s performance in

contests characterized by low innovation levels and a small supplier base. In contests involving

radical innovation or a large supplier base, however, the final product performance depends more

strongly on inherent technological uncertainties. It should be clear that, in the former case, the

buyer prefers conditions that lead suppliers to engage in high development efforts; whereas, in the

latter case, the buyer prefers high levels of aggregate uncertainty. Furthermore, the higher is the

performance correlation, the lower are the suppliers’ effort incentives and the higher is the effective

technological uncertainty.

Proposition 5 offers clear advice for managers overseeing a system contest. Whether performance

correlation is (or is not) beneficial depends on the extent of required innovation and the size of the

supplier base; it does not depend on the technological relationship between components. But how,

in practice, can a manager affect the level of performance correlation in a system contest? The

answer is fairly simple: by inviting those suppliers whose characteristics are apropos to participating

in the contest. For instance, a tightly integrated organizational structure with a single project

manager for both components will exhibit more performance correlation than will a divisional

organizational structure that undertakes essentially independent development efforts. Much the

same can be said of an approach to technology that relies on similar base technologies for both

components as compared with an approach that relies on diverse base technologies.

6. Heterogeneous Supplier Base

Until now we have assumed a homogeneous supplier base in the following sense: all suppliers

place an equal value on the supply contract promised by the buyer. Formally, we presume that

all suppliers behave as if the benefit from winning the supply contract is A. That assumption is,

of course, a simplification of real-world circumstances. In practice, each supplier’s perceived value

of winning the contract depends on the exact contract terms, and different suppliers may vary in

their adherence to these terms—because, for example, their cost positions differ. Thus suppliers

are actually heterogeneous with respect to their valuation of the supply contract. Here we study

the effect of a heterogeneous supplier base on the buyer’s choice of contest format by relaxing our

previous assumption of uniform award valuations.

We simplify the exposition by restricting our attention to the case of only two suppliers, and we

shall assume that supplier 1 values the supply contract more than does supplier 2. More precisely,:


Figure 3 Optimal Contest Format When the Supplier Base Is Heterogeneous

1

1.2

1.4

1.6

0.15 0.25 0.35 0.451

1.4

1.8

2.2

0.12 0.18 0.24 0.31

1.2

1.4

1.6

0.15 0.25 0.35 0.451

1.4

1.8

2.2

0.12 0.18 0.24 0.3

Note. The graphs show a buyer’s optimal choice of contest format for technological substitutes (left panel) and

technological complements (right panel) as a function of the heterogeneity in award valuations α. The buyer prefers

a component contest in region C but prefers a system contest in region S; no pure-strategy equilibrium exists in

region N. The parameters used here are r(x) = x, c(x) = x2, n= 2, A= 1, and ρ= 0.

supplier 1 obtains utility A from winning the supply contract whereas supplier 2 obtains utility

of only αA with α ∈ (0,1]. As a result of these heterogeneous award valuations, the two suppliers

invest different amounts of effort during the procurement contest. Supplier 1, which has a higher

valuation for winning the contest, always invests more development effort than does supplier 2;

however, both suppliers reduce their efforts as α decreases. In other words, supplier asymmetry

has a negative effect on effort provision. The question that then arises is: How does this negative

effect influence the buyer’s choice of contest format?

Given the elusiveness of closed-form results for our model when prize structures are asymmetric,

we resort to numerical simulations; Figure 3 illustrates a representative example. The figure’s

graphs plot for which levels of uncertainty σ and component asymmetry k a component contest—or

a system contest—is preferable to the buyer. It is clear that, although a shift in award asymmetry α

changes the exact numerical results, it does not qualitatively affect the shape of the regions in which

each contest format dominates. So regardless of the magnitude of supplier asymmetry α, the buyer

holds a system contest if both components are incremental innovations but holds a component

contest otherwise. Thus Figure 3 provides evidence that heterogeneity in the supplier base has

only a minor effect on the buyer’s preferences across contest formats. We conclude that our key

structural results (as presented in Propositions 1–3) are robust to heterogeneity in the supplier

base.

7. Conclusions

Procurement contests have become a popular tool for buyers seeking access to innovative products

developed by their supplier base. Yet tapping into suppliers’ innovation efforts is a challenging


endeavor. In order to exploit the benefits of holding a procurement contest, buyers must tailor

their contest designs not only to their supplier base but also to the characteristics of the desired

products. This generalization holds in particular for technologically complex innovations (e.g., an

automotive lighting system). For such complex products, the buyer’s most important question

is arguably whether to procure the entire system from a single supplier or rather to source the

product’s individual components from different suppliers. In this paper we have addressed that

question, whose answer has strong implications for how the buyer should construct its procurement

contests.

First, we find that a buyer should use a system contest when all product components require

only incremental innovation and the firm’s supplier base is small. However, if radical innovation

is required or if there is a large supplier base, then the buyer should instead hold a component

contest. In other words, managers deciding on the optimal contest format only need to consider two

dimensions: (i) the level of innovation required for each component; and (ii) the size of the buyer’s

supplier base. In practice, relying mainly on these two factors significantly reduces the managerial

decision problem’s complexity. Because the required level of innovation and the number of potential

suppliers are both well known to the buying firm, managers can worry less about factors that are

more subtle (and less tangible).

Second, our results give clear advice on how a buyer should structure its system contests and its

component contests—in particular, on identifying what type of suppliers to invite (and to avoid).

A buyer holding a component contest should favor performance correlation, which can be achieved

by inviting the same set of suppliers to participate in both contests. In a system contest, the buyer

should prefer less performance correlation when the project has two incremental components and

a small supplier base; if it has at least one innovative component or a large supplier base the buyer

should prefer greater performance correlation. These recommendations have an interesting long-

run implication. Our results suggest that managers should actively manage their supplier base; we

would then expect that, over an extended period, supplier base characteristics will come to reflect

the realities of efficient contests.

Like any model, ours is not without some limitations. In order to develop a parsimonious model,

some simplifying assumptions were made about the setting we consider. Perhaps the most salient

of these relates to the cost of component integration, for which we do not explicitly account. Thus

our model implicitly assumes that integration efforts result in a comparable cost regardless of

whether they are undertaken by the supplier or the buyer. If interpreted in a narrow sense, this

assumption limits the model’s applicability to situations in which integration tasks are relatively

less important or in which they are routine to the point where the supplier and the buyer face

about the same integration cost; examples include cases where simple (if not already standardized)


interfaces between components are sufficient for system integration. Although such settings are

common—most mechatronic systems in the automotive industry (e.g., the headlamp) fall into this

class—there are clearly many cases where this assumption is violated. Despite our model’s capacity

to accomodate somewhat more complex settings, it would need to be considerably extended in

order to accomodate highly complex integration scenarios.

In sum, we believe that our model sheds light on an increasingly utilized means of procuring

new products: the contest. Our thorough analysis of contest design yields clear managerial recom-

mendations as a function of various product contingencies. In this way we have contributed useful

insights into the critical issue of how best to source innovation.

Appendix

This Appendix comprises three parts. Section A discusses some preliminary technical results that are essential

for our analysis, and Section B characterizes the perfect Bayesian Nash equilibria of the different contest

formats. In Section C we give detailed proofs of all the mathematical results.

A. Technical Preliminaries

The Skew-Normal Distribution. Following Azzalini (1985), we refer to a random variable X with

probability density function ψ(x;α) = 2Φ(αx)φ(x), x∈R, as a Skew-Normal random variable with parameter

α∈R. The cumulative distribution function of X is given by Ψ(x;α) =∫ x−∞ψ(y;α)dy. Below we derive some

important properties of Skew-Normal random variables.

Lemma A1. (i) ψ(x; 0) = φ(x) for all x∈R.

(ii) Define Ik(α) ≡∫∞−∞Ψ(x;α)kψ(x;α)2dx for k ≥ 0. Then, Ik(α) strictly decreases in α for α < 0 and

strictly increases for α> 0.

(iii) (k+ 1)(k+ 2)Ik(0) = µ(k+2) for all k≥ 0, where µ(k) ≡∫∞−∞ xkφ(x)Φ(x)k−1.

Proof of Lemma A1. (i) See Property A in Azzalini (1985).

(ii) Taking the first-order derivative and exploiting properties of the Normal distribution yields

dIk(α)

dα=

2α

π(1 +α2)

∫ +∞

−∞φ(√

2 +α2x)Ψ(x;α)k (αxΦ(αx) +φ(αx))dx. (1)

Since αxΦ(αx) +φ(αx)> 0 for all x,α∈R, we have dIk(α)/dα< 0 for α< 0 and dIk(α)/dα> 0 for α> 0.

(iii) Ik(0) =∫∞−∞(k + 1)Φ(x)kφ(x)2dx/(k + 1) =

∫∞−∞ xφ(x)Φ(x)k+1dx/(k + 1) =

∫∞−∞ x(k +

2)φ(x)Φ(x)k+1dx/((k+ 1)(k+ 2)) = µ(k+2)/((k+ 1)(k+ 2)).

A Generalization of Slepian’s Inequality. In this section, we establish a generalization of Slepian’s

Inequality whose basic formulation can be found in, e.g., Theorem 2.1.1. in Tong (1980).

Lemma A2. Let X = (X1,X2), Y = (Y1, Y2) follow a bivariate Normal distribution with marginal distri-

butions X1, Y1 ∼N(µ1, σ21), X2, Y2 ∼N(µ2, σ

22) and correlation ρX , ρY , where ρY >ρX . Then,

(i) P(X1 ≤ u1,X2 ≤ u2)< P(Y1 ≤ u1, Y2 ≤ u2) for all u1, u2 ∈R.

(ii) P(X1 ≥ u1,X2 ≥ u2)< P(Y1 ≥ u1, Y2 ≥ u2) for all u1, u2 ∈R.


Proof of Lemma A2. (i) P(X1 ≤ u1,X2 ≤ u2) = P(X1 ≤ u1 − µ1,X2 ≤ u2 − µ2) < P(Y 1 ≤ u1 − µ1, Y 2 ≤u2 − µ2) = P(Y1 ≤ u1, Y2 ≤ u2) for all u1, u2 ∈ R, where the strict inequality follows from Theorem 2.1.1. in

Tong (1980) and the fact that Xj , Y j , j = 1,2, are centered Normal random variables.

(ii) This result is an immediate consequence of part (i).

B. Derivation of Perfect Bayesian Equilibrium

In this section, we derive the symmetric perfect Bayesian Nash equilibria (PBE) for the different contest

formats. Note that given Assumption 1, each supplier always participates in any contest because he can always

guarantee himself a strictly positive expected utility by participating and exerting zero effort. Hence we can

derive the suppliers’ equilibrium efforts by simply solving their incentive compatibility (IC) constraints.

B.1. PBE for Technological Substitutes

System contest. In a system contest, supplier i’s IC constraint is:

(esysi1 , esysi2 )∈ arg max

ei1,ei2

ui(ei1, ei2)≡AEζi1,ζi2

∏k 6=i

Fζk1+ζk2

∑j∈{1,2}

r(eij) + ζij − r(ekj)

− c(ei1)− c(ei2).

(2)

Using straightforward differentiation and the law of iterated expectations, we find that any symmetric pure-

strategy PBE satisfies the following optimality condition for j = 1,2:

c′(esysj )

r′(esysj )=A(n− 1)In−2(0)

σ√

1 + k2 + 2kρ=

Aµ(n)

nσ√

1 + k2 + 2kρ. (3)

Since c′/r′ is strictly increasing, c′(0) = 0 and c′(∞)/r′(∞) =∞, it follows that (esys1 , esys2 ) is the unique

solution to (3). Define η(x) = (r ◦ (c′/r′)−1)(x) for all x≥ 0. Then the buyer’s equilibrium expected profit is

Πsyssub = 2η

(Aµ(n)

nσ√

1 + k2 + 2kρ

)+σµ(n)

√1 + k2 + 2kρ. (4)

Component contest. Given p, supplier i’s IC constraint in each component contest j ∈ {1,2} is:

ecpoi1 ∈ arg maxei1

ui(ei1)≡ pAEζi1

[∏k 6=i

Fζk1(r(ei1) + ζi1− r(ek1))

]− c(ei1) (5)

ecpoi2 ∈ arg maxei2

ui(ei2)≡ (1− p)AEζi2

[∏k 6=i

Fζk2(r(ei2) + ζi2− r(ek2))

]− c(ei2), (6)

yielding the following optimality conditions for a symmetric PBE:

c′(ecpo1 )

r′(ecpo1 )=pA(n− 1)In−2(0)

σ=pAµ(n)

nσ(7)

c′(ecpo2 )

r′(ecpo2 )=

(1− p)A(n− 1)In−2(0)

kσ=

(1− p)Aµ(n)

nkσ. (8)

Again, since c′/r′ is strictly increasing, c′(0) = 0 and c′(∞)/r′(∞) =∞, (ecpo1 , ecpo2 ) is the unique symmetric

PBE. Given suppliers’ equilibrium efforts, the buyer chooses p to maximize expected profits:

p∗ ∈ arg maxp

Πcpo(p) = η

(pAµ(n)

nσ

)+ η

((1− p)Aµ(n)

nkσ

)+σµ(n)(1 + k). (9)

Assumption 2 ensures that η is a strictly concave function, and therefore Πcpo(p) is strictly concave in p,

implying that p∗ is unique. Moreover, the necessary and sufficient first-order condition ∂Πcpo(p)/∂p = 0

reveals that p∗(k= 1) = 1/2 and limk→∞ p∗(k) = 1, and we let Πcpo

sub ≡Πcpo(p∗).


B.2. PBE for Complementary Components

System contest. In a system contest, supplier i’s IC constraint is:

(esysi1 , esysi2 )∈ arg max

ei1,ei2

ui(ei1, ei2)≡AEζi1,ζi2

[∏k 6=i

Fsk(si)

]− c(ei1)− c(ei2), (10)

where si = min{vi1, vi2}, and since vi1 ∼N(r(ei1), σ2) and vi2 ∼N(r(ei2), k2σ2) we have

fsi(x) =

2∑j=1

1

kj−1σφ

(x− r(eij)kj−1σ

)Φ

(1

σ√

1− ρ2

(ρ(x− r(eij))

kj−1− x− r(ei,3−j)

k2−j

)), (11)

Fsi(x) = Φ

(x− r(ei1)

σ

)+

∫ ∞(x−r(ei1))/σ

Φ

(x− r(ei2)− ρkσu

kσ√

1− ρ2

)φ(u)du (12)

by eq. (46.77)-(46.78) in Kotz et al. (2000). By differentiation and the law of iterated expectations, we find

that any symmetric pure-strategy PBE satisfies the following optimality condition for j = 1,2:

c′(esysj )

r′(esysj )=A(n− 1)

∫ ∞−∞

fs(r(esysj ) +x)Fs(r(e

sysj ) +x)n−2(1−Fζ3−j |ζj (r(esysj )− r(esys3−j) +x))fζj (x)dx. (13)

Case A: k= 1. If k= 1, then there exists a solution to (13) with esys1 = esys2 , and this solution is given by

c′(esys1 )

r′(esys1 )=A(n− 1)

2σ· In−2

(−√

1− ρ1 + ρ

). (14)

Case B: k→∞. By (13), we have the following upper bound on esys2 :

c′(esys2 )

r′(esys2 )≤A(n− 1)

∫ ∞−∞

fs(r(esys2 ) +x)fζ2(x)dx (15)

≤ A(n− 1)

σ2

∫ ∞−∞

(φ

(r(esys2 )− r(esys1 ) +x

σ

)+

1

kφ( xkσ

)) 1

kφ( xkσ

)dx (16)

≤√

2A(n− 1)

kσ√π

. (17)

It follows that limk→∞ c′(esys2 )/r′(esys2 ) = 0, and thus, given the properties of c and r, limk→∞ e

sys2 = 0. We

now turn to esys1 :

limk→∞

c′(esys1 )

r′(esys1 )=A(n− 1)

σ

∫ ∞−∞

[Φ

(ρy√

1− ρ2

)φ(y)

]2 [Φ(y) +

∫ ∞y

Φ

(− ρu√

1− ρ2

)φ(u)du

]n−2dy (18)

=A(n− 1)

2nσ

∫ ∞−∞

[1 + Ψ

(y;

ρ√1− ρ2

)]n−2ψ

(y;

ρ√1− ρ2

)2

dy (19)

=A(n− 1)

2nσ

n−2∑l=0

(n− 2

l

)Il(

ρ√1− ρ2

), (20)

where the first equality is an application of Lebesgue’s Dominated Convergence Theorem. We note that the

solution (esys1 , esys2 ) is unique as k→∞.

Component contest. Equilibrium efforts in a component contest (ecpo1 , ecpo2 ) are again given by (7) and

(8) because each component contest is run separately, so the technological relationship between components

does not affect the suppliers’ equilibrium behavior. The buyer’s optimal choice of p solves:

p∗ ∈ arg maxp

Πcpo(p) = E[min{r(ecpo1 ) + maxi{ζi1}, r(ecpo2 ) + max

i{ζi2}}]. (21)

By Assumption 2, r(ecpoj ), j = 1,2, is strictly concave in p, and since concavity is preserved under the pointwise

minimization and expectation operators, it follows that Πcpo(p) is strictly concave in p. Moreover, the neces-

sary and sufficient first-order condition ∂Πcpo(p)/∂p= 0 reveals that p∗(k = 1) = 1/2 and limk→∞ p∗(k) = 1,

and we let Πcpocml ≡Πcpo(p∗).


C. Proofs

Proof of Lemma 1. (ia) The result follows directly from (9) and the discussion thereafter.

(ib) For k= 1 and ρ= 0, (3) implies that c′(esys1 )/r′(esys1 ) = c′(esys2 )/r′(esys2 ) =Aµ(n)/(nσ√

2 + 2ρ), and (7)-

(8) imply that c′(ecpo1 )/r′(ecpo1 ) = c′(ecpo2 )/r′(ecpo2 ) =Aµ(n)/(2nσ). Since c′/r′ is a strictly increasing function

and ρ< 1, it follows that esys1 = esys2 > ecpo1 = ecpo2 .

(iia) The result follows directly from (9) and the discussion thereafter.

(iib) As k → ∞, (7)-(8) together with p∗ = 1 imply that c′(ecpo1 )/r′(ecpo1 ) = Aµ(n)/(nσ) and

c′(ecpo2 )/r′(ecpo2 ) = 0, yielding ecpo1 > ecpo2 = 0. Furthermore, (3) reveals that limk→∞ c′(esys1 )/r′(esys1 ) =

limk→∞ c′(esys2 )/r′(esys2 ) = 0, and thus limk→∞ e

sys1 = limk→∞ e

sys2 = 0 by the properties of r and c.

Proof of Proposition 1. Let Πsyssub and Πcpo

sub be as given in (4) and (9), respectively, and define ∆(σ,k) =

Πsyssub−Πcpo

sub, which is a continuous function in σ and k for σ > 0 and k≥ 1.

(i) Given k = 1, Lemma 1(i) implies that Πsyssub = 2η(Aµ(n)/(nσ

√2 + 2ρ)) +

√2 + 2ρσµ(n) and Πcpo

sub =

2η(Aµ(n)/(2nσ)) + 2σµ(n). Since η is strictly increasing, η(0) = 0, and ρ < 1, it follows that limσ→0 ∆(σ,k=

1) > 0 and limσ→∞∆(σ,k = 1) < 0. By the Intermediate Value Theorem, there exist thresholds σsub, σsub,

with 0<σsub ≤ σsub <∞, such that ∆(σ,k= 1)> 0 for all σ ∈ (0, σsub), and ∆(σ,k= 1)< 0 for all σ > σsub.

The result now follows from the continuity of ∆(σ,k), the fact that a continuous mapping from a connected

subset of a metric space to another metric space yields a connected image set (Ok 2007, p. 220), and by

integrating our assumption that σ > σsub. Moreover,

σsub = min

{σ : σµ(n) =

√2(η(Aµ(n)/(nσ

√2 + 2ρ))− η(Aµ(n)/(2nσ)))√2−√

1 + ρ

}, (22)

which implies limn→∞ σsub = 0, and hence there exists n such that (σsub, σsub) = ∅ for all n> n.

(ii) Fix any σ > σsub. Then, limk→∞∆(σ,k)< 0, and thus, by the Intermediate Value Theorem, there exists

a threshold ksub <∞ such that ∆(σ,k)< 0 for all k > ksub.

(iii) Fix any k≥ 1. Then, limσ→∞∆(σ,k)< 0, and thus, by the Intermediate Value Theorem, there exists

a threshold σsub <∞ such that ∆(σ,k)< 0 for all σ > σsub.

Proof of Lemma 2. (ia) The result follows directly from (21) and the discussion thereafter.

(ib) Comparing (7)-(8) and (14) for k = 1 reveals that esys1 = esys2 > ecpo1 = ecpo2 > 0 if

In−2(−√

(1− ρ)/(1 + ρ))> In−2(0)> 0, which is true by Lemma A1(ii).

(iia) The result follows directly from (21) and the discussion thereafter.

(iib) First, by Lemma A1(ii), Ik(α)≤ limα→∞ Ik(α) = 2k+2∫∞0

(Φ(x)− 1/2)kφ(x)2dx for all α > 0. Next,

consider (7) and (20) and note that c′/r′ is strictly increasing:

limk→∞

c′(esys1 )

r′(esys1 )=A(n− 1)

2nσ

n−2∑l=0

(n− 2

l

)Il(

ρ√1− ρ2

)(23)

≤ A(n− 1)

2nσ

n−2∑l=0

(n− 2

l

)2l+2

∫ ∞0

(Φ(x)− 1/2)lφ(x)2dx (24)

=A(n− 1)

σ

∫ ∞0

[n−2∑l=0

(n− 2

l

)2l+2−n(Φ(x)− 1/2)l

]φ(x)2dx (25)


=A(n− 1)

σ

∫ ∞0

Φ(x)n−2φ(x)2dx<A(n− 1)

σIn−2(0) = lim

k→∞

c′(ecpo1 )

r′(ecpo1 ). (26)

Also, as k→∞, (8) and (17) lead to ecpo2 = 0 and esys2 = 0, respectively. Last, we note that as k→∞, (20)

reveals that the right-hand side of the optimality condition does not depend on esys1 and esys2 . Taken together

with the properties of c′/r′, this implies that (esys1 , esys2 ) is unique as k→∞.

Proof of Proposition 2. The buyer’s expected equilibrium profits in a system and component contest

are Πsyscml = E[maxi∈{1,...,n}{minj∈{1,2}{r(esysj ) + ζij}}] and Πcpo

cml = E[minj∈{1,2}{maxi∈{1,...,n}{r(ecpoj ) + ζij}}],

respectively. Define ∆(σ,k) = Πsyscml−Πcpo

cml, which is a continuous function in σ and k for σ > 0 and k≥ 1.

(i) For k = 1, Lemma 2(i) implies Πcpocml = η(A(n− 1)In−2(0)/(2σ)) + σE[minj∈{1,2}{maxi∈{1,...,n}{Xij}}]

and Πsyscml = η(A(n − 1)In−2(−

√(1− ρ)/(1 + ρ))/(2σ)) + σE[maxi∈{1,...,n}{minj∈{1,2}{Xij}}], respec-

tively, with Xij being independent standard normal random variables. Since η is strictly

increasing, η(0) = 0, In−2(−√

(1− ρ)/(1 + ρ)) > In−2(0), and E[minj∈{1,2}{maxi∈{1,...,n}{Xij}}] >

E[maxi∈{1,...,n}{minj∈{1,2}{Xij}}], it follows that limσ→0 ∆(σ,k = 1) > 0 and limσ→∞∆(σ,k = 1) < 0. By

the Intermediate Value Theorem, there exist thresholds σcml, σcml, with 0 < σcml ≤ σcml <∞, such that

∆(σ,k = 1) > 0 for all σ ∈ (0, σcml), and ∆(σ,k = 1) < 0 for all σ > σcml. The result now follows from the

continuity of ∆(σ,k), the fact that a continuous mapping from a connected subset of a metric space to

another metric space yields a connected image set (Ok 2007, p. 220), and by integrating our assumption that

σ > σcml. Moreover,

σcml = min{σ : ∆(σ,k= 1) = 0} , (27)

which implies limn→∞ σcml = 0, because limn→∞∆(σ,k= 1) =−σ limn→∞[E[minj∈{1,2}{maxi∈{1,...,n}{Xij}}]−

E[maxi∈{1,...,n}{minj∈{1,2}{Xij}}]]. Hence there exists n such that (σcml, σcml) = ∅ for all n> n.

(ii) Fix any σ > σcml. For k→∞, Lemma 2(ii) implies that Πcpocml = E[minj∈{1,2}{maxi∈{1,...,n}{r(ecpoj ) +

ζij}}]>E[minj∈{1,2}{maxi∈{1,...,n}{r(esysj )+ζij}}]>E[maxi∈{1,...,n}{minj∈{1,2}{r(esysj )+ζij}}] = Πsyscml. Thus,

limk→∞∆(σ,k) < 0, and by the Intermediate Value Theorem, there exists a threshold kcml <∞ such that

∆(σ,k)< 0 for all k > kcml.

(iii) Fix any k ≥ 1, and define r = minj∈{1,2}{r(ecpoj )} and r = maxj∈{1,2}{r(esysj )}. It

follows that Πcpocml ≥ r + σE[min{maxi∈{1,...,n}{Xi1},maxi∈{1,...,n}{kXi2}}] and Πsys

cml ≤ r +

σE[maxi∈{1,...,n}{min{Xi1, kXi2}}], where Xij are independent standard normal random variables. Since

E[min{maxi∈{1,...,n}{Xi1},maxi∈{1,...,n}{kXi2}}] > E[maxi∈{1,...,n}{min{Xi1, kXi2}}], and r as well as r are

bounded in value, it follows that limσ→∞∆(σ,k)< 0, and thus, by the Intermediate Value Theorem, there

exists a threshold σcml <∞ such that ∆(σ,k)< 0 for all σ > σcml.

Proof of Proposition 3. (i) Note that Πcposub as given in (9) is independent of ρ because µ(n) and thus also

p∗ do not depend on ρ. Hence, since Proposition 1 is true for any ρ< 1, it is also true for any ∆ρ< 1.

(iia) As a first step, we establish that Πcpocml (resp. Πsys

cml) increases (resp. decreases) in ρ for k= 1 and σ small.

We begin by verifying that Πcpocml(p

∗(ρ1);ρ1)≤Πcpocml(p

∗(ρ1);ρ2)≤Πcpocml(p

∗(ρ2);ρ2) for any fixed 0≤ ρ1 <ρ2 < 1

and any k and σ. Clearly, the last inequality follows from the optimality of p∗, so it remains to prove the first

inequality. For any given p, Πcpocml(p;ρ) = E[minj∈{1,2}{maxi∈{1,...,n}{vcpoij }}], where vcpoi = (vcpoi1 , vcpoi2 ) follows a


bivariate Normal distribution with correlation ρ and marginal distributions vcpoi1 ∼N(r(ecpo1 ), σ2) and vcpoi2 ∼N(r(ecpo2 ), k2σ2) for all i. To conclude the proof, we need to show that Πcpo

cml(p;ρ) increases in ρ for any fixed p.

This is true if the random variable V (ρ)≡minj∈{1,2}{maxi∈{1,...,n}{vcpoij }} first-order stochastically increases

in ρ; a property that we verify in the next step. Let Vj ≡ maxi∈{1,...,n}{vcpoij } for j = 1,2. For any y ∈ Rand fixed p, P(V (ρ)> y) = P(V1 > y)− P(V1 > y|V2 < y)P(V2 < y). Clearly, only the term P(V1 > y|V2 < y)

depends on ρ, and thus V (ρ) stochastically increases in ρ if and only if this term decreases in ρ. Note that

P(V1 > y|V2 < y) = 1− P(vcpoi1 < y,vcpoi2 < y)n/P(vcpoi2 < y)n, where the denominator is independent of ρ and

P(vcpoi1 < y,vcpoi2 < y) increases in ρ by Lemma A2(i). Hence P(V1 > y|V2 < y) decreases in ρ.

We now turn to Πsyscml. For given ρ, Πsys

cml(ρ) = E[maxi∈{1,...,n}{min{vsysi1 , vsysi2 }}], where (vsysi1 , vsysi2 ) follows a

bivariate Normal distribution with marginal distributions vsysi1 ∼N(r(esys1 ), σ2), vsysi2 ∼N(r(esys2 ), k2σ2) and

correlation ρ for all i ∈ {1, . . . , n}. We proceed to show that for k = 1 and σ small, Πsyscml(ρ) decreases in ρ,

and that Πsyscml(ρ) increases in ρ for k→∞ or σ→∞.

Case A: k= 1. By (14) we have esys1 = esys2 , and hence Πsyscml(ρ) = r(esys1 )+σE[maxi∈{1,...,n}{min{Xi1,Xi2}}],

where (Xi1,Xi2) follows a standard bivariate Normal distribution with correlation ρ for all i ∈ {1, . . . , n}.In addition, (14) together with Lemma A1(ii) implies that equilibrium efforts decrease in ρ, and thus

dr(esys1 )/dρ < 0. Therefore, limσ→0 dΠsyscml(ρ)/dρ < 0.

Case B: k→∞ or σ→∞. Πsyscml(ρ) = E[maxi∈{1,...,n}{min{vsysi1 , vsysi2 }}] increases in ρ if the random vari-

able W (ρ) ≡ maxi∈{1,...,n}{min{vsysi1 , vsysi2 }} stochastically increases in ρ, or equivalently, P(W (ρ) < u) =

(1 − P(vsysi1 ≥ u, vsysi2 ≥ u))n decreases in ρ for all u ∈ R. This is true if P(vsysi1 ≥ u, vsysi2 ≥ u) = P(vi1 ≥u− r(esys1 ), vi2 ≥ u− r(esys2 )) increases in ρ for all u ∈ R, where vi1 and vi2 are centered Normal random

variables. For fixed esys1 and esys2 , this probability increases in ρ by Lemma A2(ii); and for fixed ρ, it increases

obviously in esys1 and esys2 . Since both sensitivities point in the same direction, it remains to verify that esys1

and esys2 weakly increase in ρ as k→∞ (resp. σ→∞). For k→∞, this is true because esys1 increases in ρ

by (20) and Lemma A1(ii), and esys2 = 0 by (17). For σ→∞, esys1 = esys2 = 0 by (13), thereby concluding the

proof.

Given the sensitivities of Πcpocml and Πsys

cml, it is sufficient to establish that Πsyscml(ρ= 1)≥Πcpo

cml(ρ= 1) to prove

the claim. By (7), (8) and (14) it follows that for k = 1 and ρ= 1, esys1 = esys2 = ecpo1 = ecpo2 , and as a result,

Πsyscml(ρ= 1) = r(esys1 ) +E[maxi∈{1,...,n}{ζi1}] = r(ecpo1 ) +E[maxi∈{1,...,n}{ζi1}] = Πcpo

cml(ρ= 1).

(iib) For σ→∞, (7), (8) and (14) imply that esys1 = esys2 = ecpo1 = ecpo2 = 0. It follows that Πsyscml(ρ

sys) =

E[maxi∈{1,...,n}{min{ζi1, ζi2}}] and Πcpocml(ρ

cpo) = E[minj∈{1,2}{maxi∈{1,...,n}{ζij}}]. Clearly, if ρsys = ρcpo then

Πcpocml(ρ

cpo) > Πsyscml(ρ

sys); and if ρsys = 1 and ρcpo = 0 then Πsyscml(ρ

sys) = E[maxi∈{1,...,n}{ζi1}] > Πcpocml(ρ

cpo).

The result now follows immediately from the continuity of all involved functions in conjunction with the

fact that a continuous mapping from a connected subset of a metric space to another metric space yields a

connected image set (Ok 2007, p. 220).

Proof of Proposition 4. (i) The proof is already given in the proof of Proposition 3(i).

(ii) The proof is already given in the proof of Proposition 3(iia).

Proof of Proposition 5. (i) Using (4), we have dΠsyssub(ρ)/dρ = kµ(n)(nσ2(1 + k2 + 2kρ) −

2Aη′(Aµ(n)/(nσ√

1 + k2 + 2kρ)))/(nσ(1 + k2 + 2kρ)3/2), and comparing to zero proves the result.

(ii) The sensitivity of Πsyscml has already been established in the proof of Proposition 3(iia), and the result

now follows directly from the continuity of Πsyscml(ρ) and the Intermediate Value Theorem.


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