Match Your Own Price? Self-Matching as a Retailer’s Multichannel Pricing Strategy

MARKETING SCIENCE

Articles in Advance, pp. 1–23

http://pubsonline.informs.org/journal/mksc/ ISSN 0732-2399 (print), ISSN 1526-548X (online)

Match Your Own Price? Self-Matching as a

Retailer’s Multichannel Pricing Strategy

Pavel Kireyev,

Vineet Kumar,

Elie Ofek

Harvard Business School, Harvard University, Boston, Massachusetts 02163;

Yale School of Management, Yale University, New Haven,

Connecticut 06520

Contact:

pkirey[email protected] (PK); [email protected] (VK); [email protected] (EO)

Received: July 22, 2015

Revised: June 9, 2016

Accepted: August 9, 2016

Published Online in Articles in Advance:

August 3, 2017

https://doi.org/10.1287/mksc.2017.1035

Abstract. Multichannel retailing has created several new strategic choices for retailers.

With respect to pricing, an important decision is whether to oﬀer a “self-matching policy,”

which allows a multichannel retailer to oﬀer the lowest of its online and store prices to con-

sumers. In practice, we observe considerable heterogeneity in self-matching policies: There

are retailers who oﬀer to self-match and retailers who explicitly state that they will not

match prices across channels. Using a game-theoretic model, we investigate the strategic

forces behind the adoption (or non-adoption) of self-matching across a range of compet-

itive scenarios, including a monopolist, two competing multichannel retailers, as well as

a mixed duopoly. Though self-matching can negatively impact a retailer when consumers

pay the lower price, we uncover two novel mechanisms that can make self-matching prof-

itable in a duopoly setting. Speciﬁcally, self-matching can dampen competition online

and enable price discrimination in-store. Its eﬀectiveness in these respects depends on

the decision-making stage of consumers and the heterogeneity of their preference for the

online versus store channels. Surprisingly, self-matching strategies can also be proﬁtable

when consumers use “smart” devices to discover online prices in stores. Our ﬁndings

provide insights for managers on how and when self-matching can be an eﬀective pricing

strategy.

History:

Preyas Desai served as the editor-in-chief and Dmitri Kuksov served as associate editor for

this article.

Supplemental Material:

The online appendix is available at https://doi.org/10.1287/mksc.2017.1035.

Keywords:

price self-matching

•

multichannel retailing

•

pricing strategy

•

online shopping

•

omnichannel

•

price discrimination

1. Introduction

Many, if not most, major retailers today use a mul-

tichannel business model, i.e., they oﬀer products in

physical stores and online. These channels tend to

attract diﬀerent consumer segments and allow retailers

to cater to distinct buying behaviors and preferences.

Consumers are also becoming more savvy in using

the various channels during the buying process, i.e.,

researching products, evaluating ﬁt, comparing prices,

and purchasing (Neslin et al. 2006, Grewal et al. 2010,

Verhoef et al. 2015).

Retailers must attend to all elements of the market-

ing mix as they strive to maximize proﬁts. Not surpris-

ingly, pricing has always been an important strategic

variable for t hem to “get right.” When retailers were

predominantly brick-and-mortar, they had to deter-

mine the most eﬀective store price to set for their mer-

chandise. However, having embraced a multichannel

selling format, pricing decisions have become much

more complex for these retailers to navigate. Not only

do they need to price the products in their physical

stores, they also need to set prices for products in their

online outlets and consider how the prices across the

various channels should relate to one another. This

complexity in devising a comprehensive multichan-

nel pricing strategy is front and center for retailers

today, as evidenced by the myriad commentaries in

the retail trade press. As Forrester Research reports

(Mulpuru 2012), “It is imperative for eBusiness profes-

sionals in retail to adopt cross-channel best practices

including ...pricing.”

Formulating an eﬀective multichannel pricing strat-

egy can be challenging. A recent survey of lead-

ing retailers (eMarketer 2013) revealed that their top

two pricing challenges are: (1) increased price sen-

sitivity of consumers, and (2) pricing aggressiveness

from competitors. In a world where many consumers

buy online or conduct research online before enter-

ing a store, these ﬁndings suggest that the need to

manage the heightened price sensitivity and combat

intense competition are becoming even more impor-

tant. Interestingly, the above study did not ﬁnd the

item “need to provide consistency in price across

channels” to be among the top few challenges these

retailers face, underscoring that they feel they have

ﬂexibility in customizing their price to the speciﬁc

Kireyev, Kumar, and Ofek: Match Your Own Price?

channel and customer mix that chooses to shop there.

Indeed, according to Gartner’s Kevin Sterneckert (Reda

2012, p. 6), “Using a single-channel, consistent pric-

ing strategy misses important opportunities in the

marketplace ...” Consistent with these survey obser-

vations, we speciﬁcally investigate how retailers can

leverage self-matching across channels to set prices

ﬂexibly to diminish intense price competition.

With a self-matching policy, the retailer commits to

charging consumers the lower of its online and store

prices for the same product when consumers furnish

appropriate evidence of a price diﬀerence. Note that

even though self-matching can provide some degree

of price consistency, it is fundamentally diﬀerent from

committing to consistent prices and setting exactly

the same prices across all channels, as will be clear

in our analysis in Section 4. Commonly, this policy

features store to online self-matching, allowing con-

sumers to pay the typically lower online prices for store

purchases.

This policy is a novel marketing instru-

ment t hat is uniquely available to multichannel retail-

ers and not relevant in the single-channel case. The

primary objective of our paper is to understand the

strategic consequences of such self-matching policies.

We note that competitive price-matching policies have

been extensively studied by contrast (and are reviewed

in Section 2).

Examining a number of retail markets, there are two

distinct self-matching patterns that come to our atten-

tion. First, we observe considerable heterogeneity in the

adoption of self-matching policies across retailers in the

United States, including those competing for the same

market. For example, Best Buy, Sears, Staples, Oﬃce

Depot, Toys “R” Us, and PetSmart price match their

online channels in-store, whereas JCPenney, Macy’s,

Urban Outﬁtters, and Petco explicitly state that they

will not match their prices across channels.

Second, we

observe heterogeneity in self-matching across industries.

In consumer electronics and home improvement, major

players oﬀer self-matching, whereas in low-end depart-

ment stores and clothing, most or all retailers tend not

to adopt self-matching.

We aim to develop insights on when to expect dif-

ferent self-matching patterns for multichannel retailers

in a given category, i.e., all self-match, some self-match

while others do not, and none self-match. To this end,

we examine the use of a self-matching pricing policy

by multichannel retailers across a variety of compet-

itive settings, including a monopoly, a duopoly with

two competing multichannel retailers, and a mixed

duopoly in which a multichannel retailer competes

with an e-tailer. More speciﬁcally, we address the fol-

lowing research questions:

(1) What strategic mechanisms underpin the deci-

sion to implement a self-matching pricing policy?

(2) When do multichannel retailers choose to self-

match in equilibrium? How do customer and product

characteristics, and the nature of competition, inﬂu-

ence a retailer’s decision to self-match?

(3) How does self-matching aﬀect the prices charged

online and in-store?

(4) Are retailers better or worse oﬀ having access to

self-matching as a strategic tool?

To investigate these questions, we develop a model

that allows us to capture the eﬀects of self-matching

on consumer and retailer decisions. We allow for con-

sumer heterogeneity along a number of important

dimensions. These dimensions include consumers’

channel preferences, their stage in the decision-making

process (DMP), and their preference across retailers.

As to channel preferences, we allow for “store-only”

consumers who have a strong preference to purchase

in-store where they can “touch and feel” merchandise

and instantly obtain the product. By contrast, “channel-

agnostic consumers” do not have a strong preference

for any channel from which they purchase. We also

distinguish between consumers who know the exact

product they want to purchase (“Decided”) and con-

sumers who only recognize the need to purchase from

a category and require a store visit to shop around and

ﬁnd the speciﬁc version or model that best ﬁts their

needs (“Undecided”). Finally, consumers have hori-

zontal taste (or brand) preferences across retailers.

Retailers oﬀer unique products of similar value

and are at the ends of a Hotelling linear city, wit h

consumer location on the line indicating retailer pref-

erence. Retailers ﬁrst choose a self-matching pricing

policy and subsequently and simultaneously set price

levels for store and online channels. We analyze the

subgame perfect equilibria of the game.

Our analysis reveals several underlying mechanisms

that aﬀect the proﬁtability of self-matching in equilib-

rium. The ﬁrst eﬀect, termed channel arbitrage, is neg-

ative and reduces proﬁts, whereas the other eﬀects

termed decision-stage discrimination and online competi-

tion dampening increase proﬁts. Thus, the overall proﬁt

implications of implementing a self-matching pricing

policy depend on the existence and magnitude of these

eﬀects.

Consider the pricing incentives faced by a mul-

tichannel retailer absent self-matching. In the store

channel, it faces two types of consumers; those who

researched the product online before choosing a store

(decided consumers) and t hose who visit their pre-

ferred store ﬁrst to identify the product that best

matches their needs (undecided consumers). Retailers

may want to charge a higher price to the latter type,

but are unable to do so because both types purchase in

the store channel. Furthermore, consumers who shop

online tend to be informed of the online prices at both

retailers, which leads to more competitive pricing in

the online channel than in-store.

Kireyev, Kumar, and Ofek: Match Your Own Price?

Now, consider the strategic impact of self-matching

policies. With self-matching, consumers who research

the product online but purchase in-store can redeem

the lower online price; we refer to this proﬁt-reducing

eﬀect as channel arbitrage. However, consumers who

visit their preferred store ﬁrst without searching online

are unable to obtain evidence of a lower price for their

desired product before arriving at the store. These con-

sumers pay the store price even when a self-matching

policy is in eﬀect, leading to the decision-stage discrim-

ination eﬀect, thus allowing a self-matching retailer to

charge diﬀerent prices to store consumers based on

their decision stage, which can increase proﬁts.

If only one retailer self-matches, decided consumers

can redeem the lower online price at the self-matching

retailer’s store but only have access to the store price at

the rival. This induces the self-matching retailer to set

a higher online price to mitigate the negative impact of

channel arbitrage. The rival follows suit due to strate-

gic complementarity of prices and sets a higher online

price as well, thus softening online competition. We

refer to this proﬁt-increasing mechanism as the online

competition dampening eﬀect, with self-matching serving

as a commitment device to increase online prices from

the purely competitive level. It emerges only when one

of the retailers oﬀers to self-match: If both retailers self-

match, decided consumers have access to the online

prices at both stores, and intense competition in the

online channel ensues.

We also investigate the equilibrium proﬁtability of

the self-matching policy. Our analysis shows that self-

matching is not necessarily harmful. In fact, both re-

tailers can be better oﬀ by oﬀering to self-match when the

positive online competition dampening and/or deci-

sion-stage discrimination eﬀects dominate the negative

channel arbitrage eﬀect.

We investigate several model extensions in Section 5.

First, we examine how the presence of consumers

equipped with “smart” devices, who can retrieve

online price information when in the store, aﬀects

retailers’ incentives to implement a self-matching pol-

icy. Intuitively, when more consumers can retrieve the

lower online price, the negative channel arbitrage eﬀect

is more pronounced. However, we ﬁnd that the pres-

ence of “smart” consumers can allow retailers to ben-

eﬁt even more from online competition dampening by

charging higher online prices. In another extension,

we examine the eﬀects of self-matching in a mixed

duopoly, in which a multichannel retailer competes

with an online-only retailer (i.e., a pure e-tailer). We

ﬁnd that competition is dampened in the online chan-

nel when the multichannel retailer chooses to self-

match, allowing both retailers to beneﬁt.

Finally, we conducted a consumer survey that allows

us to evaluate how customer and market characteris-

tics pertain to our model setup and ﬁndings. We ﬁnd

evidence of signiﬁcant consumer heterogeneity on the

dimensions modeled. Mapping the equilibrium pre-

dictions of the model to observed self-matching poli-

cies chosen by ﬁrms is suggestive of the relevance of

our approach.

Next, we review the literature (Section 2), describe

the model (Section 3), and analyze equilibrium strate-

gies and outcomes (Section 4). We then consider sev-

eral extensions of the model (Section 5) and conclude

by discussing managerial and empirical implications

as well as future research opportunities (Section 6).

2. Literature Review

We draw from two separate streams of past research.

The ﬁrst is focused on multichannel retailing, and

the second on competitive price-matching in a sin-

gle channel. Research in multichannel retailing has

typically assumed that retailers set the same or dif-

ferent prices across channels, without examining the

incentives to adopt a self-matching policy. Liu et al.

(2006), for example, study a brick-and-mortar retailer’s

decision to open an online arm, assuming price con-

sistency, or diﬀerent prices across channels. Zhang

(2009) considers separate prices per channel and stud-

ies the retailer’s decision to operate an online arm and

advertise store prices. Ofek et al. (2011) study retail-

ers’ incentives to oﬀer store sales assistance when also

operating an online channel, allowing for identical or

diﬀerent pricing across channels. Aside from ignoring

self-matching pricing policies, this literature has not

considered or modeled heterogeneity in consumers’

DMPs, which plays an important role in their channel

choice in practice.

The key theoretical mechanisms modeling sales and

service were developed by Shin (2007) and investigated

further in the literature (e.g. Mehra et al. 2013). While

"price-matching has been suggested as a strategy to

combat showrooming, to our knowledge, t here has not

been a careful modeling and evaluation of whether and

when such policies can be eﬀective, particularly in a

competitive context.”

Competitive price-matching is an area that has been

well studied. This literature has generally focused on

a retailer’s incentives to match competitors’ prices in

a single channel setting, typically brick-and-mortar

stores. Salop (1986) argued that when retailers price

match each other, this leads to higher prices than oth-

erwise, as they no longer have an incentive to engage

in price competition, thus implying a form of tacit

collusion (Zhang 1995). However, competitive price-

matching has also been found to intensify competition

because it encourages consumer search (Chen et al.

2001). Other research in competitive price-matching

has explored its role as a signaling mechanism for cer-

tain aspects of a retailer’s product or service (Moorthy

and Winter 2006, Moorthy and Zhang 2006), the impact

Kireyev, Kumar, and Ofek: Match Your Own Price?

of hassle costs (Hviid and Shaﬀer 1999), interaction

with product assortment decisions (Coughlan and

Shaﬀer 2009), and the impact of product availability

(Nalca et al. 2010).

By contrast, self-matching pricing policies represent a

phenomenon relevant only for multichannel retailers;

recent retailing trends make self-matching an impor-

tant issue to study. First, the nature of competition

is evolving in many categories, from retailers carry-

ing similar products from multiple brands to manu-

facturers who establish their own retail stores, e.g.,

Apple, Microsoft, and Samsung. Second, many retail-

ers are moving towards establishing strong private

label brands or building exclusive product lines to

avoid direct price wars wit h competitors (Bustillo and

Lawton 2009, Mattioli 2011). For instance, 50% or more

of products sold by retailers such as JCPenney or health

supply retailer GNC are exclusive or private label;

electronics retailers such as Brookstone and Best Buy

are also increasingly focused on private label prod-

ucts.

These trends accentuate the relevance of self-

matching relative to competitive price-matching as

the product assortments retailers carry become more

diﬀerentiated.

The mechanisms underlying self-matching are also

connected to the broad literature on price discrimina-

tion. Cooper (1986) examines pricing in a two-period

model, where retailers commit to giving consumers

who purchase in the ﬁrst period the diﬀerence between

the ﬁrst and second period prices if the latter price

is lower (a form of intertemporal self-matching). The

author shows t hat this policy may increase retailer prof-

its as it reduces the incentive to lower prices in the sec-

ond period for both retailers. This eﬀect is similar to

the online competition dampening eﬀect we identify,

whereby a retailer reduces its own incentive to price

lower online by inducing channel arbitrage through

self-matching. However, whereas both retailers can

oﬀer and beneﬁt from a “most-favored-customer” pol-

icy in the intertemporal setting, the online competition

dampening eﬀect can exist only if one retailer oﬀers

to self-match. If both retailers self-match, they reignite

competition in the online market and nullify the eﬀect.

Cross-channel price-matching is thus driven by diﬀer-

ent strategic incentives.

Thisse and Vives (1988), Holmes (1989), and Corts

(1998) consider cases wherein price discrimination may

lead to lower proﬁts for competing retailers in equi-

librium. Similarly, retailers may be compelled to self-

match in equilibrium even though they would have

been better oﬀ had self-matching not been an option.

In our context, on one hand, a self-matching policy

acts as a commitment not to price discriminate decided

consumers across channels, which can lead to greater

proﬁts for both retailers because this creates an incen-

tive to increase the online price to mitigate the arbi-

trage eﬀect. On the other hand, a self-matching policy

enables price discrimination between undecided and

decided consumers who shop in-store. Depending on

the relative sizes of these segments, self-matching poli-

cies may emerge in equilibrium and lead to greater or

lower proﬁts for both retailers.

Desai and Purohit (2004) consider a competitive set-

ting where consumers may haggle over price with

retailers. Some form of haggling may occur in the self-

matching setting if retailers are not explicit about their

policies and consumers must wrangle with managers

to obtain a self-match. This interaction may induce

additional costs for consumers and for retailers when

processing a self-matching policy. In our analysis, we

focus on the case of retailers explicitly announcing

their self-matching policies and illustrate how self-

matching emerges in equilibrium in the absence of con-

sumer haggling or hassle costs. In an extension, we

consider the implications of retailer processing costs

when a consumer redeems a self-match.

3. Model

3.1. Retailers

Two competing retailers in the same category are

situated at the endpoints of a unit consumer inter-

val, or linear city, i.e., x ⇤ 0 and x ⇤ 1 (Hotelling

1929). The retailers oﬀer unique and non-overlapping

sets of products. Because they carry diﬀerent prod-

ucts, they do not have the option of oﬀering competi-

tive price-matching guarantees. For example, Gap and

Aeropostale sell apparel and operate in the same cate-

gories, but the items themselves are not the same and

reﬂect the dedicated designs and logos of each of these

retailers.

We model a two-stage game in which the retailers

must ﬁrst decide on self-matching policies and then

on prices in each channel. We denote by SM

⇤ 0 t he

decision of retailer i not to self-match and by SM

⇤ 1

the decision to self-match, leading to four possible sub-

games, i.e., (0, 0), (1, 1), (1, 0), and (0, 1) representing

(SM

, SM

). In each subgame, p

denotes t he price set

by retailer j 2{1, 2} in channel k 2{on, s}, where on

stands for the online or Internet channel and s stands

for the physical store channel. With self-matching, con-

sumers who retrieve the match pay the lowest of the

two channel prices. In the equilibrium analysis that

follows, we ﬁnd that retailers never set lower prices in-

store than online. Hence, the only relevant matching

policy to focus on is the store-to-online self-match. All

retailer costs are assumed to be zero.

3.2. Consumers

To capture important features of the shopping process

in multichannel environments, we model consumers as

being heterogeneous along multiple dimensions.

Kireyev, Kumar, and Ofek: Match Your Own Price?

Retailer Brand Preferences: Consumers vary in their

preferences for a retailer’s product, e.g., a consumer

might prefer Macy’s clothing lines to those oﬀered at

JCPenney. We capture this aspect of heterogeneity by

allowing consumers to be distributed uniformly along

a unit line segment in the preference space, x ⇠ U[0, 1].

A consumer at preference location x incurs a “mis-

ﬁt cost” ✓x when purchasing from retailer 1 and a

cost ✓ (1  x) when purchasing from retailer 2. Note

that the parameter ✓ does not involve transportation

costs, rather it represents horizontal retailer-consumer

“misﬁt” costs, which are the same across the online

and store channels. Misﬁt costs reﬂect heterogeneity in

taste over diﬀerentiated products of similar value, e.g.,

the collection of suits at Banana Republic compared to

those at J. Crew.

Channel Preferences: Channel-agnostic (A) consumers

do not have an inherent preference for either channel

and, for a given retailer, would buy from whichever

channel has the lower price for the product they pur-

chase. On the other hand, Store-only (S) consumers ﬁnd

the online channel insuﬃcient, e.g., due to waiting

times for online purchases, risks associated with online

purchases (such as product defects), etc. These con-

sumers purchase only in the store, alt hough they might

research products online and obtain online price infor-

mation for the product they plan to buy. We assume

that channel-agnostic consumers form a fraction of

size ⌘ of the market while store-only consumers form

a fraction of size 1  ⌘.

Decision Stage: Consumers can diﬀer in their deci-

sion stage, a particularly important aspect of multi-

channel shopping (Neslin et al. 2006, Mulpuru 2010,

Mohammed 2013). Undecided (U) consumers of propor-

tion  (0 <<1) need to go to the store because t hey

do not have a clear idea of the product they wish to

purchase. Decided (D) consumers of proportion 1  

are certain about the product they wish to buy and

can thus costlessly search for price information from

home. Undecided consumers ﬁrst visit a retailer’s store,

selecting the store closest to their preference location,

to ﬁnd an appropriate product that ﬁts their needs.

After determining ﬁt, they may purchase the product

in-store or return home to purchase online, depending

on their channel preference. Consumers obtain a value

v from purchasing their selected product.

Categories such as apparel, fashion, furniture, and

sporting goods are likely to feature more undecided

consumers, as styles and sizes of products are impor-

tant factors that frequently change. Because undecided

consumers do not know which product they want

before visiting a store, they do not have at their dis-

posal all prices while at the store, since keeping track

of a large number of products, models, and versions

even within a category would be impractical. Unde-

cided consumers in the model are unaware of the

Table 1. Consumer Segments and Proportions

Decision stage

Channel preference Undecided (U) Decided (D)

Store (S) (SU): (1  ⌘)  (SD): (1  ⌘)(1  )

Agnostic (A) (AU): ⌘ (AD): ⌘(1  )

exact product they wish to purchase beforehand (they

have limited ability to infer prices under diﬀerent self-

matching conﬁgurations before visiting the store).

We set the travel cost for a consumer’s ﬁrst shopping

trip to zero and assume that additional trips are suf-

ﬁciently costly. Note that if consumers have no cost to

visit multiple stores in person, then we obtain a trivial

speciﬁcation wherein there is no distinction between

decided and undecided consumers who shop in-store.

Throughout the paper, we focus on the more interest-

ing case wherein additional shopping trips are costly

enough so that store-only undecided consumers do

not shop across multiple physical stores (see proofs of

Propositions 2 and 4 in the appendix for formal condi-

tions on the travel cost). However, in all cases, decided

consumers research products and prices in advance.

Table 1 depicts the diﬀerent consumer segments in-

cluded in the model. We denote the four segments

of consumers as SU, AU, SD, and AD, depending on

their channel preference and decision stage; the size of

each segment is indicated in the corresponding cell of

the table. Each of the four segments is uniformly dis-

tributed on a Hotelling linear city of unit length, such

that consumer location on the line determines retailer

preference. In the online appendix, we present exam-

ples to illustrate the buying process of a consumer

from each segment. Table 2 details the notation used

throughout the paper.

3.3. Sequence of Events

Figure 1 details the sequence of events. First, retailers

simultaneously decide on a self-matching pricing strat-

egy. Then, after observing each other’s self-matching

decisions, they determine the price levels in each chan-

nel. Consumers, depending on their type (decided or

undecided, store or channel-agnostic, and horizontal

preference), decide on which channel and retailer at

which to shop. Decided consumers, who know the

online price before visiting the store, can ask for a

price match if the online price is lower and the retailer

has chosen to self-match. Finally, consumers make pur-

chase decisions and retailer proﬁts are realized.

3.4. Consumer Utility

We now specify the utility consumers derive under

diﬀerent self-matching scenarios. Recall that store-

decided (SD) consumers know all prices across both

retailers and channels before they make a purchase

decision. Channel-agnostic undecided (AU) consumers

Kireyev, Kumar, and Ofek: Match Your Own Price?

Table 2. Summary of Notation

Notation Deﬁnition

In-store price for retailer j

Online price for retailer j

Retailer j’s self-matching decision

⇧

, SM

Retailer j’s total proﬁt in subgame (SM

, SM

)

Consumer

v Consumer valuation of the product

✓ Retailer diﬀerentiation

1   Fraction of decided segment of consumers

 Fraction of undecided segment of consumers

⌘ Fraction of channel-agnostic consumers

1  ⌘ Fraction of store-only consumers

✓ · x for x 2[0 , 1] Measure of retailer preference for consumer

at location x

Utility for purchasing from retailer j

in channel k

have t he option of visiting a store to learn what they

want and then returning home to make an online pur-

chase, whereas store undecided (SU) consumers pur-

chase in the store they ﬁrst visit or make no purchase.

Consumers obtain zero utility when they do not make

a purchase.

Consider the case when neither retailer self-matches,

i.e., (SM

, SM

) ⇤ (0, 0). For a consumer who knows

the product she wishes to purchase, the utility for each

retailer and channel option is as follows:

⇤ v  p

 ✓x, u

⇤ v  p

 ✓x,

⇤ v  p

 ✓(1  x), u

⇤ v  p

 ✓(1  x),

(1)

Figure 1. (Color online) Sequence of Events in t he Model

Consumers make purchase decisions

Simultaneously decide on self-matching policies

Obtain product and price information from all

channel and retailer options

Visit store of retailer with closer match to

preferences to learn about products

Decided consumers Undecided consumers

Channel agnostic Store only Channel agnostic Store only

Retailers R1 and R2

Store: R1 and R2

Obtain self-match if

retailer offers policy

Choice set at purchase time

(channel and retailer)

Store: R1 and R2

Online: R1 and R2

Observing policies, simultaneously

set prices in each channel

Consumers

gather

product and

price

information

–

Store: R1 if x <

Online: R1 and R2

Does not obtain price-match

even if retailer offers policy

Store: R1 if x <

R2 o/w

–

R2 o/w

Note. o/w, otherwise.

where v is the value of the product, p

and p

(k ⇤ on or

k ⇤ s) are the prices set by retailers 1 and 2, respectively,

✓ measures the degree of consumer preferences for

retailers, and x 2[0, 1] is the consumer’s location (in

the preference space) relative to retailer 1.

Whereas t hese utilities apply to all consumers, not all

segments have access to all purchase options. Figure 1

details the choice set available to each segment. For

example, the channel-agnostic decided (AD) consumer

has access to all four options, whereas the store-only

undecided (SU) consumer only has the option of pur-

chasing from his preferred store (e.g., retailer 1). Thus,

consumer heterogeneity results in diﬀerent choice sets

available to each segment.

Undecided consumers (both SU and AU), who do

not know which speciﬁc product they need, ﬁrst visit

the retailer closer to their preference location (i.e., visit

retailer 1 if x <

and retailer 2, otherwise). After their

shopping trip, the store-only undecided (SU) segment

must decide whet her to buy the product that ﬁts their

needs at the store or make no purchase; hence only

the corresponding u

-expression in (1) is relevant for

such a consumer. Channel-agnostic undecided (AU)

consumers can purchase in the store they ﬁrst visit and

pay the store price, or return home and make an online

purchase from either retailer; the utility expressions

, u

are thus relevant for AU consumers who

prefer retailer 1, and u

, u

are relevant for AU

consumers who prefer retailer 2.

The Impact of Self-Matching Prices. We now examine

how self-matching practices by retailers impact con-

Kireyev, Kumar, and Ofek: Match Your Own Price?

sumer utilities. Decided consumers know all prices for

the speciﬁc product they want, and if they shop at

the store oﬀering self-matching, they can come armed

with the online price and request a price match. Thus,

decided consumers can obtain a price match in-store

whereas undecided consumers cannot.

When both retailers oﬀer a self-matching pricing pol-

icy, i.e., under (SM

, SM

) ⇤ (1, 1), a consumer at x 2

[0, 1] who can obtain a self-match faces the following

utilities:

⇤ v  p

 ✓x, u

⇤ v  min(p

, p

)✓x

⇤ v  p

 ✓(1  x),

⇤ v  min(p

, p

)✓(1  x).

(2)

Consumers who cannot obtain a price self-match

(i.e., undecided consumers) continue to face the cor-

responding utilities speciﬁed in Equation (1). Note

that although the utility expressions remain the same,

retailers may set diﬀerent prices under diﬀerent self-

matching scenarios. Hence, the equilibrium utilities

experienced by consumers will typically diﬀer depend-

ing on retailer self-matching policies.

Next, consider consumers’ channel preferences.

Channel-agnostic decided (AD) consumers have no

particular preference for any channel and would

choose the lower-priced channel option. Store decided

(SD) consumers choose one of the stores based on their

preferences and prices. However, they can obtain the

lower online price if the retailer oﬀers a self-matching

policy. Thus, the expressions for u

for j ⇤ 1, 2 are dif-

ferent in (2) compared to (1). Undecided (AU and SU)

consumers do not know which product they want until

they visit the store. They face the same utilities under

(1, 1) as under (0, 0) since they cannot redeem match-

ing policies when they visit a retailer’s store without

making an additional costly set of trips, i.e., back home

to determine online prices and then back to a store to

make the purchase.

Utilities in the asymmetric subgame (1, 0), where

only retailer 1 oﬀers to self-match prices, are deﬁned

similar to the (0, 0) case, with only u

changing for

decided consumers, who can obtain a self-match only

from retailer 1 but not retailer 2:

⇤ v  min(p

, p

)✓x.

4. Analysis

We begin our analysis by considering the benchmark

monopoly case, then the multichannel duopoly setting.

All proofs are in the appendix along with the deﬁned

threshold values and constants. Note that in all cases,

we derive conditions for the market to be covered in

the proof; our text discussion will focus on the region

of coverage in equilibrium.

4.1. Benchmark Monopoly: A Single Entity Owns

Both Retailers

Consider a monopolist that jointly maximizes the prof-

its of two multichannel retailers at the endpoints of a

unit segment by choosing a self-matching policy and

setting prices.

The following holds:

Proposition 1. A monopolist cannot increase proﬁts by

self-matching prices across channels.

The monopolist will price to extract the great-

est surplus from each channel. Because undecided

and decided consumers are present in both chan-

nels, the prices charged will be the same in both and

equal to the monopoly price of (v  ✓/2), regardless of

whether the monopolist oﬀers a self-matching policy.

The monopolist thus obtains no additional proﬁt when

oﬀering the policy and will not oﬀer it when it entails

a minimal implementation cost.

4.2. Multichannel Duopoly

We now consider the case of two competing multi-

channel retailers who make decisions according to the

timeline in Figure 1. We examine each of the possi-

ble self-matching policy subgames and conclude with

a result highlighting the conditions under which self-

matching emerges in equilibrium.

For notational convenience, we deﬁne the function

(p

, p

; ✓) :⇤

+ (p

 p

)/(2✓) to represent t he propor-

tion of demand obtained by retailer 1 from a speciﬁc

segment of consumers who face prices p

and p

from

the two retailers.

No Self-Matching—(0, 0). In the (0, 0) subgame wherein

neither retailer self-matches, store-only consumers (SD

and SU segments) purchase from the store channel

and pay the store price. Channel-agnostic (AD and

AU) consumers can also purchase from either retailer’s

online channel. AD consumers will begin their search

process online, whereas AU consumers will ﬁrst visit

their preferred retailer to browse products, then return

home to purchase online after they discover the speciﬁc

product they wish to purchase. Retailers compete for

these two consumer segments in the online channel.

SU consumers visit the retailer closest in preference

to them to learn about products. Recall that these con-

sumers do not purchase online and do not switch stores

because of travel costs associated with multiple store

visits. Each retailer eﬀectively has a subset (/2) of such

consumers.

On the other hand, SD consumers know the prod-

uct they want and are informed of all prices. They

purchase in-store, given their channel preference, but

make a decision on which store to visit after factoring

in their retailer preferences and prices. Thus, there is

intense competition among retailers for this segment,

since by reducing store price, a retailer can attract more

SD consumers.

Kireyev, Kumar, and Ofek: Match Your Own Price?

The proﬁt functions of retailers 1 and 2 can be writ-

ten as

⇧

0, 0

⇤ ⌘(p

, p

| {z }

Channel-Agnostic Decided and Undecided

+ (1  ⌘)



(1  )(p

, p

)

| {z }

Store-Only Decided

+ /2

|{z}

Store-Only Undecided



⇧

0, 0

⇤ ⌘(1  (p

, p

))p

+ (1  ⌘)



(1  )(1  (p

, p

)) + /2



In the (0, 0) case, the online situation is similar

to retailers competing in a horizontally diﬀerentiated

market comprised of only channel-agnostic consumers.

The resulting equilibrium prices, therefore, reﬂect the

strength of consumers’ preferences for retailers, with

⇤

⇤ ✓. We refer to a price of ✓ as the “competi-

tive” price level to reﬂect the fact that this would be the

price charged in a standard Hotelling duopoly model

with one retail channel.

Next, we turn to the store channel, where we obtain

symmetric equilibrium prices of

⇤

v 

✓



1  

✓

1  

✓

1  

. (3)

There are a few useful observations to be made here.

First, for v/✓ 

+ 1/(1  ), retailers serve the entire

market even though they charge the monopoly price

in-store. This is possible because of the existence of

SU consumers: Retailers prefer to charge the monopoly

price to extract all surplus from SU consumers if the

ratio of product value to retailer diﬀerentiation is suﬃ-

ciently low. Second, if v/✓>

+ 1/(1  ), then retailers

charge a store price of ✓/(1  ), which is larger than

the competitive price of ✓ . When v/✓ is suﬃciently

large, retailers can no longer maintain monopoly prices

in-store and prefer to compete for SD consumers. How-

ever, the existence of SU consumers enables retailers

to charge more in-store than online, and the retailers

extract more surplus from both SU and SD segments.

Symmetric Self-Matching—(1, 1). In this case, both re-

tailers implement a self-matching policy. The ﬁrst and

most obvious result of self-matching is the channel arbi-

trage eﬀect, and the intuition here is straightforward.

Recall that SD consumers shop in-store and pay

in Equation (3) absent a self-matching policy. However,

with a self-match they pay the lower online price while

shopping in-store, resulting in less proﬁt for the retailer

due to arbitrage across channels.

Although this arbitrage intuition is correct, it is

incomplete in determining whether in equilibrium a

self-matching pricing policy will be adopted. When

a multichannel retailer chooses to self-match, there

emerges an important distinction between the store-

only decided (SD) and undecided (SU) consumers.

Whereas the SD consumers can obtain a price match,

SU consumers only know which product they desire

during a store visit. Because they lack evidence of a

lower online price, they always pay the store price.

Thus, even though the two segments of store con-

sumers obtain the product in-store, they eﬀectively pay

diﬀerent prices. Self-matching thus enables the retailer

to price discriminate consumers based on their deci-

sion stage.

Retailers’ proﬁts in this self-matching setting are

thus

⇧

1, 1

⇤ (1  (1  ⌘))

, p

| {z }

Channel-Agnostic and Store-Only Decided

+ (1  ⌘)



| {z }

Store-Only Undecided

⇧

1, 1

⇤(1  (1  ⌘))(1  

, p

))p

+ (1  ⌘)



In the pricing sub-game, retailers set equilibrium on-

line prices

⇤

⇤ ✓, since there is no force to pre-

vent online prices from dropping to their competitive

level. However, retailers set store prices

⇤

⇤ (v 

✓/2) to extract surplus from their respective “captive”

sets of SU customers, who pay the store price. We refer

to the ability to extract additional surplus from SU

consumers through self-matching as the decision-stage

discrimination eﬀect.

Figure 2 illustrates pricing and purchase outcomes

across the (0, 0) and (1, 1) subgames. The dashed

regions cover the segments that pay the store price,

whereas the solid-ﬁlled regions show the segments

that pay the online price. In the (0, 0) subgame, SU

and SD consumers purchase in-store and pay the same

store price, whereas AU and AD consumers purchase

online and pay the online price. In the (1, 1) subgame,

SU and SD consumers purchase in-store, but SD con-

sumers now pay the online price. Thus, self-matching

allows a retailer to simultaneously price discriminate

consumers across decision stages in the store and de-

segment decided consumers across channels.

Asymmetric Self-Matching—(1, 0). Next, we explore

the case wherein retailer 1 oﬀers a self-matching policy

while retailer 2 does not, i.e., the (1, 0) self-matching

subgame. By symmetry (or relabeling), similar results

follow in the (0, 1) subgame. Observe t hat SD con-

sumers who visit retailer 1’s store can purchase there

and pay the lower of the online and store price,

i.e., min(p

, p

). However, if an SD consumer visits

retailer 2’s store instead, she faces a price of p

and can-

not obtain the online price in-store (since retailer 2 does

not self-match). Moreover, by oﬀering a self-matching

policy, and as long as its online price satisﬁes p

< p

retailer 1 attracts some SD consumers who are closer in

Kireyev, Kumar, and Ofek: Match Your Own Price?

Figure 2. The Eﬀects of Self-Matching

AU AD AD AU

SU SD SD SU

AU AD AD AU

SU SD SD SU

AU AD AD AU

SU SD SD SU

Retailer 1 Retailer 2

Online

In-store

Retailer 1 Retailer 2

Online

In-store

Retailer 1

Retailer 2

Online

In-store

(a) No self-matching—(0, 0)

(b) Symmetric self-matching—(1,1)

Online price Store price

preference to the competing retailer 2 but who choose

to visit retailer 1’s store in anticipation of paying the

lower online price through a self-match.

For SD consumers, under p

< p

, the store price

of retailer 1 is irrelevant (since they retrieve the price

match); the retailer can set a store price level to capture

the highest possible surplus from the SU consumers

who are closer to its location. Thus, decision-stage price

discrimination persists in the asymmetric subgame.

We obtain the following proﬁt functions:

⇧

1, 0

⇤ ⌘

, p

| {z }

Channel-Agnostic

+ (1  ⌘)



(1  )

, p

| {z }

Store-Only Decided

+ /2p

|{z}

Store-Only Undecided



⇧

1, 0

⇤ ⌘(1  

, p

))p

+ (1  ⌘)



(1  )(1  

, p

)) + /2



Solving for the second-stage pricing subgame, we ﬁnd

that the online price levels chosen by the retailers are

higher than the Hotelling competitive price of ✓ in both

channels and critically depend on the ratio of prod-

uct value v to the retailer diﬀerentiation parameter

✓ as follows. For v/✓ (

+ 1/(6(1  (1  ⌘))) + /

(2(1  ))), both retailers will extract all surplus from

SU consumers and set prices

⇤ ✓ +

(1  )(1  ⌘)(2v  3✓)

4(1  (1  ⌘))  ⌘

⇤ ✓ +

(1  )(1  ⌘)(2v  3✓)

8(1  (1  ⌘))  2⌘

⇤

⇤ v 

✓

For v/✓>(4/3 + 1/(6(1  (1  ⌘))) + /(2(1  ))),we

ﬁnd that retailers set prices

⇤ ✓

✓

3(1  (1  ⌘))

◆

⇤ ✓

✓

6(1  (1  ⌘))

◆

⇤ v 

✓

⇤

✓

2(1  )

Interestingly, equilibrium online prices in the asym-

metric self-matching (1, 0) case are greater than those

set in the no self-matching (0, 0) case and the sym-

metric self-matching (1, 1) case. The intuition follows

from the idea that although self-matching retailer 1

loses proﬁt from the SD segment of consumers who

can invoke the price self-match, the policy eﬀectively

acts like a “commitment device” to prevent online

prices from going all the way down to the compet-

itive level. More important, when only one retailer

self-matches online competition softens, which results

in channel-agnostic AD and AU consumers paying a

higher price (relative to the competitive online price of

✓ they were paying under no self-matching or sym-

metric self-matching cases). Thus, self-matching has a

positive eﬀect on proﬁts through t his third mechanism,

which we term the online competition dampening eﬀect.

Note that the situation in the asymmetric (1, 0) case

diﬀers from the case when both retailers self-match.

Under (1 , 1), SD consumers can redeem the online

price at both retailers’ stores, which forces online

prices down to their competitive level ✓. By con-

trast, Figure 2 illustrates how, in the asymmetric (1, 0)

case, SD consumers can only redeem the match from

retailer 1. Retailer 2 will price higher in-store relative

to retailer 1’s online price, as SD consumers and its

captive segment of SU consumers pay its store price,

whereas retailer 1 fully segments out its store con-

sumers through the self-matching policy invoked by

its SD consumers (while retailer 1’s SU consumers con-

tinue to pay its store price). However, to mitigate the

downside eﬀect of channel arbitrage, retailer 1 does not

set its online price as low as ✓; this move also allows

it to extract greater surplus from the channel agnostic

segments. Because online prices are strategic comple-

ments across retailers, retailer 2’s best response is to

increase its online price as well. This results in online

Kireyev, Kumar, and Ofek: Match Your Own Price?

Table 3. Eﬀects of Self-Matching for Retailer 1 in a

Multichannel Duopoly

Relevant

Eﬀect Subgames

() Channel Arbitrage: SD consumers redeem lower

online price in the store, reducing proﬁts from SD.

(1, 0)(1 , 1)

(+) Decision-Stage Discrimination: Retailer avoids

competing for SD segment on store prices; instead

letting them obtain lower online prices. This

allows higher store prices to captive SU segment.

(1, 0)(1 , 1)

(+) Online Competition Dampening: Retailer charges

higher online price to mitigate arbitrage,

increasing proﬁt from AD, AU, and SD segments.

(1, 0)

prices at both retailers being higher than the competi-

tive level, leading to online competition dampening.

The results detailed in this section are based on the

pricing subgame, taking the self-matching policies as

given. The pricing equilibria depend on the magni-

tudes of the three eﬀects induced by self-matching.

Table 3 presents a summary of the eﬀects we have iden-

tiﬁed. Note that the negative channel arbitrage eﬀect

and the positive decision-stage discrimination eﬀect

always occur for a self-matching retailer, while online

competition dampening occurs only when one self-

matches but the rival does not. We now examine the

full equilibrium results of the game beginning with the

self-matching strategy choices.

4.3. Self-Matching Policy Equilibria in a

Multichannel Duopoly

For a self-matching policy conﬁguration to emerge in

equilibrium, it must be the case that neither retailer

would be better oﬀ by unilaterally deviating to oﬀer a

diﬀerent policy. Proposition 2 details the equilibrium

conditions and the resulting choices of self-matching

policies. Across all regions of the parameter space, we

restrict our focus to Pareto-dominant equilibria.

Proposition 2. In a duopoly featuring two multichannel

retailers, self-matching policies are determined by the follow-

ing mutually exclusive regions

• Asymmetric equilibrium (1, 0). One retailer will oﬀer

to self-match its prices while the other will not when product

values are relatively low or retailer diﬀerentiation is high.

• Symmetric non-matching equilibrium (0, 0). Neither

retailer will self-match its prices when product values and

retailer diﬀerentiation are at intermediate levels.

• Symmetric matching equilibrium (1, 1). Both retailers

will self-match prices when product values are high or re-

tailer diﬀerentiation is low.

The above result indicates that all three types of

joint strategies can emerge in equilibrium depending

on the nature of the product and degree of compet-

itive interaction. To understand the intuition behind

the emergence of the diﬀerent equilibria, it is critical to

examine how the focal retailer’s best response function

evolves as the ratio of product value to retailer dif-

ferentiation (v/✓) changes. We translate best response

functions into equilibria in Figure 3. The top arrow

depicts retailer 1’s best response if retailer 2 does not

self-match. The middle arrow depicts retailer 1’s best

response if retailer 2 self-matches. The dominant eﬀects

for retailer 1 are listed below the arrows. The bottom

arrow shows the emergent Pareto-dominant equilibria.

The best response of the focal retailer depends on the

competitor’s self-matching strategy as well as the three

eﬀects we have previously described, i.e., channel arbi-

trage, decision-stage discrimination, and online competition

dampening.

Retailer 1’s Best Response to Retailer 2 Not Self-

Matching. For low v/✓, the retailer’s store price (v 

✓/2) is relatively close to its competitive online price

✓ because there is little additional surplus the retailer

can extract from its captive SU consumers by pricing

higher in-store. As a result, eﬀects that have an impact

on the store channel, i.e., the negative channel arbi-

trage eﬀect and the positive decision-stage discrimina-

tion eﬀect are negligible. However, online prices can

increase with self-matching due to the online compe-

tition dampening eﬀect. This leads retailer 1 to oﬀer

a self-matching policy to take advantage of the addi-

tional proﬁts from the online channel.

As v/✓ increases, so does the diﬀerence in prices

across channels. For intermediate values of v/✓, the

retailer can extract more surplus from SU consumers,

driving it to price higher in-store even if it does not self-

match, thus reducing the beneﬁts of decision-stage dis-

crimination. Because a self-matching policy allows SD

consumers to redeem the lower online price, the chan-

nel arbitrage eﬀect increases. As competition in the

online channel is more intense than in the store chan-

nel, the positive online competition dampening eﬀect

can no longer overcome t he negative channel arbitrage

eﬀect. Consequently, the channel arbitrage eﬀect dom-

inates the other two eﬀects, and t he retailer no longer

ﬁnds it proﬁtable to self-match as a best response.

At high v/✓ levels, retailer 1 is compelled to compete

more intensely for SD consumers closer to retailer 2 in

preference, resulting in a store price of ✓/(1  ) that

no longer grows in v, if the retailer does not self-match.

Thus, the negative impact due to channel arbitrage is

limited. However, if the retailer were to oﬀer a self-

matching policy, decision-stage discrimination would

allow it to charge the monopoly price (v  ✓/2) to cap-

tive SU consumers, which increases as v/✓ increases.

This creates a strong positive impact on proﬁts, result-

ing in retailer 1 choosing to oﬀer self-matching.

Retailer 1’s Best Response to Retailer 2 Self-Matching.

We now turn to the case wherein retailer 2 decides to

oﬀer a self-matching policy.

Kireyev, Kumar, and Ofek: Match Your Own Price?

Figure 3. (Color online) Retailer Best Responses

Best response of

retailer 1 given

retailer 2 does

not self-match

Self-match Do not self-match

Do not self-match Self-match

Equilibrium regions

(1,1)(0,1) or (1, 0) (0, 0)

Dominant effect

Online competition

dampening (+)

Channel arbitrage (–) Decision stage

discrimination (+)

Online competition

dampening (+)

Decision stage

discrimination (+)

Best response of

retailer 1 given

retailer 2 self-

matches

Self-match

–

Recall that when both retailers self-match, the online

competition dampening eﬀect ceases to exist. Because

retailer 2 is self-matching, its actions will result in

online competition dampening only if retailer 1 does

not self-match. This creates an incentive for retailer 1

to refrain from self-matching at low values of v/ ✓,to

beneﬁt from online competition dampening through

strategic complementarity in prices. Furthermore, at

low values of v/✓, the decision-stage discrimination

eﬀect is small as retailer 1 cannot extract a substantial

amount of surplus from SU consumers.

As v/✓ increases, the beneﬁt of decision-stage dis-

crimination grows because the retailer can extract

greater surplus from SU consumers if it can charge

them a diﬀerent price than SD consumers. This leads

retailer 1 to adopt self-matching for high values of v/✓.

Strategic Substitutes or Complements. We integrate

the best responses to obtain equilibrium strategies and

focus on whether self-matching strategies across retail-

ers are strategic complements or substitutes. We ﬁnd

from the best responses that at low product values,

and/or at high levels of retailer diﬀerentiation, the self-

matching strategies act like strategic substitutes, so t hat

a retailer will choose the strategy opposite to that of its

competitor. As v/✓ increases to an intermediate level,

we obtain a symmetric equilibrium where no retailer

self-matches and strategies are strategic complements.

Finally, when v/✓ is above a high threshold, the strong

impact of decision-stage discrimination leads to self-

matching being a dominant strategy regardless of what

the competitor chooses. Figure 4 shows the equilibrium

regions that emerge in the v/✓ $  space for a ﬁxed

value of ⌘ 2(0, 1) based on Proposition 2.

Next, we turn to how the equilibrium regions are

aﬀected by  and ⌘.

Corollary 1. An increase in the fraction of undecided con-

sumers  will grow the asymmetric equilibrium region and

shrink the symmetric equilibrium regions.

According to the corollary, retailer 1 has more of an

incentive to oﬀer a self-matching policy as  increases,

implying that the v/✓-region for which we can sustain

the (1, 0) equilibrium expands. To understand the intu-

ition for Corollary 1, consider the case of focal retailer

1’s best response when retailer 2 does not self-match.

As the fraction of undecided consumers increases, the

Figure 4. Equilibrium Regions

Symmetric (0, 0)

Asymmetric (1, 0)

v/à

(ratio of product

value to

differentiation)

Ç (size of undecided segment)

Symmetric (1,1)

Kireyev, Kumar, and Ofek: Match Your Own Price?

retailers stand to gain more from online competition

dampening (because of the greater fraction of AU con-

sumers). Thus, if retailer 1 self-matches, retailer 2 will

refrain from doing so (because when both self-match,

the online competition dampening eﬀect is nulliﬁed).

The next corollary examines the eﬀect of ⌘ on the

equilibrium regions.

Corollary 2. An increase in the fraction of channel-agnostic

consumers ⌘ will grow the region where retailers choose not

to self-match.

The proﬁtability of self-matching largely depends on

the existence of SU consumers. When the fraction of

store-only consumers decreases (i.e., ⌘ grows), retail-

ers can no longer beneﬁt as much from decision-stage

discrimination. Thus, the proﬁtability of oﬀering a self-

matching policy decreases as ⌘ increases.

4.4. Proﬁtability of Self-Matching

We have thus far analyzed how retailers decide

whether to adopt self-matching policies and character-

ized the strategies that can be sustained in equilibrium.

Here, we examine the proﬁt impact of having self-

matching available as a strategic option. The key issue

we seek to understand is whether retailers are com-

pelled by competitive forces to adopt self-matching,

even though it might not be beneﬁcial and could result

in lower equilibrium proﬁts were self-matching not an

option. The result in Proposition 3 addresses this issue.

Proposition 3. The proﬁt implications of self-matching,

compared to the baseline case where self-matching is not

available as an option, are as follows

(a) In the asymmetric equilibrium (1, 0). The retailer

oﬀering to self-match earns greater proﬁts, but the competing

retailer earns lower proﬁts.

(b) In the symmetric self-matching equilibrium (1, 1).

Both retailers earn higher proﬁts when product valuation is

high or retailer diﬀerentiation is low. Otherwise, they both

earn lower proﬁts.

We ﬁnd that at low values of v/✓, the proﬁt impact

of self-matching is asymmetric, with the self-matching

retailer obtaining higher proﬁts.

We ﬁnd t hat in the region of v/✓ where symmet-

ric self-matching occurs in equilibrium, when v/✓ is

close to its lower bound, self-matching reduces proﬁts

for both retailers because of the lower positive impact

of decision-stage discrimination and the increasing

negative impact of channel arbitrage. This interaction

results in a situation wherein both retailers would have

beneﬁtted had self-matching not been an option. How-

ever, when v/✓ is high, both retailers choose to self-

match and ear n higher proﬁts. This occurs because at

high v/✓, decision-stage discrimination overtakes the

negative impact of channel arbitrage. Overall, we ﬁnd

that the availability of self-matching as a strategy may

enhance proﬁts for at least one retailer and can also do

so for both retailers for a range of parameters, high-

lighting the importance of self-matching as a strategic

option.

5. Extensions

The base model analyzed in Section 4 focused on devel-

oping an understanding of the mechanisms underlying

the eﬀectiveness of self-matching and the conditions

for retailers to implement the policy in equilibrium.

Here, we have two main objectives. First, we will exam-

ine additional settings that are relevant to retailers as

they contemplate whet her to oﬀer a self-matching pric-

ing policy. Second, we relax a few key assumptions in

the baseline model, with a view towards increasing the

range of applicability of the ﬁndings. Proofs for results

presented in this section are provided in Appendix B

and the Electronic Supplement.

5.1. Impact of “Smart-Device” Enabled Consumers

The baseline model characterized undecided con-

sumers as not knowing what speciﬁc product they

want until they visit a store to evaluate which item

from the many available options best ﬁts their needs.

They could not invoke a self-matching policy because

they were in-store at the time of their ﬁnal decision,

and there was no way for them to access the Internet to

produce evidence of a lower online price.

Here, we recognize the increasing importance of

mobile devices to alter this dynamic and examine the

implications for self-matching policies. Retail Touch-

Points (Fiorletta 2013) notes that, “Ampliﬁed price

transparency—due to the instant availability of infor-

mation via the web and mobile devices—has encour-

aged retailers to rethink their omnichannel pricing

strategies.” Intuitively, one might expect that the

greater the proportion of consumers who carry smart

devices and take the trouble to check online when

in-store, the less proﬁtable self-matching should be

(because of the increased threat of cross-channel arbi-

trage). We show t hat this need not be the case.

Suppose that a fraction µ (0 <µ<1) of consumers

has access to the Internet while shopping in-store.

We refer to these consumers as “smart” to reﬂect the

notion that with the aid of Internet-enabled smart-

phone devices these consumers can easily obtain

online price information while in-store. Store-only

undecided smart consumers can now invoke a self-

matching policy if the online price oﬀered by a retailer

is lower than its store price. Channel-agnostic unde-

cided smart consumers will eﬀectively behave as we

have already modeled in the baseline model.

An increase in smart consumers can be understood

as increasing the fraction of store-only undecided (SU)

consumers who redeem the online price. However,

these consumers can only purchase from the retailer

Kireyev, Kumar, and Ofek: Match Your Own Price?

they ﬁrst visit, by contrast to store-only decided (SD)

consumers who have the option of buying from other

retailers at the outset. To see how the existence of smart

consumers impacts retailers’ strategies, consider the

proﬁts retailers earn if they both oﬀer to self-match

⇧

1, 1

⇤ (1  (1  ⌘))

, p

+ (1  ⌘)



((1  µ)p

+ µp

⇧

1, 1

⇤ (1  (1  ⌘))(1  

, p

))p

+ (1  ⌘)



((1  µ)p

+ µp

Solving for the equilibrium reveals that retailers will

set p

⇤ p

⇤ ✓(1  µ) + µ✓/(1  (1  ⌘)) and

⇤

⇤ v  ✓/2. Note that the online prices are increasing

in µ. We detail how smart consumers impact retailers’

equilibrium incentives to self-match in Proposition 4.

Proposition 4. In a duopoly with two multichannel retail-

ers, where some consumers can use a smart device in-store

to obtain online price information

(a) As the fraction of smart consumers increases, the

asymmetric equilibrium region grows, whereas the symmet-

ric self-matching equilibrium region shrinks.

(b) Retailer proﬁts can increase in the fraction of smart

consumers.

At low product values, holding ﬁxed the other

model parameters, more smart consumers enhance the

online competition dampening eﬀect in the asymmet-

ric equilibrium, which allows retailers to price higher

online when oﬀering to self-match. On the other hand,

the conditions for symmetric self-matching policies to

emerge in equilibrium for high product values become

more stringent as µ grows. That is, as µ ! 1, the sym-

metric self-matching region for high v shrinks in size to

zero. This happens because the existence of smart con-

sumers greatly erodes the positive decision-stage dis-

crimination eﬀect of self-matching, as there are fewer

SU consumers who will still pay the high store price,

while more consumers pay the lower online price,

thereby reducing retailers’ incentives to self-match.

Thus, and somewhat counterintuitively, the presence

of smart consumers need not decrease the proﬁtability

of a self-matching retailer (see proof of Proposition 4

in Appendix B for details of the proﬁt enhancing case).

On the contrary, smart consumers can enable retail-

ers to charge higher online prices, increasing the prof-

itability of self-matching policies. This suggests that

given current technology trends, horizontally diﬀeren-

tiated retailers would ﬁnd it worthwhile to more care-

fully examine whether self-matching is an appropriate

strategic option.

5.2. Mixed Duopoly: Multichannel

Retailer and E-Tailer

We consider the case of a multichannel retailer facing

a pure online e-tailer, i.e., a “mixed duopoly market.”

This market structure is becoming more important for

a number of multichannel retailers, e.g., several retail-

ers ﬁnd that Amazon and potentially other e-tailers are

their primary rivals. Past research has considered the

strategic implications of direct sellers, such as e-tailers,

competing with traditional retail channels (Balasubra-

manian 1998). Motivation for mixed channel structures

and a diﬀerent type of consumer heterogeneity across

channels has been studied by Yoo and Lee (2011). How-

ever, to our knowledge, decision-stage heterogeneity

and self-matching policies have not been examined in

this setting.

We denote the focal multichannel retailer as retailer 1

and the online-only e-tailer as retailer 2. In this setting,

only retailer 1 can oﬀer a self-matching policy in stage 1

of the game. Subsequently, both retailers set prices and

compete for demand per the timeline in Figure 1.

First, consider the case wherein the multichannel

retailer does not self-match its prices. Store-only con-

sumers can only consider retailer 1’s store channel and

are captive to this retailer, whereas channel-agnostic

consumers have the option of shopping across the two

retailers’ online sites. Proﬁts for both retailers can be

expressed as follows:

⇧

0, 0

⇤ ⌘

, p

| {z }

Channel-Agnostic Decided and Undecided

+ (1  ⌘)p

| {z }

Store-Only Decided and Undecided

, (4)

⇧

0, 0

⇤ ⌘(1  

, p

))p

| {z }

Channel-Agnostic Decided and Undecided

Retailer 1 serves as an eﬀective monopolist for store-

only consumers (SD and SU segments), who comprise

a combined segment of size 1  ⌘, and will attempt to

extract surplus from them by setting a store price of

⇤ v  ✓. Note that by contrast to the multichannel

duopoly, store-only decided consumers do not drive

down prices in the mixed duopoly case because the e-

tailer does not have a store that serves as a competitive

option. Both retailers compete online for the channel-

agnostic consumers, who form a segment of size ⌘.

We allow for channel-agnostic undecided consumers

who are closer in preference to retailer 2, the e-tailer,

to browse the product category at retailer 1’s store and

then purchase online from the e-tailer. The equilibrium

online prices are at the competitive level, with

⇤

⇤ ✓.

Next, consider the (1, 0) subgame where the multi-

channel retailer oﬀers a self-matching policy. SD con-

sumers can now retrieve the multichannel retailer’s

Kireyev, Kumar, and Ofek: Match Your Own Price?

online price in-store. Consider the retailers’ proﬁts as

given below

⇧

1, 0

⇤ ⌘(p

, p

| {z }

Channel-Agnostic Decided and Undecided

+ (1  ⌘)(1  )p

| {z }

Store-Only Decided

+ (1  ⌘)p

| {z }

Store-Only Undecided

, (5)

⇧

1, 0

⇤⌘(1  (p

, p

))p

As in the no self-matching case, online competition

places downward pressure on the price levels p

and

in the (1, 0) case. However, a portion of store con-

sumers, i.e., the (1  ⌘)(1  )-sized SD segment, now

receive the online price by invoking the self-match pol-

icy instead of paying the store price. As in the mul-

tichannel duopoly case, retailer 1 thus faces a channel

arbitrage eﬀect when it allows consumers to obtain a

price match. We might intuitively expect self-matching

to be unproﬁtable, especially since the SD consumers,

regardless of retailer preferences, cannot defect to the

e-tailer due to their preference for the store channel.

However, once again, the online competition dampening

eﬀect can act to increase proﬁtability when the multi-

channel retailer chooses to self-match. The following

proposition reﬂects the net impact of these eﬀects.

Proposition 5. In a mixed duopoly featuring a multichan-

nel retailer and a pure e-tailer, the multichannel retailer

adopts a self-matching policy when product value is rela-

tively low or retailer diﬀerentiation is high. Otherwise, the

retailer will not adopt a self-matching policy.

The intuition for Proposition 5 follows naturally

from the implications of the online competition damp-

ening eﬀect in the asymmetric case of the multichan-

nel duopoly scenario. When retailer 1 decides to self-

match, there is a cross-channel arbitrage externality. In

a bid to reduce the negative impact of channel arbi-

trage, the multichannel retailer who implements a self-

match has an incentive to raise its online price relative

to the no self-matching case. Strategic complementar-

ity in prices leads both retailers to set higher online

prices than under no self-matching. Note that there is

no decision-stage discrimination eﬀect. SU consumers

pay the same price regardless of whether there is self-

matching because the e-tailer has no rival store to

induce competition for t he SD segment and lower the

multichannel retailer’s store price.

The trade-oﬀ between the channel arbitrage and

competition dampening eﬀects depends on v/✓. For

low enough v (or high ✓), the self-matching multi-

channel retailer prices similarly across channels; thus

channel arbitrage is low. In this case, the online compe-

tition dampening eﬀect dominates and grows in v/✓.

However, the negative channel arbitrage eﬀect also

increases in v/✓ (as SD consumers redeem the lower

online price) and eventually dominates the online com-

petition dampening eﬀect. As a result, self-matching

emerges as an equilibrium outcome only for low values

of v/✓.

Turning to the proﬁt impact of self-matching on the

e-tailer, we ﬁnd the following:

Corollary 3. In a mixed duopoly, the e-tailer makes higher

proﬁts when the multichannel retailer uses a self-matching

policy.

Thus, a self-matching policy has a positive external-

ity on the e-tailer due to reduced competition in the

online channel, which allows the e-tailer to increase

prices. This holds even t hough the e-tailer’s price level

is lower than that of the multichannel retailer; t he latter

internalizes a higher beneﬁt of raising its online price

because of the positive impact on its store channel.

5.3. Additional Analyses

5.3.1. Markets with a Possibility of Expanding De-

mand. To relax t he assumption that all markets are

fully covered, we focus on a scenario where retailers

compete in a linear city and also face markets that are

not fully covered but can expand as retailers reduce

prices.

Speciﬁcally, retailers are at x ⇤ 0 and x ⇤ 1

on a Hotelling line of length

(3 + 6v/✓) such that

the distance between retailers is still equal to 1 but

they face additional monopoly (captive) consumer seg-

ments outside of the unit interval (x < 0 and x > 1).

We conduct equilibrium analysis for high levels of

v/✓ and ﬁnd that both retailers will choose to oﬀer self-

matching policies. This result coincides with Proposi-

tion 2, where we found that symmetric self-matching

emerges in equilibrium at high levels of v/✓. How-

ever, by contrast to the proﬁtability results in Proposi-

tion 3, retailers earn lower proﬁts when self-matching

than in the case where self-matching is not available

as a strategic option. This is because retailers now

have an incentive to keep store prices low, even when

self-matching, to attract consumers outside of the unit

interval. As a result, the decision-stage discrimination

eﬀect is reduced, and the proﬁtability of self-matching

suﬀers. The following proposition summarizes our

ﬁnding:

Proposition 6. In a market with demand that can expand

characterized by the existence of consumer segments beyond

the Hotelling unit interval on both sides, both retailers oﬀer

self-matching policies if the product value is high or the

level of diﬀerentiation is low. However, both retailers earn

lower proﬁts than had self-matching not existed as a strategic

option.

5.3.2. Retailer Processing Costs. Retailers may incur

a processing cost when dealing with consumers who

redeem a self-matching policy, for example, the staﬀ

time for verifying the evidence and entering it into the

Kireyev, Kumar, and Ofek: Match Your Own Price?

system. The idea here is somewhat analogous to that of

hassle costs developed in Desai and Purohit (2004). In

fully covered markets, an increase in retailer process-

ing costs will reduce the proﬁtability of self-matching

by, in eﬀect, increasing the magnitude of the channel

arbitrage eﬀect. As a result, self-matching is more dif-

ﬁcult to sustain in equilibrium and the region labeled

(1, 1) in Figure 4 shrinks as retailer processing costs

grow. In the Electronic Supplement, we illustrate the

impact a small but non-zero retailer cost of servicing

consumers who redeem a self-matching policy has on

retailer prices and proﬁts to highlight the greater chan-

nel arbitrage eﬀect and the reduced proﬁtability of self-

matching.

5.3.3. Diﬀerent Structure of Consumer Heterogeneity

in a Monopoly Setting. Proposition 1 establishes that

self-matching will never be adopted by a monopolist

and provides a benchmark for the duopoly analysis.

However, the monopolist may choose to self-match

in models that allow for a diﬀerent structure of con-

sumer heterogeneity. To illustrate how self-matching

may be proﬁtable for a monopolist, in the Electronic

Supplement we develop an alternative model where

consumers exhibit heterogeneity in their travel costs

and product valuations. All consumers are at ﬁrst

undecided and must visit the retailer’s store to iden-

tify their preferred product. Consumers have heteroge-

neous product valuations that are perfectly correlated

with their travel costs, i.e., consumers with a high prod-

uct valuation have a high travel cost, and consumers

with a low product valuation have a low travel cost.

A self-matching policy may enable the monopolist to

price discriminate by charging a higher store price and

selling to store-only consumers with a high travel cost

and a high product valuation as these consumers may

ﬁnd it costly to visit the store multiple times to redeem

a self-matching policy.

5.4. Consumer Survey

We conducted a consumer survey to characterize

market conditions across a range of retail product

categories. Our primar y goal is to evaluate whether

consumers exhibit heterogeneity along the dimensions

incorporated in the model, oﬀ-line versus online chan-

nel preference, horizontal preference across retailers

within a category, and decision-stage heterogeneity

(decided versus undecided). We also want to examine

observed market outcomes, in terms of ﬁrm behavior,

to assess the degree to which our model analysis cor-

responds to these outcomes. Full details on the survey

and its results are provided in Appendix C, and survey

questions are detailed in the Electronic Supplement.

Broadly, the survey ﬁndings point to substantial con-

sumer heterogeneity and lend support to our model

tenets.

Connection to Market Outcomes. We discuss how

the insights from our theoretical model, when com-

bined with the survey ﬁndings, compared with the

observed self-matching policies of ﬁrms across a range

of product categories (from pet supplies to electron-

ics). Obser ved Policies are detailed in Appendix D. We

emphasize that to empirically establish a causal con-

nection between our hypothesized forces and observed

market outcomes, a more thorough empirical investi-

gation is required. The survey results are intended to

provide us with preliminary evidence that may encour-

age such subsequent research.

Here, we focus on Figure 5 and t he model pre-

dictions per Proposition 2. Figure 5 shows a scatter

of product categories and indicates the self-matching

outcomes of major players above each category label

observed in the market. The ratio of value to diﬀer-

entiation is indicated on the vertical axis and the pro-

portion of undecided consumers is indicated on the

horizontal axis, as measured by the survey.

First, we observe that the pet supply market, which

was found to have few undecided consumers and low

relative product value, reﬂects an asymmetric (1, 0)

outcome in practice, consistent with what the model

predicts. Second, the apparel and low-end depart-

ment stores markets, characterized by intermediate rel-

ative value and medium to high levels of undecided

consumers, demonstrate a no-self-matching, or (0, 0)

outcome in practice, again consistent with model pre-

dictions. Finally, we examine the markets with all ﬁrms

self-matching in practice (1, 1), i.e., electronics, upscale

department stores, home improvement, and oﬃce sup-

plies. We ﬁnd (with the exception of oﬃce supply

Figure 5. (Color online) Market Characteristics and

Outcomes

10 20 30 40 50 60

0.5

1.0

1.5

2.0

2.5

Market characteristics

Proportion of undecided

Value/Differentiation

Pet supply

Electronics

Home improvement

Apparel

Upscale department

Low department

Office supply

(1, 0)

(1,1)

(0,0)

(1,1)

(0, 0)

(1,1)

Kireyev, Kumar, and Ofek: Match Your Own Price?

products) that they have a high relative value and

an intermediate proportion of undecided consumers,

which is consistent with our analysis. As for oﬃce sup-

ply ﬁrms, they may view their primary competition

as coming from pure e-tailers such as Amazon rather

than from multichannel rivals.

Whereas t his oﬀers suggestive and initial evidence

of the connection between market characteristics and

self-matching strategic choices in accordance with our

model predictions, we expect further careful exami-

nation across other product categories to be valuable.

An empirical investigation of this phenomenon would

also be useful and complementary to our theoretical

analysis.

6. Discussion, Limitations, and

Conclusion

The self-matching pricing policy has become an impor-

tant strategic aspect of multichannel retailing and is

used in a variety of markets, including consumer elec-

tronics, discount retail, and home improvement. Our

paper is, to our knowledge, the ﬁrst attempt to model

this strategic pricing policy and investigate how a com-

pany’s self-matching decision is determined by con-

sumer behavior and the competitive landscape.

Retailers in our model choose whether to oﬀer a self-

matching pricing policy in the ﬁrst stage and then set

price levels in the second stage. The retailers’ products

are horizontally diﬀerentiated, with consumers having

heterogeneous preferences over retailers. We further

allow for consumer heterogeneity along two additional

dimensions, decision stage and channel preference.

Thus, we explicitly capture a wide variety of DMPs for

consumers enabled by the multichannel setting.

The analysis illustrates how retailers in a multichan-

nel setting face downward price pressure in-store from

competition induced by the presence of store-only

decided consumers. By self-matching, a retailer relin-

quishes its ability to charge diﬀerent prices to decided

consumers across channels (desegmentation). Channel

desegmentation induces channel arbitrage, but pro-

duces another eﬀect: It can act as a commitment device

to increase online prices when only one retailer chooses

to self-match. We refer to this as the online competi-

tion dampening eﬀect. Self-matching may also enable

the retailer to charge store-only undecided consumers

a higher store price, which we call the decision-stage

discrimination eﬀect; this can result in both retailers

self-matching.

Self-matching is thus proﬁtable when the posi-

tive eﬀects of online competition dampening and/or

decision-stage discrimination overcome the negative

eﬀects of channel arbitrage. We further ﬁnd that the

proﬁtability of self-matching is determined by prod-

uct value (relative to retailer diﬀerentiation), as well as

consumer heterogeneity across diﬀerent dimensions,

such as decision stage and channel preference.

Beyond the baseline model, we consider several ex-

tensions, one of which explicitly models a setting with

smart-device enabled (“smart”) consumers, who can

look up online prices while in-store, an increasingly

prevalent phenomenon. We ﬁnd that self-matching can

increase retailer proﬁtability as the proportion of smart

consumers increases. This consumer trend may prove

to be an important issue for retailers to consider when

making pricing policy decisions going forward.

Our model yields results that are empirically

testable. First, retailers oﬀering to self-match will have

a larger online to store price discrepancy relative to

those that do not self-match. Second, we should ﬁnd

asymmetric self-matching equilibrium conﬁgurations

in markets with relatively low-valued products (or

highly diﬀerentiated retailers). Third, as the penetra-

tion of smart devices among consumers increases,

online prices set by retailers oﬀering to self-match are

expected to rise. A consumer survey we conducted pro-

vides suggestive evidence of face validity as to how

the equilibrium predictions of the model broadly cor-

respond to the emergent self-matching conﬁgurations

in practice across a number of industries.

Although we believe this to be the ﬁrst research to

rigorously examine t he idea of self-matching as a pric-

ing strategy, the present paper has several limitations

that could be addressed in future research. First, we

do not model competitive price-matching policies. Such

policies have been extensively studied in the literature,

and our focus is retailers with diﬀerentiated product

assortments, where competitive price-matching does

not play a role. It would be interesting to exam-

ine whether self-matching complements or substitutes

competitive matching policies in settings where rival

retailers sell identical products. Second, by assuming

suﬃciently large consumer travel costs for store vis-

its beyond the initial visit, we ensure that retailers can

price-discriminate their captive consumers who ﬁnd it

too costly to search additional stores for product infor-

mation. We incorporated a variety of consumer DMPs

and preference dimensions. However, it would be use-

ful to consider a richer model of consumer search, for

example, where consumers could visit a retailer’s store

and then decide whether to visit a second based on

expectations of price as well as the beneﬁts they may

obtain. Such an eﬀort would connect with the search

literature, and it would be useful to examine whether

self-matching then leads to more search and larger

consideration sets in the spirit of Diamond (1971) and

Liu and Dukes (2013). Third, the dimensions of con-

sumer heterogeneity might be correlated, e.g., con-

sumers who prefer store shopping may also be more

undecided. While we do not expect this to change our

Kireyev, Kumar, and Ofek: Match Your Own Price?

primary ﬁndings, careful modeling of these dependen-

cies might reveal additional eﬀects. Fourth, it would

be interesting to investigate how new types of compe-

tition could feature self-matching, e.g., such as Ama-

zon competing with other sellers on its platform (Jiang

et al. 2011). Finally, although we expect the mecha-

nisms detailed here to apply to the case wherein there

are ex ante diﬀerences among retailers (based on costs

or customer loyalty) beyond horizontal diﬀerentiation,

there may be additional insights obtained in modeling

the more general case.

Broadly, our ﬁndings suggest that although a self-

matching policy may initially appear to be an unprof-

itable but necessary evil, it has more subtle and pos-

itive competitive implications. Indeed, self-matching

can be proﬁtably used as a strategic lever and can

result in higher proﬁts for all retailers in the industry.

Multichannel retailers should therefore treat their self-

matching decisions as an important element of their

overall cross-channel strategy, taking into account the

products they sell, consumer characteristics, as well as

the competitive landscape.

Acknowledgments

The authors thank participants at the 2012 Marketing Science

conference for comments and helpful feedback. All errors are

the authors’.

Appendix A. Proofs of Propositions

Proof of Proposition 1

First, we determine interior and corner solutions without

self-matching. At price p, consumer demand is D(p) ⇤

min(2(v  p)/✓,1). For an interior optimal price, we have the

ﬁrst-order condition (FOC) 0 ⇤ p/✓ + (v  p)/✓ ⇤)

p ⇤ v/2.

The condition for an interior solution is v <✓. When we have

a corner solution, i.e., under v >✓, the monopolist sets a price

p ⇤ v  ✓/2. In the rest of the proof, we focus on the case

wherein the markets are covered, i.e., v >✓.

The monopolist’s proﬁt is determined as follows:

⇧

SM⇤0

⇤ (1  )(⌘p

+ (1  ⌘)p

) + (⌘p

+ (1  ⌘)p

⇧

SM⇤1

⇤ (1  )(⌘p

+ (1  ⌘) min(p

, p

))

+ (⌘p

+ (1  ⌘)p

so the demand from all segments is equal to 1.

To solve for prices, the consumer farthest from t he monop-

olist must be indiﬀerent between purchasing or not. This

yields v  p

 ✓/2 ⇤ 0 and v  p

 ✓/2 ⇤ 0 in the case

of no self-matching policy. Prices are then

⇤

⇤ v 

✓/2. Similarly, for when the retailer self-matches, we solve

v  p

 ✓/2 ⇤ 0, v  min(p

, p

)✓/2 ⇤ 0, and v  p



✓/2 ⇤ 0. Regardless of whether SD consumers choose to

redeem the self-matching policy, the multichannel retailer

will set identical prices across channels, equal to those set had

it not self-matched:

⇤

⇤ v  ✓/2. As the proﬁts under the

two conditions are equal, the monopolist will always prefer

SM ⇤ 0, which weakly dominates SM ⇤ 1.

Below, we refer to m as the cost of undertaking a second shopping

trip for store-only undecided consumers. We derive bounds on m

that ensure the consumer behavior speciﬁed in our assumptions.

Proof of Proposition 2

First, we separately consider each subgame. Then, we com-

pare the proﬁts from each subgame to derive the bounds for

the equilibrium results. The following constraints must be

imposed:

• v > 3✓/2 ensures that all markets are fully covered,

• <5/8 ensures that no retailer sets such a high online

price to earn zero demand from decided consumers in the

(1, 0) subgame;

• v <✓(1/ + 1/(4(1  ))  ⌘/(36(1  (1  ⌘))

)+ (7 11⌘)/

(36(1  (1  ⌘))) + 7/18) ensures that no retailer wants to

exclusively price for its captive segment of store-only unde-

cided consumers and forgo all demand for store-only decided

consumers;

• m > v  3✓/2 ensures that no store-only undecided con-

sumers switch stores after their ﬁrst visit, and that no con-

sumer returns home and visits the store a second time to

redeem a self-matching policy.

We focus on the case wherein >0, so that there are at

least some undecided consumers.

No Matching—(0, 0). Channel-agnostic consumers will pur-

chase online. Store-only consumers will buy in-store. All con-

sumers will pay the price set in the channel from which they

buy. The retailers will earn proﬁts

⇧

0, 0

⇤ ⌘

, p

+ (1  ⌘ )

✓

(1  )

, p

) +



◆

⇧

0, 0

⇤ ⌘(1  

, p

))p

+ (1  ⌘ )

✓

(1  )(1  

, p

)) +



◆

We solve for the FOCs @⇧

0, 0

/@p

⇤ 0 and @⇧

0, 0

/@p

⇤ 0 for

j 2{1, 2}, and check for corner solutions. We ﬁnd an interior

solution with equilibrium prices at

⇤

⇤ ✓ and

⇤

⇤ ✓/(1  ) for v/ ✓>

+ 1/(1  ), and a corner solution

in store prices with

⇤

⇤ v  ✓/2 for v/✓ 

+ 1/(1  ).

The store price is higher than the online price in all cases

as retailers have an incentive to price higher for their cap-

tive segment of store-only consumers. The binding condition

for an interior solution requires that all SU consumers pur-

chase in equilibrium. For retailer 1, this can be written as

v  p

 ✓/2 > 0 (the utility for the SU consumer farthest away

from store 1 is greater than zero). When this condition fails

(i.e., v/✓ 

+ 1/(1  )), we have a corner solution where

retailers set local monopoly prices v  ✓/2 in-store. No other

constraints apply and there are no other corner solutions. The

equilibrium proﬁts earned by retailers are

⇧

0, 0

⇤ ⇧

0, 0

⇤

[2v(1  ⌘)✓(1  3⌘)],

✓



1  

✓

1 +

(1  ⌘)

1  

◆

✓

1  

Symmetric Self-Matching—(1, 1). Channel-agnostic con-

sumers will purchase online and pay the online price.

Kireyev, Kumar, and Ofek: Match Your Own Price?

Store-only decided consumers will buy in-store, but will

redeem the online price of the store they purchase from.

Store-only undecided consumers will buy in the store they

ﬁrst visit and will pay the store price. The retailers will earn

proﬁts

⇧

1, 1

⇤ (1  (1  ⌘))(p

, p

+ (1  ⌘ )



⇧

1, 1

⇤ (1  (1  ⌘))(1  (p

, p

))p

+ (1  ⌘ )



and set prices

⇤

⇤ ✓ online and

⇤

⇤ v  ✓/2

in-store. The online price is the familiar competitive price ✓

and is an interior solution to the FOCs @⇧

1, 1

/@p

⇤ 0 for

j 2{1, 2}. Diﬀerentiating with respect to store prices yields

@⇧

1, 1

/@p

⇤ (1  ⌘)(/2) > 0, implying a corner solution. The

retailers will set the highest store price they can, ensuring

that all SU consumers purchase, which is v  ✓/2. There are

no other corner solutions. The equilibrium proﬁts earned by

retailers are

⇧

1, 1

⇤ ⇧

1, 1

⇤



✓(1  (1  ⌘)) + (1  ⌘)

✓

v 

✓

◆

Asymmetric Self-Matching—(1, 0). Channel-agnostic con-

sumers will purchase online and pay the online price. Store-

only decided consumers will buy in-store, but will redeem

the online price (as it will be lower) if they buy from the

self-matching retailer. They will pay the store price if they

buy from the non-self-matching retailer. Store-only unde-

cided consumers will buy from the store they ﬁrst visit and

pay the store price. The retailers proﬁts are then

⇧

1, 0

⇤ ⌘(p

, p

+ (1  ⌘ )

✓

(1  )(p

, p



◆

⇧

1, 0

⇤ ⌘(1  (p

, p

))p

+ (1  ⌘ )

✓

(1  )(1  (p

, p

)) +



◆

The FOCs can be written as

@⇧

1, 0

⇤ 0 ,

@⇧

1, 0

⇤ 0 for j 2{1, 2},

and

@⇧

1, 0

⇤ (1  ⌘)



> 0.

In equilibrium, there is an interior solution for online

prices and for the store price of retailer 2 and a corner solu-

tion for the store price of retailer 1, for large v. The retailers

set online prices

⇤ ✓(

+ 1/(3(1  (1  ⌘)))) and

⇤

✓(

+ 1/(6(1  (1  ⌘)))) and store prices

⇤ v  ✓/2 and

⇤

+ ✓/(2(1  )) for v/✓>

+ 1/(6(1  (1  ⌘))) +

/(2(1  )).

Otherwise, if v is small, we have a corner solution for

which yields prices

⇤ ✓ + ((1  )(1  ⌘)(2v  3✓))/

(4(1  (1  ⌘))  ⌘),

⇤ ✓ + ((1  )(1  ⌘)(2v  3✓))/

(8(1  (1  ⌘))  2⌘) online and

⇤

⇤ v  ✓/2 in-store.

The binding threshold on v for an interior solution requires

that all of retailer 2’s SU consumers purchase in equilib-

rium. In other words, v  p

 ✓/2 > 0. Substituting the inte-

rior solution equilibrium store price for retailer 2 into the

inequality shows that the corner solution holds for v/✓ 

1/(6(1  (1  ⌘))) + /(2(1  )).

The equilibrium proﬁts earned by retailers are

⇧

1, 0

⇤

v

(1  ⌘) +

✓



4(1  )



(1  17⌘) +

2(1  (1  ⌘))



⇧

1, 0

⇤

✓



9

(1  ⌘)

1  

+ (34  5 ⌘) + 29(1  ) +

1  (1  ⌘)



for v/✓>(

+ 1/(6(1  (1 ⌘)))+ /(2(1 ))). The expression

for v/✓ (

+ 1/(6(1  (1⌘))) + /(2(1 ))) can be similarly

obtained by substituting equilibrium prices into the proﬁt

functions and is available on request.

Equilibrium Analysis. A self-matching conﬁguration is a

Subgame perfect Nash equilibrium (SPNE) if no retailer has

the incentive to unilaterally deviate. Equivalently, for (0, 0) to

be an SPNE, retailer 1 must not have the incentive to deviate

to (1, 0). For (1, 1) to be an SPNE, retailer 2 must not have

the incentive to deviate to (1 , 0). For (1, 0) or (0, 1) to be an

SPNE the self-matching retailer must not prefer (0, 0) and the

non-self-matching retailer must not prefer (1, 1). By compar-

ing t he proﬁts at the equilibrium prices deﬁned above, we

can construct equilibrium regions.

Let 

⇤ (27 

25⌘

+ 448⌘ + 256)/(32(1  ⌘ )) + 5/32. The

results in Proposition 2 focus on the region where <

for

clarity of exposition. In this proof, we provide an extended

analysis, including the region where   

. Deﬁne

• z

⇤ min((27 17)/(18(1 ))+ /(9(1 )(1 (1 ⌘))),

7/(36(1  (1  ⌘))) + 1/(4(1  )) + 25/18, 3⌘/(8(1  ⌘)8 

⌘) +

• z

⇤ max((27  17)/(18(1  )) + /(9(1  )(1   ·

(1  ⌘))), 7/(36(1  (1  ⌘))) + 1/(4(1  )) + 25/18),

• z

⇤ 1/(1  )1/(9(1  (1  ⌘))) + 17/18.

We calculate equilibrium proﬁts and t he applicable thresh-

olds under all subgames. Then, for v/✓<z

, the incremental

proﬁt from self-matching for retailer 1 is positive: ⇧

1, 0

⇧

0, 0

0; retailer 2 prefers not to deviate as ⇧

1, 1

 ⇧

1, 0

< 0, so (1,0)

and (0, 1) are SPNE. For z

< v/✓<z

, (0, 0) is the unique equi-

librium for <

, as ⇧

1, 0

 ⇧

0, 0

< 0 while ⇧

1, 1

 ⇧

1, 0

< 0, and

(1,1) is the unique equilibrium for >

as ⇧

1, 0

 ⇧

0, 0

> 0

while ⇧

1, 1

 ⇧

1, 0

> 0. For z

< v/✓<z

, both (0,0) and (1,1) are

SPNE as ⇧

1, 0

 ⇧

0, 0

< 0 while ⇧

1, 1

 ⇧

1, 0

> 0, so no retailer

prefers to unilaterally deviate from either symmetric setup.

For v/✓>z

, (1, 1) is the unique SPNE as ⇧

1, 0

 ⇧

0, 0

> 0

while ⇧

1, 1

 ⇧

1, 0

> 0.

Note that for suﬃciently large , retailers prefer to oﬀer

symmetric self-matching policies at intermediate v. This is

because as  grows, retailer 1 has an incentive to match for

lower v given that retailer 2 also matches. As the critical

threshold of v becomes lower, it may cross the threshold at

which the other retailer no longer prefers to match, yielding

an equilibrium where both retailers match for intermediate v.

To summarize, for low , as v increases, there will ﬁrst be

an asymmetric solution, then (0, 0), then both (0, 0) and (1, 1),

and then uniquely (1, 1). For high , as v increases, there will

ﬁrst be an asymmetric solution, then (1, 1), then both (0, 0)

and (1, 1), and then uniquely (1, 1).

Proof of Proposition 3

A comparison of proﬁts in the (1, 0) subgame reveals that

⇧

1, 0

> ⇧

1, 0

everywhere. A comparison of proﬁts earned by

retailer 1 in the (1, 1) subgame and in the (0, 0) subgame

reveals that ⇧

1,1

> ⇧

0, 0

if v/✓>

+ 1/(1  ) ⇤ z

, which is

strictly greater than z

Kireyev, Kumar, and Ofek: Match Your Own Price?

Appendix B. Proofs for Extensions

Proof of Proposition 4

The proof of Proposition 4 proceeds just as in Proposition 2,

except with an extra parameter µ representing the fraction of

“smart” consumers, or consumers who can costlessly search

for online information while in-store. The µ segment will

be relevant for store-only undecided consumers, as they can

only claim a self-matching policy if they have access to the

Internet in-store. The remaining store-only undecided con-

sumers will be unable to claim a self-matching policy and

will have to pay the store price. We require the following

restrictions:

• v > 3✓/2 + (µ✓(1  ⌘))/(1  (1  ⌘)) ensures that all

markets are fully covered;

• <1  3/(2(4  µ)) ensures that no retailer sets such a

high online price to earn zero demand from decided con-

sumers in the (1, 0) subgame;

• Restriction

v <✓

✓

1  



4µ + 16⌘  4µ⌘ + 2

12⌘

3⌘

12(1  )⌘



(1 + 2µ)

(1  )

(1  ⌘)

36⌘(1  (1  ⌘))

)

(1 + 2µ)(1  )(1  ⌘)(2 µ + 11⌘  2µ⌘  5)

36⌘(1  (1  ⌘))

◆

ensures that no retailer wants to price exclusively for its cap-

tive segment of store-only undecided consumers and forego

all demand for store-only decided consumers;

• m > v  3✓/2 + µ✓  µ✓/(1  (1  ⌘)) ensures that no

store-only undecided consumers switch stores af ter their ﬁrst

visit, and that no consumer returns home and visits the store

a second time to redeem a self-matching policy.

No Retailers Self-Match—(0, 0). The equilibrium prices un-

der (0, 0) emerge just as in Proposition 2, as mobile con-

sumers behave just as the rest of the consumers.

One Retailer Self-Matches—(1, 0). Store-only undecided

consumers who are mobile will redeem the self-matching

policy if they ﬁrst visit t he store that oﬀers the policy. Prof-

its are

⇧

1, 0

⇤ ⌘(p

, p

+ (1  ⌘ )

✓

(1  )(p

, p



((1  µ)p

+ µp

)

◆

⇧

1, 0

⇤ ⌘(1  (p

, p

))p

+ (1  ⌘ )

✓

(1  )(1  (p

, p

)) +



◆

In equilibrium, the retailers set online prices

⇤ ✓((1 +

2µ)/(3(1  (1  ⌘))) + 2(1  µ)/3) and

⇤ ✓((1 + 2µ)/(6(1 

(1  ⌘))) + (5  2µ)/6) and store prices

⇤ v  ✓/2 and

⇤

+ ✓/(2(1  )) for v/✓>((1 + 2µ)/(6(1  (1  ⌘))) +

1/(2(1  ))  µ/3 +

). Otherwise,

⇤ v/2 + ✓/4  ✓µ/2 

(8✓µ + 9✓⌘  6v⌘  2✓⌘µ)/(4(4(1  ⌘) + ⌘  4)),

⇤



✓((6⌘  4 + µ⌘) + 4  3⌘)/(4(1  ⌘) + ⌘  4), and

⇤

v  ✓/2. The FOCs and the binding constraint are just as in

the proof for (1, 0) in Proposition 2, except for the addition of

an extra parameter µ.

Both Retailers Self-Match—(1, 1). The retailers will earn

proﬁts

⇧

1, 1

⇤ (1  (1  ⌘))(p

, p

+ (1  ⌘ )



((1  µ)p

+ µp

⇧

1, 1

⇤ (1  (1  ⌘))(1  (p

, p

))p

+ (1  ⌘ )



((1  µ)p

+ µp

and set prices p

⇤ p

⇤ ✓(1  µ) + µ✓/(1  (1  ⌘)) online

and p

⇤ p

⇤ v  ✓/2 in-store.

Equilibrium Analysis. To prove the existence of the result,

we provide an example with ⌘ ⇤

and  ⇤

. Let

⇤

2,385µ  41(529µ

+ 1,408µ + 1936)

1/2

+ 7,810

4,004

⇤

32µ + 331

198

11(2 + µ)

⇤

64µ + 395

198

88(1  µ)

, y

⇤

32µ + 427

198

22(1  µ)

Comparing proﬁts when v/✓<y

, (1, 1) is the unique SPNE.

For y

< v/✓<y

, (1, 0) and (0, 1) are SPNE. For y

< v/✓<

, (0, 0) is the unique equilibrium. For y

< v/✓<y

, (0, 0)

and (1, 1) are SPNE. For v/✓>y

, (1, 1) is the unique SPNE.

To prove the associated proposition, note that y

, y

, and

are all increasing in µ, so that holding constant v/✓, an

increase in mobile consumers shrinks the equilibrium region

that admits self-matching policies.

Increasing Proﬁts with Mobile Consumers. In the (1, 1) equi-

librium for large v, ⌘ ⇤

and  ⇤

, the retailers’ proﬁts

are increasing in µ if v/✓<

+ 8µ/11, which is possible if

µ<0.83. Furthermore, the retailers’ proﬁts are larger than

when µ ⇤ 0 if v/✓<5/2 + 4µ/11, which is possible if µ<0.72.

This shows that retailer proﬁts may increase as the fraction

of mobile consumers increases.

Proof of Proposition 5

Suppose that a multichannel retailer competes with an on-

line-only e-tailer. Assume v > 2✓ to ensure t hat all markets

are fully covered. Assume v < 4✓ and >

 3/(2(1  ⌘))

to ensure that the multichannel retailer has positive online

sales. Under (0, 0) t he retailers earn proﬁts

⇧

0, 0

⇤ ⌘(p

, p

+ (1⌘)p

, ⇧

0, 0

⇤ ⌘(1(p

, p

))p

Taking the FOCs with respect to the prices, we solve

@⇧

0, 0

⇤ ⌘



(p

, p

) + p

@(p

, p

)



⇤ 0 ,

@⇧

0, 0

⇤ (1  ⌘) > 0, implying a corner solution.

We obtain the corresponding FOCs for retailer 2 and solve

for the equilibrium corresponding to the best responses of

both retailers. All channel-agnostic consumers will purchase

online, whereas store-only consumers will buy from the mul-

tichannel retailer’s store. The retailers will set competitive

prices online

⇤

⇤ ✓, and retailer 1 will set monopoly

price in-store

⇤ ( v  ✓). That is, we obtain an interior solu-

tion for online pricing, but a corner solution for the store

Kireyev, Kumar, and Ofek: Match Your Own Price?

price where the multichannel retailer maximizes proﬁts from

all captive SU consumers.

Under the (1, 0) subgame of competition between a self-

matching multichannel retailer with an e-tailer, retailers earn

proﬁts

⇧

1, 0

⇤ ⌘(p

, p

+ (1  ⌘ )((1  )p

+ p

⇧

1, 0

⇤ ⌘(1  (p

, p

))p

As under (0, 0), channel-agnostic consumers will purchase

online and store-only consumers will purchase from the

multichannel retailer’s store. Store-only decided consumers

redeem the matching policy and pay the online price, whereas

store-only undecided consumers fail to redeem t he policy

and pay the store price. The retailers will set prices

⇤ ✓ +

(4✓(1  )(1  ⌘))/(3⌘),

⇤ ✓ + (2✓(1  )(1  ⌘))/(3⌘ ),

⇤

v  ✓ for v > 2✓ + (4✓(1  )·(1  ⌘))/(3⌘). Once again, there

is an interior solution in online pricing for v suﬃciently large

and a corner monopoly solution forthe store price. The thresh-

old for v is derived from the condition that in equilibrium

< v  ✓ for an interior solution. That is, the online price

charged by retailer 1 cannot exceed the monopoly price for SD

consumers, or equivalently, v  p

 ✓>0, ensuring that the

SD consumer farthest away from store 1 purchases in equilib-

rium for the market to remain covered. For v  2✓ + 4✓(1  )·

(1  ⌘)/(3⌘), this condition fails, and retailers will set prices

⇤ v  ✓,

⇤ v/2,

⇤ v  ✓ which corresponds to a corner

solution.

Now we substitute prices into proﬁts for the appropriate

v and identify the parameter ranges for which ⇧

1, 0

> ⇧

0, 0

see when retailer 1 would prefer to self-match. Suppose that

>

3/(4(1 ⌘)). Then, ⇧

1, 0

> ⇧

0, 0

if v < z

⇤ ✓(

+ (8(1 )·

(1  ⌘))/(9⌘)). Otherwise, if  

 3/(4(1  ⌘)), then ⇧

1, 0

⇧

0, 0

if v < z

⇤ 3✓. Hence, there exists a z

⇤ min{z

, z

}, such

that for v < z

, the multichannel retailer will prefer to self-

match.

Proof of Corollary 3

A comparison of the e-tailer’s proﬁts, ⇧

1, 0

 ⇧

0, 0

reveals that

it earns greater proﬁts when the multichannel retailer oﬀers a

self-matching policy. To see this, note that the e-tailer’s price

under (1, 0) is

⇤ ✓+ (2✓(1)(1 ⌘))/(3⌘), which is greater

than ✓, the price it would charge under (0, 0). Also in (1, 0),

the e-tailer’s price is less than

⇤ ✓ + (4✓(1 )(1 ⌘))/(3⌘),

the online price charged by the multichannel retailer. Under

(1, 0) the e-tailer sets a higher price and earns a greater frac-

tion of demand than under (0, 0). As a result, its proﬁts are

greater.

Suppose v  2✓ + (4✓(1  )(1  ⌘))/(3⌘). Then ⇧

1, 0

⇤

⌘/(8✓) and ⇧

0, 0

⇤ ✓⌘/2. The diﬀerence ⇧

1, 0

 ⇧

0, 0

⇤

 4✓

)⌘/(8✓) which is positive when v > 2✓, which is

the lower bound required for markets to be fully covered.

Now, suppose v > 2✓ + (4✓(1  )(1  ⌘))/(3⌘). Then ⇧

1, 0

⇤

✓(2(1  ⌘)2  ⌘)

/(18⌘) and ⇧

0, 0

⇤ ✓⌘/2. The diﬀerence

⇧

1, 0

⇧

0, 0

⇤ (2✓(1  )(1 ⌘)(1+ 2⌘  (1  ⌘)))/(9⌘) is g reater

than zero whenever /(1  ) > (1 + 2⌘)/(3⌘), which is

always the case as >0. Hence, the e-tailer always makes

higher proﬁts when the multichannel retailer matches.

Appendix C. Consumer Survey Across Product