Personalized pricing with heterogeneous mismatch
costs*
Noriaki Matsushima†, Tomomichi Mizuno‡, Cong Pan§
August 3, 2023
Abstract
Personalized pricing has become a reality through digitization. We examine ﬁrms’
incentives to adopt one of the three pricing schemes: uniform, personalized, or grouppricing in a Hotelling duopoly model. There are two types of consumer groups thatare heterogeneous in their mismatch costs. We show that both ﬁrms employ personalized pricing in equilibrium regardless of the heterogeneity of consumer groups. If theconsumer groups’ heterogeneity is signiﬁcant, the proﬁts are higher when both ﬁrms
use personalized pricing than when they employ uniform pricing; otherwise, the latter
proﬁts are higher than the former. Proﬁts are highest when ﬁrms employ group pricing among the three cases. The ranking of consumer welfare among the three cases is
opposite to that of proﬁts.

Running head: Personalized pricing
Keywords: Personalized pricing, Group pricing, Heterogeneous consumer types, Hotellingmodel.
JEL Classiﬁcation Numbers: L13, D43* We have beneﬁted from many constructive comments from the co-editor and anonymous referees. We
thank Rosa Branca Esteves, Akio Kawasaki, and Andrew Rhodes for their valuable discussions and comments.
We also thank the participants in EARIE2022 (Universität Wien), Japan Association for Applied Economics
2022 (Kumamoto University), and The 5th MOVE-ISER Workshop (Osaka University). We gratefully acknowledge the ﬁnancial support from the Japan Society for the Promotion of Science (JSPS), KAKENHI
Grant Numbers JP19H01483, JP20H05631, JP21H00702, JP21K01452, JP21K13300, and JP21K18430. Although Matsushima serves as a member of the Competition Policy Research Center (CPRC) at Japan Fair Trade
Commission (JFTC), our paper does not reﬂect the views of JFTC. The usual disclaimer applies.
†
Corresponding author. Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka, 567-0047, Japan. Phone: (81)-6-6879-8571. Fax: (81)-6-6879-8583. E-mail:nmatsush@iser.osaka-u.ac.jp‡
Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, Hyogo, 657-8501, Japan.
Phone: (81)-78-803-6802. Fax: (81)-78-803-7289. E-mail: mizuno@econ.kobe-u.ac.jp§
Faulty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto 603-8555, Japan.
Phone: (81)-075-705-1452. Fax: (81)-075-705-1949. E-mail: cong.pan@cc.kyoto-su.ac.jp

1

2

1 Introduction
Personalized pricing has become a reality through digitization (OECD, 2018).1 In particular,
mobile technology allows companies to oﬀer discriminatory prices because customers haveindividual mobile devices to access online services (Esteves and Resende, 2016). A famous
example of such personalized pricing is Uber’s “route-based pricing,” which depends oncustomers’ willingness to pay (Bloomberg, 2017).2 Although personalized pricing can help
ﬁrms capture each consumer’s willingness to pay, the strategic interaction with rivals can also
exert opposite eﬀects. This possibility raises questions about whether personalized pricing
can improve a ﬁrm’s proﬁts and beneﬁt the consumers.
Regarding the inﬂuence of personalized pricing on ﬁrms’ proﬁts, there has been broad
discussion since the seminal paper by Thisse and Vives (1988). They concluded that personalized pricing induces a prisoner’s dilemma where all ﬁrms become worse oﬀ than thesituation when all the ﬁrms commit to uniform pricing (Shaﬀer and Zhang, 1995; Bester andPetrakis, 1996; Zhang, 2011). However, several studies show that personalized pricing does
not necessarily result in a prisoner’s dilemma in the case of ﬁrm asymmetry (e.g., a quality
diﬀerence (Shaﬀer and Zhang, 2002), quality choice (Ghose and Huang, 2009), and initialcost diﬀerence with R&D (Matsumura and Matsushima, 2015)). Therefore, discussions on
the eﬀect of personalized pricing on ﬁrms’ proﬁts are inconclusive.
This study examines consumer heterogeneity and develops a linear market model comprising two consumer groups—one incurring high transportation costs (high-end consumers)
and the other incurring low transportation costs (low-end consumers). The model also
comprises two symmetric ﬁrms competing in this market and choosing one of the pricing
schemes: uniform, personalized, or group pricing.
The results are as follows. Both ﬁrms always employ personalized pricing in equilibrium,
which enhances both ﬁrms’ proﬁts compared with the situation where both employ uniform

3
pricing if the consumer groups are suﬃciently heterogeneous. Furthermore, ﬁrms’ proﬁts
when they both employ group pricing are always the highest compared with the other twocases mentioned before. We show that the consumer surplus in group pricing is always thelowest when comparing the aforementioned three cases. The consumer surplus in uniform
pricing is the highest if the consumer groups are suﬃciently heterogeneous.
In this regard, the intuition stems from how the low-end consumers inﬂuence the uniform prices of ﬁrms. When the low-end consumers incur a suﬃciently low transportation
cost (relative to the high-end consumers), they are price-sensitive. When ﬁrms adopt a uniform pricing scheme, they oﬀer suﬃciently low prices to capture the price-elastic consumers;
this diminishes their proﬁts from high-end consumers. However, when both ﬁrms adopt personalized pricing, they customize each consumer’s price and delink the markets for high- andlow-end consumers. This makes the competition mode similar to an asymmetric Bertrand
competition for each consumer. Unlike a uniform pricing scheme, the adoption of personalized pricing intensiﬁes the competition for low-end consumers (a negative eﬀect). Nevertheless, it enables each ﬁrm to set higher prices for high-end consumers located closer to thatﬁrm (a positive eﬀect). In summary, personalized pricing triggers market segmentation suchthat low-end consumers pay less. In contrast, some high-end consumers pay more, which
beneﬁts the ﬁrms when the heterogeneity between the two consumer groups is suﬃciently
signiﬁcant.
Group pricing is the simplest way to delink the two markets for the two consumer types
and allows ﬁrms to escape ﬁerce Bertrand competition under personalized pricing at eachpoint. However, group pricing is not attainable in equilibrium. The comparison between
group and personalized pricing is equivalent to that between uniform and personalized pricing in Thisse and Vives (1988) when we focus on each consumer group. We can directly
apply the conclusion in Thisse and Vives (1988) and ﬁnd that each ﬁrm has an incentive toemploy personalized pricing, given that the rival employs group pricing. Therefore, group4
pricing is not sustainable as an equilibrium outcome, although ﬁrms are better oﬀ if they
employ group pricing.
In a recent study, Esteves (2022) considers a static setting where personalized pricing canbecome a winning strategy for competing ﬁrms.3 The intuition relies on the heterogeneity
in the quantity demanded by consumers. As in Shin and Sudhir (2010), some consumers
demand more products than others. We capture consumers’ heterogeneity by their horizontal
valuation of products (i.e., the transportation cost) instead of the quantity purchased. We
also consider the endogenous choices of pricing schemes. Given these diﬀerences, our study
complements Esteves (2022).
Esteves and Shuai (2022) also discuss the proﬁtability of personalized pricing in a Hotelling
duopoly with a CES type elastic demand (Gu and Wenzel, 2009).4 They show that personalized pricing is more proﬁtable than uniform pricing if the elasticity is high. The reason isthat personalized prices are independent of the elasticity, although uniform prices decrease
with the elasticity. Contrasting to their paper, we consider the endogenous choices of pricing
schemes in addition to the proﬁt comparison. Also, our paper diﬀers from theirs regarding
the relationship between price elasticities and personalized prices. Given these diﬀerences,
our study complements Esteves and Shuai (2022).
Rhodes and Zhou (2022) investigate the eﬀects of personalized pricing on proﬁts andwelfare, using a generalized oligopoly model based on Perloﬀ and Salop (1985).5 The authors generalize the number of ﬁrms and the degree of market coverage and show that in
the short run (e.g., when ﬁrms’ number is exogenously given), personalized pricing would
beneﬁt ﬁrms and harm consumers compared with uniform pricing, only when the market
coverage is low. By incorporating consumers’ heterogeneity in their mismatch costs, weshow that the same argument could hold even under a full market coverage. Therefore, we
provide a complement to Rhodes and Zhou (2022).
The remainder of this paper is organized as follows. Section 2 formulates the model.

5Then, section 3 analyzes the model and presents the main results. Finally, section 4 concludes the paper. The detailed mathematical procedures are available in the Online Appendix.

2 ModelWe consider a product characteristic space with interval [0, 1]. There are two consumer types
called consumer k (= L, H). We refer to the markets with consumers H and L as markets H
and L. We normalize the number of consumers to 1 and assume that the shares of consumers
H and L are λ and 1 − λ, respectively. The distribution of consumers in each market isuniform on the interval [0, 1].
There are two ﬁrms—ﬁrms 0 and 1 located at 0 and 1, respectively. We assume that ﬁrm
i can choose one of the three pricing schemes—uniform, personalized, or group pricing. In
uniform pricing, the ﬁrm oﬀers the same price to all consumers regardless of their location
or type. In personalized pricing, the ﬁrm can customize the price for each consumer locatedat x ∈ [0, 1] in market k. Finally, in group pricing, the ﬁrm oﬀers discriminatory prices to
consumers H and L.
The utility function for a consumer located at x ∈ [0, 1] in market k isbuying from ﬁrm 0, v − p0 (x, k) − tk x
U(x, k) =  v − p1 (x, k) − tk (1 − x) buying from ﬁrm 1,
where v is the willingness to pay for the ideal product, pi (x, k) is the customized price oﬀered
to consumers located at x in market k, and tk is the per-distance transportation (mismatch)cost in market k, as in Armstrong (2006, pp.116-117). Note that, for any x ∈ [0, 1] andk = H, L, pi (x, k) is constant when ﬁrm i uses uniform pricing. We denote the uniform priceoﬀered by ﬁrm i by pi . When ﬁrm i uses group pricing, it oﬀers a uniform price pi (k) to6market k. Assume tH > tL > 0 and deﬁne τ ≡ tL /tH . We perform comparative statics on τ
keeping tH ﬁxed throughout the study.
Our model eﬀectively captures a feature of app-based taxi services, which is price discrimination based on users’ characteristics. Some ﬁrms can oﬀer group pricing based onusers’ mobile phone types, such as iPhones versus other mobile phones.6 Users of iPhones
tend to be less price elastic because of their higher wealth levels compared to users of othermobile phones.7 This diﬀerence in price elasticity can be captured by heterogeneity of transportation costs, which represent price elasticity. Additionally, personalized pricing can be
implemented by considering consumers’ preferences for taxi services, such as their urgency
or willingness to consider alternative transportation options, as exempliﬁed by Uber’s routebased pricing. To capture these preferences, we use locations on the interval [0, 1].
Each ﬁrm produces a product without cost. Then, ﬁrm i’s proﬁt is as follows:
π0 [p0 (x, k), p1 (x, k)] ≡ λ

∫ xH

∫ xL

p0 (x, H)dx + (1 − λ)p0 (x, L)dx,
0
∫ 1
∫ 1
π1 [p0 (x, k), p1 (x, k)] ≡ λp1 (x, H)dx + (1 − λ)p1 (x, L)dx,
0

xH

xL

where xk is the location of indiﬀerent consumers in market k.
We follow the timing structure of Thisse and Vives (1988). In the ﬁrst stage, each ﬁrm
selects one of the three pricing schemes. In the second stage, a ﬁrm employing a uniform
or group pricing oﬀers its observable uniform price. Subsequently, a ﬁrm employing personalized pricing oﬀers personalized prices, pi (x, k), determined by the consumer types andlocations. If the two ﬁrms adopt the same pricing scheme, they simultaneously determine
their prices. The timing of pricing oﬀers are in line with those considered in the literature(e.g., Thisse and Vives, 1988; Shaﬀer and Zhang, 2002; Clavorà Braulin and Valletti, 2016;
Choe et al., 2018; Chen et al., 2020; Esteves, 2022). Using backward induction, we solve
this game.

7

3 Analysis
We need to solve six types of subgames: (i) Both ﬁrms choose uniform pricing; (ii) Bothﬁrms choose personalized pricing; (iii) Both ﬁrms choose group pricing; (iv) One ﬁrm
chooses uniform pricing, and the other one chooses personalized pricing; (v) One ﬁrm
chooses group pricing, and the other one chooses uniform pricing, and (vi) One ﬁrm chooses
group pricing, and the other one chooses personalized pricing. As the calculations are simple, we provide only the results in the subgames.

3.1 Both ﬁrms choose uniform pricing
We consider the case where both ﬁrms employ uniform pricing. They simultaneously oﬀer
their uniform prices. We use the superscript “UU” to denote this case.
We classify the outcome in this subgame into two cases: (1) a pure strategy equilibrium
exists; (2) no pure strategy equilibrium exists. The former case appears if τ satisﬁes thefollowing inequality, otherwise the latter appears:√−(1 − λ)(2 + λ) + 2 1 − λτ≥≡ τuu .
λ(3 + λ)(1)

In the ﬁrst case (τ ≥ τuu ), we obtain each ﬁrm’s price and proﬁt as follows:
p≡

tH τ
tH τ
, π≡
.
1 − λ(1 − τ)
2 − 2λ(1 − τ)(2)In the second case (when τ < τuu ), we instead consider a mixed strategy equilibrium
where, (i) ﬁrm 0 chooses a low price p0l with probability α and a high price p0h with probability 1 − α, and (ii) ﬁrm 1 chooses a low price p1l with probability β and a high price p1h
with probability 1 − β. We use the superscript m to indicate an outcome in a mixed strategy

8
τ
1.0

0.8

0.6
A pure strategy NE exists in UU

0.4

0.2
No pure strategy NE exists in UU

0.0

0.2

0.4

0.6

0.8

λ

1.0

Figure 1: The area in which a pure strategy NE exists in UU
equilibrium if necessary.
We focus on a symmetric mixed strategy NE such that xL (p0h , p1l ) < 0, xL (p0l , p1h ) > 1,
xL (p0l , p1l ) = xL (p0h , p1h ) = 1/2, xH (p0l , p1h ) ∈ (0, 1), xH (p0h , p1l ) ∈ (0, 1), xH (p0l , p1l ) =xH (p0h , p1h ) = 1/2, and α = β ∈ (0, 1). In this equilibrium, the resulting outcome in market
L is the corner solution in which the low-price ﬁrm obtains all type L consumers’ demand
if the rival oﬀers the high price. We can numerically obtain a symmetric mixed strategy
equilibrium with pUUm
, pUUm, and αUU = βUU ∈ (0, 1) that satisfy (the detail is available inilih(A.1) in the Appendix):
The expected proﬁt of ﬁrm i when it oﬀers pUUm
il
= The expected proﬁt of ﬁrm i when it oﬀers pUUm
= πUUm.
iih
Given the complexity of the last two simultaneous equations, we cannot explicitly derive
the values of αUU and βUU but numerically show those values in Figure 2. The region where
αUU (λ, τ) ∈ (0, 1) coincides with that of the pure strategy equilibrium is not sustainable.
The relationship between αUU and λ can be positive and negative, depending on λ (seeαUU (λ) in Figure 2): it is negative if λ takes on an intermediate value; it is positive if λ is

9
αUU
1.0

αUU
1.0
αUU

αUU

0.8

0.8

0.6

0.6

0.4

0.4
τ=1/20

0.2

λ=1/2

0.2

τ

λ
0.0

αUU (λ, τ)

0.2

0.4

0.6

0.8

αUU (λ) (τ = 1/20)

1.0

0.00

0.02

0.04

0.06

0.08

αUU (τ) (λ = 1/2)

Figure 2: Mixed strategy in which both ﬁrms employ uniform pricing
large enough but is not near 1. When λ is near zero, the ﬁrms oﬀer a low uniform price,which is close to τtH (see (2)), with probability 1, resulting in a pure strategy equilibrium.
However, when λ takes on an intermediate value, a pure strategy equilibrium is not sustainable due to the incentive for ﬁrms to deviate by oﬀering higher prices to achieve higher
price cost margin. To eliminate this deviation incentive, ﬁrms need to adopt a mixed strategy involving probabilistic oﬀers of high prices, thereby making low prices more proﬁtablethrough demand expansion of type L consumers in the case where the rival oﬀers such a highprice. As λ increases, the deviation incentive becomes stronger because the gain from type
H consumers increases. The increased gain induces ﬁrms to oﬀer high prices with higher
probabilities (a decrease of αUU ).
In the second case where λ is large enough but not near 1, the importance of supplying
to type L consumers is low enough, inducing ﬁrms to increase pUUm
along with an increase
ilof αUU to target type H consumers. However, when λ is near 1, the ﬁrms mainly focus ontype H consumers and oﬀer p, close to tH , in (2) with probability 1.

10
We can summarize the discussion as follows:

pUU
i

πUU
i


tH τ






1 − λ(1 − τ)



 


= 

pUUm
in (A.1) with probability αUU (βUU ),




il









 pUUm in (A.1) with probability 1 − αUU (1 − βUU ).
ih


tH τ




 2 − 2λ(1 − τ) if τ ≥ τuu ,
= 




 πUUmin (A.1), if τ < τuu .
i

if τ ≥ τuu ,if τ < τuu ,

(3)

(4)

Conducting comparative statics, we ﬁnd that when λ is large and τ is smaller than but
suﬃciently close to τuu , a decrease in τ enlarges the proﬁts of the two ﬁrms. If τ < τuu , eachﬁrm sets a high price, pUUm
, with a positive probability to escape from ﬁerce competition inihmarket L and to concentrate on market H (see αUUm (λ, τ) in Figure 2). The strategic eﬀect
mitigates price competition, although the direct eﬀect of a decrease in τ intensiﬁes price competition. If τ is smaller than but suﬃciently close to τuu , the marginal impact of the strategic
eﬀect is larger than the direct eﬀect because the probability of oﬀering pUUm
becomes strictly
ihpositive, implying that the strategic eﬀect begins to be eﬀective and has a ﬁrst-order impact
on competition. Otherwise, the direct eﬀect of a decrease in τ dominates the strategic eﬀect,aggravating competition. We summarize the above ﬁnding in the following proposition:
Proposition 1 Suppose that both ﬁrms employ uniform pricing. When τ is smaller than but
suﬃciently close to τuu and λ is larger than a threshold value, a decrease in τ increases the
expected proﬁts of the ﬁrms.

3.2 Both ﬁrms choose personalized pricing
We consider the case where both the ﬁrms use personalized pricing. They simultaneously
oﬀer personalized prices for distinct consumers. We use the superscript “PP” to denote this
case.

11
In each market, the ﬁrms face an asymmetric Bertrand competition at each location,x ∈ [0, 1]. The ﬁrms choose the following prices at location x in market k (= H, L).





 tk (1 − x) − tk x
PP
p0 (x, k) ≡ 



 0





 0
PP
p1 (x, k) ≡  tk x − tk (1 − x)

for x ∈ [0, 1/2],
for x ∈ (1/2, 1],
for x ∈ [0, 1/2],
for x ∈ (1/2, 1].

Then, consumers in [0, 1/2] and (1/2, 1] buy from ﬁrms 0 and 1, respectively. We obtain the
proﬁt of each ﬁrm as follows:πPP ≡tH (λ + τ(1 − λ))
.
4

(5)

When λ = 1 or τ = 1, the outcome is the same as that with the personalized pricing in
Thisse and Vives (1988). When τ , 1, πPP decreases when λ decreases. However, piPP (x, k)(k = H, L) remains the same because the two markets are independent. Similarly, when
λ , 1, πPP and piPP (x, L) decrease when τ decreases, and piPP (x, H) remains the same.

3.3 Both ﬁrms choose group pricing
When both ﬁrms adopt group pricing, they oﬀer uniform prices to each consumer group.
Therefore, they compete in markets H and L, respectively. We use the superscript “GG” todenote this case.
The prices and the resulting proﬁts arepGG (H) ≡ tH , pGG (L) ≡ tL = τtH , πGG ≡A decrease in λ and a decrease in τ lower proﬁts.

tH (λ + (1 − λ)τ)
.
2

(6)

12

3.4 One ﬁrm chooses uniform pricing, and the other one chooses personalized pricing
We consider the case where one ﬁrm (say ﬁrm 0) employs uniform pricing and the other (sayﬁrm 1) adopts personalized pricing. In the second stage, ﬁrms 0 and 1 become the leader andthe follower, respectively. We use the superscript “UP” to denote this case.
We check the optimal pricing of the ﬁrms. Since ﬁrm 1 becomes the follower, given p0 ,ﬁrm 1 chooses pUP
1 (x, k) = max{p0 + tk x − tk (1 − x), 0} at x ∈ [0, 1] in market k. Next, weconsider ﬁrm 0’s decision. A consumer who is indiﬀerent between buying from ﬁrm 0 or
ﬁrm 1 must satisfy p0 + tk x = p1 + tk (1 − x), from which we have p1 (x, tk ) = p0 + (2x − 1)tk ,which must be non-negative. By solving p0 +(2xkUP −1)tk = 0, for xkUP , we obtain the locationof the indiﬀerent consumer in market k, xkUP (p0 ) ≡ (tk − p0 )/(2tk ). Since xkUP (p0 ) ∈ [0, 1],
ﬁrm 0’s proﬁt function is
 {
}



p0 λxUP(p0 ) + (1 − λ)xUP(p0 ) if 0 ≤ p0 ≤ τtH ,

H
L





UP
π0 (p0 ) ≡ 
p0 λxUPif τtH ≤ p0 ≤ tH ,H (p0 )






 0
if tH ≤ p0 .
By solving the proﬁt maximization problem of ﬁrm 0, we obtain the following optimal price:

τ
tH τ



if 0 < λ ≤or 1/2 ≤ τ,
 2 − 2λ(1 − τ)1−τUP
p0 = 


t
τ


 H
if
≤ λ < 1 and τ < 1/2.
2
1−τIf λ is large and τ is small, ﬁrm 0 will abandon the supply to type L consumers, and ﬁrm 1 will
monopolize the market for type L consumers. Otherwise, ﬁrm 0 will serve both consumer
types by oﬀering pUP
0 , which is lower than tH /2 unless τ = 1. This pricing of ﬁrm 0 implies
intense price competition when the ratio of type H consumers is low (λ is small).

13
Using this result, we obtain the proﬁt of ﬁrm 0:

tH τ
τ



if 0 < λ ≤or 1/2 ≤ τ,
 8 − 8λ(1 − τ)1−τUP
π0 = 


t λ
τ


 H
if
≤ λ < 1 and τ < 1/2.
8
1−τ

(7)

We obtain the proﬁt of ﬁrm 1 is as follows:tH [9τ + 4λ(1 − τ)2 − 4λ2 (1 − τ)2 ]τ
if 0 < λ ≤or 1/2 ≤ τ,



16
−
16λ(1
−
τ)
1
−
τ
UP
π1 = t (8 + λ)
τ


 H
if
≤ λ < 1 and τ < 1/2.
16
1−τ

(8)

A decrease in λ and a decrease in τ lower proﬁts.

3.5 One ﬁrm chooses group pricing, and the other one chooses uniform
pricing
We consider the case where one ﬁrm (say ﬁrm 0) employs group pricing and oﬀers p0 (k) tomarket k and the other (say ﬁrm 1) adopts uniform pricing and oﬀers p1 . They simultaneously
oﬀer their prices. We use the superscript “GU” to denote this case. The indiﬀerent consumer
in market k is denoted by xGU
k (p0 (k), p1 ) = (−p0 (k) + p1 + tk )/(2tk ).
We need to classify the outcome in this subgame into the following two cases: (1) a purestrategy equilibrium exists; (2) no pure strategy equilibrium exists. The former case appears
if τ satisﬁes the following inequality, otherwise the latter appears:√−8 + 3λ + 9λ2 + 4 4 − 3λτ≥≡ τgu .
3λ(5 + 3λ)(9)In the ﬁrst case (τ ≥ τgu ), ﬁrms solves max p0 (H),p0 (L) λp0 (H)xGU
H (p0 (H), p1 ) + (1 −GUGUλ)p0 (L)xGU
L (p0 (L), p1 ) and max p1 p1 {λ(1 − xH (p0 (H), p1 )) + (1 − λ)(1 − xL (p0 (L), p1 ))},

14
τ
1.0

0.8

0.6
A pure strategy NE exists in GU

0.4

0.2
No pure strategy NE exists in GU

λ
0.0

0.2

0.4

0.6

0.8

1.0

Figure 3: The area in which a pure strategy NE exists in GU
from which we havetH (1 − λ + τ + λτ) GU∗
tH (2 − λ + λτ) GU∗tH τ, p0 (L) ≡, p1 ≡
2(1 − λ + λτ)2(1 − λ + λτ)
1 − λ + λτ
tH λ(1 − λ + τ + λτ)2 tH (1 − λ)τ(2 − λ + λτ)2 GU∗
tH τ
πGU∗
≡
+
, π1 ≡.
02
2
8(1 − λ + λτ)8(1 − λ + λτ) )2(1 − λ + λτ)
pGU∗
(H) ≡
0

(10)

In the second case (τ < τgu ), we instead consider a mixed strategy equilibrium where,(i) ﬁrm 0 employs a pure strategy and oﬀers p0 (H) and p0 (L), and (ii) ﬁrm 1 chooses a low
price p1l with probability β and a high price p1h with probability 1 − β.
GU
We focus on a mixed strategy NE such that xGU
L (p0 (L), p1l ) ∈ (0, 1), xL (p0 (L), p1h ) > 1,
GUxGU
H (p0 (H), p1l ) ∈ (0, 1), xH (p0 (H), p1h ) ∈ (0, 1), and β ∈ (0, 1). In this mixed strategy

equilibrium, the resulting outcome in market L is the corner solution where ﬁrm 0 obtains alltype L consumers if the rival oﬀers the high price. We can obtain a mixed strategy NE withpm0 (H), pm0 (L), pm1l , pm1h , and βGU ∈ (0, 1) that satisﬁes (the detail is available in (A.2) in the

15
Appendix):
The expected proﬁt of ﬁrm 1 when it oﬀers pm1l
= The expected proﬁt of ﬁrm 1 when it oﬀers pmih = πGUm.
1
Although we can explicitly derive the value of β which is complex, we numerically show
this value in Figure 4. The region where βGU (λ, τ) ∈ (0, 1) coincides with that of the purestrategy equilibrium is not sustainable.
βGU
1.0

βGU
1.0
τ=1/10

0.8

0.8

λ=1/2
βGU

0.6

0.6
βGU
0.4

0.4

0.2

0.2

0.0

β

GU

(λ, τ)

0.2

β

GU

0.4

0.6

0.8

(λ) (τ = 1/10)

1.0

λ0.00 0.05 0.10 0.15 0.20 0.25 0.30

β

GU

τ

(τ) (λ = 1/2)

Figure 4: Mixed strategy in which one ﬁrm chooses group pricing and the other chooses
uniform pricing
The relationship between βGU and λ is always negative (see βGU (λ) in Figure 4). We canapply the intuition in the case of UU here. Given the group pricing by ﬁrm 0, ﬁrm 1 deviates
by raising its price because supplying only to type H consumers is proﬁtable. To eliminate
this deviation incentive, p0 (H) should be lower and ﬁrm 1 needs to induce ﬁrm 0 to set sucha low p0 (H) by oﬀering a low price with probability βGU . However, as λ increases, ﬁrm 1’s
incentive to oﬀer a high price becomes stronger because the gain from type H consumers
increases. The increased gain induces ﬁrm 1 to oﬀer a high price with higher probabilities(a decrease of βGU ). Because no pure strategy exists for any high values of λ if τ < τgu , thenegative relationship remains.

16
We can summarize the discussion as follows:pGU
0 (H) =pGU
0 (L) =

=
pGU
1

πGU
=
0

πGU
=
1



tH (1 + τ − (1 − τ)λ)if τ ≥ τgu , 2(1 − λ(1 − τ))
(11)





 pm0 (H) in (A.2)
if τ < τgu ,tH (2 − (1 − τ)λ) 2(1 − λ(1 − τ)) if τ ≥ τgu ,
(12)




m

 p0 (L) in (A.2)
if τ < τgu ,

tH τ



if τ ≥ τgu ,



1
−
λ(1
−
τ)




 

m
GU
(13)




 pil in (A.2) with probability β ,



if
τ
<
τ
,


gu





 pmih in (A.2) with probability 1 − βGU , tH λ(1 − λ + τ + λτ)2 (tH (1 − λ)τ(2 − λ + λτ)2+if τ ≥ τgu ,
8(1 − λ + λτ)28(1 − λ + λτ)2


(14)


πGUm
≡ pm0 (H)λ(βm xH (pm0 (H), pmil ) + (1 − βm )xH (pm0 (H), pmih ))


0





+pm0 (L)(1 − λ)(βm xL (pm0 (L), pmil ) + (1 − βm )), if τ < τgu .


tH τ




 2 − 2λ(1 − τ) if τ ≥ τgu ,
(15)




GUm

 π1 in (A.2), if τ < τgu .

We ﬁnd that when τ < τgu , a decrease in τ increases ﬁrm 1’s expected uniform price,
mitigating price competition. A decrease in τ has direct and strategic eﬀects on proﬁts. First,
a decrease in τ directly intensiﬁes competition in market L, which harms ﬁrm 1. Second,
when τ < τgu , the lower τ, the lower the probability of setting p1l , βGU (see Figure 4),
because the market for type L consumers becomes less proﬁtable. The change of βGU induces
ﬁrm 0 to oﬀer a higher price in market H because of the strategic complementarity of bothﬁrms’ prices, which beneﬁts ﬁrm 1. The relative signiﬁcance of the two contrasting eﬀects
depends on the pricing scheme of ﬁrm 0. When ﬁrm 0 adopts group pricing, ﬁrm 0 can oﬀer
discriminatory prices in the two markets, implying that ﬁrm 0 can keep its price in market H
high for any τ (see pm0 (H) in (A.2)). Such a high price of ﬁrm 0 in market H allows ﬁrm 1 toconcentrate on market H to obtain a positive proﬁt even if τ is suﬃciently small (see βGU in17
Figure 4). Therefore, for ﬁrm 1’s proﬁt, the strategic eﬀect dominates the direct eﬀect.
However, when τ is suﬃciently small, the mitigation of price competition becomes weak.
Firm 0 is less likely to compete with ﬁrm 1 in market L because of a low βGU , weakening
the strategic complement eﬀect on pm0 (L). In fact, pm0 (L) increases in τ if τ is suﬃciently
small; otherwise, pm0 (L) decreases in τ. Therefore, when τ is smaller than τ̄ deﬁned in Figure
5, a reduction in τ reduces pm0 (L) and such a price reduction consequently causes a proﬁt
reduction for ﬁrm 0, which captures the entire demand in market L with probability 1 − βGU .
We summarize the above ﬁnding in the following proposition:
Proposition 2 Suppose that one ﬁrm (ﬁrm 0) employs group pricing and the other one (ﬁrm
1) employs uniform pricing. When τ̄ < τ < τgu , a decrease in τ increases the expected proﬁt
of ﬁrm 0. When τ < τgu , a decrease in τ increases the expected proﬁt of ﬁrm 1.
τ
1.0
0.8
τgu

0.6

τ

0.4
0.2
λ
0.0

0.2

0.4

0.6

0.8

1.0

Figure 5: The area in which a decrease in τ increases proﬁts

3.6 One ﬁrm chooses group pricing, and the other chooses personalized
pricing
We consider ﬁrm 0 employs group pricing and ﬁrm 1 employs personalized pricing. In
the second stage, ﬁrms 0 and 1 become the leader and the follower, respectively. We use

18
the superscript “GP” to denote this case. Given ﬁrm 0’s price p0 (k), ﬁrm 1 oﬀers p1 =p0 (k) + (2x − 1)tk . The indiﬀerent consumer in each market is denoted by xGP
k (p0 (k)) ≡(tk − p0 (k))/(2tk ). Firm 0 then solves max p0 (k) xGP
k (p0 (k))p0 (k), from which we havepGP
0 (k) ≡

tk GP tH ((1 − λ)τ + λ) GP 9tH ((1 − λ)τ + λ), π ≡, π1 ≡
.
2 0
8
16

(16)

Each market outcome is the same as in Thisse and Vives (1988) except for the market size.
As in the previous arguments, a decrease in λ and a decrease in τ lower proﬁts.

4 Equilibrium pricing scheme
First, we derive the equilibrium pricing schemes in the ﬁrst stage and show that both ﬁrms
employ personalized pricing in equilibrium. Second, we discuss how pricing schemes inﬂuence proﬁts and welfare.

4.1 Decisions on pricing schemes
The discussions in the previous subsections provide the payoﬀ matrix in the ﬁrst-stage game.
Firm 0/Firm 1
uniform

uniform
πUU in (4)πUU in (4)personalized

πUP
0 in (7)πUP
1 in (8)

group

πGU
1 in (15)πGU0 in (14)personalized
πUP
1 in (8)πUP
0 in (7)πPP in (5)πPP in (5)πGP
1 in (16)πGP
0 in (16)

group
πGU
0 in (14)πGU
1 in (15)πGP
0 in (16)πGP
1 in (16)πGG in (6)πGG in (6)We solve for the ﬁrms’ equilibrium pricing schemes. We ﬁnd that both ﬁrms employ
personalized pricing in equilibrium.

19
Proposition 3 Both ﬁrms employ personalized pricing as a unique equilibrium outcome.
The resulting outcome is expected given the outcome in Thisse and Vives (1988). If wepick up only personalized and group pricing, personalized pricing is the dominant strategy.
Similarly, if we pick up only personalized and uniform pricing, personalized pricing is still
the dominant strategy. Therefore, we ﬁnd that personalized pricing is attainable in equilibrium. The result implies that the ﬁrms can be better oﬀ if they have coarser information
about consumers, aligning with the ﬁnding in Laussel, Long, and Resende (2020).8

4.2 The eﬀects of pricing schemes on proﬁts and welfare
We numerically compare πUU in (4), πPP in (5), and πGG in (6), due to the mathematicalcomplexity.
Proposition 4 πUU < πPP < πGG if and only if the parameter pair (λ, τ) is in the gray areain Figure 6; otherwise, πPP < πUU < πGG . The border curve of the gray area consists of thepair (λ, τ) that satisﬁes πPP = πUU .
1.0

0.8

0.6

τuu
τ
0.4

1+2 λ 2 λ2
1 4 λ 4 λ2
2λ 1 λ

0.2

0.0
0.0

0.2

0.4

0.6

0.8

1.0

Horizontal axis: λ; Vertical axis: τ.

Figure 6: The condition that πUU < πPP
When the mismatch costs of the two consumer groups are suﬃciently heterogeneous,20
personalized pricing is better than uniform pricing. This contrasts with the standard result in
the context of personalized pricing.
The intuition behind Proposition 4 is explained as follows. Because type L consumers
incur a lower transportation cost compared with type H consumers, the former is more pricesensitive. When both ﬁrms adopt uniform pricing, they must oﬀer suﬃciently low prices tocapture the price-elastic consumers, detrimental to the proﬁt obtained from type H consumers. In contrast, adopting personalized pricing enables ﬁrms to delink the markets forthe two types of consumers by oﬀering customized prices to each individual. The competition mode becomes similar to an asymmetric Bertrand competition for each consumer. Then,personalized pricing brings each ﬁrm the following trade-oﬀs: on the one hand, it intensiﬁes
the competition for type L consumers (a negative eﬀect); on the other hand, each ﬁrm can set
higher prices for some of type H consumers who located closer to it compared with the rival(a positive eﬀect). The positive eﬀect dominates when the heterogeneity between the two
types is suﬃciently large. In sum, under personalized pricing, there is a market segmentation with the outcome that both ﬁrms exploit less surplus from type L consumers but exploit
more from some type H consumers who show strong brand preferences towards a particularﬁrm (located geographically closer to one ﬁrm).
We discuss how the demand condition is important to evaluate the proﬁtability of personalized pricing, comparing our result with Esteves (2022), who derives a reversal result inwhich personalized pricing is more proﬁtable than uniform pricing, which is a reversal result
to Thisse and Vives (1988). Esteves (2022) introduces two types of consumers in a Hotellingmodel: those who demand only one unit (type L consumers) and those who demand q(> 1)units of the product (type H consumers). As q increases, type H consumers become moreprice elastic and more proﬁtable for ﬁrms. The reversal result occurs only if the ratio of type
H consumers is smaller than 50% and q is larger than 5. As q increases, the possibility ofachieving the reversal result increases (the feasible ratio of type H becomes wider), with 20%

21
of type H consumers being highly desirable to achieve the reversal result.9 The relationshipbetween q (higher price elasticity) and the higher possibility of achieving the reversal resultaligns with ours (see Figure 6). However, the preferable ratio of type H to achieve the reversal result (around 20%) diﬀers from ours (around 50%) (see Figure 6) because of the double
meaning of q in Esteves (2022): high price elasticity and high demand. The latter meaning
distorts the ratio in Esteves (2022).
We investigate consumer welfare and numerically show that the ﬂip side of Proposition
4 holds.
Proposition 5 The consumer surplus in UU is the highest, and that in GG is the lowest if
and only if the parameter pair (λ, τ) is in the gray area in Figure 6; otherwise, the consumer
surplus in PP is the highest and that in GG is the lowest. The border curve of the gray area
consists of the pair (λ, τ) that satisﬁes πPP = πUU .
Because of the inelastic demand in the Hotelling model, the welfare ranking between thepricing schemes is opposite to the proﬁt ranking.
Case GG is always the worst for consumer welfare among the three cases, UU, PP, and
GG. Although UU can be the best for consumer welfare when the group heterogeneity ishigh, it is not easy to intervene in personalized pricing because such an intervention could
facilitate group pricing, which is acceptable in practice, leading to the worst case.
Finally, we mention the total surplus in each case. In GG and PP, there is a unique purestrategy equilibria in which the ﬁrms equally split the market demand, achieving the ﬁrstbest allocation. However, in UU, there is a mix strategy equilibrium if τ < τuu because theex-post outcome can be ineﬃcient because of the ex-post asymmetric prices. Otherwise, apure strategy equilibrium in UU is also the ﬁrst-best.
Proposition 6 PP and GG achieve the ﬁrst-best allocation regardless of τ. If τ ≥ τuu ,UU also achieves the ﬁrst-best allocation; otherwise, the ex-post outcome in UU can be22
ineﬃcient.

4.3 Additional pricing schemes
We discuss a case in which each ﬁrm has more pricing schemes than in the model we haveconsidered. We add the following two pricing schemes: (i) uniform pricing for type H andpersonalized pricing for type L; (ii) uniform pricing for type L and personalized pricing for
type H. The additions expand the payoﬀ matrix from 3 × 3 to 5 × 5.
Firm 0/Firm 1

uni

personal

group

uni H
personal L

uni L
personal H

uni

✓

✓

✓

(∗∗) Firm 0changes(∗∗) Firm 0changes

personal✓

Equilibrium
pair✓(∗) Firm 1changes(∗) Firm 1changes

group

✓

✓

✓

(∗) Firm 1changes(∗) Firm 1changesuni H
personal L(∗∗) Firm 1changes(∗) Firm 0changes(∗) Firm 0changes(∗) Firm ichanges(∗) Firm ichangesuni L
personal H(∗∗) Firm 1changes(∗) Firm 0changes(∗) Firm 0changes(∗) Firm ichanges(∗) Firm ichanges

We can show that the additional 16 pricing pairs are not achievable in equilibrium. First,
without detailed calculations, we can show that the pairs of the pricing schemes with (∗)
in the cells are not sustainable as an equilibrium. For those pricing pairs, the markets forthe two types of consumers are delinked because the ﬁrms can oﬀer diﬀerent prices for thetwo types of consumers. The market segmentation implies that we can apply the result in
Thisse and Vives (1998) in each market segment. Concretely, at least one of the ﬁrms has anincentive to change its pricing scheme to (complete) personalized pricing. Second, we canalso show that the pairs of the pricing schemes with (∗∗) in the cells are not sustainable as anequilibrium because the ﬁrm that employs uniform pricing deviates to personalized pricing(the detail is available in the Online Appendix).

23

5 Conclusion
We examine ﬁrms’ incentives to adopt one of the three pricing schemes: uniform, personalized, or group pricing in a Hotelling duopoly model. There are two types of consumer
groups that are heterogeneous in their mismatch costs.
We show that regardless of the heterogeneity of consumer groups, both ﬁrms employ
personalized pricing in equilibrium. Moreover, the proﬁts when both ﬁrms employ personalized pricing are higher than those when they use uniform pricing if the heterogeneity of
consumer groups is signiﬁcant. The result is aligned with that of Esteves (2022). However,proﬁts are always the highest among the three cases when the ﬁrms employ group pricing.
Additionally, if we focus on the three cases where both ﬁrms employ (i) uniform pricing(UU), (ii) personalized pricing (PP), and (iii) group pricing (GG), the consumer surplus in
UU is the highest and that in GG is the lowest among the three cases if the heterogeneity of
consumer groups is signiﬁcantly high; otherwise, the consumer surplus in PP is the highest
and that in GG is the lowest among the three cases. Therefore, group pricing is always themost harmful from the consumer welfare perspective. Furthermore, this result implies that
it is not easy to intervene in personalized pricing because this could facilitate group pricing,leading to the worst case.
We can incorporate consumers’ privacy concerns into our model by using the demand
structure in Montes et al. (2019), where there are old and new consumers in a Hotelling
duopoly model. In their model, a monopolistic data broker has the locational information ofold consumers and can sell it to at least one of the ﬁrms. They consider each old consumer’s
incentive to delete the locational information from the database. As there are two dimensionsof consumer characteristics in our model, location x and type tk , each consumer has four options to manage the information characteristics. The richness of the consumer characteristics
may lead to interesting results. Considering the extension can be future research.

24
Also, we can also consider ﬁrms’ optimal degree of data collection. In our paper, wehave assumed that ﬁrms have ﬁner consumer information to implement personalized pricing.
As in Laussel, Long, and Resende (2020), having coarser consumer information can be
proﬁtable for ﬁrms. Considering data collection in our model can be future research.

25

Notes
1

OECD (2018) “Personalised pricing in the digital era,” 28 November 2018https://one.oecd.org/document/DAF/COMP(2018)13/en/pdf (Last access: 13 March 2022).
2

Bloomberg (2017) “Uber starts charging what it thinks you’re willing to pay,” 19 May 2017https://www.bloombergquint.com/markets/uber-s-future-may-rely-on-predicting-how-much-you-re-willing-to-pay (Last access: 13 March

2022).
3

Chen et al. (2020) examine a model wherein consumers’ identity management allows them to choose theuniform prices set only for new customers by ﬁrms. The model shows that personalized pricing can allow ﬁrms
to fully extract the surpluses of targeted consumers.
4

Lu and Matsushima (2023) discuss the eﬀect of personalized pricing on proﬁts and welfare in a Hotellingmodel in which consumers can simultaneously purchase from both ﬁrms (so-called multi-homing). They show
the possibility that personalized pricing improves consumer welfare as well as ﬁrms’ proﬁts. However, they do
not consider heterogeneity in terms of mismatch costs.
5

The Perloﬀ and Salop (1985) model is also used in the context of mixed bundling (see Zhou, 2021).

6

Wu (2021) mentions anecdotal evidence about price discrimination based on mobile phone types.

https://kr-asia.com/researchers-took-over-800-trips-using-chinese-ride-hailing-apps-heres-what-they-found7

A survey conducted by Slickdeals, a US crowdsourced shopping platform, found that iPhone owners tend
to have higher incomes and spend more compared to Android users.
https://www.marketingdive.com/news/survey-iphone-owners-spend-more-have-higher-incomes-than-android-users/541008/
8

By incorporating consumers’ identity management, Laussel, Long, and Resende (2022) further extend
Laussel, Long, and Resende (2020).
9This ratio, 20%, aligns with the Pareto Principle in marketing (Twedt, 1964).

26

Appendix
The mixed strategy in Section 3.1

We can numerically obtain a symmetric mixed strategy

equilibrium:10(2λτ + αUU (1 − λ)(1 + λτ) + (αUU )2 (1 − λ)2 )tH τ,
2λτ(1 − λ(1 − τ)) + αUU (1 − αUU )(1 − λ)(1 − λ − 2λτ)(λ(1 + τ − λ(1 − τ)) + αUU (1 − λ)(1 − λ(2 − τ)) − (αUU )2 (1 − λ)2 )tH τ
=
pUUm
,
ih
2λτ(1 − λ(1 − τ)) + αUU (1 − αUU )(1 − λ)(1 − λ − 2λτ)
pUUm
=
il

αUU = βUU ∈ (0, 1) that satisfy the following two equations:
πUUm≡ pUUm[λ{βUU xH (pUUm, pUUm) + (1 − βUU )xH (pUUm, pUUm
0
0l
0l
1l
0l
1h )}
+(1 − λ){βUU xL (pUUm, pUUm) + (1 − βUU )}]
0l
1l
UU
UUm
UUm
UUm
UU
UUm
= p0h [λ{β xH (p0h , p1l ) + (1 − β )xH (pUUm
0h , p1h )}UUm+(1 − λ)(1 − βUU )xL (pUUm
0h , p1h )] and

(A.1)

UUm
πUUm
≡ pUUm[λ{αUU (1 − xH (pUUm, pUUm)) + (1 − αUU )(1 − xH (pUUm
))}
1
1l
0l
1l
0h , p1l
UUm
UUm
UU
UU
+(1 − λ){α (1 − xL (p0l , p1l )) + (1 − α )}]
UU
UUm
UUm
= pUUm(1 − xH (pUUm)) + (1 − αUU )(1 − xH (pUUm
1h [λ{α0h , p1l
0h , p1h ))}UUm+(1 − λ)(1 − αUU )(1 − xL (pUUm
0h , p1h ))].

Given the complexity of the last two simultaneous equations, we cannot explicitly derive thevalues of αUU and βUU but numerically show those values in Figure 2.

27
The mixed strategy in Section 3.5

We can obtain a mixed strategy NE:11

tH (9(1 − λ) + 4(1 + 2λ)τ − βGU (1 − λ)(3 − 2τ)),3(3 − (3 − 4τ)λ + βGU (1 − λ))tH τ(12 + βGU − (βGU )2 − 12λ + 5βGU λ + (βGU )2 λ + 12λτ − 6βGU λτ)
pm0 (L) =,3(3 − (3 − 4τ)λ + βGU (1 − λ))tH (6 + 5βGU + (βGU )2 − 6λ + 7βGU λ − (βGU )2 λ)τ
pm1l =,
3(3 − (3 − 4τ)λ + βGU (1 − λ))tH (9 − 9λ + 2τ + βGU τ + 10λτ − βGU λτ)
pm1h =,
3(3 − (3 − 4τ)λ + βGU (1 − λ))pm0 (H) =(A.2)

βGU ∈ (0, 1) that satisﬁes the following equation:πGUm≡ λpm1h (1 − xH (pm0 (H), pm1h ))1= pm1l [λ(1 − xH (pm0 (H), pm1l )) + (1 − λ)(1 − xL (pm0 (L), pm1l ))].
Although we can explicitly derive the value of β which is complex, we numerically show
this value in Figure 4.

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A1

A Online Appendix
A.1 Both ﬁrms choose uniform pricing (UU)
We ﬁrst show condition under which no ﬁrm unilaterally deviates from (pUU , pUU ). By
symmetry, assume that ﬁrm 0 deviates. Since ﬁrm 0 must obtain positive share, we exclude
the case with xH = xL = 0. i.e., p0 ≥ pUU + tH . In addition, for any p0 ≤ pUU − tH , we have
xH = xL = 1, which means that demand for ﬁrm 0 is constant. Then, for p0 ∈ [0, pUU − tH ],ﬁrm 0’s proﬁt is maximized at p0 = pUU − tH . Hence, it suﬃces to consider two types ofdeviation: (i) pUU − tH ≤ p0 ≤ pUU − τtH and (ii) pUU + τtH ≤ p0 < pUU + tH .
First, we consider case (i) in which xL = 1. Given p1 = pUU , π0 (p0 , pUU ) = λp0 xH + (1 −λ)p0 . The ﬁrst-order condition yields the deviation price (with superscript UD1).
pUD1
=
0

tH [λ2 (1 − τ) − 3λ(1 − τ) + 2].
2λ[1 − λ(1 − τ)]The necessary condition for this deviation is pUD1
< pUU − tL , which is always violated0
because(pUU − tL ) − pUD1=−
0tH [2λ2 τ2 + 3(1 − λ)λτ + (2 − λ)(1 − λ)]< 0.
2λ[1 − λ(1 − τ)]Hence, ﬁrm 0 does not deviate in case (i).
Next, we consider case (ii) in which xL = 0. With p1 = pUU , π0 (p0 , pUU ) = λp0 xH . The
ﬁrst-order condition yields the deviation price (with superscript UD2).
pUD2
=
0

tH [1 + τ − λ(1 − τ)],
2 − 2λ(1 − τ)The deviation proﬁt is
πUD2
=
0

tH λ[1 + τ − λ(1 − τ)]2.
8[1 − λ(1 − τ)]2

A2
yieldsComparing πUU with πUD20
πUU − πUD2=0tH (1 − λ)[λ2 (1 − τ)2 − λ(1 + 2τ − 3τ2 ) + 4τ],
8[1 − λ(1 − τ)]2

which is strictly positive if√
2 1 − λ − (2 − λ − λ2 )≡ τUU < τ < 1.
λ(3 + λ)Notice that under this condition, pUU + tL − pUD2
≤ 0 always holds. Hence, (pUU , pUU ) is a0pure strategy NE if τ > τUU .
For 0 < τ ≤ τUU , we derive the mixed NE as described in Equation (A.1). We solve four
simultaneous equations derived from the following maximization system:
max πUU
0l ≡ p0l λ(βxH (p0l , p1l ) + (1 − β)xH (p0l , p1h )) + p0l (1 − λ)(βxL (p0l , p1l ) + (1 − β)),p0l

max πUU
0h ≡ p0h λ(βxH (p0h , p1l ) + (1 − β)xH (p0h , p1h )) + p0h (1 − λ)(1 − β)xL (p0h , p1h ),p0h

max πUU
1l ≡ p1l λ(α(1 − xH (p0l , p1l )) + (1 − α)(1 − xH (p0h , p1l ))p1l+p1l (1 − λ)(α(1 − xL (p0l , p1l )) + (1 − α)),max πUU
1h ≡ p1h λ(α(1 − xH (p0h , p1l )) + (1 − α)(1 − xH (p0h , p1h )))p1h+p1h (1 − λ)(1 − α)(1 − xL (p0h , p1h )).
UU
Next, we let πUU0l = π0h and α = β, and solve for α and β. The numerical outcome is in

Figure 2.

A3

A.2 One ﬁrm chooses uniform pricing and the other one chooses personalized pricing (UP)We ﬁrst consider the case with 0 ≤ p0 ≤ τtH . The ﬁrst-order condition yields
pUPl
≡
0

tH τ
tH τ
tH [9τ + 4λ(1 − τ)2 − 4λ2 (1 − τ)2 ]UPl, πUPl
≡
,
π
≡
,
0
1
2 − 2λ(1 − τ)
8 − 8λ(1 − τ)
16 − 16λ(1 − τ)

where the superscript UPl denotes the case of pUPl
∈ [0, τtH ], which can be reduced to0
0 < λ < 1/(2 − 2τ) ≡ λUPl or τ ≥ 1/2.
Next, we consider the case with τtH ≤ p0 < tH . The ﬁrst-order condition yields:
pUPh
≡
0

tH
tH λ
tH (8 + λ)
, πUPh≡, πUPh
≡
,
0
1
2
8
16

where the superscript UPh denotes the case of p0 ∈ [τtH , tH ]. Here, we derive the condition
such that τtH ≤ pUPh
≤ tH , from which we have τ ≤ 1/2.
0
UPl
From the above, we have two equilibrium candidates, pUPl
and pUPhand00 , if 0 < λ < λUPl
0 < τ < 1/2, whereas there is a unique candidate, pUPh
with
0 , otherwise. Comparing π0πUPhyields0πUPl− πUPh=00τtH (1 − λ)[τ − λ(1 − τ)]> 0 ⇐⇒ λ <≡ λUP .
8 − 8λ(1 − τ)1−τSince λUPl > λUP for 0 < τ < 1/2, the optimal p0 is

τ
tH τ



if 0 < λ ≤or 1/2 ≤ τ,
 2 − 2λ(1 − τ)
1−τ
pUP
=

0


τ
t


 H
if
< λ < 1 and 0 < τ < 1/2.
2
1−τ

A4
Using this result, we obtain the proﬁt of ﬁrm 0.

tH τ
τ



if 0 < λ ≤or 1/2 ≤ τ,
 8 − 8λ(1 − τ)1−τUP
π0 = 


t λ
τ


 H
if
< λ < 1 and 0 < τ < 1/2.
8
1−τ

A.3 One ﬁrm chooses group pricing and the other one chooses uniform
pricing (GU)Due to symmetry, let ﬁrm 0 choose group pricing and oﬀer p0 (H) to market H and p0 (L) to
market L. Let ﬁrm 1 choose uniform pricing and oﬀer p1 . The indiﬀerent consumer in eachmarket is given by xGU
k (p0 (k), p1 ) = (tk − p0 (k) + p1 )/(2tk ). Then, ﬁrm 0 and ﬁrm 1 solveGU
max p0 (H),p0 (L) p0 (H)λxGU
H (p0 (H), p1 ) + p0 (L)(1 − λ)xL (p0 (L), p1 ),GU
max p1 p1 {λ[1 − xGU
H (p0 (H), p1 )] + (1 − λ)[1 − xL (p0 (L), p1 )]},

which yields the equilibrium prices pGU∗
(H), pGU∗(L), pGU∗
as in Eq. (10). Since ﬁrm 0’s
001
proﬁt function with respect to p0 (H) and p0 (L) is globally concave, it has no incentive to deviate. However, given pGU∗
(H) and pGU∗(L), ﬁrm 1’s proﬁt function is not globally concave,00(1−λ)(1−τ)< 0,and may therefore unilaterally deviate from pGU∗
. It can be conﬁrmed that tH2(1−λ+λτ))1

given pGU∗
(H), pGU∗(L) and any p1 ∈ (0, 1), xGU
H > 0. Moreover, ﬁrm 1 will never oﬀer a00
price such that both xH = 1 and xL = 1, because its proﬁt would become zero. Therefore,
given pGU∗
(H), pGU∗(L) ﬁrm 1’s proﬁt function is
0
0
GU∗
πGU
(L), pGU∗(H), p1 )
1 (p0
0




< p1 ≤ tH (3−3λ+τ+3λτ),
p1 λ[1 − xGU
if tH τ(4−3λ+3λτ)
H (p1 )]2(1−λ+λτ)2(1−λ+λτ)




 {
}
tH λ(1−τ)τ
GU≡ p1 λ[1 − xGU
if 2(1−λ+λτ))≤ p1 ≤ tH τ(4−3λ+3λτ),
H (p1 )] + (1 − λ)[1 − xL (p1 )]2(1−λ+λτ)tH λ(1−τ)τ
 p {λ[1 − xGU (p )] + (1 − λ)}
if 0 < p1 < 2(1−λ+λτ))
.
1
1
H

A5
The second case induces the equilibrium price pGU
1 , so we discuss whether ﬁrm 1 deviates totH λ(1−τ)τ
remaining two cases: (i) tH τ(4−3λ+3λτ)< p1 ≤ tH (3−3λ+τ+3λτ), and (ii) 0 < p1 < 2(1−λ+λτ)).
2(1−λ+λτ)2(1−λ+λτ)First, we consider case (i) in which xL = 1. Solving the ﬁrst-order condition for p1 yields
the deviation price and proﬁt (with superscript GU1):
=
pGU1
1

tH (3 − 3λ + τ + 3λτ) GU1 tH λ(3 − 3λ + τ + 3λτ)2, π1 =
4(1 − λ + λτ)32(1 − λ + λτ)2Since pGU1
always satisﬁes case (i), we compare πGU1with πGU
1
1
1 , from which we have√4
4 − 3λ − (8 − 3λ − 9λ2 )
GU1
⇐⇒
πGU∗
>
π
≡ τGU < τ < 1,
1
1
3λ(5 + 3λ)
Next, we consider case (ii) in which xL = 0. Solving the ﬁrst-order condition for p1
yields the deviation price (with superscript GU2)
pGU2
=
1

tH [(4 − 5λ + λ2 + 5λτ − λ2 τ)].
4λ(1 − λ + λτ)

It can be conﬁrmed that pGU2
always violates case (ii). To summarize, (pGU∗(L), pGU∗(H), pGU∗)1001
is a pure strategy NE if τ > τGU .
For 0 < τ ≤ τGU , we derive the mixed NE as described in Equation (A.2). We solve four
simultaneous equations derived from the following maximization system:
max p0 (H),p0 (L) p0 (H)λ(βxH (p0 (H), p1l ) + (1 − β)xH (p0 (H), p1h ))+p0 (L)(1 − λ)(βxL (p0 (L), p1l ) + (1 − β));max p1l πGU
1l ≡ p1l (λ(1 − xH (p0 (H), p1l )) + (1 − λ)(1 − xL (p0 (L), p1l )));
max p1h πGU
1h ≡ p1h (λ(1 − xH (p0 (H), p1h ))).
GU
Next, we let πGU
1l = π1h and solve for β. The numerical outcome is Figure 4.

A6

A.4 Equilibrium pricing schemes
We ﬁrst show that GG, GP (PG), UP (PU), GU(GU) and UU cannot be an NE.
GG is not an NE: Given ﬁrm 0 choosing group pricing, ﬁrm 1’s proﬁt change fromchoosing group pricing to choosing personalized pricing is
GG
πGP
=
1 −π

tH (τ + (1 − τ)λ)> 0.
16

GP (PG) is not an NE: Due to symmetry, we only consider GP. Given ﬁrm 1 choosing personalized pricing, ﬁrm 0’s proﬁt change from choosing group pricing to choosing
personalized pricing is
π0PP − πGP =tH (τ + (1 − τ)λ> 0.
8

UP (PU) is not an NE: Due to symmetry, we only consider UP. From the results in UP(see Proof A.2), we have (i) UPl when 0 < λ ≤ τ/(1 − τ) or τ ≥ 1/2, and (ii) UPh whenτ/(1 − τ) < λ < 1 and 0 < τ < 1/2. In case (i), given ﬁrm 1 choosing personalized pricing,ﬁrm 0’s proﬁt change from choosing uniform pricing to choosing personalized pricing isπPP

tH [2λ(1 − λ)τ
− πUPl=0+ (1 − 2λ)2 )τ + 2λ(1 − λ)],8(1 − λ + λτ)

2which is always positive for any τ ∈ (0, 1) and λ ∈ (0, 1). In case (ii), given ﬁrm 1 choosing personalized pricing, ﬁrm 0’s proﬁt change from choosing uniform pricing to choosing
personalized pricing is
π0PP − πUPh=0tH (λ + 2(1 − λ)τ)> 0.
8

UU is not an NE: when τ ≥ τuu , we have pure strategy NE. We want to conﬁrm

A7
that whether ﬁrm 1 would unilaterally deviate from choosing uniform pricing to choosing personalized pricing. From the results in UP (see Proof A.2), we have (i) UPl when0 < λ ≤ τ/(1 − τ) or τ ≥ 1/2, and (ii) UPh when τ/(1 − τ) < λ < 1 and 0 < τ < 1/2. Incase (i), given ﬁrm 0 choosing uniform pricing, ﬁrm 1’s proﬁt change from choosing uniform
pricing to choosing personalized pricing is
πUPl− πUU =
1

tH [4λ(1 − λ)τ2 + (1 − 8λ + 8λ2 )τ + 4λ(1 − λ)],16(1 − λ + λτ)

which is always positive for any τ ∈ (0, 1) and λ ∈ (0, 1). In case (ii), given ﬁrm 0 choosing
uniform pricing, ﬁrm 1’s proﬁt change from choosing uniform pricing to choosing personalized pricing is
πUPh− πUU =
1tH [(8 − 7λ − λ2 ) − (8 − 8λ − λ2 )τ]> 0,
16(1 − λ + λτ)

which is always positive for any τ ∈ (0, 1) and λ ∈ (0, 1).
When τ < τuu , we have mixed strategy NE. Due to mathematical complexity involved,
we can numerically show that each ﬁrm always has an incentive to deviate to choosing personalized pricing.
GU is not an NE: When τ ≥ τgu , we have pure strategy NE. We want to conﬁrm thatwhether ﬁrm 0 would unilaterally deviate from choosing group pricing to choosing personalized pricing. From the results in PU (see Proof A.2), we have (i) PUl when 0 < λ ≤ τ/(1−τ)
or τ ≥ 1/2, and (ii) PUh when τ/(1 − τ) < λ < 1 and 0 < τ < 1/2. In case (i), given ﬁrm
1 choosing uniform pricing, ﬁrm 0’s proﬁt change from choosing group pricing to choosing
personalized pricing is
π0PUl − πGU
0 =tH [2λ(1 − λ)τ2 + (1 − 2λ)2 τ + 2λ(1 − λ)],16(1 − λ + λτ)A8which is always positive for any τ ∈ (0, 1) and λ ∈ (0, 1). In case (ii), given ﬁrm 1 choosing
uniform pricing, ﬁrm 0’s proﬁt change from choosing group pricing to choosing personalized
pricing is
π0PUh − πGU
0 =tH [−2λ(1 − λ)τ2 − (8 − 12λ + 3λ2 )τ + (8 − 9λ + λ2 )],16(1 − λ + λτ)

which is always positive for any τ ∈ (0, 1) and λ ∈ (0, 1).
When τ < τgu , we have mixed strategy NE. Due to mathematical complexity involved,
we can numerically show that each ﬁrm always has an incentive to deviate to choosing personalized pricing.
PP is an NE: We show that given ﬁrm 1 choosing personalized pricing, ﬁrm 0 does notdeviate to choosing uniform pricing or to choosing group pricing. From the results in UP(see Proof A.2), we have (i) UPl when 0 < λ ≤ τ/(1 − τ) or τ ≥ 1/2, and (ii) UPh whenτ/(1 − τ) < λ < 1 and 0 < τ < 1/2. In case (i), ﬁrm 0’s proﬁt change from choosing
personalized pricing to choosing uniform pricing is
πUPl− πPP =0tH (−2λ(1 − λ)τ2 − (1 − 2λ)2 τ − 2λ(1 − λ)< 0.
8In case (ii), ﬁrm 0’s proﬁt change from choosing personalized pricing to choosing uniform
pricing is
πUPh− πPP = −0tH (λ + 2(1 − λ)τ
< 0.
8Moreover, ﬁrm 0’s proﬁt change from choosing personalized pricing to choosing group pricing is
πUPh− πPP = −0tH (λ + (1 − λ)τ
< 0.
8

A9

A.5 Proof for Subsection 4.3
Without a loss of generality, we consider the following two cases: (i) ﬁrm 0 chooses uniform
pricing, whereas ﬁrm 1 chooses uniform pricing for type H and personalized pricing for type
L (hUlP); (ii) ﬁrm 0 chooses uniform pricing, whereas ﬁrm 1 chooses uniform pricing for
type L and personalized pricing for type H (hPlU).
(i) hUlP: Suppose xH (p0 , p1 (H)) ∈ (0, 1) and xL (p0 , p1 (x, L)) ∈ (0, 1). That is, p1 (H) −tH ≤ p0 < tH τ. Firm 1’s personalized prices oﬀered to type L consumers are given by
p1 (x, L)) = p0 + (2x − 1)tH τ. Then, ﬁrms solve the following maximization system:
max p0 π0 = (xH λ + xL (1 − λ))p0 ,∫ 1

max p1 (H) π1 = (1 − xH )λp1 (H) + (1 − λ)(p0 + (2x − 1)tH τ)dx,
xL

s.t., p1 (H) − tH ≤ p0 < tH τ,from which we have the equilibrium prices:
phUlP =0tH (2 + λ)τ
tH (2 − 2λ + τ + 2λτ), phUlP (H) =,4 − 4λ + 3λτ 1
4 − 4λ + 3λτand the corresponding equilibrium proﬁts πhUlP, πhUlP. To derive the existence condition, it
0
1
suﬃces to ensure that ﬁrm 0 does not deviate given phUlP (H). First, ﬁrm 0 never deviates by1

letting xL ≥ 1, because its proﬁt increases in p0 for p0 ≤ phUlP (H)−tH , and p0 = phUlP (H)−tH
1

is dominated by phUlP .
0

Then, ﬁrm 0 can only deviate by letting xL ≤ 0 through oﬀering adeviation price pd such that
0

pd =
0

1

max πd0 = λxH (p0 , phUlP (H), p1 (x, L))p0 ,1

p0s.t. pd ∈ [tH τ, phUlP(H) + tH ).
1
0

A10
from which we have
pd =0

tH (6 − 6λ + τ + 5λτ) d tH λ(6 − 6λ + τ + 5λτ)2, π0 =.
2(4 − 4λ + 3λτ)8(4 − 4λ + 3λτ)2Such a deviation does not happen if πd0 < πhUlP, or pd does not fall into the interval [tH τ, phUlP (H)+001tH ), from which we have the condition
√
2
2(−4
−
λ
+
14λ
)
+
2
16 + 8λ + 24λ2 + 26λ3 + 7λ4
τ > τhUlP ≡.
3λ(5 + 7λ)Now, suppose xH (p0 , p1 (H), p1 (x, L)) ∈ (0, 1) and xL (p0 , p1 (H), p1 (x, L)) ≤ 0. That is,
tH τ ≤ p0 < p1 (H) + tH . Firm 1’s personalized prices oﬀered to type L consumers are givenby p1 (x, L)) = p0 + (2x − 1)tH τ. Then, ﬁrms solve the following maximization system:
max p0 π̄0 = xH λp0 ,∫ 1

max p1 (H) π̄1 = (1 − xH )λp1 (H) + (1 − λ)(p0 + (2x − 1)tH τ)dx,0s.t., tH τ ≤ p0 < p1 (H) + tH ,from which we have the equilibrium prices:p̄hUlP= p̄hUlP(H) = tH ,01
= tH (2−λ). To derive the existence
and the corresponding equilibrium proﬁts π̄hUlP= tH2λ , π̄hUlP
1
02
condition, it suﬃces to ensure that ﬁrm 0 does not deviate given p̄hUlP(H). First, ﬁrm 0 never1
deviates by letting xL ≥ 1, because its proﬁt increases in p0 for p0 ≤ p̄hUlP(H) − tH , and1. Then, ﬁrm 0 can only deviate by letting xL ∈ (0, 1)(H) − tH is dominated by p̄hUlP
p0 = p̄hUlP
0
1

A11
through oﬀering a deviation price p̄d0 such thatp̄d0 =

max π̄d0 = (λxH (p0 , phUlP(H), p1 (x, L)) + (1 − λ)xL (p0 , p1 (H), p1 (x, L)))p0 ,1
p0s.t. p0 ∈ ( p̄hUlP(H) − tH , tH τ).
1from which we havep̄d0 =tH (1 + λ)τ
tH (1 + λ)2 τ, π̄d0 =.
2(1 − λ + λτ)8(1 − λ + λτ)

Such a deviation does not happen if π̄d0 < π̄hUlP, or p̄d0 does not fall in the interval ( p̄hUlP
(H) −
0
1
tH , tH τ), from which we have the conditionτ < τhUlP ≡4λ.
1 + 3λTo summarize, we have








hUlP
π
=







πhUlP
i f τ > τhUlP ;0
πhUlPi f τ < τhUlP ,0

where τhUlP < τhUlP for any λ ∈ (0, 1).
Now, let us consider the case in which ﬁrm 0 deviates by choosing personalized pricing
for both groups. Then,xH =tH + p1 (H)1, xL = .
2tH
2Firm 1 solves the following maximization problem:∫ 1
p̂hUlP(H) = max π1 = λ(1 − xH )p1 (H) + (1 − λ)1p1 (H)H= (9λ+4τ−4λτ)tfrom which we have π̂hUlP.
0
16((2x − 1)tH τ)dx =
1/2

tH
,
2

A12
> πhUlP, for ﬁrm 0 uniform pricing is dominated by personalized pricing.
Since π̂hUlP00(ii) hPlU: Suppose xH (p0 , p1 (x, H)) ∈ (0, 1) and xL (p0 , p1 (L)) ∈ (0, 1). That is, p1 (L) − tH τ <p0 < min{p1 (L)+tH τ, tH }. Firm 1’s personalized prices oﬀered to type L consumers are givenby p1 (x, H)) = p0 + (2x − 1)tH . Then, ﬁrms solve the following maximization system:
max p0 π0 = (xH λ + xL (1 − λ))p0 ,∫ 1
max p1 (L) π1 = +λ(p0 + (2x − 1)tH )dx + (1 − xL )(1 − λ)p1 (L),xH

s.t., p1 (L) − tH τ < p0 < min{p1 (L) + tH τ, tH },from which we have the equilibrium prices:
phPlU
=
0

tH (3 − λ)τ
3tH τ − 2tH λτ + 2tH λτ2, phPlU(L)=,
3 − 3λ + 4λτ 1
3 − 3λ + 4λτand the corresponding equilibrium proﬁts πhPlU, πhPlU. With the equilibrium prices, phPlU
(L)+
0
1
1
tH τ < (>)tH if τ < (>)1/2.
First, let τ < 1/2 such that xH (p0 , p1 (x, H)) ∈ (0, 1) and xL (p0 , p1 (L)) ∈ (0, 1) for
phPlU(L) − tH τ < p0 < phPlU(L) + tH τ. To derive the existence condition, it suﬃces to ensure
1
1that ﬁrm 0 does not deviate given phPlU(L). Firm 0 never deviates by letting xL ≥ 1, because1
its proﬁt increases in p0 for p0 ≤ phPlU(L) − tH τ, and p0 = phPlU(L) − tH τ is dominated by
1
1
phPlU
. Then, ﬁrm 0 can only deviate by letting xL ≤ 0 through oﬀering a deviation price pd00such that
pd0 = max πd0 = λxH (p0 , phPlU(L), p1 (x, H))p0 , s.t. p0 ∈ [phPlU(L) + tH τ, tH ),11p0

from which we havepd0 =tH d tH λ
, π =
.
2 0
8

A13
, or pd0 does not fall into the interval [phPlU(L)+
Such a deviation does not happen if πd0 < πhPlU01
tH τ, tH ), from which we have the condition
τ > τhPlU ≡−9 + 6λ + 5λ2 +√81 − 108λ + 27λ2 + 6λ3 − 2λ4.
6λ(3 + λ)For τ < τhPlU we have a mixed-strategy NE in which ﬁrm 0 chooses a low price phPlU
0lwith a probability γ such that xL ∈ (0, 1) and a high price phPlU
such that xL ≤ 0. Then, ﬁrms
0hsolve the following maximization system:
max p0l π0l = p0l (λxH (p0l ) + (1 − λ)xL (p0l , p1 (L))),max p0h π0h = λp0h xH (p0h ),[ ∫ 1]
max p1 (L) π1 = γ λ(p0l + (2x − 1)tH )dx + (1 − γ)(1 − λ)(1 − xL (p0l , p1 (L)))p1 (L)[ ∫ 1+(1 − γ) λxH (p0l )](p0h + (2x − 1)tH )dx + (1 − γ)(1 − λ)p1 (L) ,xH (p0h )from which we havetH (2 + γ − 2λ + γλ)τ,γ(3 − 3λ + 4λτ)
tH
phPlU
= ,
0h
2
tH τ(4 − γ − 4λ + 2γλ + 4λτ − 2γλτ)
phPlU
(L) =
,
1
γ(3 − 3λ + 4λτ))tH (2 + γ − 2λ + γλ)2 τ(1 − λ + λτ)
π0l (phPlU)=,
0l
2γ2 (3 − 3λ + 4λτ)2tH λ
π0h (phPlU.
0h ) =
8
phPlU
=
0l

hPlU
hPlU
, we have
The probability γhPlU solves π0l (phPlU0l ) = π0h (p0h ). Substituting γ

hPlU
πhPlUm= γπ0l (phPlU00l ) + (1 − γ)π0h (p0h ) =

tH γ
.
8

A14
From 0 < γhPlU < 1, we have the existence condition for the mixed-strategy NE,τ < τhPlU .
Now, let τ < 1/2 such that xH (p0 , p1 (x, H)) ∈ (0, 1) and xL (p0 , phPlU(L)) ∈ (0, 1) for1phPlU(L) − tH τ < p0 < tH . To derive the existence condition, it suﬃces to ensure that ﬁrm 0
1
does not deviate given phPlU(L). Firm 0 never deviates by letting xL ≥ 1, because its proﬁt
1
increases in p0 for p0 ≤ phPlU(L) − tH τ, and p0 = phPlU(L) − tH τ is dominated by phPlU. Then,1
10
ﬁrm 0 can only deviate by letting xL ≤ 0 through oﬀering a deviation price pdd
0 such thatddhPlU(L) + tH τ),
pdd
0 = max π0 = λxH (p0 , p1 (x, H))p0 , s.t. p0 ∈ [tH , p1p0from which we havepdd
0 =tH dd tH λ
, π =
.
2 0
8

hPlU
Such a deviation does not happen if πdd, or pdd
0 < π00 does not fall into the interval[tH , phPlU(L) + tH τ), which is always true provided that τ > 1/2.
1Now, let us consider the case in which ﬁrm 0 deviates by choosing personalized pricing
for both groups. Then,tH τ + p1 (L)1
xH = , xL =
.
2
2tH τ
Firm 1 solves the following maximization problem:∫ 1
p̂hPlU(L) = max π1 = λ
1p1 (L)((2x − 1)tH )dx + (1 − λ)(1 − xL )p1 (L) =
1/2

tH τ
,
2

from which we have π̃hPlU= tH (4λ+9τ−9λτ).
016for τ < τhPlU , for ﬁrm 0 uniform> πhPlUmfor τ > τhPlU and π̃hPlU> πhPlU
Since π̃hPlU0000pricing is dominated by personalized pricing.