1

Deal or No Deal? : Advertizing and Sampling Effects on Two-sided Daily Deal Platforms

Shivendu Shivendu Zhe Zhang

sshivend@uci.edu zhez1@uci.edu

The Paul Merage School of Business, University of California Irvine, Irvine, CA 92697

Draft: 11/29/2012

Abstract

With the advent of Groupon.com in 2008, daily deal platforms have seen phenomenal growth. But surprisingly there is very sparse analytical research that has studied the economics of daily deal platforms which are two-sided in the sense that they connect merchants on the one side to consumers on the other side. We develop a stylized two-period game-theoretic model to analyze the strategic interaction between heterogeneous merchants and consumers. In our conceptualization, a monopolist daily deal platform is revenue maximizer who not only takes into consideration the cross-side network effect leading to chicken-and-egg problem, but also sampling and advertizing effect due to the presence of the platform. Keeping in view the real-world market conditions, we do not impose the restriction that the two-sides can interact only through the platform and allow the two-sides to transact outside the platform too. We model the strategic interaction between the daily deal website (platform) and merchants as a Stackelberg leader-follower game where merchants choose discount rate, keeping that in view the platform’s choice of the fixed fee rate or revenue sharing rate. Our result shows the merchants offer higher discount rate on the daily deal website than outside the website, and surprisingly, merchants’ optimal choice of discount rate is independent from the platform’s fixed fee rate. We also show that merchants that are new in the market place and therefore, less well-known gain more from offering a deal on the daily deal website. Some of merchants never offer a deal on the platform even if offering a deal on the platform is free. Keywords: Daily deal website, two-sided platform, cross-side network externality, experience goods, sampling effects, advertizing effects,

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1. Introduction

Daily deal business models came to limelight with the advent of Groupon in 2008

which became the fastest online business to reach one billion dollar valuation in history (Steiner,

2010). Though online daily deal industry is considered to be only in its fourth year (Dholakia,

2012) of evolution, it is estimated that consumers will spend $ 3.6 billion on daily deal websites

in 2012, an increase of nearly 87% over 2011 spend (Rueter, 2012) and around 60% of all online

shoppers subscribe to a daily deal website in the US (Freed & Berg, 2012). While daily deal

industry has seen phenomenal growth in the last four years, market analysts have raised

concern about its sustainability, growth and profitability (Clifford & Miller, 2012) and business

models pursued by these websites have been questioned (Cohan, 2012; Etter & McMillan, 2012).

Coupon industry has been around for more than a century in the US (Slater. 2001) and

with the advent of the Internet, digital coupons (coupons offered online) have outpaced printed

coupons by 10 to 1 in the year 2010 (Kruger, 2010). It is estimated that shopper’s saved more

than $3.7 billion dollars in 2010 through the use of coupons (NCH, 2011). Though the daily deal

websites are similar to coupons in that the daily deals also provide discounts, there are some

key differences. First, deals offered on daily deal websites like Groupon.com or LivingScoial.com

often offer discounts of more than 50% while traditional coupons are more heterogeneous with

relatively lower discount rates and many times coupons have a dollar value. Second, daily deals

offered by these websites have to first purchase by the consumer and then used to get the

service or product from the merchant and these deals are non-refundable. On the other hand a

coupon or a voucher, digital or printed, is redeemed only at the point of purchase. Third, daily

deals are offered on a platform and the consumer purchases the deal from the platform. The

merchant shares a significant proportion (often more than 50% of the value) of the deal value

with the platform. Fourth, while daily deal websites have characteristics of a two-sided

platform, traditional coupon distributors like PennySaverUSA or PennySaverUSA.com facilitate

delivery of coupons or vouchers to the prospective consumers.

Given these key differences, it is not clear as why merchants offer huge discounts on

daily deal websites. What framework the platform and merchants should use to determine the

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payoffs? Should merchants offer higher discounts on daily deal websites compared to coupons

through traditional channels? Can offering a deal on the daily deal website be profitable for a

merchant? How does the profitability of these deals vary with merchant characteristics? What

types of merchants join these websites? What are the tradeoffs that a merchant needs to

consider in offering a deal on these platform? What is the optimal discount rate that a merchant

should offer? Should daily deal platform subsidize consumers and merchants to join the

platform? Should daily deal platform offer a fixed fee contract or a revenue sharing contract? In

this paper, we develop a model, first to our knowledge that takes into account the two-

sidedness of the daily deal platform, with merchants who are heterogeneous in market size and

consumers who have heterogeneous taste for quality.

In our model, consumers can transact with merchants through the daily deal platform or

outside the platform, though outside the platform less number of consumers know about the

merchant. Two key aspects of our conceptualization are: First, unlike most of the theoretical

work in Economics and Marketing in the two-sided platform literature which assumes that the

two sides can transact with each other only through the platform (Rochet & Tirole, 2003), in our

setting merchants and consumers can transact outside the platform too. In this sense, merchants

actions on the platform impact his revenue outside the platform. Second, daily deal website has

large number of consumers as registered members and by offering a deal on the daily deal

platform the merchant is able to access larger market. We term this expansion in market size as

advertizing effect.

Merchants sell an experience good whose true quality is revealed to a consumer only

after consumption (Nelson, 1974; Milgrom & Roberts, 1986). We develop a two-period model

similar to Bils (1989) where merchants offer discounts on the daily deal platform as well as

outside the platform to first-time consumers and extract surplus in the second period. Our

modeling approach is also similar to models of introductory offers of Shapiro (1983) and

Bagwell (1990). Merchants are heterogeneous in market size that is the number of consumers

who know about the merchant, and without loss of generality, consumers form a priori

expectation about the quality of good based on merchant characteristic. This conceptualization

is similar to the treatment of advertizing by daily deal websites like Groupon in Edelman et al.,

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(2010) where some consumers are simply not aware about the merchant’s existence but is

different from Shapiro (1983) where some consumers over-estimate the quality of the good. In

our setting consumers consistently underestimate the quality of the good and realize the true

quality only upon consumption. This is justified in our context because a merchant whose

product has been overestimated by the consumers will never offer a deal and excluding all such

merchants does not impact our results. Since consumers underestimate the true quality of the

product, daily deal platform allows merchant to offer discounts at a different rate to the

consumers on the platform which captures the sampling effect of the platform.

We model the cross-side network effects on the lines of Yoo et al., (2003) and Bakos &

Katsamakas (2008) where consumers are heterogeneous in network externality. Similar to

Rochet and Tirole (2006), the daily deal platform drives benefit from membership on both sides,

that is merchants and consumers, but there are only pure membership externalities and gains

from interaction between merchants and consumers arises only when a transaction takes place.

Note that our daily deal platform is a monopolist and therefore, we do not allow for any multi-

homing.

Our analysis on merchants’ decision to offer a deal on the platform provides some

interesting insights: 1) merchants should always offer higher discount rate on the daily deal

website than outside the website, 2) the merchants’ choice of optimal discount rate is

independent of the daily deal platform’s fixed fee contract, 3) merchants with lower consumer

awareness and high uncertainty about the quality of the goods will gain more from offering a

deal on the platform, 4) some merchants will never offer a deal on the platform even if the

platform charges zero fixed fee. We also show that the daily deal website’s optimal fixed fee

depends on the characteristics of cross side network externality.

The rest of the paper is organized as follows. In §2, we present the model. In §3, we

examine the optimal discount rate offered by a merchant of a particular type and revenue to the

merchant in the base case where there is no daily deal website. In §4, we present the model of

daily deal website and analyze optimal discount rates offered by heterogeneous merchants and

optimal fixed fee announced by the platform. In §5, we discuss our results and identify suitable

theoretical and managerial implications.

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2. Model

We consider a monopolist daily deal website as a two-sided platform which connects to

merchants on one side and consumers on the other side. We assume a given mass, mN and cN

of potential participants for the merchants and consumers. The merchants and consumers who

join the daily deal website enjoy network externality from the participants in the opposite side,

which makes the daily deal website a two-sided platform. The network externality can mean

increased variant of the goods or the increasing probability of matching consumers to

merchants’ offering (Bakos & Katsamakas 2008). We consider merchants and consumers are

heterogeneous with regard to their valuation of the network externality benefit, with

corresponding types cg and mg for the consumer and merchant side, where [0,1]c Ug : and

[0,1]m Ug : . The platform set mf and cf as the participation prices that merchants and

consumers pay. They choose to join or not to join the daily deal website platform based on their

respective participation utility m m m c mU n fg a= - and c c c m cU n fg a= - , where cn and mn

are the proportion of consumers and merchants that join the platform. The parameters ma and

ca are parameters of positive real number and captures the strength of the cross side network

externality for merchants and consumers on the platform.

Merchants and consumers also enjoy benefit from transacting on the platform besides

the network externality. We consider the consumers are heterogeneous in taste for quality of the

good. The consumer taste parameter, θ , is their private information though it’s distribution

which is uniform on support 0 to 1, that is [0,1]Uθ , is common knowledge. Each consumer can

buy one good or nothing from merchants in each period. A consumer buys the good only if he

derives non-negative surplus from purchase of the good. 1 When a consumer of type θ

considers purchasing a good of expected quality q , his expected utility is ( ),U q qθ θ= .

The merchants sell one experience good which may be a product or a service. The true

quality of the good is q which is exogenously given, and without loss of generality merchants’

marginal cost of good is normalized to zero. Merchants are heterogeneous in terms of 1 Our merchant is “she”, consumer is “he” and the daily deal website or platform is “it” throughout the paper.

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consumers’ awareness about the merchant and the heterogeneity is captured by a parameter

[ ]0,1δ ∈ , which is common knowledge. It implies that consumers as well the platform knows

the value of the parameter δ and there is no information asymmetry. Further, though the

merchant type δ is public information, it is non-verifiable, and therefore, non-contractible. In

order to keep the model tractable, we assume that parameter δ captures two characteristics of

the merchant. For a merchant of type δ , the market size or proportion of potential consumers is

δ , and consumers’ a priori expectation of quality of merchant’s good is [ | ]E q qδ δ= . Note that a

merchant with higher δ has larger market size and also has closer expected quality of the good

to the true quality. Furthermore, we consider that consumers’ and merchants’ decisions to

participate the daily deal website platform are independent from consumers’ knowledge about

the merchants and merchants’ characteristics.

Merchants gain exposure to consumers who did not know the merchants by

participating the platform. Consumers who participates the platform can observe the

characteristic d of all merchants on the other side of the platform. The proportion of the

consumers who are on the platform is highlighted in grey circle in Figure 1. For those

consumers who had known the merchant of type d , their a priori expectation of the quality is

still ( | )E q qd d= . They are the existing consumers and highlighted in green circle in Figure 4.

For those consumers who were not aware of the merchant of type d , their a priori expectation of

the quality is ( | )nE q qd bd= , where (0,1]b Î . Those consumers are the new consumers that a

merchant does not access to in the absence of the platform. The proportion of these new

consumers is highlighted by the overlapping part between the green and grey circle. The

platform characteristic b is exogenous parameter which may represent the state of the

technology of the platform which facilitates the merchants to communicate the good’s quality to

consumers. Greater b implies that new consumers form a higher expectation about the quality,

though their expected quality is always less or equal to the expectation of the existing

consumers.

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Figure 1: Expansion of consumer base

Daily deal website serves as a two-side platform which connects merchants and

consumers. Since merchant type δ is non-verifiable, the platform2 cannot write merchant type

contingent contracts. Platform can generate revenue from two sources: (a) participation prices

by either side or both sides, and (b) a fixed fee for merchants to offer a deal on the platform. The

platform maximizes these two revenue sources independently.

The sequence of the two-period game is that the merchants and consumers decide to join

the platform before the first period. At the beginning of the first period, merchants release new

experience goods of quality q and decide the price and discount rate, while consumers form a

priori expectations about the quality and decide to purchase based on merchants’ characteristic

d . Consumers who bought the experience goods find out the true quality q , while consumers

who did not buy maintains the same quality expectation in the first period. Merchants only sell

the experience goods at a full price in the second period and consumers will decide to purchase

in the second period. Thus, the utility for a θ type consumer given the merchant’s awareness

characteristic d in the second period is:

, if has not bought it

( , | ), if bought it beforeq

U qqd

dìïï= íïïî

θθ

θ (1)

2 We use the term “daily deal website” and “platform” interchangeably.

Proportion δ

E(q|δ)= f(δ)q

Proportion δ Proportion nc

E(q|δ)= f(δ)q E(q|δ)=βf(δ)q

Proportion δ Proportion nc

E(q|δ)= f(δ)qE(q|δ)=βf(δ)q

Proportion δ

E(q)= q

Proportion Buy

E(q)=f(δ)q

Proportion Buy

E(q)= q

No platform:

Platform:

1st period 2nd period

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We will consider the base case where the daily deal website platform is absent to

examine the merchant’s optimal pricing and discount strategy in section 4. We turn to the case

where the daily deal website services as a platform of connecting merchants and consumers and

discuss the platform’s optimal access pricing strategy in the section 5. We will summarize the

key finds, discuss the implications of the results and offer managerial recommendations in the

section 6.

3. Base Case: no platform

To better understand the effect of selling the experience goods in a discounted price by

the merchants, we consider consumers and merchants are connected via traditional channels in

the absence of the platform in the base case. In the period zero, no consumer in the market

knows about the true quality of the experience good that is offered by merchants. Thus

consumers form expectation about the quality based on merchants’ awareness characteristic. In

the first period, the merchants offer a discount rate, based , to price, 1p , and then consumers

make purchase decision. The consumers who bought the goods find out the true quality of the

good, ( )E q q= , while others’ expectation of the quality is still based on merchant’s awareness

characteristics. In the second period, the merchants offer no discount on price, 2p , and then

consumers make purchase decision. The merchants commit to the same price in both periods,

1 2p p p= = . And merchants’ awareness characteristics do not change over these periods.

Figure 2: Sequence of game of the base case with no platform

Consumer uncertainty about quality of the goods and coupon’s sampling effect

Consumer form expected value of q

1. Merchants offer q at discounted price pbase(1-dbase)

2. Consumers decide to purchase q and then update expected value of q

Period 1

1. Merchants only offer qat full price pbase

2. Consumers decide to purchase q

t= 0 t= 1 t= 2Period 2

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In Shivendu & Chellappa (2005), the match between product and consumer tastes is not

only known a priori, thus the true value of an experience good is only known after consumption.

However, in their model, the uncertainty lies in consumer and product fit, over and under

estimation for the valuation both are possible whereas in our model, the uncertainty lies in the

quality of the experience good, such that expectation is always less to the true quality. This is

referred as the pessimistic case in Shapiro (1983), which also assumed that all consumers have the

same expectation of quality. It is optimal to have a two-step regime which makes an introductory

offer lower than the full price in the first period and benefit from additional and more informed

consumers in the second period (Shapiro 1983).

Merchants could have a lower price in the first period by offering a discounted price

(1 )base based p- in the first period and a full price basep in the second period. It implies that the

merchants are committed to a price in both period but achieve the same outcome as adopting

two-step regime.

Lemma 1: The d type merchant will adopt a two-step pricing regime by selling the experience good of

quality q at price *1 2

qp d= in the first period and at price *2 2

qp = in the second period, or committing

to a single price by selling the experience good at price *

2baseqp = and offers a discount rate

* 1based d= - in the first period in a market where consumers’ expectation about the quality is qd

before consumption and q after consumption. The optimal revenues of both case are the same:

* * (1 )4c

baseN

R qR d= = + .

All proofs are in Appendix A.

Lemma 1 is consistent with the Shapiro (1983)’s account of the two-step pricing regime

which consists of a lower introductory price in the first period followed by a higher price in a

market where consumers are pessimistic about the quality of experience goods. It is easy to see

that the consumers who bought the good in the first period also buy in the second period and

the demand is half of the market for both periods. The consumers’ uncertainty about the quality

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translates to a discount on their willingness-to-pay and a reduction of the price by the same

proportion.

Furthermore, merchant’s the optimal discount rate decreases and the optimal revenue

increases if the consumers’ uncertainty about the quality of the quality, 1 d- , decreases.

Merchant’s optimal price is independent of his characteristic d and optimal discount rate of the

first period is 1 d- in the case where there is no platform. It implies merchants should set the

price as the optimal price as if all consumers know about the true quality of the goods q . It

maximizes the revenue in the second period. More importantly, merchants use the discount rate

to offset the consumers’ uncertainty about the quality based on the merchant characteristic d in

the first period, so that the consumers that would have bought the goods in the case of no

uncertainty about the good’s quality will buy in both the first and second period.

4. In the Presence of Platform

In the presence of the platform, merchants can join the platform to gain access to a large

consumer base. Similarly, consumers can join the platform to face a larger assortment of

merchants. Most of the literature (Armstrong, Tirole, Yoo et al. Bakos, etc) either bundle the

membership benefit with the transaction benefit or only consider the membership benefit. In

this paper, we consider the case where merchants and consumers make decision of participating

in the platform independently from the transaction benefit for the following reasons.

Consumers and merchants can be active users of the platforms only if the consumers and the

merchants can derive positive surplus from transacting on the platform. It is for tractability.

Consumers do not gain any valuation for additional consumer consumption of the goods. The

externality only exists in membership but not in transactions. (more….)

Consumers who join the platform before the first period can observe all merchants on

the platform and thus potentially buy goods from them. The consumer base of the merchants

who participate in the platform is the total number of consumers who also participate in the

platform and consumers who are not on the platform but knew the merchants. The consumers

who are on the platform but did not know the merchants before are considered new consumers

acquired for the merchants while the consumers who are on the platform and knew the

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merchants before are existing consumers. These newly acquired consumers form lower

expectation about the quality based on the merchant’s awareness characteristic than the existing

consumers. Both the new and the existing consumers buy the good from the d type merchant

only if they derive non-negative surplus based on the expectation of the quality in the first

period. And these consumers who bought the goods will find out the true quality before the

second period. Consumers maintain the same expectation about the quality if they did not

purchase in the first period. Therefore, in the second period, consumers decide to purchase if

they derive non-negative surplus based on the expectation of the quality for those did not

purchase it or on the true quality for those bought it in the first period. The sequence of

decisions by the platform, merchants and consumers in the presence of platform is illustrated in

Figure 3. The consumers’ expectations about the quality and purchase decisions are illustrated

in Figure 4.

Figure 3: Sequence of game of the platform case

1. Platform announces pm and pc

2. Consumers and merchants decide to join the platform

3.Merchants release new goods of quality q

4. Consumer form expected value of q

1. Merchants offer q at the discounted price p(1-d) on the platform and at the price p(1-dnp) off the platform

2. Consumers decide to purchase and update expected value of q on and off the platform

1. Merchants only offer q at full price p off the platform

3. Consumers decide to purchase q

Period 1t= 0 t= 1 t= 2

Period 2

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Figure 4: Consumer's Decision Tree

4.1 Membership decision by merchants and consumers:

Merchants and consumers decide to participate in the daily deal website platform

independently from their likely gain from transactions on the platform. Consumers are

motivated to join the platform because there are also a large number of merchants on the

platform which presents a large assortment of goods. The benefit associated with an additional

merchant to a consumer can be understood as large variety of goods.

Consumers and merchants are heterogeneous in transportation cost of participating in

the platform in Yoo, et al. (2003). However, we see it is not a good match to the daily deal

website where transportation cost is significantly reduced by the wide use of internet and

availability of auxiliary services provided by the website. We consider the model in Bakos

(2008) where consumers and merchants are heterogeneous in valuation of cross-side network

externality. Utility of joining a platform for merchants and consumers are: m m m c mU n pg a= - ,

c c c m cU n pg a= - , where ma and ca measures the strength of cross-side network externality,

cn and mn are numbers of merchants and consumers that participate in the platform. Thus, the

U(q, θ)>=p

knows about δ type merchant

Does not know about δ type merchant, not on platform

U(E(q), θ)>p(1-d)

U(E(q),θ)<p(1-d)

U(q, θ)<p

time= 0 time= 1 time= 2

Does not knows about δ type merchant, but on the platform

U(En(q), θ)>p(1-d)

U(En(q) (q),θ)<p(1-d)

U(E(q), θ)>=p

U(E(q), θ)<p

U(q, θ)>=p

U(q, θ)<p

U(En(q), θ)>=p

U(En(q), θ)<p

Form expectation β(q|δ) if

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platform’s revenue of the membership is platform membership c c m mR n p n p- = + and we solve for sub-

game perfect Nash equilibrium:

( , ; , )

( , ; , 0), if 22 2

( ) ( ) ( )(2 ) ( )(2 )( , ; , ),

3 3 9 9

if 22

( , ; 0, ), if 2 2 2

c m c m

c m cm c m

c c m m c m m c m c m c c m m c

c m m c

mc m

m c m mc c

n n p pN N

N

N N N N

N NN

aa a

a a a a a a a a a a a aa a a a

aa a

a aa

ìïï >ïïïï + + + - + -ïïïïï= íïïï £ <ïïïïï <ïïïî

, (2)

When 2c ma a> or 2m

c

aa < , it implies that one side has a dominant cross-side network

externality over the other side. Thus, it reaches the limiting case when merchants’ valuation for

consumers is much more than consumers’ valuation for merchants and it leads to an

equilibrium that all merchants and half of the consumers will participate in the platform, and

the optimal membership fee is zero for consumers and is a function of total number of

consumers and strength of cross-side network externality for merchants. Readers can find the

complete proof for the equilibrium in (2) in Bako (2008).

4.2 Merchants’ benefit of offering a deal on the platform

Merchants made independent decision whether to offer a deal for merchants and

similarly consumers make purchase decision independently. It implies that consumers and

merchants who participate in the platform may not transact on the platform. We assume that

the platform only allows transactions if merchants offer a deal with a discount rate 0pd > .

Merchants who joined the platform may find it unprofitable to offer a deal with a discount rate

pd on the platform as they incur revenue loss from the existing consumers. Therefore,

merchants only offer a deal if it is more profitable than selling the good via the traditional

channel which is depicted in the base case. Similarly, consumers who are attracted onto the

platform may find the deal is not attractive and choose not to purchase on the platform.

A merchant can offer a deal with a discounted price on and outside the platform, thus

generate revenue from consumers both on and outside the platform. There are two sources of

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revenue from the consumers who are on the platform: the existing consumers who had already

known the merchant and the new consumers who were only acquired by the merchant on the

platform. The quality expectation is ( )E q qd= for the existing consumers, and ( )E q qbd= for

the new consumers where [0,1]b Î . The merchant still sells the good via the traditional

channel, therefore, that is another source of revenue from the consumers who are outside the

platform but knew the merchant. Consumers will update their expectation about the quality of

the good after they made purchases in the first period and buy the goods if they derive non-

negative surplus in the second period.

Thus, the revenue function of type d merchant who offers p as the price, pd as the

discount rate on the platform and npd as the discount rate outside the platform is:

_ 1 _ 1 _ 1 _ 2 _ 2 _ 2 deal e p n p np p e p n p np pR R R R R R R= + + + + +

(3)

, where _ 1

(1 ) ( )(1 ) (1 ) ,( )

( )

if 1 1

0, if 0 1

pc p c

p

p

p

e

p d f qN p d n df q

f qd

pR

p

ddd

d

ìïï - £ <ïïï= íïï £ < -

-

î

--

ïïï

,

_ 1

(1 ) ( )(1 ) (1 ) ( if 1 1

0, i

1 ),(

0

)( )f 1

pc p c p

p

n pp

p d f qN p d n dR q

f qdp

fb dd

b db d

ìïï - £ <ïïï= íïï £ <

-- -

î

-

-ïïï

,

_ 1(1 )

(1 ) (1 )(1 )( )

npc nnp p p c

p dN p d n

fR

qd

d-

- - -= ,

_ 2

if

if

(1 ) , 1 ( ) 1

(1 ) ( )(1 1

0, if

) , 1 ( )( )

( )10

c c p

pc ce p p

p

pN pn f dqp d f qN pn d fR

ff q

pq

pd

d d

dd dd

d

- -ìïïïïïïï

³ ³

-- ³ > -ïï= -íïïïïï -

î³ >ïïïï

,

15

_ 2

(1 ) (1 ), ( )

(1 ) ( )(1 ) (1 ), ( )( )

if 1 1

if 1 1

0, f 0 1 )i (

c c p

pcp cn p

p

pN pn f dqp d f qN pn d f

f qf qd

Rp

p

d b d

b dd b db

b d

ìïï - £ £ïïïïïïï= - £ < -íïïïïï £ <

- -

-- -

-ïïïïî

,

_ 2 (1 ) (1 )cn cp pp pq

R N n d- -= , cn is the number of the consumers on the platform and f is the

fixed fee for offering a deal on the platform.

For tractability, we assume that the merchants adopt consistent pricing strategy on and

outside the platform that *

2baseqp p= = and the existing consumers’ uncertainty about the

quality of the product is ( )f d d= .

From Lemma 1, it is easy to see that the merchant should set the discount rate d to

1 bd- if all consumers are new on the platform, and set the discount rate d to 1 d- if all

consumers are existing on the platform. Thus, intuitively, if the merchant maximizes the total

revenue from both the new and existing consumers, the optimal discount rate should be

between 1 d- and 1 bd- , and it depends on the relative numbers of the existing and new

consumers. The optimal discount rate, *pd , is closer to 1 d- if there are relatively more existing

consumers or closer to 1 bd- if there are relatively more new consumers. Moreover, the

optimal discount rate should not lead to negative revenue from the new consumers in the first

period.

Proposition 1: When offering a deal on the platform is free, a merchant of type dwill set the optimal

discount rate * [max{ (1 2 ),(1 )} ,(1 )]pd bd d bdÎ - - - on the platform.

As illustrated in Figure 5A, when 102

pq

b< < = , none of the new consumers purchase

the goods if the merchant set the discount rate is inadequate ( *pd is between 1 d- and

1 1 2qpbd bd- = - ). It implies that by offering an inadequate discount rate, 1 2pd bd< - , none

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of the new consumers will purchase from the merchants. Therefore, the feasible region for the

discount rate should be from max(1 2 ,1 )bd d- - to 1 bd- .

Figure 5: Revenues from existing and new consumers in the first period, A) 102

b< < , and B)

1 12

b< £ .

The Proposition 1 says that the optimal discount rate is bounded in a feasible region. The

merchant balances the gain in revenue in the second period and also loss in the first period with

subject to the relative mass of new and existing consumers on the platform. Any discount rate

outside that feasible region will result in sub-optimal revenue for the merchant.

Lemma 2: When offering a deal on the platform is free, the optimal discount rate on the platform offered

by a merchant of type d is *

1 , if 3 1 , if 2 1 (1 )1 , if 0

1

pd

ddb d

bd

d d

d d d

d d

ì -ïïïï -ï= íï - -ïïï -ïî

£ £

£ <

£ <

, where 1 8(1 ) 14(1 )

b bb

db

+ - --

= ,

and 1 12(1 ) 2 1,mi 1]8(1 )

n[ b bd bb b

+ - + --

= . The optimal discount rate outside the platform for all

merchants of type d is * 1npd d= - .

We show that the merchant’s discount strategy is independent from the fixed fee

charged by the platform. This implies that the merchants optimal discount rates on and outside

the platform is invariant from the platform’s fixed fee contract. Lemma 2 also says that for those

merchants whose d is relatively small, the optimal discount rate is * 1d d= - , while for those

merchants whose d is moderate, the optimal discount rate on the platform is

Revenue

d1 d- 1 bd-1 2bd-

Exist ing Consumers

New Consumers

0

Revenue

d1 d- 1 bd-1 2bd-

Exist ing Consumers

New Consumers

0

10 <2

b<1 12

b< £

1 1

17

* 3 12 1 (1 )

d db d

-=- -

, and for those merchants whose d is relatively large, the optimal discount

rate on the platform is * 1d d= - .

A relatively small d implies a little mass of existing consumers and thus, the revenue

gain from the existing consumers in the first period does not compensate the revenue loss from

the new consumers in the second period by decreasing discount rate from * 1d bd= - to

* 1d d= - . It only makes sense for the merchant to decrease the discount rate from

* 1d bd= - to * 1d d= - when there is a greater mass of existing consumers ( d is relatively

large), and the optimal discount rate is * 3 32 2( 1 )

d db d

-=+ - +

which decreases with d . The

merchant’s optimal discount rate outside the platform is * 1npd d= - which is identical in the

base case.

The optimal discount rates with respect to b are illustrated in Figure 6. The left graph of

the Figure 6 illustrates the case where * 1

2pq

b < = , and the optimal discount rate, *d , is

between *

1 1 2qpbd bd- = - and 1 bd- , and it always dominates * 1npd d= - . The d%

corresponds to the marginal merchant whose optimal discount rate is such that none of the new

consumers will purchase the good in either of the two periods. The right graph of the Figure 6

depicts the case where 12

b > , and the optimal discount rate, *d , is between 1 d- and 1 bd- ,

and it always dominates * 1npd d= - . The optimal *d will converge with the *npd at 1d =

where every consumers knew the merchant and have no uncertainty about the quality of the

good even before purchasing it.

18

Figure 6: Optimal discount rates on the platform and outside the platform

Proposition 2: No merchant should offer a discount rate lower than the discount rate offered outside the

platform.

The proposition 2 implies that the merchant should offer a higher discount rate to

induce the consumers, especially the new consumers, to buy the experience goods in the first

period, so that it leads to maximum revenue from both the new and existing consumers in the

second period. It is consistent the antidotal evidence that merchants who offer deals on daily

deal website may incur a loss in the short run but they expect consumers will return even

without coupons in the long run. It always holds as long as the merchant characteristic d is less

than 1.

After substituting the optimal discount rates on the platform and outside the platform

and the optimal price into the revenue function (3), we get the optimal revenue of the merchant

of offering a deal on the platform as *dealR . When a merchant does not offers any deal on the

platform, the revenue is identical to the optimal revenue in the base case where the merchant

maximizes only the revenue from the existing consumers via the traditional channel, thus,

* * no deal baseR R= . The optimal revenues of a merchant who offers a deal and no deal are

illustrated in Figure 7. Any merchant whose d is below the threshold d will set the discount

rate as * 1pd bd= - on the platform, as any further increase in discount rate will not lead to any

increase in the second period revenue but only decrease the first period revenue. The left graph

of the Figure 7 illustrates the case where 12

b < , where the merchant of the threshold d type

discount

d1d

*pd

0

*npd

1

d

discount

dd

*pd

0

*npd

1

= 1d

10 <2

b<1 12

b< £

19

will set the optimal discount rate as * 1 2pd bd= - and generate zero sale from the new

consumers at either period. Thus, the merchants whose d is greater than d will not offer a deal

on the platform.

The right graph of the Figure 7 illustrates the case where 12

b > and the optimal revenue

of offering a deal dominates the optimal revenue of having no deal. In either case, we can

clearly that the gap between the optimal revenue of offering a deal and no deal increases as the

merchant characteristic d decreases. It implies that the merchants with lower d gain more from

offering a deal on the platform if the platform charges a single fixed fee.

Figure 7: Optimal revenue of offering a deal and no deal

Proposition 3: When the new consumers’ uncertainty about the quality is relatively large ( 12

b < ),

merchants whose d is greater than d will never consider offering a deal even when platform charges zero

from transactions.

The Proposition 3 says about merchants’ motivation of offering a deal does not outweigh

the revenue loss from the existing consumers when the new consumers have a significantly

high uncertainty about the quality of the experience goods. The great revenue loss from the

existing consumer by offering a higher discount rate cannot be compensated by the gain from

the new consumers. Therefore, those merchants will offer no deal on the platform even when it

is free to offer deal on the platform.

5. Platform Fee strategies of charging merchants

Revenue

d1d

deal

0 d

no deal

Revenue

d= 1dd

deal

0

no deal

10 <2

b< 1 12

b< £

20

The merchants can gain from offering a discount on the full price to induce both the new

and existing consumers to purchase in the first period and find out the true quality of the

experience goods. It leads to maximum revenue for the merchants in the second period where

consumers who experienced the goods in the first period make purchase decision based on the

true quality of the goods and the full price. In that sense, the daily deal website is not only a

two-sided platform which connects the two sides of participants, but also facilitates sampling

and advertising effect of offering deals by merchants.

5.1 A fixed fee contract

The platform could have implemented a customized contract which specifies the

amount of fixed fee F as the gain of offering a deal on the platform by the merchant3, if the

merchant characteristic d is contractible. We can easily see from Figure 8 that the gain of

offering a deal compared to offering no deal increases as merchant characteristic d decreases.

Therefore, the fixed fee F for a merchant of type d is:

* * deal no dealF R R= - (4)

It is the first best strategy to charge different fixed fee for merchants according to their

characteristic. However, the merchant characteristic d is not contractible and the platform has

to decide a fixed fee contract F for all merchants to maximize the revenue. The platform’s

revenue function from the fixed fee for offering a deal is:

_platform F m mR n q F= , (5)

where mn is the total number of merchants who participated on the platform, and mq is the

proportion of merchants who offer a deal.

In order to get the proportion of merchants who offer a deal, we consider the marginal

merchant with ( )Fd% who are indifferent from offering a deal or no deal on the platform given a

fixed fee F . In order words, the marginal merchant is the ( )Fd% type who derive zero benefit

from offering a deal after paying the fixed fee f to the platform, * * ( ) ( )deal no dealR F Rd d- =% % .

3 We do not assume a particular bargaining strategy in this merchant-platform relationship for the following reason: the merchants’ goal is to get access to new consumers and induce consumers to try the goods, instead of maximizing the revenue from the transactions on the platform.

21

Lemma 3: When the platform charges a fixed fee, F , from any merchant who offers a deal on the

platform, then the optimal fixed fee is

* ( )(7 (8 7 ) (1 ) (4 (5 4 )))108 (1 (1 ) )

c c m

c

N qF

a a b b b b ba b b

+ - - + - - -=

- - .

Since we showed that merchants with lower d gain more by offering a deal on the

platform, any merchant whose d is less than the marginal ( )Fd% will offer a deal on the

platform. And furthermore [0,1]d Î and then is from Thus, the proportion of merchants who

will offer a deal is ( )mq Fd= % since [0,1]Ud : . The Lemma 3 says that lower proportion of

merchants will offer a deal on the platform as the fixed fee increases. The threshold 1F

corresponds to the case where all merchants who will offer a deal set * 3 12 1 (1 )pd d

b d-=

- - on the

platform if the fixed fee is 1[0, )F FÎ . The threshold 2F is the optimal revenue that the merchant

with 0d = can get from the new consumers in the second period. The threshold 2F

corresponds to the case where the platform can charge the maximum fixed fee and even the

merchant with 0d = is indifferent from offering a deal on the platform. All merchants who

will offer a deal set * 1pd bd= - on the platform if the fixed fee is 1 2[ , ]F F FÎ .

The proportion of merchants who offer a deal on the platform is illustrated in Figure 8.

The left graph depicts the case where 12

b > and the merchant with very high d will always

gain by offering a deal to maximize revenue from both the new and existing consumers. Thus,

all merchants join the platform, 1mq = at 0F = and the proportion of merchants who offer a

deal decreases as fixed fee increases. The middle graph depicts the case where b is moderate,

and some merchants with high d will not offer a deal on the platform even when the fixed fee is

zero. It is because for those merchants it is never optimal to maximize revenue from both the

new and existing consumers by offering a deal on the platform which we explained in

Proposition 3. The right graph depicts the case where b is very small, that all the merchants

who would like to offer a deal will always set the discount rate as * 1pd bd= - .

22

Figure 8: Proportion of merchants who will offer a deal on platform with respect to

platform’s fixed fee

The platform maximizes the fixed fee revenue from transactions in (5). The platform

needs to balance the effect of fixed fee on the number of merchants who will offer a deal and

also on the amount of fee per merchant.

Proposition 5: When the platform charges a fixed fee from the merchants who offer a deal then (i) only

the merchants of type *( )Fd d<)

, where *( ) [0, )Fd dÎ)

will offer a deal *_ 1p Fd bd= - , (ii) no merchant

of type [ ,1]d dÎ will offer a deal.

“The marginal merchant corresponding to this optimal F will offer 1-bd as discount rate,

which implies, all merchants who will offer a deal on the platform will offer discount rate as 1-

bd. It clearly screen out the high type merchants who may have good number of transactions

and good price, but gain little from offering a deal on the platform”

Lemma 4 illustrates the platform’s balancing the number of the merchants who are

willing to offer a deal on the platform and the fixed fee per merchant depends on the number of

consumers on the platform, the strength of the cross-side network externality, and the new

consumers’ uncertainty about the quality of the experience goods on the platform.

mq

2f1f0

1mq

02f1f0

1mq

f2f

1

f f

23

Figure 9: Comparison of *F where 300Nc = , 600Nm = and 1q =

5.2 A revenue sharing contract

The platform can also adapt a strategy of sharing a percentage of revenue per

transaction. Groupon.com normally has a revenue sharing agreement with merchants such that

Groupon.com collects sale of deals and only passes the 50% to the merchants. In other words,

Groupon will take 50% of the revenue.

The platform will take a percentage, s , per transaction from merchants in the first

period. The platform’s revenue from a merchant is:

_ _ 1 _ 1( )platform s e p n pR s R R= + ,

where _ 1e pR and _ 1 n pR are merchant’ revenues of existing and new consumers in the first

period.

Thus, the merchant’s optimization problem in (3) becomes:

_ _ 1 _ 1 _ 1 _ 2 _ 2 _ 2 _ 1 _ 1( )deal s e p n p np p e p n p np p e p n pR R R R R R s R Rp = + + + + + - + (6)

Lemma 4: When the platform charges a revenue sharing percentage, 0 1s£ < , the optimal discount

rate offered by a merchant of type d on the platform is

*

_

1 , if (3 2 )(1 ) , if

2(1 )(1 (1 ) )1 , if 0

1

s

s sp s

s

sds

d d

d d d

d d

ddb d

bd

ì -ïïïï - -ï= íï - -

£ £

£ <

£ <-ïïï -ïî

, where 1 8(1 )(1 ) 1

4 (1 )(1 )ss

sb b

bd

b+ - - -

- -= , and

1 4(1 ) ((3 ) 3) 1 2 (1 )max[ ,1](1 )(1 )s

s s ss

b b bdb b

- - + - - + -=- -

. The optimal discount rate offered by any

0.2 0.4 0.6 0.8 1.0

30

32

34

36

38

0.2 0.4 0.6 0.8 1.0

30

32

34

36

38

40*F *F

b b

0.8ca = 1ma =0.9ma =

1ma =

1.1ma =

0.7ca =

0.8ca =

0.9ca =

24

merchant outside the platform is *_ 1np sd d= - . When 1s = , the optimal discount rate offered by a

merchant on the platform is any value less than *_ 1p sd bd£ - .

Here 11/24

Figure 10: Comparison of optimal discount rate with different b

The merchant’s profit is impacted by the platform’s revenue sharing strategy, s .

However, we notice that the merchant’s optimal discount rate outside the platform is

independent of the platform’s revenue sharing strategy, s , such as * *_ 1np s npd d d= = - . It

explains the merchant’s revenues of offering no deal under the fixed fee and revenue sharing

regimes are the same, e.g. * * _

*_

1 (1 )4no deal no deal s deal sR qR p d d= += = , for min( ,1) 1sd d£ £ .

The merchant’s optimal discount rate will increases as the revenue sharing percentage, s ,

increases.

Figure 11: Comparison of maximum profit of deal or no deal with different .1s = and .8s =

In Figure 11, Proposition 3 holds even when the platform adopts a revenue sharing

strategy. The marginal merchant, sd%, has the same profits of offering deal or no deal, thus will

discount

d1d

*_p sd

0

*_np sd

1

d

discount

dd

*_p sd

0

*_np sd

1

1

p

d1 sd

deal

0

12

sd

no deal

p

d1 sd

deal

0

12

sd

no deal

sd% sd%

25

not offer a deal on the platform. And sd% is impacted by the platform’s revenue sharing

percentage, s .

Lemma 6: When 1[0, )s sÎ , the marginal merchant who is indifferent from offering a deal or no deal is

1( )s sd)

, which is the solution to the equation:

2 2(1 (2 4(1 ) (1 ( ( 3 2 ) 3))))0

16(1 ) ((1 ) 1)cqn s s

sd d b d d d b d d

bd b d- - - - + + - - + + + -

=- - -

, and

1 ( ) [ ,min( ,1))s s ssd d dÎ)

; when 1( ,1]s sÎ , the marginal merchant who is indifferent from offering a deal or

not is 22

1 (1 ) 5 (1( )

) (6 (5 ) )s s s ss

b b bd

- - + - - - -=

) , and where 2 [0, )s sd dÎ

). The threshold is

the solution to the equation:

2 2

(1 (2 4(1 ) (1 ( ( 3 2 ) 3))))0

16(1 ) ((1 ) 1)c s s s s s s s

s s

qn s ss

d d b d d d b d dbd b d

- - - - + + - - + + + -=

- - -.

Furthermore, the marginal merchant when 1s = is 2( 1) 21 5s sd = =

+

).

Figure 12: Comparison of proportion of merchants who offer a deal on the platform with

different b

We can see that 22 0.618 0

1( 1

5)s sd =

+= » >

). It implies that even when the

platform extracts all of the 1st period revenue from merchants, some merchants whose

2( 1)s sd d< =)

will offer a deal on the platform.

mq

11s0

1

s

21 5+

mq

11s0

1

s

21 5+

mq

10

1

s

21 5+

26

And since the platform cannot offer customized revenue sharing contract based on

merchants’ characteristic, platform will have a single revenue sharing percentage for all

merchants. It implies, that only the merchants whose sd d< % will offer a deal as they can get

higher profit by offering a deal than no deal when the platform adopts a revenue sharing

strategy.

By announcing a revenue sharing strategy, s , the platform will attract merchants whose

( )s sd d< % to offer a deal. And thus, the platform’s revenue function from the fixed fee for

offering a deal is:

( )

_ _ _0( ) ( ) ( )s s

platform S m s p m sR s n s R dd

d d d= ò%

% , (7)

where mn is the total number of merchants who participated on the platform, and _ _p m sR is

the platform’s revenue from transactions by d type merchant.

Lemma 7: The optimal revenue sharing percentage that the platform will charge a merchant who offers a

deal *s is the solution to the equation:

2 2 (3 (3 ) )(25 (25 9 ) )7 3 ( (10 (10 3 ) ) 7(2 ) 42 (1 ) 15 ) s s sB B s s sB

b b bb b b b b - - - -+ + - + - - - - - =

where 5 (1 ) (6 (5 ) )B s sb b= - - - - .

In Lemma 7, the platform’s optimal revenue sharing percentage, *s , is bounded by (0, 1). It

implies that there is a lower limit b beyond which that b b< , the optimal * 1s = . And it is

easy to see that the optimal *s is not impacted by ma , ca , mN and cN .

6. Discussion

In this paper, we developed a stylized game model to capture the daily deal website’s

strategic interaction between merchants and consumers as a two-side platform. Different from

other two-sided platform study, we examine not only the daily deal website’s access pricing

strategy on membership, but also the pricing strategy on transactions by merchants and

consumers. We focus on the daily deal website’ role of facilitating sampling and advertising

27

effects in a two-period game setting where the merchants can offer a deal to induce both the

new and existing consumers to purchase the experience goods in the first period so that it leads

to maximum revenue for both types of consumers who find out the true quality of the goods as

the merchants charge the full price on the second period.

We showed that the merchants should always offer a steeper discount on the daily deal

website than on other traditional channel regardless the consumers’ awareness about the

merchant and uncertainty about the quality of the goods. It lends support to the existing

practice of offering significant discount on daily deal websites, like Groupon and Livingsocial.

We also showed that the merchants’ discount decision is independent from the daily deal

website’s fixed fee contract.

We discussed the merchants’ decision to join the daily deal website and offer a deal on

the website. The decision to offer a deal on the platform depends on the new consumers’

uncertainty about the quality of the goods. Merchants who are well-known and their existing

consumers have little doubt about the quality of their goods will not offer a deal on the daily

deal website if the newly acquired consumers have high uncertainty about the quality of the

goods even when the platform charges zero for offering a deal. Furthermore, the merchants

with lower consumer awareness will gain more from offering a deal on the daily deal website

given a fixed fee contract by the website.

We also examined the platform’s optimal fixed fee contract through the lens of two-

sided platform. We showed that the optimal fixed fee depends on the number of consumers on

the platform, the strengths of the cross-side network externality and the new consumers’

uncertainty about the quality of the experience goods as the daily deal website has to balance

the number of merchants who are willing to offer a deal on the website and the fixed fee paid

per merchant.

Our results offer some insights on daily deal website industry which is still considered a

new industry. One of managerial recommendation is that the daily deal website should reduce

the new consumers’ uncertainty about the quality of the experience goods. It may be carried by

providing consumer reviews about the merchants to the new consumers who did not the

merchants. Another managerial recommendation is that the daily deal website should have

28

customized contract with merchants as we showed that the merchants’ incentive to offer a deal

depends on consumer awareness about the merchants and uncertainty about the quality of the

goods. The market is more efficient if the daily deal website can contract on those merchants’

characteristics.

Our stylized two-period game has some limitations. We do not consider the competition

effect on daily deal websites’ strategy. Competition between rival daily deal website will

potentially exert upward pressure on discount rate. We also assume away the same side

negative network externality effect, particularly among the merchants.

We can study the daily deal website’s optimal contract structure based on merchants’

characteristics as future extension of the paper. Furthermore, we can study the daily deal

website’s the impact of membership fee structure on his optimal fixed fee contract on

transactions by the merchants.

Reference

Bagwell, K. (1990). Informational product differentiation as a barrier to entry. International Journal of Industrial Organization 8 (2), 207-223. Bils, M. (1989). Pricing in a customer market. Quarterly Journal of Economics 104 (4), 699-718. Bakos, Y and Katsamakas, E. (2008). Design and Ownership of Two-Sided Networks: Implications for Internet Platforms. Journal of Management Information Systems 25 (2), 171 – 202. Clifford, S and Miller, C. (2012 ). Merchants and Shoppers Sour on Daily Deal Sites. The New York Times. Cohan, P. (2012). Why Groupon Is Over and Facebook And Twitter Should Follow. Forbes. August 20, 2012. Dholakia, UM. (2012). How Businesses Fare with Daily Deals as They Gain Experience: A Multi-Time Period Study of Daily Deal Performance, Available at SSRN: http://ssrn.com/abstract=2091655 Edelman, B., S Jaffe and S Kominers. (2011). To Groupon or Not to Groupon: The Profitability of Deep Discounts. Harvard Business School NOM Unit Working Paper No. 11-063. Available at SSRN: http://ssrn.com/abstract=1727508

29

Etter, L and MacMillan, D. (2012). The Education of Groupon CEO Andrew Mason. Businessweek, July 12, 2012, http://www.businessweek.com/articles/2012-07-12/the-education-of-groupon-ceo-andrew-mason Freed, L. (2012). Daily deal websites and emails bring in new and existing customers for retailers. Foresee daily deal commentary. Milgrom, P. and Roberts, J. (1986). Price and advertising signals of product quality. Journal of Political Economy 94 (4), 796-821. Nelson, P. (1974). Advertising as information. Journal of Political Economy 82 (4), 729-754. Rochet, JC and Tirole, J. (2006). Two-sided markets: a progress report. The RAND Journal of Economics 37(3), 645–667. Steiner, C. (2010). Meet the Fastest Growing Company Ever. Forbes Magazine . Shapiro, C. (1983). Optimal pricing of experience goods. Bell Journal of Economics 14 (2), 497-507. Yoo, B, Choudhary, V and Mukhopadhyay, T. (2002). A Model of Neutral B2B Intermediaries. Journal of Management Information Systems 19 (3), 43 – 68

30

Appendix A: Proofs of Lemmas and Propositions

Lemma 1: The d type merchant will adopt a two-step pricing regime by selling the experience

good of quality q at price *1 2

qp d= in the first period and at price *2 2

qp = in the second

period, or committing to a single price by selling the experience good at price *

2baseqp = and

offers a discount rate * 1based d= - in the first period in a market where consumers’ expectation

about the quality is qd before consumption and q after consumption. The optimal revenues of

both case are the same: * * (1 )4c

baseN

R qR d= = + .

Proof of Lemma 1:

In the case of merchant’s adopting two-step pricing scheme:

In the first period, consumers form expectation about the quality based on the merchant

awareness characteristic d and buy the good only if their utility is greater or equal to the price 1p

. Thus, the proportion of consumers buy in the first period is:

11 1Pr( ( , | ) ) 1

pD U q p

qd

d= ³ = -θ .

In the second period, for consumers who bought the good and knew the true quality, all

of them buy if the price 2p is lower than 1p , or some of them buy if the price 2p is greater or

equal to 1p . For consumers who did not buy the good and still maintain the same expectation

about the quality as in the first period, none of them buy if the price 2p is greater or equal to 1p ,

or some buy if 2p is lower than 1p . Thus, the proportion of consumers buy in the second period

is:

2

2 2 1

22

1 2 1 2 1

Pr( ( , ) ) 0 1 ,

Pr( ( , | ) ( , | ) ) 1 ,

pU q p if p p

qDp

D U q p U q p if p pq

d dd

ìïï ³ + = - ³ïïï= íïï + ³ Ç < = - <ïïïî

θ

θ θ

The merchant’s revenue function is:

31

1

1 2 2 2 1

1 1 2 21 2

1 2 2 1

((1 ) (1 ) ), ( )

((1 ) (1 ) ),

c

c

c

pN p p p if p p

qR N D p D pp p

N p p if p pq q

d

d d

ìïï - + - ³ïïï= + = íïï - + - <ïïïî

After solving the optimization problem for the revenue with respect to 1p and 2p , we get

two sets of solutions,:

* *1 2 2 1

* *1 2 2 1

, , 2 2

, , 2 2

q qp p if p p

q qp p if p p

d

d d

ìïï = = ³ïïïíïï = = <ïïïî

And the corresponding revenue is:

2 1*

2 1

,

, 2

(1 )4c

c

Nif p p

RN q

p p

q

ifd

dìïï ³ïïï= íïï <ïïïî

+

Since (0,1)d Î , the d type merchant should always have 2 1p p³ , where *1 2

qp d= and *2 2

qp = , and

the revenue * (1 )4c q

NR d+= . It implies that the optimal first period price is less than the second

period price.

In the case of merchant’s offering discounted prices in the first period and full price in the

second period:

Consumers form expectation about the quality based on the merchant awareness

characteristic d and buy the good only if their utility is greater or equal to the price (1 )base based p-

:

1(1 )

Pr( ( , | ) (1 ) ) 1 base basebase base

d pD U q d p

qd

d-

= ³ - = -θ

Since (1 )base base based p p- < , none of the consumers who did not buy the good will buy in

the second period. For the consumers who bought the good in the first period, only those have

higher utility based on the true quality of the good than the price will buy in the second period:

2 Pr( ( , ) ) 0 1 basebase

pD U q p

q= ³ + = -θ

32

The merchant’s revenue function is:

(1 )((1 )(1 ) (1 ) )base base base

base c base base based p p

R N d p pq qd

-= - - + - . After solving for optimization

problem for revenue, ,

maxbase base

basep dR , we get: *

2baseqp = , * 1based d= - and * (1 )

4c

base qN

R d= + .

We show Lemma 1.∎

Proposition 1: When offering a deal on the platform is free, a merchant of type dwill set the

optimal discount rate * [max{ (1 2 ),(1 )} ,(1 )]pd bd d bdÎ - - - on the platform.

Proof of Proposition 1:

From merchant’s revenue function (3), the 1st and 2nd period revenue from the new

consumers will be zero when ( )1 1 2pf qp

d b d bd< - = - , given 2qp = and ( )f d d= , therefore,

the merchant’s optimal discount rate * 1 2pd bd³ - . Furthermore, from the merchant’s revenue function (3), the 2nd period revenue from the existing consumer will be less if 0 1 ( )pd f d³ > - ,

as (1 )

(1 ) (1 )( )

pc c

p dp pn pnq f q

d dd-

- > - . Thus, * 1pd d³ - , given ( )f d d= . Therefore,

*

11 2 , 211 , 2

p

whend

when

bd b

d b

ìïï - £ïï³ íïï - >ïïî

.

Considering the joint revenue from the new consumers in two period _ 1n pR and _ 2n pR , the merchant cannot generate higher revenue by offering a discount higher than 1 bd- . Therefore, * 1pd bd£ - . We prove Proposition 1.∎ Lemma 2: When offering a deal on the platform is free, the optimal discount rate on the

platform offered by a merchant of type d is *

1 , if 3 1 , if 2 1 (1 )1 , if 0

1

pd

ddb d

bd

d d

d d d

d d

ì -ïïïï -ï= íï - -ïïï -ïî

£ £

£ <

£ <

, where

1 8(1 ) 14(1 )

b bb

db

+ - --

= , and 1 12(1 ) 2 1,mi 1]8(1 )

n[ b bd bb b

+ - + --

= . The optimal discount rate

outside the platform for all merchants of type d is * 1npd d= - .

Proof of Lemma 2:

33

From proposition 1, we show that * { max[(1 2 ),(1 )],1 }pd bd d bdÎ - - - . It has three cases.

Case 1 is that * (max(1 2 ,1 ),1 )pd bd d bdÎ - - - , then the merchant’s revenue from the new and

existing consumers are _ 1(1 )

(1 ) (1 )pc p ce p

p dN p d nR

qd

d-

-= - , _ 2 (1 )p c cep nR N pq

d-= ,

_ 1(1 )

(1 ) (1 ) (1 )pc pp cn

p dN p

qR d n d

bd-

- - -= , and _ 2(1 )

(1 ) (1 )pn p c c

p dN n

qR p d

bd-

- -= , thus

the total revenue for the merchant who offers a deal on the platform is:

_ 1 _ 1 _ 1 _ 2 _ 2 _ 2

(1 ) (1 )

( (1 ) (1 ) (1 ) (1 ) (1 )

(1 ) + (1 ) (1 )(1 ) (1 )

(1 ) + (1 ) (1 ) (

1

= p pc p c p

deal e p n p np p e p n p np

c

npnp c

p

c

pc

R Rp d p d

N p d n p d nq q

p d pp d n pnq q

p dpn

q

R R R R R

d dd bd

d dd

dbd

- -- - + - - -

-- - - + -

-- -

= + + +

+

+ +

) (1 ) )cp p nq

d- -

To solve the _,max

p npm pd d

R , we take the two FOCs with respect to pd and npd , and we have:

* 3 12 1 (1 )pd d

b d-=

- - and * 1npd d= - . When 1

2b ³ , then 1 1 2d bd- ³ - , and,

* (1 ,1 )pd d bdÎ - - , therefore, the values of d corresponding to these two limits are 1d = and

1 8(1 ) 14(1 )

d bb

d bb

+ - --

= = . When 12

b < , then 1 1 2d bd- < - , and * (1 2 ,1 )pd bd bdÎ - - ,

therefore, the values of d corresponding to these two limits are

1 12(1 ) 2 1,1]8(1 )

min[ b b bb b

d d= - + --

= + and 1 8(1 ) 14(1 )

d bb

d bb

+ - --

= = , thus min( ,1)d d d< < .

Case 2a: when * (0, max(1 2 ,1 ))pd bd dÎ - - and 12

b < , then 1 2 1bd d- > - , and the

merchant’s revenue from the new and existing consumers are

34

_ 1(1 )

(1 ) (1 )pc p ce p

p dN p d nR

qd

d-

-= - , , _ 2 (1 )p c cep nR N pq

d-= and

_ 1 _ 2(1 )

(1 ) (1 ) (1 )( )

pn p n p c p c

pR

f qR

dN p d n d

b d-

- - -= = , for (1 ,1 2 )pd d bdÎ - - ; and

_ 1(1 )

(1 ) (1 )pc p ce p

p dN p d nR

qd

d-

-= - , _ 1 0n pR = ,

_ 2(1 )

(1 )cep

cpRp d

N pnq

dd-

-= , and

_ 2 0n pR = for (1 2 ,1 )pd d dÎ - - ; _ 1 _ 2 _ 1 _ 2 0e p e p n p n pR R R R= = = = for (0,1 2 )pd dÎ - and

thus the total revenue for the merchant is:

(if [1 ,1 2

(1 ) (1 )(1 ) (1 ) +(1 ) (1 )(1 )

(1 ) (1 ) (1 ) ),

(1 ) (1 )(1 ) (1 ) +(1 ) (1 )(1 )

(1 )(1 ) (1 ) (1 ) ),

]

(if

c

p

c

d

p npp c np c

c c

p npp c np c

pc c

eal

p d p dp d n p d n

q qp ppn p nq q

p d p dp d n p d n

q

Nd

NR q

p d ppn p nq q

d bd d

d d

d d

d dd

d dd

d

d

- -- - - - -

+ - + - -

- -- - - - -

-+ - + - -

Î - -

= [1 2 ,1 )

0, if [0,1 2 )

p

p

d

d

d d

d

ìïïïïïïïïïïïïïïïïïï Î - -íïïïïïïïï Î -ïïïïïïïïïïî

To solve the _,max

p npm pd d

R , we take the two FOCs with respect to pd and npd , and we have: when

(1 ,1 2 )pd d bdÎ - - , then * 1pd d= - and * 1npd d= - . And when (1 2 ,1 )pd d dÎ - - , then

* 32pd d= - and * 1npd d= - . However, 31 1

2d d- < - < , then * 3

2pd d= - is not a solution

and * 1pd d= - when (1 2 ,1 )pd d dÎ - - .

Case 2b: When * (0, max(1 2 ,1 ))pd bd dÎ - - and 12

b ³ , then 1 2 1bd d- £ - , then the

merchant’s revenue from the new and existing consumers are:

35

_ 1(1 )

(1 ) (1 )pc p ce p

p dN p d nR

qd

d-

-= - , _ 1(1 )

(1 ) (1 ) (1 )pc pp cn

p dN p

qR d n d

bd-

- - -= ,

_ 2(1 )

(1 )cep

cpRp d

N pnq

dd-

-= and _ 2(1 )

(1 ) (1 )pn p c c

p dN n

qR p d

bd-

- -= for

(1 2 ,1 )pd bd dÎ - - ; for _ 1(1 )

(1 ) (1 )pc p ce p

p dN p d nR

qd

d-

-= - , _ 2(1 )

(1 )cep

cpRp d

N pnq

dd-

-=

and _ 1 _ 2 0n p n pR R= = for (1 2 ,1 2 )pd d bdÎ - - ; _ 1 _ 2 _ 1 _ 2 0e p e p n p n pR R R R= = = = for

(0,1 2 )pd dÎ - and thus the total revenue for the merchant is:

(1 ) (1 )(1 ) (1 ) (1 ) (1 ) (1 )

(1 ) (1 )+(1 ) (1 )(1 ) (1 )

(1 )(1 ) (1 ) (1 ) (1 ) )

(

if [1 2 ,1 ]

(

,

(1 ) (1(1 ) (1 ) +(1

p pp c p c

np p

c

p

c

d

np c c

pc c

pc

eal

p

p d p dp d n p d n

q qp d p d

p d n pnq q

p

N

d

d ppn p nq q

p d pp

RnN d

q

b

d dd bd

d dd d

d dbd

d

d

d

d

- -- - + - - -

- -- - - + -

-+ - - + - -

- -- - -

Î - -

=

)) (1 )(1 )

(1 )(1 ) (1

if [1 2 ,1 2 )

0, i

) (1

f [0,1 2 )

) ),

npnp c

p

p

pc c

dp d n

qp d ppn p n

d

qd

q

d bd

d

d dd

d

bd

ìïïïïïïïïïïïïïïïïïïïïïïïïï Î - -íïïïïïïïï Î -ïïïïïïïïï

- -

-+ - + - -

ïïïïïïïïîTo solve _,

maxp np

m pd dR , we take the two FOCs with respect to pd and npd , and we have:

* 3 ( 3 )2 2( 1 )pd b d

b d+ - +=+ - +

and * 1npd d= - , for [1 2 ,1 ]pd bd dÎ - - . But, 3 ( 3 )12 2( 1 )

b ddb d

+ - +- <+ - +

,

therefore, * 1pd d= - and * 1npd d= - for [1 2 ,1 ]pd bd dÎ - - . And * 32pd d= - and

36

* 1npd d= - , for [1 2 ,1 2 ]pd d bdÎ - - . However, 312

d d- < - , therefore, * 1pd d= - and

* 1npd d= - for [1 2 ,1 2 ]pd d bdÎ - - .

Therefore, * 1pd d= - and * 1npd d= - when * (0, max(1 2 ,1 ))pd bd dÎ - - . And the value of d

corresponding to the limit is 1 12(1[ ) 2 1,1]8(1 )

b bd bb b

d + - + --

= = , thus 1d d£ < .

Case 3: When * (1 ,1)pd bdÎ - , then the merchant’s revenue from the new and existing

consumers are: _ 1(1 )

(1 ) (1 )pc p ce p

p dN p d nR

qd

d-

-= - ,

_ 1(1 )

(1 ) (1 ) (1 )pc pp cn

p dN p

qR d n d

bd-

- - -= , _ 2 (1 ) (1 )cn p cpN pnRq

d- -= and

_ 2 (1 )p c cep nR N pq

d-= , thus the total revenue for the merchant is:

(1 ) (1 )(1 ) (1 ) (1 ) (1 ) (1 )

(1 ) + (1 ) (1 )(1 ) (1 )

(1 ) (1 ) (1 ) (1 )

(

)

p pp c p c

npn

d

p c c

c

e l

c

a cp d p d

p d n p d nq q

p d pp d n pnq q

p ppn p nq

R

q

N d dd bd

d dd

d d

- -- - + - - -

-- - - + -

+ - - + - -

=

where (1 ,1)pd bdÎ - .

To solve the _,

maxp np

m pd dR , we take the two FOCs with respect to pd and npd , and we have:

* 11 ( 1 )pd d

b d-=

+ - + and * 1npd d= - . However, 1 1

1 ( 1 )d bd

b d- < -

+ - +, then * 1pd bd= - and the

value of d corresponding to the limit is 1 8(1 ) 14(1 )

d bb

d bb

+ - --

= = , thus 0 d d£ < .

To combine all three cases, we have:

37

*

1 , if 3 1 , if 2 1 (1 )1 , if 0

1

pd

ddb d

bd

d d

d d d

d d

ì -ïïïï -ï= íï - -ïïï -ïî

£ £

£ <

£ <

,

where 1 8(1 ) 14(1 )

b bb

db

+ - --

= , and 1 12(1 ) 2 1,1]8(1

[)

b b bb b

d + - + --

= . And * 1npd d= - .

We prove Lemma 2.∎ Proposition 2: No merchant should offer a discount rate lower than the discount rate offered

outside the platform.

Proof of Proposition 2:

From Lemma 2, we show:

*

1 , if 3 1 , if 2 1 (1 )1 , if 0

1

pd

ddb d

bd

d d

d d d

d d

ì -ïïïï -ï= íï - -ïïï -ïî

£ £

£ <

£ <

where 1 8(1 ) 14(1 )

b bb

db

+ - --

= , and 1 12(1 ) 2 1,1]8(1

[)

b b bb b

d + - + --

= . And * 1npd d= - .

Since 0 1b< £ , then 1 1bd d- £ - . Furthermore, when min( ,1)d d d£ < , then

3 11 12 1 (1 )

dbd db d

-- £ < -- -

. Therefore, * *p npd d£ for 0 1d£ £ .

We prove Proposition 2.∎ Proposition 3: When the new consumers’ uncertainty about the quality is relatively large (

12

b < ), merchants whose d is greater than or equal to d will never consider offering a deal

even when platform charges zero from transactions.

Proof of Proposition 3

38

From Lemma 2, we show that when 102

b< < , then the merchants whose [ ,1]d dÎ will offer a

discount rate on the platform same as outside the platform, * * 1p npd d d= = - . It implies

* *_ 1* * |

p npde p dal m dR R d= = -= and * * no deal baseR R= . And since * *

*1

*_ |

p npm p ad b sedR Rd= = - = , then * * deal no dealR R= .

Therefore, the merchants have no incentive to offer a deal on the platform even when the platform charges zero from transactions. We prove Proposition 3.∎ Proposition 4: When the platform charges a fixed fee for a merchant to offer a deal, the merchants’ optimal discount rates on the outside the platform remains the same as given in Lemma 2. Proof of Proposition 4: The merchant’s of type d pays a fixed fee, F , to the platform for offering a deal, thus, the merchant’s profit function is:

_ _ 1 _ 1 _ 1 _ 2 _ 2 _ 2m p deal e p n p np p e p n p np pR F R R R R R R F= - = + + + + + -π .

Since the fee is fixed, then the marginal analysis on optimal *_p Fd and *

_np Fd is independent of F . Thus, the merchant’s optimal discount rates when the platform charges a fixed fee are identical as the optimal discount rate offered in the benchmark case, * *

_p F pd d= and * *_np F npd d= .

We prove Proposition 4.∎ Lemma 3: When the platform charges a fixed fee, F , from any merchant who offers a deal on the platform, then the optimal fixed fee is

* ( )(7 (8 7 ) (1 ) (4 (5 4 )))108 (1 (1 ) )

c c m

c

N qF

a a b b b b ba b b

+ - - + - - -=

- - .

Proof of Lemma 3:

We can see that the merchants with smaller d benefit more than merchants with a larger d from

Figure 7. From Lemma 1, we show: * (1 )4base

cNqR d+= , thus *

(1 )4no deal

cqR

Nd= +

And From Lemma 2, we show:

*

1 , if 3 1 , if 2 1 (1 )1 , if 0

1

pd

ddb d

bd

d d

d d d

d d

ì -ïïïï -ï= íï - -ïïï -ïî

£ £

£ <

£ <

Therefore, when 0 d d£ < , * 2 2 21 , 1 ( (1 ) (| 1 ) )

4p npd d cdea c cl c n n nqR Ndb d d b d d b b d= - = - + - - + - - += ;

when d d d£ < , then

39

*3 1 , 121 (1 )

(1 )(1 ( 1 4 (1 (2 (3 )))))(1 ) )4 16 (1 (1

|) )

(p np

deal cc

d dR

qnqNd db d

d d b d d b dd dbd b d-= = -

- -

- + - + - + - ++- -

= + ; when

1d d£ £ , then *1 , 1|

(1 )4p np

cd deal d

N qR d d

d d= - = -

+= .

When 1d d£ £ , then * * deal no dealR R= , so no merchant will offer a deal for any 0F ³ , as we

showed in Proposition 3.

When 1[0, )F FÎ where merchants whose [ , )d d dÎ have higher revenue from offering a deal

than no deal after subtracting the fixed fee, then the marginal merchant, 1( )Fd% , who is indifferent

of offering a deal and no deal is:

3 1 , 121 (1

* * *

) |

p npdeal deal no deald d

R F Rd db d

-= = -- -

= - =π .

It follows (1 )(1 ( 1 4 (1 (2 (3 ))))) (1 )(1 ) )4 16 (1 (1 ) )

(4

c ccN

qn N qq Fd d b d d b d d dd d

bd b d- + - + - + - + ++ -

-+ =

-

% % % % % % %% %% % , and

(1 )(1 ( 1 4 (1 (2 (3 )))))16 (1 (1 ) )

c cN qnF

d d b d d b dbd b d

- + - + - + - +=

- -

% % % % %% % . Therefore, the 1( )Fd% is the solution to

the equation.

When 1 2[ , )F F FÎ where merchants whose [0, )d dÎ have higher revenue from offering a deal

than no deal after subtracting the fixed fee, then the marginal merchant, ( )Fd% , who is indifferent

of offering a deal and no deal is:

* 1 , 1

* *|p npdeal deal no ded d alR F Rbd d= - = -= - =π .

It follows 2 2 2 (1 )( (1 ) (1 ) )

4 4c

c c c cN q

N n n nq Fd d

d b d d b b d+

+ - - + - - + - =% %

% % % . Therefore, the 2( )Fd%

is the solution to the equation. And we have:

2( 16 (1 (1 ) ) (5 (6 5 ))) (1 )

2 (1 1(

()

) )c c c c c c

c cF

N n F N n N nN n

b b b bd

bb b

- - - + - - - -=

- -%

40

Since the both *dealπ and *

no dealR are continuous for 0 min( ,1)d d£ £ . The solutions of 1F to

*1 , 1 ,

* *1 |

p npdeal deal no deald dR F Rbd d d d= - = - == - =π and

* * *3 1 , 1 ,21 (1 )

1 | |p np

deal deal no dead ldR F Rd d dd d d

b d- == = - =

- -

= - =π should be the same. Thus, we have:

2 2 21

(1 )( (1 ) (1 ) )

4 4c

c c ccN q N q

n n n Fd d

d b d d b b d+

+ - - + - - + - = . And solve for the equation,

we have: 21 2

((1 ) (2 (5 6 ) (1 (1 ) (3 2 )) 1) 1)32(1 )

c cN n AF

q b b b b b b b

b b

- - + - - - - --

= , where

2 2

1 8(1 )(1 )

A b bb b

+ --

= .

We also have: 1 , 1 ,* *

0 0*

2 | |p npd ddeal deal no dealR F Rbd d d d= - = - = == - =π . And we have: 2 4

c cNF

n q= .

When the fixed fee 1[0, )F FÎ , we have that the marginal merchant 1( )Fd% is the solution to the

equation (1 )(1 ( 1 4 (1 (2 (3 )))))16 (1 (1 ) )

c cqN nF

d d b d d b dbd b d

- + - + - + - +=

- -

% % % % %% % . After solving the equation, we

substitute ( )3

c mc

cn

a aa+

= , ( )3

m c mm

m

Nn

a aa

=+

from merchants’ and consumers’ decisions to join

the platform (2), and 1( )mq Fd= % into platform’s revenue function (5), and we have

1( )platform mR n F Fd= % . However, the solution to the maximization problem is outside the support

*1[0, )F FÎ .

When the fixed fee 1 2[ , )F F FÎ , from Lemma 3, we have the marginal merchant

2( 16 (1 (1 ) ) (5 (6 5 ))) (1 )

2 (1 (1 ) )( ) c c c c c

c c

N n F nF

N nN nb b b

bd

b bb

- - - + - - - -=

- -% . By substituting

( )3

c mc

cn

a aa+

= and ( )3

m c mm

m

Nn

a aa

=+ (from (2)), and 2( )mq Fd= % into platform’s revenue

41

function (5) and we have:

2( )( ( )(5 (6 5 )) 48 (1 (1 ) ))

6 (1 (1 ) )( )(1 )

6 (1 (

( )

1 ) )

m c c m c c m c

c m

m c c m

c

platform

m

mFN N q N q F

N qFN N q

R n

N q

F Fa a a a b b a b b

a b ba a b

a b b

d+ + - - - -

= =-

- -+ -

-- -

%

By solving the FOC with respect to F , the optimal fixed fee that the platform should charge per

merchant is: * ( )(7 (8 7 ) (1 ) (4 (5 4 )))108 (1 (1 ) )

c c m

c

N qF

a a b b b b ba b b

+ - - + - - -=

- -, which is within the

1 2[ , )F F FÎ .

Thus, we show Lemma 3. ∎

Proposition 5: When the platform charges a fixed fee from the merchants who offer a deal then

(i) only the merchants of type *( )Fd d<)

, where *( ) [0, )Fd dÎ)

will offer a deal *_ 1p Fd bd= - , (ii)

no merchant of type [ ,1]d dÎ will offer a deal.

Proof of Proposition 5:

From the proof of Lemma 3, we show that the platform’s optimal fixed *F is within 1 2[ , )F F FÎ

and the corresponding *2( )Fd

) is within *

2( ) [0, )Fd dÎ)

. Therefore, some merchants of type d less

than *2( )Fd

) will offer a deal, and no merchant of type *

2( )Fd d d³ >)

will offer a deal.

Thus, we show Proposition 5. ∎

Lemma 4: When the platform charges a revenue sharing percentage, 0 1s£ < , the optimal

discount rate offered by a merchant of type d on the platform is

*

_

1 , if (3 2 )(1 ) , if

2(1 )(1 (1 ) )1 , if 0

1

s

s sp s

s

sds

d d

d d d

d d

ddb d

bd

ì -ïïïï - -ï= íï - -

£ £

£ <

£ <-ïïï -ïî

, where 1 8(1 )(1 ) 1

4 (1 )(1 )ss

sb b

bd

b+ - - -

- -= , and

1 4(1 ) ((3 ) 3) 1 2 (1 )max[ ,1](1 )(1 )s

s s ss

b b bdb b

- - + - - + -=- -

. The optimal discount rate offered by

42

any merchant outside the platform is *_ 1np sd d= - . When 1s = , the optimal discount rate

offered by a merchant on the platform is any value less than *_ 1p sd bd£ - .

Proof of Lemma 4: Similar to the proof of Lemma 2: that the optimal discount rate offered on the platform is within the boundary *

_ [max(1 2 ,1 ),1 ]p sd bd d bdÎ - - - , when 0 1s£ < .

Case 1 is that *_ (max(1 2 ,1 ),1 )p sd bd d bdÎ - - - , the merchant’s profit function is:

_ _ 1 _ 1 _ 1 _ 2 _ 2 _ 2 _ 1 _ 1( )

(1 ) (1 )

((1 ) (1 ) (1 ) (1 ) (1 )

(1 ) + (1 ) (1 )(1 ) (1 )

(1

=

+ (1

p pc p c p

deal s e p n p np p e p n p np p e p n p

c

npnp c c

p

p d p dN p d n p d n

q qp d pp d n pn

q qp d

R R R R R R s R R

d dd

d

p

bd

dd

- -- - + - - -

-- - - + -

= + -

-

+ + +

-

+ +

)) (1 ) (1 ) (1 )

(1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ))

c c

p pp c p c

ppn p nq q

p d p ds p d n s p d n

q q

d dbd

d dd bd

- + - -

- -- - - - - - -

To solve the _ _

_,max

p s np sdeal sd d

p , we take the two FOCs with respect to _p sd and _np sd , and we have:

*_

(3 2 )(1 )2(1 )(1 (1 ) )p s

sds

db d

- -=- - -

and *_ 1np sd d= - . When 1

2b ³ , then 1 1 2d bd- ³ - , and since

*_ (1 ,1 )p sd d bdÎ - - , therefore, the values of d corresponding to these two limits are 1d = and

1 8(1 )(1 ) 1

4 (1 )(1 )ss

sb b

b bd d + - -

-= -

-= . When 1

2b < , then 1 1 2d bd- < - , and since

*_ (1 2 ,1 )p sd bd bdÎ - - , therefore, the values of d corresponding to these two limits are

1 4(1 ) ( 3 (3 ) ) 1 2 (1 )min[ ,1](1 )(1 )s

s s ss

d d b b bb b

- - - + + - + -- -

= = and

1 8(1 )(1 ) 1

4 (1 )(1 )ss

sb b

b bd d + - -

-= -

-= , thus min( ,1)s sd d d< < .

Case 2a: when *_ (0, max(1 2 ,1 ))p sd bd dÎ - - and 1

2b < , then 1 2 1bd d- > - : for

_ (1 ,1 2 )p sd d bdÎ - - , then * 1pd d= - and * 1npd d= - ; And when _ (1 2 ,1 )p sd d dÎ - - , then

43

*_

112 2p sd

sd= + -

- and *

_ 1np sd d= - . However, since [0,1]s Î , then 11 12 2s

d d- < + --

, then *_

112 2p sd

sd= + -

- is not a solution. It means *

_ 1p sd d= - when _ (1 2 ,1 )p sd d dÎ - -

.

Case 2b: When *_ (0, max(1 2 ,1 ))p sd bd dÎ - - and 1

2b ³ , then 1 2 1bd d- £ - :

*_

3 2 (1 ) (3 )2(1 )(1 (1 ) )p s

sds

d b db d

- - - -=- - -

and *_ 1np sd d= - , for _ [1 2 ,1 ]p sd bd dÎ - - . But,

3 2 (1 ) (3 )12(1 )(1 (1 ) )

ss

d b ddb d

- - - -- <- - -

, for , [0,1]s s" Î and , [0,1]b b" Î therefore, *_ 1p sd d= - and

*_ 1np sd d= - for _ [1 2 ,1 ]p sd bd dÎ - - . And *

_11

2 2p sds

d= + --

and *_ 1np sd d= - , for

_ [1 2 ,1 2 ]p sd d bdÎ - - . However, 11 12 2s

d d- < + --

,for , [0,1]s s" Î and , [0,1]b b" Î ,

therefore, *_ 1p sd d= - and *

_ 1np sd d= - for _ [1 2 ,1 2 ]p sd d bdÎ - - .

Case 3: When *_ (1 ,1)p sd bdÎ - , and we have: *

_1

1 (1 )p sd db d

-=- -

and *_ 1np sd d= - .

However, 1 11 (1 )

d bdb d

- < -- -

, then *_ 1p sd bd= - for *

_ (1 ,1)p sd bdÎ - .

To combine all three cases, we have:

*

_

1 , if (3 2 )(1 ) , if

2(1 )(1 (1 ) )1 , if 0

1

s

s sp s

s

sds

d d

d d d

d d

ddb d

bd

ì -ïïïï - -ï= íï - -

£ £

£ <

£ <-ïïï -ïî

,

44

where 1 8(1 )(1 ) 1

4 (1 )(1 )ss

sb b

bd

b+ - - -

- -= , 1 4(1 ) ( 3 (3 ) ) 1 2 (1 )[ ,1]

(1 )(1 )ss s s

sb b b

b bd - - - + + - + -=

- -.

And *_ 1np sd d= - .

Thus, we show Lemma 5. ∎

Proposition 6: When the platform charge a revenue sharing percentage, s , both the boundary values, sd and sd increase as s increases. And the optimal discounted rate offered by the merchants increases as s increases for merchants of type [ , ]s sd d dÎ . Proof of Proposition 6:

Check that 0sddsd

> and 0sddsd

> for [0,1)s Î and [0,1]b Î .

And for [ , ]s sd d dÎ , the optimal discount rate on the platform is *_

(3 2 )(1 )2(1 )(1 (1 ) )p s

ss

d db d

- -- - -

= and

check that _2

* 1 02(1 ) (1 (1 ) )

p sdd

s sdd

b d- >

- - -= .

Thus, we show Proposition 6. ∎

Note: Proposition 6 also implies that all merchants whose optimal discount rate is 1 bd- in the benchmark and fixed fee cost cases, also will offer 1 bd- in this revenue sharing case; however,

some merchants whose optimal discount rate is 3 12 1 (1 )

db d

-- -

in the benchmark and fixed fee cost

cases, will offer 1 bd- or (3 2 )(1 )2(1 )(1 (1 ) )

ss

db d

- -- - -

in the revenue sharing case; some merchants

whose optimal discount rate is 1 d- in the benchmark and fixed fee cost cases, will offer 1 bd- , (3 2 )(1 )

2(1 )(1 (1 ) )s

sdb d

- -- - -

or continue offer 1 d- in the revenue sharing case.

That means that merchants who are between 1sd and 2sd will offer a higher discount rate, merchants who is higher than 2sd will still offer *

_ 1p sd d= - , merchants who is lower than 1sd will still offer *

_ 1p sd bd= - , where 1 2s s< . Proposition 7: When the platform charge a revenue sharing percentage, the optimal sharing percentage is the solution to the equation:

2 2 (3 (3 ) )(25 (25 9 ) )7 3 ( (10 (10 3 ) ) 7(2 ) 42 (1 ) 15 ) s s sB B s s sB

b b bb b b b b - - - -+ + - + - - - - - =

where 5 (1 ) (6 (5 ) )B s sb b= - - - - . Proof of Proposition 7: When 1[0, )s sÎ , the marginal merchant who is indifferent from offering a deal or no deal is

1( )s sd)

, which is the solution to the equation:

45

2 2(1 (2 4(1 ) (1 ( ( 3 2 ) 3))))0

16(1 ) ((1 ) 1)cqn s s

sd d b d d d b d d

bd b d- - - - + + - - + + + -

=- - -

, and

1 ( ) [ ,min( ,1))s s ssd d dÎ)

; when 1( ,1]s sÎ , the marginal merchant who is indifferent from offering a

deal or not is 22

1 (1 ) 5 (1( )

) (6 (5 ) )s s s ss

b b bd

- - + - - - -=

) , and where 2 [0, )s sd dÎ

) . The

threshold is the solution to the equation: 2 2

(1 (2 4(1 ) (1 ( ( 3 2 ) 3))))0

16(1 ) ((1 ) 1)c s s s s s s s

s s

qn s ss

d d b d d d b d dbd b d

- - - - + + - - + + + -=

- - -.

Furthermore, the marginal merchant when 1s = is 2( 1) 21 5s sd = =

+

).

Proof of Lemma 6:

From Lemma 5, we Substituting *_p sd to merchant’s profit function (6), and have:

*2 2_

2 2

1 (1 ), if 41 (1 )4 if (1 (2 4(1 ) (1 ( ( 3 2 ) 3))))

,16(1 ) ((1 ) 1)

1 ( (1 ) (1 (1 )(1 ) ) ), if

min( ,1) 1

min( ,1)

04

s

deal sc

c c

s s

sc

q

q

qn s ss

q n n s n s

d d

d dp d d b d d d b d d

bd b d

d b b

d

d d

d

d d

b

d

b d dd

ìïï +ïïïïïï +ïïï= í - - - - + + - - + + + --

- - -

+ - - + + - - -

£ £

£ <

- £ <î

ïïïïïïïïïïïSimilar to Lemma 3, that we can see that the merchants with smaller d benefit more than merchants with a larger d from Figure 11. When 1[0, )s sÎ , where merchants whose [ , min( ,1))d d dÎ have higher revenue from offering a deal than no deal after subtracting the fixed fee, then the marginal merchant, 1( )s sd% , who is indifferent of offering a deal and no deal is:

* *_ d neal ls o deaRp = .

It follows 2 2(1 (2 4(1 ) (1 ( ( 3 2 ) 3))))

016(1 ) ((1 ) 1)

cqn s ss

d d b d d d b d dbd b d

- - - - + + - - + + + -=

- - -.

Therefore, the 1 ( )s sd% is the solution to the equation.

When 1[ ,1]s sÎ where merchants whose [0, )d dÎ have higher revenue from offering a deal than

no deal after subtracting the fixed fee, then the marginal merchant, 2( )s sd)

, who is indifferent of

offering a deal and no deal is:

46

* *_ d neal ls o deaRp = .

It follows 2 21 1( (1 ) (1 (1 )(1 ) ) ) (1 )4 4c c cq n n s n sd b b d d b b d d d+ - - + + - - - = +- . By

solving the equation, we have: 22

1 (1 ) 5 (1( )

) (6 (5 ) )s s s ss

b b bd

- - + - - - -=

).

To get the threshold 1s , it is the platform sharing percentage where the marginal merchant is also the lower threshold merchant, sd d= . Therefore, we have:

* * _ | |

s sdeal s no dealRd d d dp = == .

It follows 2 2

(1 (2 4(1 ) (1 ( ( 3 2 ) 3))))0

16(1 ) ((1 ) 1)c s s s s s s s

s s

qn s ss

d d b d d d b d dbd b d

- - - - + + - - + + + -=

- - -,

and therefore, the 1s s= is the solution to this equation. To marginal merchant who is indifferent from offering a deal or no deal on the platform given

1s = : 2( 1) 21 5s sd = =

+

).

We prove Lemma 6. ∎

Lemma 7: The optimal revenue sharing percentage that the platform will charge a merchant who offers a deal *s is the solution to the equation:

2 2 (3 (3 ) )(25 (25 9 ) )7 3 ( (10 (10 3 ) ) 7(2 ) 42 (1 ) 15 ) s s sB B s s sB

b b bb b b b b - - - -+ + - + - - - - - =

where 5 (1 ) (6 (5 ) )B s sb b= - - - - . Proof of Lemma 7: When 1[0, )s sÎ , from Lemma 6, the marginal merchant 1s sd d=

) ), which is the solution to the

equation: 2 2(1 (2 4(1 ) (1 ( ( 3 2 ) 3))))

016(1 ) ((1 ) 1)

cqn s ss

d d b d d d b d dbd b d

- - - - + + - - + + + -=

- - -, and

1 [ ,min( ,1))s s sd d dÎ)

. It implies that the merchants whose [0, ]sd dÎ will offer *_ 1p sd bd= - and

*_ 1np sd d= - ; and merchants whose 1( , ]s sd d dÎ

) will offer *

_(3 2 )(1 )

2(1 )(1 (1 ) )p ss

sd d

b d- -

- - -= and

*_ 1np sd d= - . Therefore, the platform’s revenue from transactions from merchants who offer a

deal given 1[0, )s sÎ is: 1

**_ _

_ _ _ _ _ _ (3 2 )(1 )12(1 )(1 (1 ) )

0 0( ) ( ) | ( ) |s s s

p s s p sp m s p m s p m sd d s

s

R d R d R dd d

dbdb

d

dd

d d d d d d- ---

=-

=-

= +ò ò ò)%

(1).

After substituting it into the platform’s revenue function (5) and we try to solve the

optimization problem that: _max ( )platform SsR s . However, since _ ( )

0platform SdR s

ds> , no solution

is found.

47

When 1[ ,1]s sÎ , from Lemma 6, the marginal merchant

22

1( )

(1 ) 5 (1 ) (6 (5 ) )s s ss

s sb b bd d

- - + - - - -= =

) ), and 2 [0, )s sd dÎ

). It implies that the

merchants whose 2[0, ]sd dÎ)

will offer *_ 1p sd bd= - and *

_ 1np sd d= - . Therefore, the

platform’s revenue from transactions from merchants who offer a deal given 1[ ,1]s sÎ is: 2

*_ 1_ _ _ _0 0

( ) ( ) |s s

p sp m s p m s dR d R db

d

d

dd d d d= -=ò ò

)%.

After substituting it into the platform’s revenue function (5) and we try to solve the optimization problem that: _max ( )platform Ss

R s , we have the optimal s as the solution to:

2 2 (3 (3 ) )(25 (25 9 ) )7 3 ( (10 (10 3 ) ) 7(2 ) 42 (1 ) 15 ) s s sB B s s sB

b b bb b b b b - - - -+ + - + - - - - - =

where 5 (1 ) (6 (5 ) )B s sb b= - - - - . And we have 2 3 3 3 4

2 2

2

2 3

156 708 4(246 ) 3(236 ) 3(52 )3 3 4( ) 3 2 2( )3 4( ) ( 1 )

12s

dg g gg c d e g c d e gc d e g

b b b

bd

b

b

- + - - + + - +- + - + + + - + + + +- + + - +=

)

where 2 3 4

2 216 61 126 61 16

3( 1 )d b b b b

b b- + - +

- += ,

2 3 4

2 216 61 126 61 16

9 (1 2 )e b b b b

b b b- + - +

- += ,

23 3( 1 )

g b bb b

- + -=- +

,

𝑏 = (21 3⁄ (1745𝛽4 − 13876𝛽5 + 46151𝛽6 − 89854𝛽7 + 114260𝛽8 − 89854𝛽9 + 46151𝛽10 − 13876𝛽11 +1

1821 3⁄ (−1+𝛽)2𝛽4(16𝛽6(16 − 61𝛽 + 126𝛽2 − 61𝛽3 + 16𝛽4)3 + 324𝛽6(−3 + 4𝛽 − 4𝛽2 +

3𝛽3)(16 − 61𝛽 + 126𝛽2 − 61𝛽3 + 16𝛽4)(−13 + 72𝛽 − 141𝛽2 + 141𝛽3 − 72𝛽4 + 13𝛽5) +1458𝛽6(1 − 2𝛽 + 𝛽2)(−13 + 72𝛽 − 141𝛽2 + 141𝛽3 − 72𝛽4 + 13𝛽5)2 + 972𝛽6(−3 +4𝛽 − 4𝛽2 + 3𝛽3)2(5− 66𝛽 + 142𝛽2 − 198𝛽3 + 142𝛽4 − 66𝛽5 + 5𝛽6) − 864𝛽6(1 − 2𝛽 +𝛽2)(16 − 61𝛽 + 126𝛽2 − 61𝛽3 + 16𝛽4)(5 − 66𝛽 + 142𝛽2 − 198𝛽3 + 142𝛽4 − 66𝛽5 +5𝛽6) + √(68826726756𝛽12 − 1331264302560𝛽13 + 12420967608072𝛽14 −75335392232208𝛽15 + 335745480218220𝛽16 − 1171746828299520𝛽17 +3322658699366760𝛽18 − 7834066859656656𝛽19 + 15588084625243740𝛽20 −26426624599796688𝛽21 + 38400770390351184𝛽22 − 47996528474402928𝛽23 +51691768858351656𝛽24 − 47996528474402928𝛽25 + 38400770390351184𝛽26 −26426624599796688𝛽27 + 15588084625243740𝛽28 − 7834066859656656𝛽29 +3322658699366760𝛽30 − 1171746828299520𝛽31 + 335745480218220𝛽32 −75335392232208𝛽33 + 12420967608072𝛽34 − 1331264302560𝛽35 +68826726756𝛽36))1 3⁄ . Therefore, the optimal *s is the solution to the equation: s as the solution to:

2 2 (3 (3 ) )(25 (25 9 ) )7 3 ( (10 (10 3 ) ) 7(2 ) 42 (1 ) 15 ) s s sB B s s sB

b b bb b b b b - - - -+ + - + - - - - - =

where 5 (1 ) (6 (5 ) )B s sb b= - - - - .

48

We prove Lemma 7. ∎