Group Buying Via Social Network ∗

Group Buying via Social Network ∗

Fei Shi†

Yong Sui‡

Jianxia Yang§

May, 2014

Abstract

This paper studies a special selling strategy, group buying, under which consumers enjoy a

discounted group price if the committed purchases exceeds the speciﬁed minimum size within

a certain time period. Introducing social network among consumers, we analyze group buying

from social advertising perspective. We characterize the optimal group buying strategy for a

ﬁrm and investigate the effect of social network on group buying strategies. We provide sufﬁ-

cient conditions under which group buying strategy is more proﬁtable than traditional private

selling strategies and more effective than traditional advertising.

Keywords: Group Buying, Advertising, Social Network

JEL Classiﬁcation: D85, L12, D83

∗We are grateful to Cheng-Zhong Qin for helpful comments. Yang gratefully acknowledges ﬁnancial support from National Natural Science Foundation of China (Grant No. 71103065), 2011 Shanghai Young College Teacher Training Subsidy Scheme, the Humanities and Social Sciences Campus Fund of East China University of Science and Technology (Grant No. YN0157129), and the Fundamental Research Funds for the Central Universities (Grant No. WN1122003). †Antai College of Economics and Management, Shanghai Jiao Tong University. E-mail: [email protected] ‡Antai College of Economics and Management, Shanghai Jiao Tong University. E-mail: [email protected] §School of Business, East China University of Science and Technology. E-mail: [email protected]

1 1 Introduction

Group Buying, as a new selling mechanism, has generated increasing attention in both online business industry and academia. While group buying has long existed in certain industries1, the Internet and recent emergence of social media dramatically reduced the transaction cost for different buyers to form a purchase group so as to qualify the discount price. In a typical group buying mechanism, a good or service is offered for sale under a discount for group purchase if the committed purchases exceeds the specified minimum size within a certain time period. In recent years, many Internet sites have emerged that allow sellers to offer group buying campaigns, where consumers purchasing with group coupons receive substantial discounts when the minimum group size is reached. Launched in November 2008, Groupon was a pioneer in this industry and has grown into a major public company.2 The existing literature mainly considers group buying to be an efficient selling mechanism under either demand uncertainty (Chen and Zhang, 2012) or demand certainty (Jing and Xie, 2011). In this paper, we investigate group buying from the perspective of social advertising. Many firms, es- pecially small business, launch group buying campaign in order to advertise their goods or services through social interactions among consumers. On the one hand, the minimum size requirement for group discount will provide an incentive for the informed consumers to transmit the information about the sale in their social networks, thus stimulating the dissemination of product information and expanding market demand. On the other hand, after the committed buyers consume the products or services, they may further diffuse among their social networks the product information, including price, quality, evaluation, and so on. The ex post network diffusion works again as advertising for new products or services. In essence, group buying serves as a marketing strategy to attract an ideal number of consumers to purchase by discount for further social advertising. The advertising channels may vary in many ways such as word-of-mouth communication and online

1In a Business to business context, groups of independent grocers, convenience stores, or retail hardware stores have long existed in the United States as well as in Europe. In some industries, such as food services and health care, group purchasing organizations have a dominant presence. See Chen and Roma (2011) for more examples. 2There are many group buying sites around the world, including Lashou, Meituan in China, MyCityDeal in Europe and Spreets in Australia.

2 referral.3 In this study, we investigate the optimal group buying strategy for a monopolistic seller and how the optimal strategy is affected as the consumer network structure changes. Moreover, we examine under what conditions group buying strategy is more effective than traditional advertising strategy. Assuming consumers may value the product or service either high or low, we focus our discussion on the minimum group size: how the optimal minimum size is determined, whether the seller announces or conceals it, and how it varies with the changes of consumers’ network structure. The optimal minimum group size results from the balance of two effects: group sale effect and social advertising effect. When the informed or committed consumers communicate with sufﬁciently large number of their social neighbors about the product or service, the social advertising effect tends to dominate, leading to a relatively smaller minimum group size. Otherwise, the seller would set a relatively larger minimum group size. Moreover, we show that it is weakly dominant for the seller to conceal the minimum group size in order to take better advantage of social advertising effect. This result provides a rationale for the stylized fact of most practice of group buying strategy. We also examine how the change of consumers’ network structure affects the optimal minimum group size. When consumers’ network turns less dispersed, the social advertising effect of group buying becomes strengthened, resulting in a smaller optimal minimum group size. In addition, we show that group buying is more proﬁtable than traditional advertising strategy with higher ratio of high-value consumers or larger mean of conditional degree distribution.

1.1 Related Literature

In the presence of demand uncertainty, the group buying is shown to outperform posted pricing or other pricing schemes under demand heterogeneity, economies of scale and risk-seeking sellers. For instance, Anand and Aron (2003) study group buying as a a menu of quantity discounts in markets with demand uncertainty. They show that group buying outperforms posted price under heterogeneous demand. Chen et al. (2007) analyze an optimal pricing strategy when the ﬁrm adopts

3On LivingSocial, a consumer receives a free deal if she gets three persons to buy the product.

3 a group-buying auction. They show that group buying outperforms a fixed-price mechanism when there are economies of scale or the firm is risk-seeking. Chen et al. (2010) compare the uniform group buying price with nonuniform-price group buying mechanisms and show that buyers may over purchase in exchange for a lower unit price. Assuming the firm has adopted group buying mechanism, Hu et al. (2013) focus on the deal’s success rate under distinct information disclosure schemes. In a dynamic game where consumers are engaged in sequential decisions, they investigate the benefit of revealing the cumulative number of sign-ups in increasing deal success rates. Reveal- ing the sign-up information can reduce consumer’s reluctance to sign up to deals for fear that the deals fail in reaching the thresholds. To isolate the impact of information disclosure, they assume that the group buying price and the minimum size for the deal to be valid are exogenously determined, whereas we assume that that prices and the minimum size are optimized by the seller. Wu et al. (2013) empirically identify two threshold effects in online group buying diffusion, induced by the minimum sign-up quantity of a deal. All of these studies do not address the perspective of social interaction or advertising effect among consumers. Group buying’s function of social advertising has been investigated in the following literature. Jing and Xie (2011) study group buying as a mechanism for selling through social interactions. They explore the role of group buying in facilitating social interaction among consumers with different levels of product information. They show that group buying can dominate traditional promotional schemes as it can motivate informed consumers to market products to less-informed consumers. Chen and Zhang (2012) consider group buying as interpersonal bundling and compare it with traditional bundling. They characterize the conditions to show that group buying is a profitable strategy in the presence of demand uncertainty, and how it may further boost profits by stimulating product information dissemination. Subramanian (2012) studies when and why (or why not) the firm finds profitable to prescribe a minimum group size for the deal to be valid. His find- ings provide a rationale for the observed trends that more and more firms no longer set minimum limits. Edelman et al. (2014) examine how the firms can potentially use group buying as a mechanism for price discrimination and advertising. The profitability of group buying generally depends

4 on the heterogeneity of consumers’ valuation and the level of firms’ recognition. In addition, they analyze certain challenges facing group buying in various business environment and characterize the narrow conditions in which group buying is likely to increase firm profits. However, the aforementioned studies do not explicitly investigate the effect of consumers’ social network. Our study contributes to the literature by explicitly analyzing consumers’ social network and its influence on firms’ optimal group buying strategy. There are quite a few studies that examine group buying as buyers’ strategic commitment. Chen and Roma (2011) study group buying in a distributional channel and characterize the competition between two retailers, with and without group buying. With symmetric retailers, group buying is always preferable under linear demand curves. With asymmetric retailers, group buying is ben- eficial to the weak or smaller players. Focusing on competition of duopoly firms, Chen and Li (2013) study how the presence of group buying affects firms’ incentives to invest in quality im- provement. One interesting finding in their study suggests that group buying may discourage firms from increasing investments in quality if firms are asymmetric. Dana (2012) studies buyer groups as strategic commitments and assumes that sellers have complete information about buyers’ preferences so that sellers can price discriminate against each individual buyer. He shows that buyer groups composed of buyers with heterogeneous preferences can increase price competition among rival sellers and curtail sellers’ ability to price discriminate by committing to purchase exclusively from one seller. Group buying mechanism adds to an expanding list of new Internet institutions, including tar- geted advertising, paid placement, and online auctions. Our study joins many recent contributions to the economies of Internet markets.4 Our analysis will shed light on the economic forces behind the rapid growth of companies such as Groupon by explaining the profitability of group buying strategy compared to traditional separate selling strategies. The remainder of this paper is organized as follows. In Section 2, we set up the model. In Section 3, we characterize the optimal group buying strategy and provide conditions under which

4See Levine (2012) for a nice survey.

5 group buying is more proﬁtable than private selling and traditional advertising. We also examine the effect of network structure on group buying strategies. Section 4 concludes with a review of our main ﬁndings and a discussion of potential future research. All proofs are included in the Appendix.

2 The Model

We consider a market where a firm promotes a product or service via group buying mechanism. In the promotion stage, the firm sets a stand-alone price p, a group purchase discounted price pG < p, and a minimum group size m for the discounted price to take effect. Each consumer can separately purchase the good at price p, but consumers who use the group coupon can purchase at the discounted price pG if and only if the number of group participants reaches or exceeds the minimum group size. After the promotion period, consumers can only purchase the good at price p. We assume that the committed consumers during the promotion period will disseminate the product information to their social neighbors via social communications, say word-of-mouth process. The firm’s marginal cost is assumed to a constant, which is normalized to zero. There is an infinite set of consumers N in the market. Each consumer is in a undirected social network with degree distribution P. For simplicity, assume that a random sampled consumer draws k neighbors with probability P(k) ≥ 0, k ∈ K = {kl,kh}, where 1 ≤ kl < kh < ∞ and P(kl)+P(kh) = 1. Note that the infinity of N and the finite of k together imply that the intersection of the neighbors of any two randomly picked consumers is empty. The average degree of the network is zP =

2 ∑k∈K P(k)k, and the variance of P is σP. In addition, denote

P(k)k P˜(k) = (1) zP to be the probability of the degree of a consumer conditional on that he is at the end of a randomly

6 chose link in the network. We denotez ˜P as the mean of P˜, which can be calculated as

2 + z2 ˜ σP P z˜P = ∑ P(k)k = (2) k∈K zP

Clearly,z ˜P is strictly increasing (decreasing) under a first-order shift of P when zP > σP (zP < σP). In order to examine the impact of changes in network structure, we introduce the following definitions. Let P and P0 be distinct degree distribution defined on K.

Deﬁnition 1. P0 ﬁrst-order stochastically dominates (FOSD) P if and only if k P0(d) ≤ k P(d) ∑d=kl ∑d=kl for every k ∈ K.

As a consequence, the mean under P0 is larger than the mean under P. Note that the degree of each consumer in the network has only two values, so we use a modiﬁed version of mean preserving spread (MPS).

0 0 0 0 0 Deﬁnition 2. P is a mean preserving spread (MPS) of P if and only if K = {kl,kh} under P such 0 0 0 that kl < kl, kh > kh, and P and P have the same mean.

After the promotion stage, a newly-informed consumer privately learn his valuation v about the product. We assume that v = H with probability λ, and v = L with probability 1 − λ, where 0 < L < H, λ ∈ (0,1). Each consumer desired to purchase at most one unit of the product. In each period, we assume that the information is obsolete only after one step diffusion.5 Namely, the diffusion process happens only among the informed consumers and their neighbors in stage 1, and among the neighbors of the informed consumers and their neighbors at the beginning of stage 2 if group buying succeeds. Otherwise, the informed consumer diffuses the product information only in stage 2. For simplicity, we normalize the diffusion cost to zero.6 We consider two alternatives of the ﬁrm’s group buying strategy: announcing the minimum group size or concealing it.

5Galeotti and Goyal (2012) allow the information to be transmitted by r steps. In essence, such generalization won’t change our results qualitatively. 6It can be shown that our main results remain to hold when a slight diffusion cost occurs.

7 At the beginning of group buying stage, there is only 1 informed consumer who values high.

Naturally, we have m ≥ 1. Without loss of generality, we concentrate on the case of pG ≤ L. Each informed consumer can just share the information of group buying campaign without persuasion and entail no cost. Then, we can safely conﬁne our attention on the ﬁrm’s design of minimum group size to m ∈ {1 + kl,1 + kh}.

3 Analysis and Results

3.1 Optimal Group Buying Strategy

3.1.1 Announcing Minimum Group Size m

We start with a preliminary inquiry into the optimal design of (pG,m, p) and discuss how the network structure influences the design. Let the intertemporal discounting factor be δ. Under group buying, for k neighbors to participate in the group buying campaign, all consumers will purchase at pG if 1 + k ≥ m and pG ≤ L. Otherwise, these consumers will buy at p if L < p ≤ H. When m = 1 + kh and the informed consumer has only kl neighbors, he does not have incentive to share information with his neighbors. As a result, only the informed consumer himself will purchase the product at p in stage 1. In addition, the firm has no incentive to set m > 1 + kh for similar reason. Then the firm’s problem is to maximize its two-stage expected profit:

max πG = Pr(1 + k ≥ m)[(1 + k)pG + δk(z˜P − 1)p] + Pr(1 + k < m)(1 + δλk)p (3) pG,m,p

If m = 1 + kl, we have

πG = (1 + zP)pG + δλzP(z˜P − 1)p (4)

If m = 1 + kh, we have

πG = P(Kh)[(1 + kh)pG + δλkh(z˜P − 1)p] + [1 − P(kh)](1 + δλkl)p (5)

8 The first part of the above equation is the expected profit induced by group buying, and the second part is the expected profit resulting from failure of group buying campaign. Since πG strictly ∗ ∗ increases in pG and p for any m, we have pG = L and p = H. Moreover, the optimal minimum group size m∗ can be set as following:

∗ (1+δλkl)H−(1+kl)L Lemma 1. The ﬁrm sets the optimal minimum group size to be m = 1+kl if z˜P > 1+ , δλklH ∗ otherwise, m = 1 + kh.

3.1.2 Concealing Minimum Group Size m

Since the informed consumer does not know the minimum group size m for group buying, he always diffuses the information to his neighbors without any cost, which is his dominant strategy.

Note that the firm can privately design a minimum group size impossibly reached, i.e., m > 1 + kh, in order to lure the informed consumer into transmitting information to his neighbors but ending with paying regular price. Consequently, the firm’s expected profit is as follows: If m = 1 + kl, we have

πG = (1 + zP)pG + δλzP(z˜P − 1)p (6)

If m = 1 + kh, we have

2 πG = P(Kh)[(1 + kh)pG + δλkh(z˜P − 1)p] + [1 − P(kh)](1 + λkl + δλ kl(z˜P − 1))p (7)

If m > 1 + kh, we have 2 πG = (1 + λzP)p + δλ zP(z˜P − 1)p (8)

By similar argument, we obtain the ﬁrm’s optimal group buying strategy:

∗ ∗ ∗ Lemma 2. The ﬁrm optimally sets pG = L, p = H, and m = 1 + kl if and only if z˜P > 1 +

(1+λkl)H−(1+kl)L ∗ , m = 1 + kh if and only if δλ(1−λ)klH

(1 + λkh)H − (1 + kh)L (1 + λkl)H − (1 + kl)L 1 + < z˜P ≤ 1 + δλ(1 − λ)khH δλ(1 − λ)klH

9 ∗ otherwise, m > 1 + kh.

To summarize, in these two cases, whenz ˜P is large enough, the firm sets a relatively small minimum group size, otherwise a relatively large one. The result is due to the interaction of two effects. The first effect, group promotion effect, is captured by the expected profit of group buying, and the second effect, social advertising effect, is captured byz ˜P − 1. When the committed consumers communicate with sufficiently large number of their neighbors about the product, the second effect dominates the first one, leading to a relatively small minimum group size. In contrast, when the social advertising effect is not dominating, i.e.,z ˜P is relatively small, the firm will optimally set larger minimum group size to balance these two effects.

Lemma 3. Concealing minimum group size is at least as proﬁtable as announcing it for the ﬁrm.

The proof of Lemma 3 follows from the proofs of Lemma 1&2, so is omitted. Even though many firms announce the minimum group size during their group buying campaigns, more and more firms choose to conceal it in practise. Our Lemma 3 provides a rationale for this stylized fact regarding group buying strategy. Intuitively, without knowing the minimum group size, the informed consumer would like to share the product information with his neighbors without thinking about whether the activity will fail or not. This unconditional pre-purchase social communication increases the firm’s expected profit on average. In addition, ignoring the diffusion cost, the firm can manipulate the outcome of group buying campaign, that is, whether to launch group buying campaign. Such flexibility facilitates the firm to get the second-best profit rather than that induced by selling through regular price. Therefore, concealing strategy will increase the social advertising effect both ex ante and ex post.

3.2 Effects of Network Structure

In this section, we analyze how the change of network structure affects the minimum size of group buying designed by the firm, as well as the expected profit. According to Lemma 3, we focus on the firm’s strategy of concealing minimum size.

10 Proposition 1. The optimal minimum group size is decreasing under a ﬁrst order shift of P if

∗ zP > σP, and vice versa. Meanwhile, in the case of m = 1+kl, the expected proﬁt of group buying is strictly increasing under a ﬁrst order shift of P if

s L + δλH(z˜P − 1) 1 + zP > σP δλHz˜P and vice versa, and also strictly increasing under a mean preserving spread of P.

Proposition 1 indicates that asz ˜P rises under the ﬁrst order change of degree distribution if zP > σP, the social advertising effect is strengthened, resulting in the weakly declining of minimum size. This result implies that a smaller minimum size for the ﬁrm may survive under a less dispersed network structure.

3.3 Advantage of Group Buying

In this section, we compare the profitability of group buying strategy with that of traditional advertising to identify the conditions under which group buying is more advantageous than traditional advertising. Consider the firm may, alternatively, take traditional advertising strategy to target T consumers, where T ≥ 0. When T = 0, it corresponds to the trivial case that the firm does not advertise in the first period. Advertising makes the firm entail the cost of C(T). Without loss of

0 generality, assume that C(·) is increasing and convex, C(0) = 0, and limT→∞ C (T) = ∞. The other settings are the same as the case of group buying. For comparison, we confine our attention to the scenario of T ≤ kh since the information is obsolete after one-step diffusion. Then, the firm’s profit-maximizing problem is

  (1 + T)p1 + δPr(u ≥ p2)(1 + T)zP p2 −C(T), if p1 = L max πM = T,p1,p2  (1 + λT)p1 + δPr(u ≥ p2)(1 + λT)zP p2 −C(T), if p1 = H

Deﬁne πM(L) and πM(H) to be the expected proﬁt with p1 = L and p1 = H, respectively.

11 ∗ Intuitively, p2 ∈ {L,H}, and the relationship between πM(L) and πM(H), depends on the range of λ. The following Lemma summarizes this intuitive.

Lemma 4. The optimal price in stage 2 and the expected proﬁt are determined as

L ∗ H−L • If 0 < λ ≤ , then p = L, and πM(L) ≤ πM(H) when T ≤ T ≡ . H L 1 L+δLzP(1−λ)−λH

L L ∗ H−L • If < λ < +δλ(1−λ)zP, then p = H, and πM(L) ≤ πM(H) when T ≤ T ≡ . H H 2 2 L+δλHzP(1−λ)−λH

L ∗ • If λ ≥ H + δλ(1 − λ)zP, then p2 = H, and πM(L) < πM(H)

We are interested in when group buying strategy is more profitable than the traditional advertising strategy for the seller, and how the network structure influences its profitability. For convenience, define πA as the additional expected profit generated by advertising, and πG as the expected profit induced by group buying neighbors’ advertising to each neighbor. Let ρ = πA , capturing AG πG the ratio of aggregate effect of traditional advertising to the average one of group buying. The following theorem offers the sufficient conditions for πG > πA.

Theorem 1. The group buying strategy is more proﬁtable than traditional advertising strategy if z˜P is sufﬁciently large. Furthermore, let δ > H , and LzPkh

φAG ≡ [P(kh) − [1 − p(kh)]λkl]H − P(kh)(1 + kh)L

ωG ≡ [λzP + (1 − λ)P(kh)kh]δλK

L ∗ • The case of 0 < λ ≤ H . If T ≤ T1, the group buying strategy dominates if

H − [1 + (1 − λ)zP]L δzPL + φAG λH − δL z˜P > min{1 + ρAG + ,1 + ρAG + ,1 + ρAG + } δλHzP ωG δλH

∗ If T > T1, it dominates if

(1 − δ)L (1 + δzP)L − H + φAG (1 + δzP)L − (1 − λzP)H z˜P > min{1+ρAG − ,1+ρAG + ,1+ρAG + 2 } δλH ωG δλ HzP

12 L L ∗ • The case of H < λ < H + δλ(1 − λ)zP. If T ≤ T2, the group buying strategy dominates if

H − (1 + zP)L δλzPH + φAG 1 − δ z˜P > min{2 + ρAG + ,1 + ρAG + ,1 + ρAG − } δλHzP ωG δλ

∗ If T > T2, it dominates if

L (δλzP − 1)H + L + φAG L − (1 + (1 − δ)λzP)H z˜P > min{2+ρAG − ,1+ρAG + ,1+ρAG + 2 } δλH ωG δλ HzP

L • The case of λ ≥ H +δλ(1−λ)zP. The group buying strategy dominates if the same condition ∗ under T ≤ T2 holds.

We can view the advertising effect generated by group buying campaign in stage 2 as the conditional advertising effect, because it is determined by the conditional degree distribution of the informed consumer’s neighbors. The higherz ˜P is, the greater the advertising effect is, and group buying strategy generates more expected profit than traditional advertising strategy, which is what Theorem 1 mainly implies. Meanwhile, we also find out that the relative advantage of group buying depends on the value of λ. In essence, the conditional advertising effect can be measured by λ(z˜P − 1) in the sense of expectation since p∗ = H. As stated above, the seller should tradeoff the group-scale effect and conditional advertising effect when taking both strategies. When λ is relatively small, the stage- 1 expected profit is more prominence. Hence, the domination of the first effect requires more vigorous second effect for the firm to select group buying, or in other words, requires a higherz ˜P as support, on average. By contrast, when λ is relatively large, the second effect dominates, a small z˜P can let group buying survive. To conclude, there are two factors affecting the superiority of group buying: the ratio of high- value consumers, and the mean of conditional degree distribution, or the network structure. Larger

λ andz ˜P make group buying more proﬁtable. In this paper, we are more interested in the impacts of network structure on the proﬁtability of group buying, so ignore the comparative statics of λ, unless analyzing the effects ofz ˜P simultaneously. The following Proposition investigates how the

13 change of network structure affects the proﬁtability of group buying strategy.

0 L Proposition 2. Let P be a mean preserving spread of P. If 0 < λ ≤ H , if the ﬁrm chooses group buying strategy rather than traditional advertising under P, then it chooses the same strategy under

0 L H−L P . If H < λ < 1, when kh ≥ [1+(1−P(kh)) 2δH ]zP, if the ﬁrm chooses group buying strategy under P, then it chooses the same strategy under P0.

The above result provides an implication that group buying better matches highly dispersed network in degree. When λ is relatively small, the conditional advertising effect is not so effective. Then in a more dispersed network, group buying strategy will amplify its conditional advertising effect, leading to its domination to traditional advertising. However, when λ is relatively large, the conditional advertising effect and expected prot of group buying failure weighs more when m∗ =

1 + kh. Thus, a more dispersed network may induce a diminishing conditional advertising effect under group buying, which reduces its total expected prot. In order to offset this prot reduction, the conditional advertising effect should be augmented once group buying succeeds, which is ensured by a higher kh relative to zP. Note that as a whole, given λ, or the dispersion of willingness to pay, group buying has a power to concentrate the dispersion of degree of consumer network, while traditional advertising does not. From this perspective, we can argue that the superiority of Group-Buying not only lies in concentrating the dispersed valuations among consumers, but also the dispersed degree in consumer network. Consequently, we can view Group-Buying as a dual-concentration marketing instrument. In Chen and Zhang (2012), Group-Buying works as interpersonal bundling. In our theory, we go further to endow it with a new insight: degree bundling in the consumer network.

3.4 Group Buying as Optimal Selling Strategy

In current theoretical literature, group buying was often captured by an optimal selling strategy (Jing and Xie, 2011). Our framework can also incorporate such conclusion in the presence of demand uncertainty. The demand uncertainty menifests that the ﬁrm only knows the degree distribution of consumer network.

14 Under the environment of advertising, when T ∗ = 0, the firm equivalently takes a separate selling strategy. There are four candidates for the firm, (L,L), (L,H), (H,L),(H,H), and which one will be adopted primarily depends on the value of λ. Intuitively, the group buying strategy is more profitable than separate selling strategy in most cases. Such intuition is subtly reflected below.

Theorem 2. Given T ∗ = 0, group buying strategy dominates the separate selling strategy if δ = 1.

L L In addition, if δ < 1, group buying strategy dominates in the case of λ ∈ (0,δ H ]∪( H ,1), and in the L L case of λ ∈ (δ H , H ) if

λH − δL H − [1 + (1 − δ)zP]L δzPL + φAG z˜P > 1 + min{ , , } δλH δλHzP ωG

By contrast to the separate selling, group buying campaign promotes ex ante spontaneous social interactions among potential consumers, and aggregates the demand of both high and low-value

∗ consumers. Such advantage disappears in separate selling. Even when δ < 1 and T > T1 under λ ∈ L L (δ H , H ]. Group buying strategy can also overwhelm separate selling through generating powerful ex post conditional advertising effect whenz ˜P is relatively large. This may be the reason who online group buying campaign is so popular. Subsequently, we analyze the effects of network structure change on the proﬁtability of group buying as a selling strategy.

Proposition 3. Let P0 be a mean preserving spread of P. If the ﬁrm uses group buying strategy rather than separate selling under P, then it takes the same strategy under P0.

This proposition points out that group buying has its own advantage over separate selling when the ﬁrm faces more dispersed consumer network with the same mean in degree. Hence, group buying, working as selling strategy, is somewhat like the idea of interpersonal bundling proposed by Chen and Zhang (2012).

15 4 Discussion and Conclusions

In this paper, we examine an important nature of group buying strategy which works via social advertising among consumers’ network. In a simple model, we focus our discussion on the design of minimum group size. Based on our characterization of optimal group buying strategy, we find that it is weakly dominant for the seller to conceal the minimum group size, which provides a rationale for most practice of group buying campaigns all over the world. When consumers’ network tends to less dispersed, the social advertising effect of group buying becomes strengthened, resulting in a smaller optimal minimum group size. In addition, we show that group buying is more profitable than traditional advertising strategy with higher ratio of high-value consumers or larger mean of conditional degree distribution. Some limitations in our model are worth to point out. First, the setup of consumer network in our model could be extended to a more general formalization using general degree distribution as the current literature of social network does. Second, the diffusion preferences of consumers are ignored in our model. Although some diffusion articles utilize similar diffusion setup, such preferences do influence their transmitting behaviors and the seller’s strategy. However, we conjecture that the aforementioned extensions may not qualitatively change our main results in the superiority of group buying.

16 Appendix

Proof of Lemma 2

∗ Proof. We take two steps. First, ﬁnd out the condition of m = 1 + kl among the three strategies. ∗ ∗ ∗ Comparing the proﬁts under m = 1 + kl and m = 1 + kh, we have m = 1 + kl if

(1 + λkl)H − (1 + kl)L z˜P > 1 + δλ(1 − λ)klH

∗ ∗ In parallel, the comparison between the proﬁts under m = 1 + kl and m > 1 + kh implies that ∗ m = 1 + kl if (1 + λzP)H − (1 + zP)L z˜P > 1 + δλ(1 − λ)zPH

∂ (1+λx)H−(1+x)L δλ(1−λ)xH L−H (1+λkl)H−(1+kl)L ∗ Since = 2 < 0, ∀x > 0, thenz ˜P > 1+ results in m = 1+kl. ∂x δλ(1−λ)Hx δλ(1−λ)klH ∗ Second, we derive the condition of m = 1 + kh for the remaining two strategies. According to ∗ the ﬁrm’s maximization problem, we obtain that m = 1 + kh if

(1 + λkh)H − (1 + kh)L z˜P > 1 + δλ(1 − λ)khH

∗ Therefore, m = 1 + kh if and only if

(1 + λkh)H − (1 + kh)L (1 + λkl)H − (1 + kl)L 1 + < z˜P ≤ 1 + δλ(1 − λ)khH δλ(1 − λ)klH

Proof of Proposition 1

∗ Proof. Based on Lemma 2, m becomes weakly increasing with biggerz ˜P. Sincez ˜P turns larger under a ﬁrst order shift of P if zP > σP, then the result follows. At the mean time, zP is increasing

17 ∗ under a ﬁrst order shift of P. When m = 1 + kl,

2 ∂πG σP = L + δλH(z˜P − 1) + δλHzP(1 − 2 ) ∂zP zP which implies that πG increases if

s L + δλH(z˜P − 1) 1 + zP > σP δλHz˜P

Proof of Lemma 4

∗ Proof. Apparently, if L < λH, then p2 = H, and vice versa for either p1 = L or p1 = H. Holding T,

πM(H) − πM(L) = n(H − L) + T[λH − L − δλHzP(1 − λ)]

n(H−L) L L so when T > and < λ < +δλ(1−λ)zP, we have πM(L) > πM(H), otherwise, L+δλHzP(1−λ)−λH H H

πM(L) ≤ πM(H). ∗ If L ≥ λH, then p2 = L. For given T,

πM(H) − πM(L) = n(H − L) + T[λH − L − δLzP(1 − λ)]

n(H−L) so when T > , we have πM(L) > πM(H), otherwise, πM(L) ≤ πM(H). L+δLzP(1−λ)−λH

For the case of p2 = H, the ﬁrst order condition of πM(L) with respect to T is

0 L + δλHzP = C (TL)

while of πM(H) is 0 λH(1 + δλzP = C (TH)

18 Proof of Theorem 1

Proof. Based on Lemma 4, we can partition the proﬁt comparison between group buying and traditional advertising strategy into three scenarios which depend on the value of λ.

L ∗ (i) The case of 0 < λ ≤ H . Now p2 = L under advertising. To make the problem interesting, H ∗ ∗ suppose T < k , or alternatively, δ > . When T ≤ T , πM = πM(H), if m = 1 + k , then 1 h LzPkh 1 l

∗ ∗ ∗ ∗ πG(m ) − πM = (1 + zP)L + δλzP(z˜P − 1)H − [(1 + λT )H + δ(1 + λT )zPL −C(T )]

Therefore, group buying is more proﬁtable if

∗ ∗ H − [1 + (1 − δ)zP]L λT (H + δzPL) −C(T ) z˜P > 1 + + (9) δλHzP δλHzP H − [1 + (1 − δ)zP]L = 1 + ρAG + (10) δλHzP

≡ z˜P1 (11)

∗ If m = 1 + kh, then group buying is more proﬁtable if

∗ ∗ [δzP − P(kh)(1 + kh)]L + [P(kh) − (1 − P(kh))λkl]H λT (H + δzPL) −C(T ) z˜P > 1 + + (12) [λzP + (1 − λ)P(kh)kh]δλH λzP + (1 − λ)P(kh)kh]δλH δzPL + φAG = 1 + ρAG + (13) ωG

≡ z˜P2 (14)

∗ When m > 1 + kh, then group buying is more proﬁtable if

∗ ∗ λH − δL λT (H + δzPL) −C(T ) λH − δL z˜P > 1 + + 2 = 1 + ρAG + ≡ z˜P3 (15) δλH δλ HzP δλH

Then group buying is more proﬁtable if

∗ ∗ (1 − δ)L T (1 + δzP)L −C(T ) (1 − δ)L z˜P > 1 − + = 1 + ρAG − (16) δλH δλHzP δλH

19 ∗ When m = 1 + kh, group buying is more proﬁtable if

∗ ∗ [1 + δzP − P(kh)(1 + kh)]L − [1 − P(kh)](1 + λkl)H T (1 + δzP)L −C(T ) z˜P > 1 + + (17) [λzP + (1 − λ)P(kh)kh]δλH λzP + (1 − λ)P(kh)kh]δλH (1 + δzP)L − H + φAG = 1 + ρAG + (18) ωG

∗ Finally, when m > 1 + kh, group buying is more proﬁtable if

∗ ∗ (1 + δzP)L − (1 + λzP)H T (1 + δzP)L −C(T ) z˜P > 1 + 2 + 2 (19) δλ HzP δλ HzP (1 + δzP)L − (1 + λzP)H = 1 + ρAG + 2 (20) δλ HzP

L L ∗ ∗ (ii) The case of H < λ < H + δλ(1 − λ)zP. Now p2 = H under advertising. When T ≤ T2, ∗ πM = πM(H). When m = 1 + kl, group buying is more proﬁtable if

∗ ∗ H − (1 + zP)L T (1 + δλzP)λH −C(T ) z˜P > 2 + + (21) δλHzP δλHzP H − (1 + zP)L = 2 + ρAG + (22) δλHzP

∗ When m = 1 + kh, group buying is more proﬁtable if

∗ ∗ [δλzP + P(kh) − (1 − P(kh))λkl]H − P(kh)(1 + kh)L T (1 + δλzP)λH −C(T ) z˜P > 1 + + (23) [λzP + (1 − λ)P(kh)kh]δλH [λzP + (1 − λ)P(kh)kh]δλH δλzPH + φAG = 1 + ρAG + (24) ωG

∗ When m > 1 + kh, group buying is more proﬁtable if

∗ ∗ 1 − δ T (1 + δλzP)λH −C(T ) 1 − δ z˜P > 1 − + 2 = 1 + ρAG − (25) δλ δλ zPH δλ

∗ ∗ Next consider the case of T > T2. When m = 1 + kl, group buying is more proﬁtable if

∗ ∗ L T (L + δλHzP) −C(T ) L z˜P > 2 − + = 2 + ρAG − (26) δλH δλHzP δλH

20 ∗ When m = 1 + kh, group buying is more proﬁtable if

[δλzP − (1 − P(kh))(1 + λkl)]H + [1 − P(kh)(1 + kh)]L z˜P > 1 + (27) [λzP + (1 − λ)P(kh)kh]δλH T ∗(L + δλHz ) −C(T ∗) + P (28) [λzP + (1 − λ)P(kh)kh]δλH (δλzP − 1)H + L + φAG = 1 + ρAG + 2 (29) δλ zPH

∗ When m > 1 + kh, group buying is more proﬁtable if

∗ ∗ L − [1 + (1 − δ)λzP]H T (L + δλHzP) −C(T ) z˜P > 1 + 2 + 2 (30) δλ HzP δλ HzP L − [1 + (1 − δ)λzP]H = 1 + ρAG + 2 (31) δλ HzP

L ∗ (iii) The case of λ ≥ H + δλ(1 − λ)zP. Since p2 = H and πM(L) < πM(H), group buying is more proﬁtable if (21)-(25) hold.

∗ To summarize, since the relationship among the thresholds ofz ˜P under different m in each case of λ is ambiguous, based on Lemma 2, group buying strategy dominates ifz ˜P is larger than ∗ the corresponding threshold under m = 1 + kl.

Proof of Proposition 2

∂zP 0 Proof. By the deﬁnition of MPS, kh turns larger, kl becomes smaller, and = 0 under P . Then ∂kh based on the proof of Theorem 1, we partition our proof into two cases.

L ∗ (i) The case of 0 < λ ≤ H . First, consider the case of T ≤ T1. The RHS of holds, butz ˜P rises 0 under P relative to under P, so the result follows. At the mean time, we can rewrite φAG as

0 φAG = P(kh)(H − L) − zPL + [1 − P(kh)]kl(L − λH)

L Since λ ≤ H , φAG is weakly decreasing in kl, and ωG is increasing in kh. Therefore, the RHS of (A2) is decreasing under P0 if ρ + δzPL+φAG ≥ 0. When ρ + δzPL+φAG < 0, then the RHS of (A2) AG ωG AG ωG

21 is less than 1, under which the result trivially holds. Analogously, we can support the results in the

∗ case of T > T1 through similar logic. L ∗ (ii) The case of H < λ < 1. In the case of T ≤ T2, the results follow in both (A7) and (A9) by ∗ the deﬁnition of MPS. When m = 1 + kh, since

[1 − P(kh)]kl = zP − P(kh)kh we can rewrite (A8) as

zP(δλH − L) + P(kh)(H − L) [zP − P(kh)kh](L − λH) z˜P > 1 + (ρAG + ) + (32) ωG ωG

= 1 + (ρAG + A) + B (33)

Moreover,

∂(ρAG + A + B) P(kh)(λH − L)zP = 2 (34) ∂kh δλH[λzP + (1 − λ)P(kh)kh] ∗ ∗ (1 − λ)P(kh)[zP(δλH − L) + P(kh)(H − L) + T (1 + δλzP)λH −C(T )] − 2 (35) δλH[λzP + (1 − λ)P(kh)kh] P(kh)(λH − L)zP − (1 − λ)P(kh)∆π = 2 (36) δλH[λzP + (1 − λ)P(kh)kh]

On the one hand, if ∆π ≥ 0, then

∂(ρ + A + B) lim AG ≤ 0 L ∂k λ→ H h and ∂(ρ + A + B) lim AG > 0 λ→1 ∂kh

Thus, we have ∂(ρAG + A + B) L limarg max ≡ λh ∈ { ,1} (37) L ∂k H H <λ<1 h

L for the continuity of (A14) in λ. If λh = H , then we return to corresponding result in the case of

22 L 0 < λ ≤ H . If λh = 1, then ∂(ρ + A + B) P(k )(H − L) AG = h ∂kh δHzP

zP−P(kh)kh By the deﬁnition of variance, when kl = , we have 1−P(kh)

2 2 P(kh) 2 σP = ∑P(k)(k − zP) = (kh − zP) (38) k 1 − P(kh)

Then ∂z˜P P(kh) 2(kh − zP) = 2 ∂kh 1 − P(kh) zP Therefore, ∂z˜P ∂(ρAG + A + B) H − 1 ≥ if kh ≥ [1 + (1 − P(kh)) ]zP ∂kh ∂kh 2δH under which the result follows.

On the other hand, if ∆π < 0, then the result trivially follows by similar logic to that in part (i). Meanwhile, in the case of T ∗ > T , denote the RHS of (A11) to be ϕ = 1+ρ + (δλzP−1)H+L+φAG . 2 AG ωG

Take derivative with respect to kh yields

∂ϕ P(kh)(λH − L)zP = 2 (39) ∂kh δλH[λzP + (1 − λ)P(kh)kh] ∗ ∗ (1 − λ)P(kh)[zP(δλH − L) − (1 − P(kh))(H − L) + T (L + δλzP) −C(T )] − 2 (40) δλH[λzP + (1 − λ)P(kh)kh]

Comparing with, we can prove corresponding results by similar logic, so omit the rest of proof.

Proof of Theorem 2

Proof. Given T ∗ = 0, we partition the proﬁt comparison of group buying with separate selling into three cases.

L (i) The case of λ ∈ (0,δ H ]. By (A3),

λH − δL π > π ifz ˜ > 1 + ≡ z˜ under m > 1 + k G M P δλH P1 h

23 L Since λ ≤ δ H , we havez ˜P1 ≤ 1, implying the condition is naturally satisﬁed forz ˜P > zP > 1. Then ∗ the result follows from πG(m ) ≥ πG(m > 1 + kh). L L (ii) The case of λ ∈ (δ H , H ]. Based on the proof of Theorem 1, we obtain that group buying ∗ ∗ ∗ dominates if πG > πM under either case of m = 1+kl, m = 1+kh, or m > 1+kh. Therefore, the sufﬁcient condition is

z˜P > min{z˜P1 ,z˜P2 ,z˜P3 }

L (iii) The case of λ ∈ ( H ,1). By (a9),

(1 − δ) π > π ifz ˜ ≥ 1 − ≤ 1 under m > 1 + k G M P δλ h

Then for similar logic, the result directly follows.

Proof of Proposition 3

L L Proof. Based on Theorem 2, we only need to prove the result in the case of λ ∈ (δ H , H ] when ∗ L δ < 1. Since the current case is the T = 0 version of Theorem 1 in the case of 0 < λ ≤ H with ∗ T ≤ T1, the result immediately follows by similar proof method to that in the proof of part (i) of Proposition 2.

24 References

Anand KS, Aron R (2003) Group buying on the Web: A comparison mechanism of price-discovery. Management Science. 49(11): 1546-1562.

Chen J, Chen X, Song X (2007) Comparison of the group-buying auction and the ﬁxed pricing mechanism. Decision Support Systems. 43(2):445-459.

Chen RR, Li C, Zhang RQ (2010) Group buying mechanisms under quantity discount. Working paper, University of California at Davis.

Chen RR, Roma P (2011) Group buying of competing retailers. Production and Operations Man- agement. 20(2): 181-197.

Chen Y, Zhang T (2012) Group coupon: Interpersonal bundling on the internet. Working paper, University of Colorado, Boulder.

Chen Y, Li X (2013) Group buying commitment and sellers’ competitive advantages. Journal of Economics and Management Strategy. 22 (1): 164-183

Dana JD (2012) Buyer groups as strategic commitments. Games and Economic Behavior. 74: 470-485

Economides N (1996) The economics of networks. International Journal of Industrial Organiza- tion. 14(2): 673-699.

Edelman B, Jaffe S, Kominers SD (2014) To groupon or not to groupon: The proﬁtability of deep discounts. Working paper, Harvard University, Cambridge, MA.

Ellison G, Fudenberg D (1995) Word-of-mouth communication and social learning. Quarterly Journal of Economics. 110(1): 93-125.

Galeotti A, Goyal S (2009) Inﬂuencing the inﬂuencers: A theory of strategic diffusion. Rand Journal of Economics. 40(3): 509-532

25 Galeotti A, Goyal S (2012) Network multipliers: The optimality of targeting neighbors. Review of Network Economics. 11(3):1-11

Hu M, Shi M, Wu J (2013) Simultaneous vs. sequential group-buying mechanism. Management Science. 59(12): 2805-2822.

Jing X, Xie J (2011) Group buying: A new mechanism for selling through social interaction. Management Science. 57(8): 1354-1372.

Levine J (2012) The economics of internet markets. Advances in Economics and Econometrics. Acemoglu D, Arellano M, Dekel E ed.

Marvel HP, Yang H (2008) Group purchasing, nonlinear tariffs, and oligopoly. International Journal of Industrial Organization. 26: 1090-1105

Subramanian U (2012) A theory of social coupons. Working paper, University of Texas at Dallas.

Surasvadi N, Tang C, Vulcano G (2014) Operating a group-buy mechanism in the presence of strategic consumers. Working paper, New York University

Wu J, Shi M, Hu M (2013) Threshold effects in online group buying. Working paper, Rotman School of Management, University of Toronto, Canada.