Simultaneous Vs. Sequential Group-Buying Mechanisms

Simultaneous vs. Sequential Group-Buying Mechanisms

Ming Hu, Mengze Shi, Jiahua Wu Rotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada {[email protected], [email protected], [email protected]}

his paper studies the design of group-buying mechanisms in a two-period game where cohorts of consumers Tarrive at a deal and make sign-up decisions sequentially. A firm can adopt either a sequential mechanism where the firm discloses to second-period arrivals the number of sign-ups accumulated in the first period, or a simultaneous mechanism where the firm does not post the number of first-period sign-ups and hence each cohort of consumers faces uncertainty about another cohort’s size and valuations when making sign-up decisions. Our analysis shows that, compared with the simultaneous mechanism, the sequential mechanism leads to higher deal success rates and larger expected consumer surpluses. This result holds for a multiperiod extension and when the firm offers a price discount schedule with multiple breakpoints. Finally, when the firm can manage the sequence of arrivals, it should inform the smaller cohort of consumers first. Key words: promotion; group buying; coordination game History: Received August 4, 2011; accepted February 27, 2013, by J. Miguel Villas-Boas, marketing. Published online in Articles in Advance July 19, 2013.

1. Introduction commitment through escrow payment systems. Most A group-buying mechanism is a scheme designed of the representative group-buying websites that to help coordinate a group of interested buyers so became popular in the late 1990s, including Mercata, that they can reach their common purchase goals. Mobshop, and Letsbuyit, either ceased operating or In a typical group-buying mechanism, no transac- changed their business models a few years later tion will take place unless the total number of (Kauffman and Wang 2002). Interestingly, despite committed purchases exceeds a specified thresh- the failure of these pioneering group-buying sites, old within a certain time period. Various forms of a decade later another generation of social buying group-buying mechanisms have been observed on websites like Groupon and LivingSocial emerged. Led a wide range of occasions and for a long time. by the market leader, Groupon, these newcomers typ- One version is street performer protocol (SPP), an ically offer “a deal a day” tailored to each local agreement between an artist and a group of poten- market (Wortham 2009). The market enthusiasm for tial users. The artist does not start the creative online group buying peaked when Groupon declined work until the potential funders have pledged a a $6 billion offer from Google (Weiss 2010). required amount of support. Renowned musicians This paper investigates the information manage- like Beethoven and Mozart used such arrangements ment strategy for group-buying mechanisms, specif- to ensure that enough tickets were sold for their ically, whether or not the sponsor should ask concerts. The recent emergence of social media has participants to make decisions without knowing the popularized these funding concepts and generated decisions of others. We take the perspective of third- similar concepts like threshold pledge systems and party group-buying platforms like Groupon and crowd funding. Websites like Kickstarter.com allow Kickstarter and investigate the impact of alternative artists, museums, and entrepreneurs to post project information management mechanisms on deal suc- proposals and seek funding from interested donors. cess rates. Knowing how to improve the success rates In another version of the group-buying mechanism, a is important because group-buying firms typically number of buyers pool their purchases, often through earn revenues from successful listings only and not the facilitation of a third party, to obtain a quan- all group-buying deals succeed. For example, from tity discount from sellers. Online group-buying web- the launch of Kickstarter.com in April 2009 to March sites first appeared in the late 1990s, as part of the 2011, a total of 20,371 projects were proposed. Among wave of innovative online market-based mechanisms. them, 7,496 projects attracted enough funds, lead- Usually the consumers had to make the purchase ing to a success rate of 43% (blog.kickstarter.com).

Zhang and Liu (2012), who studied requests for cohort, it is better to inform the potentially larger microloans listed in Prosper.com, reported a success cohort of consumers later because the confidence of rate of 12.3% among 49,693 listings. Without careful consumers arriving earlier can be boosted by a subse- analysis, the firm’s decision does not appear straight- quent stronger cohort who does not suffer discount- forward because of the uncertainty about the number ing in belief due to information asymmetry. of consumer arrivals and their individual valuations. To further understand the dominance of sequen- Looking forward, it can be beneficial to post the num- tial mechanisms over simultaneous mechanisms, we ber of sign-ups if a large cohort of consumers with extend our model in a number of directions. First, we high individual valuations turn out in the early stage, extend our model to one with more than two peri- but it can be detrimental if the first cohort of con- ods, where the threshold of group buying is equal sumers turn out to be small and have low individual to the number of arrivals. Such a model allows a valuations. sequential mechanism to have different reporting fre- To investigate the influence of information manage- quencies. We find that the mechanism with the high- ment mechanisms on deal success rates, we develop est reporting frequency, that is, updating after each a two-period model where two cohorts of consumers period, leads to the highest success rate. Because all arrive at the deal sequentially. The two-period model of the consumers have to commit to purchases for is a stylized capture of the fact that earlier arrivals the deal to be on, the intuition behind the backward are faced with more uncertainty in the deal’s suc- induction described earlier still applies when we trace cess rate than later arrivals. The firm being stud- along the sole sign-up trajectory that leads to the suc- ied chooses between a “sequential mechanism” where cess. Second, we extend the single-level scheme to the firm posts the number of sign-ups at the end a multilevel pricing schedule where the participants of the first period, and a “simultaneous mechanism” can receive a greater benefit when the total number where the firm does not post the first-period out- of sign-ups reaches a higher level of threshold. Our come. Somewhat surprisingly, our analysis shows analysis shows that, all else being equal, the sequen- that the deal’s success rate is always higher under tial mechanism still always yields higher success rates the sequential mechanism. To see the reason for than the simultaneous mechanism. Third, when the this result requires a backward-inductive reasoning, firm faces a capacity constraint, the expected number starting with the second period and then moving of sign-ups may be lower under the sequential mech- back to the first period. A sequential mechanism anism than that under the simultaneous mechanism. increases the ex ante expected sign-up rates of the This is because the capacity constraint introduces second cohort of consumers by eliminating the uncer- negative externality as the chance to receive service tainty facing them. The increased expected sign-up decreases when more people have signed up to the rates of the second cohort enhance the confidence deal. Knowing that a large number of people have of the first cohort of consumers, thereby increas- already signed up can simply depress the expected ing the ex ante expected sign-up rates of the first values of a group-buying deal. cohort. This result underscores the importance of modeling and investigating the dynamics of sign-up 1.1. Literature Review behavior under group-buying mechanisms. The result Our paper is related to research on private provi- also offers a potential explanation for why firms like sion of public goods. Like the group-buying deals, Groupon and Kickstarter display the updated number provision of public goods requires successful coor- of sign-ups along with the minimum number required dination among the private contributors. In the lit- to unlock the deals. erature on private provision of public goods, most When implementing the information management papers examine the simultaneous mechanism (e.g., mechanism, a firm may control the sequence of con- Palfrey and Rosenthal 1984, Bagnoli and Lipman sumer arrivals by, for example, contacting different 1989, Tabarrok 1998). Work on sequential mechanisms groups of consumers separately. Since the sequen- includes Varian (1994), Gächter et al. (2010), and tial mechnism always leads to a higher success rate Romano and Yildirim (2001). Varian (1994) shows that over the simultaneous mechanism, it is better off for a simultaneous decision-making mechanism leads the firm to have cohorts of consumers arrive sequen- to more supplied public goods than a sequential tially rather than simultaneously. Interestingly, our mechanism where the outcome of early decisions is analysis also finds that the firm should first inform revealed to those participants who make decisions the cohort with a potentially smaller number of con- later. In Varian (1994), one player can free ride on sumers, and then the potentially larger cohort of con- another’s contributions under the sequential mecha- sumers. The reason for this result can again be seen nism. This result was further supported by evidence from the backward-inductive reasoning described ear- from lab experiments (Gächter et al. 2010). In contrast, lier. When the firm is uncertain about the size of each Romano and Yildirim (2001) show that a sequential Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2807 mechanism could perform better than a simultane- to the network externality literature by examining the ous mechanism with more general utility functions unique form of externality effect in the group-buying that include psychological values associated with the context. A group-buying deal is on if and only if warm-glow or snob appeal effect. In their model, an a minimum number of consumers sign on. Moreover, individual may derive a positive value from donating a group-buying deal can consist of multiple levels more money than others to support causes. Such psy- with the deal offering a greater value when the num- chological values mitigate the incentive to free ride, ber of sign-ups reaching a higher level of threshold. which is the key driving force in Varian (1994). Our Finally, network externality effect can also be negative results are similar to Romano and Yildirim (2001), but when a group-buying deal is offered by a service firm with different reasons in a different context. Unlike constrained by some service capacity. In this case, it is the research on public goods, in our paper free rid- more difficult for a consumer to secure a space when ing on contributions does not exist. A consumer can- a larger number of people are interested in signing on not benefit from the deal without signing up with to the deal. it. Since group-buying arrangements reimburse consumers when a deal is not on, this type of mechanisms belongs to assurance contracts for discrete public 2. A Two-Person Model goods (Bagnoli and Lipman 1989, Tabarrok 1998). Our In this section we develop a two-person and two- paper contributes to this literature by examining the period model to study the firm’s design of a assurance contracts in a dynamic model and, more group-buying mechanism and subsequent consumer specifically, analyzing how the sequential order of responses. In practice, firms may use variations sign-up decisions can affect the success rate. of group-buying formats depending on the specific Our paper is also related to the small but growing products or services being promoted. For instance, theoretical literature on group-buying mechanisms. Kickstarter sets the total amount of demanded dollar In the presence of demand uncertainty, the group- commitment as the threshold, and the project sign-up buying mechanism is shown to outperform posted horizon typically can last for several weeks. We base pricing under demand heterogeneity, economies of our description of the model on group-buying web- scale (Anand and Aron 2003), and risk-seeking sell- sites like Groupon; nevertheless, the results also apply ers (Chen et al. 2007). Chen et al. (2010) compare to settings like Kickstarter where the individual sign- the uniform group-buying price with nonuniform- up amount is not binary. price group-buying mechanisms. Jing and Xie (2011) 2.1. An Illustrating Example explore the role of group buying in facilitating con- We start with a numerical example to illustrate our sumer social interaction and show that group buy- core results and the driving forces behind. Consider ing can dominate other related promotional schemes. a firm selling a group-buying deal to two consumers. Edelman et al. (2010) examine how the vendors The firm earns a positive profit whenever the deal is can potentially use group buying as a mechanism on. If both consumers sign up to the deal, the deal for price discrimination and advertising. Wu et al. s (2013) empirically identify two threshold effects in will be on and each consumer will receive a surplus online group-buying diffusion, induced by the mini- (which is the difference between the deal value and mum sign-up quantity of a deal. Unlike these papers, deal price). Otherwise, the deal is off and each con- we assume the firm has adopted the group-buying sumer receives zero surplus. Each consumer has pri- mechanism and focus on the deal’s success rate under vate information regarding her surplus s; that is, the distinct information disclosure schemes. consumer knows the value of her surplus s, but the Our model and analysis bear some similarities with firm and another consumer only know the probabil- the literature on technology adoption and network ity distribution of s in the market. The distribution externality (e.g., Farrell and Saloner 1985; Katz and follows a four-point discrete distribution as follows: Shapiro 1985, 1986). Research on network external-  sH = $18 with prob. 1 1 ity studies the markets such as telecommunications  4 sM = $7 with prob. 1 1 and computer softwares where a user’s product val- = 4 S 1 uation increases with the number of other users who sL = $5 with prob. 1  4 adopt the same product. The uncertainty on other  0 = 1 s $0 with prob. 4 0 users’ adoptions and hence the uncertainty on the relative value of new technologies may create inertia Finally, a consumer incurs an opportunity cost c = $4 against the adoption of new technologies (Farrell and when signing up to a deal. When a consumer faces Saloner 1985). Positive network effect exists in our uncertainty in the deal’s success, we assume that the setting because a deal is more likely to be on when consumer uses the other consumer’s sign-up likeli- more consumers sign up. Thus, our paper contributes hood as the belief in making sign-up decisions. This Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2808 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS approach is consistent with the equilibrium concepts will sign up as long as her surplus is no less than in multiplayer sign-up games that we analyze in §§3 the opportunity cost, i.e., s ≥ c, with the probability and 4. P4S ≥ c5. Given the sequential nature of the process, The firm chooses from two alternative mechanisms: the deal success rate is the product of the likelihood the simultaneous mechanism under which each con- that the first mover signs up and the conditional sumer signs up separately without knowing the deci- probability that the second mover signs up given the sion of the other, or the sequential mechanism under first mover has already signed up, i.e., P4S ≥ c/Q5 · which one consumer makes a decision first and the P4S ≥ c5. Under the equilibrium, the first mover’s decision is revealed to the second consumer. Before estimate should be consistent with the second con- ∗ analyzing each mechanism, it is useful to provide a sumer’s behavior, i.e., Q = P4S ≥ c5 = 3/4. With a ∗ = 3 benchmark case where two consumers know the val- belief of Q 4 as the second consumer’s sign-up ues of each other’s surplus. Such a full-information probability, the first mover will sign on to the deal H M case leads to the first-best solution, under which the when her surplus turns out to be s or s with a 1 ∗ 3 3 3 9 · H − = · deal’s success rate is equal to · = because the sign-up probability being 2 because Q s c 4 4 4 16 − ∗ · M − = 3 · − ∗ · deal will be on unless at least one consumer has a $18 $4 > 0 and Q s c 4 $7 $4 > 0 but Q i zero surplus. In the first-best solution, whenever both s − c < 0 for i = L1 0. Overall the probability that both consumers will sign up is 1 · 3 = 3 . consumers can gain from the deal (i.e., the positive 2 4 8 surplus s is above the opportunity cost c), the deal We now compare the success rates under alter- will be on. In other words, two consumers can per- native mechanisms. The above results indicate that fectly coordinate on signing up for the deal. the sequential mechanism yields a higher expected success rate than the simultaneous mechanism. First, Under the simultaneous mechanism, when con- under the sequential mechanism, given that the first sumers make the sign-up decision, each needs to esti- mover has signed up to the deal, the second mover’s mate the sign-up likelihood of the other consumer. sign-up probability is 3 . This is higher than individ- Since two consumers are identical, we denote such a 4 ual consumer’s sign-up probability 1 in the simulta- belief by q. A consumer will sign up if her expected 4 neous mechanism. Clearly the second mover under surplus is no less than the opportunity cost, i.e., the sequential mechanism benefits from the certainty. q · s ≥ c. Then the likelihood for each consumer to sign Second, under the sequential mechanism, the first up is P4S ≥ c/q5, and the likelihood for deal success 1 mover’s sign-up probability is 2 , which is higher is P4S ≥ c/q5 · P4S ≥ c/q5, where P denotes probabil- 1 than either consumer’s sign-up probability 4 in the ity. In the equilibrium, the consumer’s sign-up like- simultaneous mechanism. Thus, the sequential mech- lihood needs to be consistent with the consumer’s anism, by eliminating the uncertainty facing the sec- ≥ = original estimate, i.e., P4S c/q5 q. This consistency ond mover, also boosts the confidence of the first condition yields to the equilibrium sign-up likelihood mover. Intuitively, the superiority of the sequential ∗ = 1 ∗ = 1 ∗ · q 4 . To verify it, given q 4 , one can see that q mechanism stems from the need for consumers to H − = 1 · − ∗ · M − = 1 · − s c 4 $18 $4 > 0 but q s c 4 $7 $4 < 0 coordinate their sign-up decisions. Consumers’ sign- ∗ i L 0 and q · s − c < 0 for i = L1 0, because s and s are up decisions are strategic complements with a common M even smaller than s . Thus, a consumer will sign up goal of benefiting from the deal. In the first-best to the deal if and only if the consumer’s surplus turns solution under full information, two consumers can H 1 out to be s with probability 4 . The likelihood that perfectly coordinate. Both the firm and two con- both consumers sign up to make the deal succeed is sumers can benefit from the deal as long as neither 1 · 1 = 1 ∗ = 1 thus 4 4 16 . One can further verify that q 4 is consumer has a zero surplus. Under the sequential the unique consistent estimate under the simultane- mechanism, the second mover faces certainty as in ous mechanism.1 the full-information case, but the first mover still Under the sequential mechanism, one consumer faces uncertainty regarding the second mover’s sur- moves before the other. We denote the first mover’s plus. As a result, coordination fails when the first estimate of the second consumer’s sign-up likelihood mover has S = sL = $5. Finally, under the simultane- by Q. The first mover will sign up if and only ous mechanism, both consumers face the uncertainty if this consumer’s expected surplus is no less than that clearly exacerbates the coordination. In this case, the opportunity cost. This implies the first mover’s even when both consumers have S = sM = $7, the likelihood of signing up as P4S ≥ c/Q5. Next, know- coordination fails and the deal will be off. Thus, the ing that the first mover signs up, the second mover sequential mechanism provides a second-best solution that leads to a better coordination than the simultaneous mechanism by reducing uncertainty for both 1 Note that zero is always a consistent belief under the simultaneous mechanism. It is self-fulfilling that the deal will fail if both consumers. Next we propose analytical models to consumers are convinced that the other consumer will not sign up. investigate the differences between simultaneous and However, this result is trivial. sequential group-buying mechanisms. We conduct Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2809 formal equilibrium analysis with belief perturbations. from the deal at least as high as that of not signing We also study multiple directions of model extensions on to the deal. To sign on to the deal is a commitment to identify the potential boundary conditions for our to purchase if the deal is on. In making the decision, results.2 a consumer takes into consideration her own valuation for the deal as well as her expectation about 2.2. Model Setup the success rate of the group-buying deal. In the two- We consider a market where a firm (either a pro- person case, one consumer uses the other consumer’s ducer or a broker) uses the group-buying format to sign-up likelihood as the belief in making sign-up promote a product or service to consumers. At the decisions, as conditional on signing on to the deal beginning of the first period, the firm posts its group- by herself, the deal’s success is a direct consequence buying deal, which is characterized by three elements: of the other consumer’s sign-up decision. We assume group-buying price w, minimum number of buyers N that consumers form rational expectations and focus required, and a time horizon of two periods for sign- on the pure strategy equilibria of the game. The use ing up. When firms such as Groupon use group- of the rational expectations hypotheses is standard buying mechanisms for promotion, the firms also post in the literature (Frydman 1982, Jerath et al. 2010). a regular price p that consumers would otherwise Specifically, we denote by Hi4q−i5 the likelihood of an have to pay in the market. We assume that the thresh- individual consumer i’s signing on to a group-buying old number N and the group-buying price w are deal, where q−i is consumer −i’s sign-up likelihood exogenously determined. The threshold may be the that consumer i expects. This notation emphasizes minimum number at which the product or service that the likelihood of signing up depends on the provider can enjoy economies of scale and offer the other consumer’s sign-up likelihood, but suppresses discount. Naturally the group-buying price should its dependence on the characteristics of the deal. We be at a discount to the regular price, i.e., w < p. assume the following relation between Hi4q−i5 and q−i. Alternatively, a firm may replace the deal price with Assumption 1 (Sign-Up Likelihood). For all i and the discount level in its announcement. For example, q−i ∈ 601 17, we assume a group-buying deal may offer a 50% discount for a (i) Hi4q−i = 05 = 0; lunch buffet at a popular Japanese sushi restaurant, (ii) Hi4q−i5 is nondecreasing in q−i. regularly priced at $20 per person, on condition that Assumption 1(i) states that if a consumer expects the number of buyers surpasses 150 within one day. with certainty that the other consumer will not sign This is equivalent to a deal price of $10. up, then she expects zero benefits from signing on We consider a two-person model as an efficient way to the deal and will definitely not sign up. Assump- to capture the sequential nature of consumer arrivals tion 1(ii) indicates that the higher the likelihood that a and the interdependence between purchase decisions consumer expects the other consumer to sign up, the of early and late arrivals. The longer the sign-up hori- higher the probability that the consumer will sign up. zon, the more time discounting the early arrival takes If both consumers can fully communicate with each into account in calculating the expected surplus at the other before making sign-up decisions, the first-best end of the horizon. In practice the length of the time solution of the deal’s success rate can be obtained horizon can vary from one day to several months. at H1415 · H2415. However, a full-information structure For example, most deals offered by Groupon expire is uncommon in reality where typical consumers do within 24 hours. In the two-person model, we assume not reveal their own private information to strangers that one consumer arrives in the first period, and the in online group-buying settings. Instead, the firm other in the second period. Each consumer demands considers two alternative group-buying mechanisms: and may purchase up to one unit of the product a sequential mechanism and a simultaneous mech- or service. We limit the threshold N to be 2 in this anism. The firm’s goal is to achieve a higher suc- section, but generalize it in the extensions. The deal cess rate. Higher success rates lead to higher expected succeeds if and only if both consumers sign up for profits when the firm receives a fixed lump-sum the deal, and they receive the product or service at profit for each successful deal and/or a variable price w. Otherwise, the deal is off and no transac- profit for each committed purchase given the deal is tion takes place. The individual product valuation for successful. Given any group-buying mechanism, the consumer i is denoted by Vi, which is drawn from a game consumers play is in nature a coordination game. cumulative distribution function (CDF) Fi4 · 5. To achieve a higher success rate, the firm aims at A consumer decides to sign on to the group-buying achieving a better coordination between consumers. deal if and only if she expects a discounted utility We interpret the success rate in the following sense. Definition 1 (Success Rate). The success rate mea- 2 We thank an anonymous reviewer for offering a similar example sures the ex ante likelihood that a group-buying deal to illustrate the key insights on coordination. is successful before the consumers arrive. The higher Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2810 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS the success rate, the higher the expected payoffs for the desired product at a bargain price, but take the the firm and the higher the expected individual and risk that the deal may not be successful and they total surpluses for consumers. will have to go elsewhere to buy the same product at a regular price and with a delay. For exam- Henceforth it suffices to compare the success rates ple, a consumer who shops around for a camcorder under the two different mechanisms. The distinc- before a family vacation may want to sign up for tion between these two mechanisms is created by a group-buying bargain if time permits, making a the firm’s decision on whether to reveal informa- trade-off between buying from a local store with the tion to the consumer who arrives in the second ability to enjoy it right away, versus waiting for the period; specifically, the second consumer can see the bargain to be realized at a later time with a possi- decision of the consumer who arrives in the first bility that the deal may not be successful. In such period under a sequential mechanism, but not under a case, when a consumer arrives at a group-buying a simultaneous mechanism. When a firm adopts the deal, she will sign up if and only if the present simultaneous mechanism, although the second con- value of the expected surplus resulting from signing sumer arrives and makes the decision later than the on with the group-buying deal is no less than the first consumer, she does not gain any information present value of buying the same product immedi- advantage by arriving late. Not disclosing the deci- ately through another channel. The expected surplus sion of the first consumer makes the process equiv- from signing up for a group-buying deal has two alent to one where consumers simultaneously make components: if the deal turns out to be successful, the decisions under appropriate time discounting adjust- consumer enjoys the product at the bargain price w; ment. Thus, we define the simultaneous and sequen- otherwise, the consumer purchases the product else- tial mechanisms from an information perspective, not where at a later time after the sign-up period ends. based on the sequence of arrivals. Our approach fol- For any consumer i holding a belief of the other con- lows the tradition of Varian (1994). The distinction sumer’s sign-up likelihood being q and an individ- between simultaneous and sequential mechanisms in −i ual product valuation v , the mathematical form of this paper is similar in nature to that between sealed i this trade-off statement is to sign on if and only if and sequential auctions. We assume away observa- vi ≥ w and 6q−i4vi − w5 + 41 − q−i54vi − p57i ≥ 4vi − p51 tional learning where consumers may draw quality where is a discounting factor used by consumer i inferences from observation of peer choices, though i in calculating the value of signing on to the deal. the sign-up pattern under the sequential mechanism The discounting factor can incorporate time discount- is similar to information cascades (Zhang 2010). ing, since the outcome of a success resolves at a lat- 2.3. Micromodeling of Consumer Decisions ter time. From the above condition, we can derive the range of consumer i’s valuation within which To gain granularity on exactly how consumers’ pur- she will sign up for the group-buying deal as fol- chase decisions may depend on the success rate, lows: w ≤ v ≤ p +4 /41 − 554p −w5q 0 If consumer i we define and examine two distinct types of prod- i i i −i has a product valuation higher than p + 4 /41 − 55 · ucts or services that fit into our framework: necessity i i 4p − w5q , then the consumer will resort to an imme- goods and luxury goods. We define necessity goods −i diate access to the product through the alternative as those products considered to be essential for a cer- channel instead of going after the group-buying deal. tain consumption purpose. More specifically, if a con- Let consumer i have the valuation v drawn from sumer does not buy from the group-buying deal, she i a continuous distribution with CDF F 4 · 5, then we will buy the product elsewhere, for example, from a i can derive the probability of consumer i signing up local store. We define luxury goods as those products for the group-buying deal as follows: H 4q 5 = F 6p + or services that consumers do not have to consume. i −i i 4 /41 − 554p − w5q 7 − F 4w5, q ∈ 401 17, which is Examples of such luxury goods might be a dinner at i i −i i −i nondecreasing in q . We see that the consumer belief an expensive restaurant, an hour of pampering at a −i plays an important role in sign-up decisions. fancy spa, or a donation to a music project proposed From the seller’s perspective, H 4q 5 is the sign- by a remote museum. Note that we do not follow the i −i up likelihood for an individual consumer i. The standard income-elasticity approach to define neces- market for necessity goods tends to be very competi- sity versus luxury goods. Our definition is based on tive because retailers as alternative purchase channels whether or not the no consumption is a viable option. may have already enjoyed economies of scale. As a We define these two categories for the purpose of result, the benefit of group-buying deals over pur- illustration, and our results are by no means limited chasing from a retailer as measured by price discount to these special cases. p −w is likely to be limited, and therefore the individ- Example 1 (Necessity Goods). Consumers sign up ual sign-up likelihood can be low. In §2.8, we show for group-buying deals of necessity goods to obtain that the expected low individual sign-up likelihood Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2811 may reduce other consumers’ confidence in the deal’s consumer i’s types, whose distribution is public infor- success, eventually leading to a low success rate of mation. For any consumer i, si4vi5 ∈ 801 19 denotes the group buying for necessity goods. decision rule that consumer i takes to decide whether or not to sign up, given her realized type being v . Example 2 (Luxury Goods). Consumers sign up for i Given consumer i’s sign-up decision s 4v 5 and the group-buying deals of luxury goods to obtain more i i other consumer −i’s sign-up decision s 4v 5, the sur- valuable ways to spend their discretionary income. −i −i plus of consumer i is denoted as 4s 4v 51 s 4v 55. Specifically, we assume that any consumer i will buy i i i −i −i luxury goods if and only if the present value of Definition 2 (Bayesian Equilibrium). The Bayes- ∗ the expected surplus from signing on to a group- ian equilibrium strategy 8si 4vi59 for the simul- buying deal is not less than a predetermined thresh- taneous game is defined by the best-response old a , which is the surplus of her no-purchase option. ∗ i strategy played by each consumer i, si 4vi5 ∈ In practice, the luxury goods sold through group- ∗ arg maxs 4v 5∈801 19EV−i 8i4si4vi51 s−i4V−i559 for all vi0 buying sites are usually the try-it-for-the-first-time i i products or services. The expected surplus of sign- The existence of Bayesian equilibria follows stan- ing up has two components: if the deal turns out dard arguments (Tabarrok 1998). When consumers to be successful, the consumer enjoys the product make sign-up decisions under the simultaneous at price w; otherwise, the consumer takes the no- mechanism, the consumer who arrives in the second purchase option at a later time after the deal turns period faces the same uncertainty in the sign-up prob- off. The sign-up decision for consumer i for this type ability of the other consumer as the one arrives in the first period. The solution scheme for this type of product can be characterized as 6q−i4vi − w5 + 41 − q−i5ai7i ≥ ai. Similar to the case of necessity of game is a threshold strategy: there exists a valu- goods, i is a broadly defined discounting factor for ation range for consumer i such that the consumer consumer i used in calculating the value of sign- signs up if and only if her valuation falls into such ing on to the deal. In the case of buying luxury a range. At the beginning of the game, the ex ante goods through group-buying discounts, the regular belief of the firm on the success rate under a Bayesian ∗ price of this type of product or service is usually equilibrium strategy 8si 4vi59 can be characterized by

∗ P2 ∗ higher than the valuation of consumers. In other q = P4 i=1 si 4Vi5 = 251 which is the success rate at words, for the target consumer i, we have vi − p < equilibrium. Since the revenue of a group-buying site ai, which indicates that the consumer would not depends on the success of deals, from this point purchase the luxury goods at the regular price. We on we compare different mechanisms by the seller’s can then derive the range of the consumer’s val- expected deal success rate. uation of signing up for the group-buying deal as Denote qi the belief of consumer i’s sign-up likeli- ai61 − 41 − q−i5i7/4q−i · i5 + w ≤ vi < p + ai. Given hood held by consumer −i. As a result, consumer 1 the distribution of valuations, Fi4 · 5, for consumer i, signs up with probability H14q25, and consumer 2 we can specify the consumer i’s sign-up likelihood signs up with probability H24q15. Equilibrium is char- ∗ ∗ function for luxury goods as Hi4q−i5 = Fi4p + ai5 − acterized by a pair of beliefs 4q1 1 q2 5 that satisﬁes the Fi4ai61 − 41 − q−i5i7/4q−i · i5 + w5, q−i ∈ 401 17, which is following conditions: nondecreasing in q−i. Note that ai61 − 41 − q−i5i7 ≥ 0 ∗ ∗ ∗ ∗ is the loss of surplus from signing up to a deal that H14q2 5 = q1 1 H24q1 5 = q2 0 turns out to be off at the end. The larger such a loss, ∗ the less likely the consumer will sign on to the deal. Or equivalently, qi is given by

2.4. Equilibrium Analysis Under q = Hi4H−i4q551 i = 11 20 Simultaneous Mechanism When the firm adopts the simultaneous mechanism, The above equations characterize equilibria in the neither consumer is informed of the sign-up deci- sense that any Bayesian equilibrium results in a pair ∗ ∗ sion of the other. As a result, each consumer bases of beliefs 4q1 1 q2 5 that satisfies the equations, and any ∗ ∗ her sign-up decision on her belief in the valuation pair of beliefs 4q1 1 q2 5 that satisfies the equations cor- of the other one. The game is a Bayesian game in responds to a Bayesian equilibrium where consumer i the Harsanyi sense (Harsanyi 1968) where “types” are behaves as if consumer −i signs up for the deal with ∗ defined by valuations. Specifically, the realized type probability q−i. The existence of such a pair of beliefs ∗ ∗ of consumer i is defined by its realized valuation vi, 4q1 1 q2 5, and equivalently, the existence of a Bayesian which is private information known to the consumer equilibrium in pure strategies, is a direct consequence herself but not to the other consumer. Similarly, we of Tarski’s fixed point theorem, since Hi4H−i4q55 is denote by Vi the corresponding random variable of nondecreasing in q ∈ 601 17, i = 11 2. Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2812 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS

Following the preceding discussion, the success rate second consumer makes her decision, there is no more q∗ at equilibrium can be characterized by uncertainty about the future. Given the decision of the first consumer, the optimal strategy for the sec- ∗ ∗ ∗ ∗ ∗ q = P4X14q2 5 + X24q1 5 = 25 = H14q2 5 · H24q1 51 (1) ond consumer can be characterized by a valuation range: to sign up if and only if (a) the first con- where Xi4q5 is a Bernoulli random variable with suc- sumer signs up and (b) her own valuation falls into cess probability Hi4q5, i = 11 2. the range for signing up with the likelihood of the ∗ Note that qi is given by q = Hi4H−i4q55, i = 11 2, so first consumer’s sign-up being 1. After solving the there may exist multiple equilibria. Similar to Jackson best response from the second consumer, we move and Yariv (2007), we categorize equilibria into two backward to the first consumer. Suppose consumer i types, stable equilibria and tipping points, depend- expects that consumer −i will sign up with proba- ing on their sensitivity to minor perturbation in belief. bility q−i. Then consumer i signs up with probabil- Next we provide a formal definition and include an ∗ ity Hi4q−i5 in the first period. At equilibrium q−i is illustration in Figure 1, where 4A11 A25 is a tipping consistent with consumer −i’s sign-up likelihood, i.e., point, and 4B 1 B 5 and 4B0 1 B0 5 are stable equilibria.3 ∗ 1 2 1 2 q−i = H−i415. ∗ Definition 3 (Stable Equilibrium and Tipping From the seller’s perspective, the success rate Qi at ∗ ∗ equilibrium can be characterized by Point). A pair of beliefs 4q1 1 q2 5 constitutes a stable equilibrium (tipping point) if for all i = 11 2, there ∗ ∗ ∗ Q = P4X 4q 5 + X 415 = 25 = H 4q 5 · H 4151 (2) exists 0 > 0 such that for all ∈ 401 05, when con- i i −i −i i −i −i sumer i expects that the other consumer makes her where Xi4q5, X−i415 are Bernoulli random variables decision with the belief q∗ − , consumer i’s sign-up i with success probability Hi4q5 and H−i415, respec- ∗ likelihood will be higher (lower) than qi − ; when tively. It is noteworthy that the equilibrium under consumer i expects that the other consumer makes the sequential mechanism is guaranteed to be unique, ∗ her decision with the belief qi + , consumer i’s sign- and hence it is stable. up likelihood will be lower (higher) than q∗ + . 2.6. Mechanism Design: Simultaneous or 2.5. Equilibrium Analysis Under Sequential?

Sequential Mechanism Given the potential presence of multiple equilib- Under the sequential mechanism, at the beginning ria under the simultaneous mechanism, how can of the second period the firm posts the decision of one compare the success rates under simultaneous the consumer who arrives in the first period. Because and sequential mechanisms? In this paper, we adopt two consumers make decisions sequentially, the first the approach proposed in Jackson and Yariv (2007) consumer needs to make a prediction on the sign- because it allows us to compare the set of equilib- up probability of the second consumer. The sequen- ria regardless of equilibrium multiplicity. First, we tial game we analyze follows the concept of rational formalize our criteria for comparisons of each con- expectations (RE) equilibrium. sumer’s belief regarding the other consumer’s sign- up likelihood by the following definition. Definition 4 (RE Equilibrium). For any realization vi of the valuation for consumer i who moves Definition 5 (Higher Beliefs). For each individ- ∗ first, an RE equilibrium q−i4vi5 in the sequential game ual consumer, one scenario or mechanism generates a satisfies: (i) Consumer i plays an optimal strategy higher belief than another if, for any belief at a stable ∗ of whether to sign up Si 4vi5 ∈ 801 19, given belief equilibrium of the latter, there exists a higher belief at q−i about the sign-up likelihood of consumer −i. a stable equilibrium of the former, and for any belief ∗ (ii) Given the decision Si 4vi5 from consumer i at a tipping point of the latter there is a lower belief and any realization v−i of the valuation of con- at a tipping point of the former or no lower tipping sumer −i. consumer −i plays a best-response strategy points of the former at all. ∗ ∗ ∈ S−i4v−i3 Si 4vi55 801 19. (iii) The belief is consistent We illustrate the comparison criteria defined above − = with the sign-up likelihood of consumer i: q−i in Figure 1. In the figure, given all others the same, ∗ = ∗ ∗ = q−i4vi5 P4S−i4V−i3 Si 4vi55 15. the solid line and the dashed line correspond to Suppose consumer i arrives in the first period and the right-hand side of the equilibrium characteriza- consumer −i arrives in the second period. When the tion equations under the simultaneous mechanism, and the sequential mechanism, respectively. Equi- libria are the intersections of these lines with the 3 In Figure 1, 401 05 is also an equilibrium under the simultaneous mechanism. It is self-fulfilling that the deal will fail if everyone is diagonals. Figure 1(a) illustrates the belief regarding convinced that the other consumer will not sign up. However, this consumer 2’s sign-up likelihood held by consumer 1 result is trivial and thus is not examined in this paper. who moves first under the sequential mechanism, and Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2813

Figure 1 Comparison Between Simultaneous and Sequential mechanism, while there is no tipping point under the Mechanisms in the Two-Person Model sequential mechanism. (a) The first consumer The reason why the sequential mechanism yields 1.0 higher beliefs for both consumers is twofold. First, B 1 with no tipping point, it is more likely for consumers’ 0.9 expectations to move upward to the beliefs at the stable equilibrium. Second, when the stable equilibrium 0.8 B1 is reached, the beliefs regarding the other consumer’s 0.7 sign-up likelihood under the sequential mechanism is higher than that under the simultaneous mechanism 0.6 for both consumers. If both consumers have higher expectations at equilibrium, then each individual is q 0.5 more likely to sign up, and thus the deal is more 0.4 likely to succeed. Consequently, we can compare the deal’s success rate by comparing the belief held by 0.3 each individual consumer. (0, 0) A1 0.2 Definition 6 (Higher Success Rates). One scenario or mechanism generates a higher success rate 0.1 than another if the belief of each consumer in the former is higher than the belief of the consumer in the 0 0 0.2 0.4 0.6 0.8 1.0 latter in the sense of Definition 5. q To formally compare the belief held by each indi- (b) The second consumer vidual under alternative mechanisms, consider f 4q5 1.0 and g4q5 as two functions parameterized by q ∈ 601 17. B 2 Suppose f 4q5 ≥ g4q5 for any q ∈ 601 17. That is, f 4q5 0.9 is always higher than g4q5 pointwisely for any belief q ∈ 601 17. As illustrated in Figure 1(a), for any left-to- 0.8 right crossing point (e.g., point B1) of curve g4q5 with 0.7 the 45 degree line, there always exists a higher cross- B2 0 ing point (e.g., point B1) of curve f 4q5; for any right- 0.6 to-left crossing point (e.g., point A1) of curve g4q5, there exists no lower crossing point of curve f 4q5. q 0.5 Consequently, if f 4q5 ≥ g4q5 for all q ∈ 601 17, we can conclude that mechanism with equilibrium character- 0.4 A2 ization f 4q5 = q yields higher belief than mechanism 0.3 with equilibrium characterization g4q5 = q. Recall that when consumer i arrives first and (0, 0) 0.2 consumer −i arrives later, the belief held by consumer −i, q∗, under the simultaneous mechanism 0.1 i is given by q = Hi4H−i4q55, and the belief held by 0 consumer −i under the sequential mechanism is 0 0.2 0.4 0.6 0.8 1.0 simply 1. As 1 ≥ Hi4H−i4q55 for any q ∈ 601 17, the q sequential mechanism yields a higher belief than the Notes. The parameters are specified as follows: V , i 11 2, follows a uni- i = simultaneous mechanism for consumer i. Similarly, form distribution with support 6151 307, and p = 301 w = 101 = 0041 = 1 2 for the first mover consumer i, as H 415 ≥ H 4H 4q55 005, and a = 3. In this numerical example, 401 05 is a trivial equilibrium, −i −i i for any q ∈ 601 17, the sequential mechanism yields 4A11 A25 is a tipping point, and 4B11 B25 is a stable equilibrium under the 0 0 higher belief than the simultaneous mechanism for simultaneous mechanism; 4B11 B25 is a stable equilibrium under the sequential mechanism. consumer i as well. Given the preceding discussions, we have the following proposition. Proposition 1 (Mechanism Comparison for the Figure 1(b) illustrates the belief of consumer 2. The Two-Person Game). Given all others the same, the figure shows that the simultaneous mechanism yields sequential mechanism always yields higher success rates a lower belief B1 (B2) at the stable equilibrium than the than the simultaneous mechanism. 0 0 stable equilibrium B1 (B2) under the sequential mech- The driving force behind this result is that in anism for consumer 1 (consumer 2). Moreover, there the simultaneous mechanism, each consumer faces exists a tipping point 4A11 A25 under the simultaneous uncertainty about the other consumer when making Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2814 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS decisions. However, in the sequential mechanism, the Table 1 The Payoff Matrix second consumer decides after the uncertainty about Consumer 2 Consumer 2 the first consumer has been resolved. Moreover, in selects leader selects follower anticipation that the second consumer will sign up Consumer 1 selects leader 4q∗1 q∗5 4Q∗1 Q∗5 for the deal without discounting belief in the sign-up 1 1 Consumer 1 selects follower 4Q∗1 Q∗5 4q∗1 q∗5 likelihood, the first consumer’s confidence about the 2 2 second consumer’s sign-up likelihood is consequently boosted. This result may explain why the prevalent If the firm can manage the sequence of arrivals, practice of online group buying nowadays is sequen- ∗ ∗ 4 the firm will compare Q1 and Q2, and investigate the tial mechanisms. conditions under which one of the equilibria leads to higher success rates. Recall that Q∗ = H 4H 4155 · 2.7. Endogenous Sequencing i i −i H 415, i = 11 2. Following Assumption 1, we identify So far we have assumed that the sequence of arrivals −i the following condition under which success rates are of two consumers is exogenously determined. How- −i ever, in some cases group-buying firms can man- higher when consumer moves first than the other age the sequence of arrivals of different cohorts by, way around. (See the proof in Appendix B.) for example, contacting one cohort ahead of another. Proposition 3 (Discounting and Endogenous In such cases, a natural question is which sequence of Sequencing). If Hi415 = H−i415 and Hi4q5 ≤ H−i4q5 for arrivals will yield the higher success rate? To investi- all q ∈ 601 15, then the scenario where consumer −i acts as gate this problem, we add a stage 0 before our current the leader has higher success rates than the scenario where two-person game. In stage 0, two consumers simulta- consumer i acts as the leader. neously decide between making the sign-up decisions first or second. If both consumers decide to move The condition Hi415 = H−i415 says that if the other first or move later, then the simultaneous mechanism consumer is expected for sure to sign up, both con- applies. If one decides to move first and the other sumers share the same likelihood of signing up. The decides to move second, then the sequential mecha- condition Hi4q5 ≤ H−i4q5, q ∈ 601 15 indicates that given nism applies and the game plays out according to the everything else being identical, consumer −i is less determined sequence. When making these decisions, averse to the possibility that the other consumer may the uncertainties regarding consumer valuations and not sign up. In the context of necessity goods (Exam- preferences have not been resolved yet. Finally, we ple 1) and luxury goods (Example 2), one sufficient assume that the individual consumer’s sign-up likeli- condition is that the two consumers have the same hood function Hi4q−i5 is independent of decision time. valuation distribution, i.e., Vi =st V−i, but consumer i In other words, the time discounting does not play a discounts time to a larger extent than consumer −i, significant role as in today’s a-deal-a-day practice. i.e., i ≤ −i. The proposition then predicts that the We summarize the payoff matrix at stage 0 in consumer who discounts less is preferred by all to Table 1. Given the nondecreasing one-to-one corre- move first, given that all the other characters are iden- spondence between the success rate and the payoffs tical between two consumers. In other words, if con- facing consumers and the firm, we can use the success sumer valuations have the identical distribution, the rate as a proxy for the payoff and compare success weaker consumer in the sense of discounting more is rates in the sense of Definition 6. As an immediate preferred by all as the information free riders in the result of Proposition 1, we can identify the equilibria sequential coordination game. This is because if one of stage 0 game. of the two consumers has the chance to decide later when uncertainty regarding the other is resolved, it Proposition 2 (The Two-Person Game). There exist is mutually beneficial to have the consumer who dis- two equilibria 4Q∗1 Q∗5 and 4Q∗1 Q∗5, both of which cor- 1 1 2 2 counts more to enjoy such a luxury to minimize the respond to sequential mechanisms. chances of both walking away from the deal due to uncertainty discounting. 4 Dominance of the sequential mechanism over the simultaneous mechanism can be strengthened if the late consumer can learn 2.8. Comparative Statics for about the value of deal from the sign-up decision of the first person. In such a case, if the first person does not sign on, then the Sequential Mechanism deal is off and any information disclosure from the firm will not In keeping with the prevalent adoption of sequen- affect the outcome. However, if the first person signs up, under tial mechanisms in the current group-buying sites, we the sequential mechanism, the second person may increase the have shown that the sequential mechanism always perceived value of deal and thus the group-buying deal is more likely to succeed. Because such a positive learning effect does not yields higher success rates than the simultaneous exist under the simultaneous mechanism, overall the benefit of the mechanism. But how does the success rate vary sequential mechanism will be even stronger. across different types of products? Which products Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2815 are more suitable for the group-buying selling for- 3. A Multiperiod Extension mat? To explore these issues, we conduct compar- So far we have used a two-person model to demon- ative statics analysis on the success rate under the strate that a sequential mechanism leads to higher sequential mechanism. To derive specific insights, we expected success rates than a simultaneous mecha- resort to the necessity and luxury goods defined nism. In reality a group-buying deal faces a large earlier in Examples 1 and 2. Next we discuss the number of potential consumers arriving at the deal results, leaving formal statements and proofs in in sequence. To implement a sequential mechanism, Appendix A. the firm could periodically post the cumulative num- Our analysis shows that, first, the success rate is ber of sign-ups, with the reporting frequency ranging higher if the products are of less immediate need. from never updating to updating after every period. Luxury goods are more suitable for group buying In this section, we extend our base model to a multi- because consumers are more tolerant of waiting for period model and show that it is optimal for the firm the goods not considered to be necessities. A shorter to update as frequently as possible. deal duration can also help boost the success rate Consider an N -period model where, similar to the because the potential payoff from the deal is less two-person game, exactly one consumer will arrive discounted from waiting. This may partially explain during each period and the preset threshold is equal why we observe lower success rates in the earlier to N ; that is, everyone has to sign up for the deal to wave of group-buying websites a decade ago, whose be on. The firm can choose to release the cumulative deals usually lasted for one to two months (Tang number of sign-ups after every periods, where ≥ 1. 2008). Because the consumers were reluctant to sign In one extreme case, the firm can update the cumula- on due to a long waiting period, the success rate tive number of sign-ups after every period, i.e., = 1. would remain low in spite of the long duration In another extreme case, the firm follows the simulta- for signing up. In contrast, today’s successful firms neous mechanism where = N . like Groupon usually offer deals that expire within We let Vi, i = N 1 N − 11 0 0 0 1 1, denote the valuation 24 hours. Second, a group-buying deal is more likely of the 4N + 1 − i5th arrival, which is drawn from a to succeed when the deal offers a deeper discount. given CDF Fi4 · 5. We let Hi4q5 denote the likelihood For both types of goods, the number of consumers of the 4N + 1 − i5th arrival to sign up for the deal who may potentially sign up to a deal increases with as a function of her belief q on the likelihood that the size of the deal discount. In general, the dis- all remaining consumers sign up. Whereas the valu- count p − w tends to be small for necessity goods ation of each consumer is private information, sign- such as digital cameras because the retail market is up likelihood functions are public information. Under already very competitive. It is hard for group-buying the information updating scheme with frequency , websites to offer a lower price than what is avail- every consumer can be viewed as a cohort because able at retail giants like Walmart or Best Buy that these consumers, although arriving and making deci- have already enjoyed buying power and economies sions sequentially, do not know the valuations or of scale. In contrast, luxury local services like spa sign-up decisions of the other consumers within the and leisure activities are more differentiated and asso- same cohort. Consequently, within each cohort, to any ciated with low variable costs. As a result, the dis- individual, there exists uncertainty on the valuations counts offered by group-buying sites tend to be larger. of the other consumers, and the consumers behave The typical discount level offered by current group- as if the firm implements the simultaneous mech- buying firms, like Groupon and LivingSocial, is 50% anism within each cohort. Effectively the N people off regular prices for service products. Combining the are divided into a number of cohorts, indexed by above results, we conclude that necessity goods are j = 11 21 0 0 0 1 N /, where x is the smallest integer less suitable than luxury goods for the group-buying greater than or equal to x. Given that consumers in all format. These findings may explain the stark contrast previous cohorts sign up, let qj denote the deal’s suc- between the fate of the first-generation group-buying cess likelihood before j more cohorts arrive, namely, sites and the enormous success enjoyed by their the likelihood that all consumers in the remaining counterparts in the current generation. This view is cohorts sign up. Again given that consumers in all echoed in a recent speech by Groupon’s CEO Andrew previous cohorts sign up, denote qj1 i, the belief of Mason: deal’s success rate held by consumer i, i = 4j − 15 + 11 0 0 0 1 min8j1 N 9 within cohort N / + 1 − j. Note I thought about why collective buying sites had failed in the past. 0 0 0 Mercata was from the dot-com era. The that qj1 i is derived from the sign-up likelihoods of more people bought, the lower the price would go. The consumers within the same cohort, and those con- trouble was it took a week to get enough people to sumers from later cohorts. Since the threshold is equal drive the price down. They might buy a camera, but to the number of arrivals, all consumers need to sign they’d have to wait a week. (Cutler 2010) up for the deal to be on. Hence, conditional on that Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2816 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS all consumers in previous cohorts sign up, the deal’s they value the product and that their valuations are success likelihood before j more cohorts arrive can be not influenced by other people’s sign-up decisions. characterized by We further assume that when one cohort of consumers arrive at the deal and need to make sign-up min8j1 N 9 Y decisions, these consumers find out the size of the qj = Hi4qj1 i5 · qj−11 j = 21 0 0 0 1 N /1 (3) i=4j−15+1 cohort and its members’ valuations. Such information becomes available to all consumers within the cohort, Q  i=1 Hi4q11 i51 ≥ 23 but not to the other cohort. This assumption on intra- = q1 (4) cohort communication can be restrictive, but substan-  = H14151 10 tially simplifies our analysis. The model with periodic The belief held by each consumer at equilibrium information disclosure in §3 can be viewed as a sce- should be self-enforced by other consumers’ behav- nario where consumers within a cohort do not share Qmin8j1 N 9 private information. ior. That is, qj1 i = qj−1 · l=4j−15+11 l6=i Hl4qj1 l5, j ≥ 2, i = Q Recall that in the two-person case, each consumer 4j − 15 + 11 0 0 0 1 min8j1 N 9, and q11 i = l=11 l6=i Hl4q11 l5, i = 11 0 0 0 1 . For the firm, the ex ante success rate of a forms a belief in the likelihood of the other consumer group-buying deal with threshold N before the con- signing up for the deal when making the sign-up decision. In the case of the two cohorts, consumers sumers arrive is qN /, which can be computed recur- sively by Equation (3) with the initialization step (4). form beliefs on the number of sign-ups from the 5 We can prove the following result by induction. other cohort. We represent such a consumer belief within cohort i by a discrete probability distribu- Proposition 4 (Periodic Updating). In the multi- tion, denoted by Q−i. Let Hi4Q−i5, i = 11 2, denote period model, given all others being equal, the sequen- the individual sign-up likelihood of cohort i given tial mechanism with = 1 always yields higher success the individual consumer’s belief being Q . Because rates than any periodic updating mechanism with a lower −i it is more convenient to work with tail distributions, frequency. we let qi1 l ≡ P4Qi ≥ l5, l = 11 0 0 0 1 mi, and for conven- The above proposition indicates that in the multi- ≡ tion qi1 mi+1 0. The belief variable Qi can be char- period model, the firm should implement a sequen- acterized by its tail distribution, namely, the vector tial mechanism that updates the cumulative number E = T qi 4qi1 11 qi1 21 0 0 0 1 qi1 mi 5 . (We interchangeably use the of sign-ups after each period. This result is consistent random variable Q and its tail distribution vector qE with observations in practice. when referring to a belief distribution.) We generalize Assumption 1 to the following assumption. 4. A Two-Cohort Extension Assumption 2. The individual sign-up likelihood To further explore the impact of information manage- H 4 5, i = 11 2, is nondecreasing in , in the ment on group-buying deal success rates, in this sec- i Q−i Q−i sense of the first-order stochastic dominance (Shaked and tion we extend the two-person model to a two-cohort model with multiple consumers within each cohort. Shanthikumar 2007). In particular, there are two cohorts of consumers, indexed by i = 11 2. At the beginning of the game, 4.1. Simultaneous vs. Sequential Mechanism the firm knows the probability distribution of the size 4.1.1. Simultaneous Mechanism. Under the sim- of each cohort, but does not know the exact num- ultaneous mechanism, consumers in each cohort base ber. Consumers within each cohort may have different their sign-up decisions on the realized size and val- valuations for the firm’s product or service. Although uations of their own cohort, as well as their beliefs the firm does not know an individual consumer’s val- in the size and valuations of the other cohort. Thus, uation, the firm knows the distribution of valuations cohort sizes and valuations define the “types” for among consumers within each cohort. We denote the the Bayesian game that has been studied. Similar to number of consumers of cohort i by a discrete random the argument in the two-person game, equilibrium is variable Mi, with support 811 21 0 0 0 1 mi9. The individual product valuation for consumers in cohort i, denoted by Vi, is drawn from a given CDF Fi4 · 5. These 5 For donors at Kickstarter.com, we can define the expected individ- two cohorts of consumers can differ in both size and ual donation amount H4Q5 as a function of the other donor’s dona- individual valuations. tion amount Q, according to the distribution of potential donation The distributions for the two cohorts’ sizes and val- amounts and the characteristics of the proposed project. As long as Assumption 2 is satisfied for H4Q5, all results in the subsequent uations of consumers in each cohort are public infor- sections apply except for those specifically related to the Groupon mation, known to the firm as well as to all consumers. context, for example, the coexistence of the discount price and reg- We assume that consumers know exactly how much ular price for a deal. Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2817

∗ ∗ characterized by a pair of belief distributions 4Q11 Q25 4.1.3. Mechanism Comparison. In the two-cohort that satisfies the following conditions: case, the belief held by consumers of cohort i is characterized by a discrete random variable Q−i. M1 Consequently, we need to extend the pointwise dom- X ∗ ≥ = ∗ = P X11 k4Q25 l q11 l1 l 11 0 0 0 1 m11 (5) inance used for comparing mechanisms in the two- k=1 person case to the first-order stochastic dominance. M2 Denote 4Q∗ 1 Q∗ 5 and 4Q∗ 1 Q∗ 5 the pairs of X ∗ ≥ = ∗ = 11 se 21 se 11 sm 21 sm P X21 k4Q15 l q21 l1 l 11 0 0 0 1 m20 (6) equilibrium beliefs held by consumers under the k=1 sequential mechanism and the simultaneous mech- The existence of such a pair of belief distributions is anism, respectively. First, it is easy to show that a direct consequence of the multidimensional version the number of sign-ups from cohort −i under the of Tarski’s fixed point theorem. The success rate q∗ at sequential mechanism always stochastically domi- equilibrium can be characterized by nates the number of sign-ups under the simultane- PM−i PM−i ∗ ous mechanism, i.e., k=1 X−ik4e5E ≥st k=1 X−ik4Qi1 sm5. M1 M2 The inequality is due to Assumption 2 and the fact ∗ X ∗ X ∗ q = P X1k4Q25 + X2k4Q15 ≥ N 1 (7) that the belief of the deal being on with certainty k=1 k=1 is the most promising belief distribution that con- ∗ PM−i sumers can possibly have. As Q−i1 se = k=1 X−ik4e5E ≥st where Xik4Q−i5, i = 11 2, all k, are independent, and PM−i ∗ ∗ PMi ∗ X− 4Q 5 = Q , we have X 4Q 5 ≥ Xik4Q−i5, all k, i = 11 2, are identically distributed k=1 ik i1 sm −i1 sm k=1 ik −i1 se st PMi ∗ Bernoulli random variables with success probability k=1 Xik4Q−i1 sm5 again by Assumption 2. Conse- quently, we obtain that the total number of sign-ups Hi4Q−i5. We use the same notation Xik4Q−i5 for individual sign-up random variables throughout the rest from the two cohorts under the sequential mecha- of the paper. nism is greater than the total number of sign-ups under the simultaneous mechanism in the sense of 4.1.2. Sequential Mechanism. Suppose cohort i first-order stochastic dominance, i.e., for all i = 11 2, arrives in the first period and cohort −i arrives PMi ∗ PM−i PM1 ∗ k=1 Xik4Q−i1 se5 + k=1 X−ik4e5E ≥st k=1 X1k4Q21 sm5 + in the second period. Under the sequential mech- PM2 ∗

k=1 X2k4Q11 sm5, and thus the sequential mechanism anism, the equilibrium behavior can be solved by leads to higher success rates. Therefore, the sequential backward induction. When individual consumers of mechanism should generate higher expected profit for cohort −i make decisions, there is no more uncer- the firm and yield higher expected individual and tainty about the decisions made by consumers of total consumer surpluses. We summarize the above cohort i. Given the number of sign-ups from cohort discussions in the next proposition. i revealed as n, the subgame equilibrium strategy in the second period depends on the number of poten- Proposition 5 (Mechanism Comparison for Two tial subscribers in cohort −i, that is, those consumers Cohorts). Given all others the same, compared with the simultaneous mechanism, the sequential mechanism yields who will sign up with a belief qEi = eE, where eE is a vector with all entries equal to 1. Specifically, if the higher success rates. potential subscribers are no fewer than N − n, then 4.1.4. Endogenous Sequencing. Similar to the these consumers will sign up and the belief for the two-person game, we can investigate the impact of deal to be on will be fulfilled. Otherwise, nobody in endogenous sequencing by adding a stage 0 before the second cohort will sign up and the deal is off. our current two-period game. In stage 0, the firm After solving the subgame equilibrium in the second decides whether two cohorts should arrive simultane- period, we move backward to the first period when ously or sequentially and if sequentially, which cohort cohort i makes sign-up decisions. At equilibrium, the should arrive first. In practice, group-buying firms belief Q−i held by cohort i should be self-enforced. can predict the purchase likelihood and cohort size of ∗ their members for any given product by tracking their That is, any equilibrium Q−i satisfies the condition purchase histories. M−i At stage 0, the firm and cohorts face the payoff ∗ X P4Q−i = l5 = P X−ik4e5E = l 0 (8) matrix exactly the same as the one shown in Table 1. k=1 As a result, in stage 0 of the two-stage game, there ∗ exist two equilibria, both of which correspond to From the seller’s perspective, the success rate Qi at equilibrium can be characterized by sequential mechanisms. Comparing these two equilibria, we identify the conditions under which the equi-

Mi M−i librium where cohort −i should move before cohort i. ∗ X ∗ X Qi = P Xik4Q−i5 + X−ik4e5E ≥ N 0 (9) We summarize the condition in the following propo- k=1 k=1 sition and relegate the proof to Appendix B. Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2818 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS

Proposition 6 (Heterogeneous Cohort Size and of the cumulative number of sign-ups exceeding the

Discounting). If Mi ≥st M−i, Hi4e5E = H−i4e5E and discount schedule Nj can be characterized by Hi4Q5 ≤ H−i4Q5 for all belief distribution Q, then the equi- M1 M2 librium where cohort −i acts as the leader Pareto dominates ∗ X ∗ X ∗ q = P X 4 5 + X 4 5 ≥ N 1 j = 11 0 0 0 1 J 0 the other equilibrium where cohort i acts as the leader. j 1k Q2 2k Q1 j k=1 k=1 Recall that in the two-person game, the consumer 4.2.2. Sequential Mechanism. Suppose cohort i who discounts time to a less extent should be chosen arrives in the first period and cohort −i arrives in to move first. Proposition 6 shows that a similar result ∗ the second period. The equilibrium belief Q held by ≥ −i still holds here. Moreover, the condition Mi st M−i cohort i should satisfy the same condition as that in indicates that cohort i is the larger cohort in the proba- the single-threshold model, which is given by Equa- bilistic sense. Thus, given all others the same, the firm tion (8). The likelihood of the cumulative number of should choose the cohort that is potentially smaller to ∗ sign-ups exceeding the discount schedule Nj , Qij , can move first. This result is somewhat surprising because be characterized by one may want to have the potentially larger cohort to move first to assure the smaller cohort. However, our Mi M−i ∗ X ∗ X result indicates that the firm prefers to have the larger Qij = P Xik4Q−i5 + X−ik4e5E ≥ Nj 1 j = 11 0 0 0 1 J 0 cohort to be the information free riders in the second k=1 k=1 period. Such an arrangement can eventually benefit 4.2.3. Mechanism Comparison. Invoking the both cohorts by boosting the confidence of the ear- same proof as that in §4.1, we can show that the lier cohort, who is in anticipation of potentially larger sequential mechanism results in a higher likelihood later cohort. of the cumulative number of sign-ups exceeding each

threshold level Nj , j = 11 0 0 0 1 J . That is, when a firm 4.2. Multilevel Price Schedule offers a multilevel group-buying deal, a sequential Our main analysis investigates group-buying deals mechanism not only leads to a higher likelihood of with a single threshold. In practice, a group-buying exceeding the lowest threshold and unlocking the deal can have multiple levels of thresholds, and deal but also increases the chance to achieve a higher a higher threshold associated with a deeper dis- level of threshold. As a result, the sequential mech- count. Such multilevel price schedules were com- anism leads to larger expected consumer surpluses. monly offered by the first-generation group-buying We summarize the result in the proposition below. websites (Kauffman and Wang 2002). In this section we extend our analysis of simultaneous ver- Proposition 7 (Mechanism Comparison for sus sequential mechanisms to the group-buying deals Multilevel Price Schedule). If the deal is in the form with multilevel price schedules. Without a careful of a multilevel price schedule, the sequential mechanism analysis, it is unclear if the benefit of sequential mech- always yields a higher likelihood of reaching any threshold anisms on deal success rates will remain when a level than the simultaneous mechanism. deal consists of multiple levels of thresholds. Overall, 4.3. Capacity Limit the analysis of multilevel deals generates additional Group-buying deals are often offered by service firms insights on how the mechanism choice may affect the with capacity constraints. Examples of such service likelihood of achieving different levels of thresholds. firms include restaurants, salons, and art schools. Consider a J -level price schedule where the level Because these businesses earn much higher profit mar- of discounts depends on the total number of cumu- gins from regular customers who do not use group- lative sign-ups X. Specifically, the group-buying price buying deals, they allocate only the excess capacities w4X5 = w if N ≤ X < N , j = 11 0 0 0 1 J , where N ∈ j j j+1 j for group-buying customers. As a result, when a large for all j = 11 0 0 0 1 J , N = and w > w > ··· > w . J +1 1 2 J number of consumers are anticipated to sign up to a Similar to the single-threshold case, consumers form deal that has a capacity limit, the expected value of beliefs on the number of sign-ups from the other the deal for each consumer may decrease because of cohort while making decisions. However, a deal’s the possibility of being rationed. To incorporate such success rate needs to be extended to a vector with a negative externality effect into our model, we con- J elements, which characterizes the likelihood of the sider a group-buying deal with a capacity limit C and cumulative number of sign-ups exceeding each level a minimum threshold N . If the total number of sign- of the discount schedule N , j = 11 0 0 0 1 J . j ups at the end of the second period is greater than C, 4.2.1. Simultaneous Mechanism. Similar to the we assume that C units of services will be randomly single-threshold case, the pair of belief distributions allocated to the subscribers with equal probability. ∗ ∗ at equilibrium 4Q11 Q25 satisfies Equations (5) and (6). In the case with a capacity limit, the character- Under the simultaneous mechanism, the likelihood izations of equilibria and deals’ success rates are Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2819 similar to our basic two-cohort model. That is, the Figure 2 Comparisons of Expected Numbers of Sign-Ups and Success success rate q∗ under the simultaneous mechanism Rates with Various Surpluses of Nonpurchase Option a ∗ PM1 ∗ PM2 ∗ is given by q = P4 k=1 X1k4Q25 + k=1 X2k4Q15 ≥ N 5, (a) 2.5 ∗ ∗ where the pair of equilibrium belief 4Q11 Q25 satisfies Equations (5) and (6). When cohort i arrives in the first period and cohort −i arrives in the sec- 2.0 ∗ ond period, the success rate Qi under the sequential ∗ PMi ∗ mechanism is characterized by Qi = P4 k=1 Xik4Q−i5+ PM−i ∗ k=1 X−ik4e5E ≥ N 5, where Q−i is given by Equation (8). 1.5 However, Assumption 2 no longer holds when a capacity limit is imposed to the deal. With a capacity limit, there are two opposing forces driving the sign- 1.0 up decisions. One is the positive network externality due to the need to coordinate to meet the minimum threshold, and the other is a negative network exter- Expected number of sign-ups 0.5 nality due to the capacity limit. When a consumer makes the sign-up decision, she bases her decision not only on the probability of the deal being on but 0 also on the chance of eventually receiving the service. 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Consequently, for any individual consumer, knowing a that a larger number of people in the other cohort (b) 1.0 will sign up to the deal does not always increase her intention to sign up. 0.9

Under the sequential mechanism, the relative 0.8 strength of these two forces depends on the number of sign-ups in the first cohort. Specifically, if only a 0.7 small number of consumers sign up in the first period,

0.6 the positive network externality dominates. In this case, since rationing is unlikely to occur, consumers in 0.5 the second cohort do not need to discount their per- 0.4 ceived values for the deal. However, if a large number of consumers have signed up for the deal in the Deal’s success rate 0.3 first period, then the negative externality effect due 0.2 to the capacity limit will be much stronger, depressing the expectations of the second cohort. Compared with 0.1 the simultaneous mechanism, the sequential mecha- 0 nism amplifies both types of network effects because 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 consumers of the second cohort observe the decisions a of the first cohort and adjust their expected valuations Notes. The solid and dashed lines correspond to simultaneous and sequen- accordingly. tial mechanisms, respectively. The parameters are specified as follows: Vi , We use the numerical results in Figure 2 to illus- i = 11 2, follow uniform distributions with support 6151 307, and p = 30, w = 10, N = 21 C = 21 1 = 004, and 2 = 007, and the surpluses of nonpur- trate the intuitions regarding the capacity limit. Our chase option a vary from 3 to 7. numerical example concerns a group-buying deal for luxury goods as specified in Example 2. We consider number of sign-ups under the simultaneous mech- three potential consumers, two of them arriving in the anism can exceed that under the sequential mech- first period, and one arriving in the second period. anism. Thus, the sequential mechanism no longer Both the deal threshold and capacity limit are equal dominates the simultaneous mechanism in terms of to 2, i.e., N = C = 2. We plot the expected number the number of sign-ups. In essence, our model with- of sign-ups and the deal success rate in Figures 2(a) out capacity limits is a pure coordination game with and 2(b), respectively. Given all other parameters the positive externality effect only. Capacity constraints same, the negative externality effect increases when introduce negative externalities through competition the surplus of the nonpurchase option a decreases. among consumers that reduce the information benefit Figure 2(b) shows that the success rate under the associated with the sequential mechanism. The extent sequential mechanism still dominates that under the of competition among consumers and the resulting simultaneous mechanism. However, as shown in Fig- negative externality effect is stronger with a higher ure 2(a), when a is sufficiently small, the expected sign-up likelihood from each individual consumer. Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2820 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS

5. Conclusion sequential mechanism, which may amplify the nega- This paper studies optimal group-buying mechanisms tive externality effect, could lead to a lower number in a two-period game where cohorts of consumers of sign-ups when the firm faces a stringent capacity arrive at the deal sequentially. The two cohorts can- constraint. not fully communicate with each other about their sizes and consumer valuations. Instead, the firms Acknowledgments use group-buying mechanisms to coordinate. In par- The authors thank the department editor, the associate edi- ticular, we examine the success rate of a group- tor, and two anonymous reviewers for their guidance and buying deal under two alternative mechanisms: a constructive comments. The authors also thank Juanjuan sequential mechanism and a simultaneous mecha- Zhang, Kinshuk Jerath, Ashutosh Prasad, Cuihong Li, Ying-Ju Chen, seminar participants at the University of nism. Our analysis shows that, all other things being Toronto Operations Management seminar series, and con- equal, a sequential mechanism dominates a simul- ference participants at the Manufacturing and Service Oper- taneous mechanism. Interestingly, posting the cumu- ations Management Conference 2011 at the University of lative number of sign-ups from the first period can Michigan, the Revenue Management Conference 2011 at reduce uncertainty and thus increase the expected Columbia University, the Summer Institute in Competitive sign-ups among the second cohort of consumers. The Strategy Conference 2011 at the University of California at increased second-period sign-ups can in turn improve Berkeley, and the Frontiers of Research in Marketing Sci- the confidence among the first cohort of consumers ence Conference 2012 at the University of Texas at Dallas and lead to a higher expected number of sign-ups in for their helpful suggestions on this paper. the first period, thus further increasing the deal’s suc- Appendix A. Comparative Statics for Sequential cess rate. This backward-inductive perspective, start- Mechanism in a Two-Person Model ing from the second period and going back to the Consider two sets of signing up probability functions H = first period, is crucial to understanding the intu- ˜ ˜ 4Hi5 and H = 4Hi5, which are sustained by two distinct ition behind our result. The driving force behind our ˜ group-buying deals. Suppose that Hi4q5 ≥ Hi4q5 for all q ∈ result is that consumers essentially play a coordina- ˜ ∗ ˜ ˜ ˜ 601 17 and i = 11 2, then we have Qi = Hi4H−i4155H−i415 ≥ ∗ tion game, and the sequential mechanism, with infor- Hi4H−i4155H−i415 = Qi , i = 11 2. This implies that one can mation revealed by one cohort to the other, allows for compare the success rates of two group-buying deals by a better coordination. simply comparing the individual sign-up probability. We This result is consistent with the observed domi- summarize the results in the following corollaries. nance of sequential mechanisms in practice. Broadly Corollary 1 (Time Discounting). Consider either neces- speaking, the superiority of the sequential mechanism sity goods or luxury goods. A deal H˜ yields higher success rates can also be implemented by exploiting communica- than another deal H under the sequential mechanism, if, ceteris tion among consumers who are adjacent in a social paribus, the deal H˜ has higher values of the discount rate (less network. Furthermore, our analysis provides useful discounting) than those of the deal H for each cohort, i.e., ˜i ≥ i, guidance on how a group-buying firm may properly i = 11 2. arrange the sequence of consumer arrivals to increase Proof of Corollary 1. By Assumption 1(i), it is suffi- ˜ the success rate and the expected number of sign-ups. cient to show that Hi4q5 ≥ Hi4q5, i = 11 2 for all q ∈ 401 17. Having the smaller cohort come first and the larger Recall that, for the necessity goods, Hi4q5 is in the form cohort come later, or having the cohort who discounts of Hi4q5 = Fi4p + 4i/41 − i554p − w5q5 − Fi4w5 for q ∈ 401 17. ˜ time to a less extent come first and the cohort who If ˜i ≥ i, ceteris paribus, then Hi4q5 ≥ Hi4q5 for q ∈ 401 17, discounts more come later, can increase the deal’s suc- since the function Fi4p + 4i/41 − i554p − w5q5 is increas- cess rate. ing in i. For the luxury goods, Hi4q5 is in the form = + − − − · + We enhance our key insights through a number of of Hi4q5 Fi4p ai5 Fi4ai61 41 q5i7/4q i5 w5 for ∈ ˜ ≥ ˜ ≥ model extensions. First, when a group-buying deal q 401 17. If i i, ceteris paribus, then Hi4q5 Hi4q5 for q ∈ 401 17, since the function −F 4a 61 − 41 − q5 7/4q · 5 + w5 requires the entire set of consumers to participate in i i i is increasing in . a multiperiod model, we find that it is optimal for i the firm to update the sign-up numbers as frequently Corollary 2 (Price Discount). Consider either necessity as possible. Second, when the firm offers a multilevel goods or luxury goods. A deal H˜ yields higher success rates group-buying deal that hands out greater benefits for than another deal H under the sequential mechanism, if, ceteris reaching higher levels of thresholds, the sequential paribus, the deal H˜ has a higher regular price and a lower group- mechanism not only leads to a higher deal success buying price than those of the deal H, i.e., p˜ ≥ p and w˜ ≤ w. rate, but also increases the chance to reach higher lev- Proof of Corollary 2. By Assumption 1(i), it is suf- els of thresholds. Third, when the group-buying firm ˜ ficient to show that Hi4q5 ≥ Hi4q5, i = 11 2 for all q ∈ faces a capacity limit, negative externality may arise 401 17. Recall that, for the necessity goods, Hi4q5 is in because the limited capacity has to be rationed among the form of Hi4q5 = Fi4p + 4i/41 − i554p − w5q5 − Fi4w5 the consumers who have signed up. As a result, the for q ∈ 401 17. If p˜ ≥ p and w˜ ≤ w, ceteris paribus, then Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms Management Science 59(12), pp. 2805–2822, © 2013 INFORMS 2821

6w1˜ p˜ + 4i/41 − i554p˜ − w5q7˜ ⊇ 6w1 p + 4i/41 − i554p − w5q7. reduced to our basic two-person model, and thus we have ˜ 1 2 1 N Consequently, Hi4q5 ≥ Hi4q5 for all q ∈ 401 17. For the lux- q2 ≥ q1 . Suppose we have qN ≥ q1 for some N ≥ 2, we 1 N +1 ury goods, Hi4q5 is in the form of Hi4q5 = Fi4p + ai5 − want to show that qN +1 ≥ q1 . By the recursive deﬁnitions, N +1 QN +1 N +1 N +1 N +1 Fi4ai61 − 41 − q5i7/4q · i5 + w5 for q ∈ 401 17. If p˜ ≥ p and w˜ ≤ we have q1 = i=1 Hi4q11 i 5 = q11 N +1HN +14q11 N +15 and ˜ ˜ 1 1 1 N N w, ceteris paribus, then 6ai61 − 41 − q5i7/4q · i5+w1 p+ai7 ⊇ qN +1 = HN +14qN 5qN ≥ HN +14q1 5q1 , where the last inequality ˜ 6ai61 − 41 − q5i7/4q · i5 + w1 p + ai7. Consequently, Hi4q5 ≥ is given by the induction result in the previous step. To this N N +1 Hi4q5 for all q ∈ 401 17. end, all we need to show is that q1 ≥ q11 N +1. From the deﬁ- nitions, we have Corollary 3 (Nonpurchase Option). Consider the lux- N ury goods. A deal H˜ yields higher success rates than another deal N +1 Y N +1 q11 N +1 = Hi4q11 i 51 and H under the sequential mechanism, if, ceteris paribus, the deal i=1 H˜ has a higher sum of the regular price and nonpurchase value, N and a lower nonpurchase value than those of the deal H for each N +1 N +1 Y N +1 q11 i = HN +14q11 N +15 · Hl4q11 l 51 i = 11 0 0 0 1 N 1 cohort, i.e., p˜ + a˜i ≥ p + ai and a˜i ≤ ai, i = 11 2. l=11 l6=i

Proof of Corollary 3. By Assumption 1(i), it is suffi- N ˜ N = Y N cient to show that Hi4q5 ≥ Hi4q5, i = 11 2 for all q ∈ 401 17. q1 Hi4q11 i51 and i=1 Recall that, for the luxury goods, Hi4q5 is in the form of = + − − − · + ∈ N Hi4q5 Fi4p ai5 Fi44ai61 41 q5i75/4q i5 w5 for q Y qN = H 4qN 51 i = 11 0 0 0 1 N 0 401 17. If p˜+a˜i ≥ p+ai and a˜i ≤ ai, ceteris paribus, then 6a˜i61− 11 i l 11 l l=11 l6=i 41 − q5i7/4q · i5 + w1 p˜ + a˜i7 ⊇ 6ai61 − 41 − q5i7/4q · i5 + ˜ w1 p + ai7. Consequently, Hi4q5 ≥ Hi4q5 for all q ∈ 401 17. N +1 N N +1 N As q11 i ≤ q11 i1 i = 11 21 0 0 0 1 N , we have q11 N +1 ≤ q1 , and thus we obtain the announced result. Appendix B. Proofs For any , we prove by induction on j that q1 ≥ Proposition 3 is a special case of Proposition 6, where M , min8j1 N 9 i q for j = 11 0 0 0 1 N /. The desired result can be obtained i = 11 2, are both equal to 1, and the predetermined thresh- j when j = N /. If j = 1, the last cohort under the informa- old N is 2. As a result, we include proof for Proposition 6, tion release scheme with frequency behaves exactly the but not for Proposition 3 to avoid repetition. same as under the simultaneous mechanism. Consequently, Proof of Proposition 4. For convenience, we focus on 1 the dominance of qmin81 N 9 over q1 is a direct consequence the largest stable equilibrium; nevertheless, the result holds 1 of Corollary 4. Suppose we have qmin8j1 N 9 ≥ qj for some generally in the sense of Definition 6, because what we will 1 j ≥ 1, we want to show that q + ≥ qj+1. By Equa- verify is essentially the pointwise dominance needed for min84j 151 N 9 tion (3), q1 = Ql H 4q1 5 · q1 1 l = j + 11 0 0 0 1 min84j + 15 · the equilibrium comparison. We start by showing the dom- l i=j+1 i i−1 j = Qmin84j+151 N 9 · = inance of the sequential mechanism with = 1 over the 1 N 9 and qj+1 i=j+1 Hi4qj+11 i5 qj 1 where qj+11 i Qmin84j+151 N 9 simultaneous mechanism (i.e., = N ) in the multiperiod qj · l=j+11 l6=i Hl4qj+11 l5. Following the proof of Corollary 4, model, which is summarized in the following corollary. This we can show, by induction on each individual consumer result will be repeatedly invoked in the rest of the proof. If l = 11 0 0 0 1 within the 4N / + 1 − 4j + 155th cohort, that 1 the firm adopts the sequential mechanism with = 1, Equa- qmin84j+151 N 9 ≥ qj+1. 1 1 1 tions (3) and (4) reduce to qi = Hi4qi−15qi−1, for i = 21 0 0 0 1 N 1 1 Proof of Proposition 6. It is easy to verify that the and q1 = H1415. When the firm adopts the simultaneous mechanism (i.e., = N ), the ex ante success rate is character- stipulations of Proposition 6 provide a sufficient condi- N QN N 1 ≥ ized by q1 = i=1 Hi4q11 i5, for N ≥ 21 and q1 = H1415, where tion for the following conditions: Mi st M−i, and for all N QN N Q, H 4e56E 1 − H 4Q57 ≥ H 4e56E 1 − H 4Q57 as inequality (H1), q11 i = l=11 l6=i Hl4q11 l51 i = 11 0 0 0 1 N . i i −i −i 61 − Hi4e57HE i4Q5 ≤ 61 − H−i4e57HE −i4Q5 as inequality (H2). Corollary 4. In the multiperiod model, given all others Hence, it suffices to prove the desired result under these 1 N being equal, qN ≥ q1 . more general conditions. Recall that Xik4Q5 is a Bernoulli Proof of Corollary 4. First, we make the following random variable with success probability Hi4Q5 and Xik4e5E is N N +1 E observation: q11 i ≥ q11 i , i = 11 0 0 0 1 N . That is, for any con- a Bernoulli random variable with success probability Hi4e5. sumer i, her belief regarding the likelihood that the rest will Hence, sign up for deals is nonincreasing in the market size N .  1 with prob. H 4e56E 1 − H 4Q571 This is intuitive in the sense that if the 4N + 15th con-  i i sumer signs up the deal with certainty, then the uncer- Xik4e5E − Xik4Q5 =st −1 with prob. 61 − Hi4e57HE i4Q51  tainty facing the other consumers is the same as the case 0 otherwise. when the market size is N . However, generally, the likelihood that consumer N + 1 signs up for the deal is less Given inequality (H1), we know that Xik4e5E − Xik4Q5 has a than 1, so the other consumers’ beliefs regarding the deal’s larger mass in value of 1 than X−ik4e5E − X−ik4Q5. By inequal- success likelihood at their stage of decision making given ity (H2) alone, Xik4e5E −Xik4Q5 has a smaller mass in value of that all consumers in previous cohorts have signed up, −1 than X−ik4e5E − X−ik4Q5; equivalently, Xik4e5E − Xik4Q5 has will decrease with the introduction of consumer N + 1. a larger mass in values of 0 and 1 combined than X−ik4e5E 1 N Next we prove by induction on N that qN ≥ q1 . It is − X−ik4Q5. Hence, under conditions (H1) and (H2), we obvious for N = 1. If N = 2, the multiperiod model is have Xik4e5E − Xik4Q5 ≥st X−ik4e5E − X−ik4Q5. Furthermore, if Hu, Shi, and Wu: Simultaneous vs. Sequential Group-Buying Mechanisms 2822 Management Science 59(12), pp. 2805–2822, © 2013 INFORMS

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