Duopoly Competition with Network Effects in Discrete Choice Models

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Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org Duopoly Competition with Network Effects in Discrete Choice Models

Ningyuan Chen, Ying-Ju Chen

To cite this article: Ningyuan Chen, Ying-Ju Chen (2021) Duopoly Competition with Network Effects in Discrete Choice Models. Operations Research 69(2):545-559. https://doi.org/10.1287/opre.2020.2079

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Crosscutting Areas Duopoly Competition with Network Effects in Discrete Choice Models

Ningyuan Chen,a Ying-Ju Chenb,c a Rotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada; b School of Business and Management, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; c School of Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Contact: [email protected], https://orcid.org/0000-0002-3948-1011 (NC); [email protected], https://orcid.org/0000-0002-5712-1829 (Y-JC)

Received: August 9, 2018 Abstract. We consider two firms selling products to a market of network-connected cus- Revised: March 9, 2019; October 25, 2019 tomers. Each firm is selling one product, and the two products are substitutable. The cus- Accepted: August 5, 2020 tomers make purchases based on the multinomial logit model, and the firms compete for their Published Online in Articles in Advance: purchasing probabilities. We characterize possible Nash equilibria for homogeneous network February 24, 2021 interactions and identical firms: When the network effects are weak, there is a symmetric fi fi equilibrium that the two rms evenly split the market; when the network effects are strong, Subject Classi cations: games/ fi noncooperative; marketing/choice models; there exist two asymmetric equilibria additionally, in which one rm dominates the market; marketing/competition interestingly, when the product quality is low and the network effects are neither too weak nor Area of Review: Revenue Management too strong, the resulting market equilibrium is never symmetric, although the firms are ex ante symmetric. We extend these results along multiple directions. First, when the products have https://doi.org/10.1287/opre.2020.2079 heterogeneous qualities, the firm selling inferior product can still retain market dominance in Copyright: © 2021 INFORMS equilibrium due to the strong network effects. Second, when the network effects are heterogeneous, customers with higher social influences or larger price sensitivities are more likely to purchase either product in the symmetric equilibrium. Third, when the network consists of two communities, market segmentation may arise. Fourth, we extend to the dynamic game

when the network effects build up over time to explain the ﬁrst-mover advantage.

Funding: The research of N. Chen is partially supported by Research Grants Council of Hong Kong [Early Career Scheme 26201617]. The research of Y-J. Chen is partially supported by Research Grants Council of Hong Kong [General Research Fund 16503918]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2020.2079.

Keywords: duopoly competition • network effects • multinomial logit model • Cournot competition

1. Introduction externalities, the aforementioned industries are Network effects are widely observed when human highly profitable, and network effects lift the po- beings engage in social interactions. They arise from tential and competition there to the next level. the “payoff externality,” in the sense that an indi- Competition over networks gives rise to various vidual’s payoff depends not only on her own action, phenomena that could not be observed in conven- butalsoontheactionschosenbyothers.Examples tional industry structures. In particular, it is widely abound, from the classical videotapes, keyboards, believed that network effects create the winner- operating systems, and telecommunication networks, take-all phenomenon (Bruner 2014): A product to recent online games, cloud services, and mobile (such as Dropbox) may capture a dominant market apps. For instance, Dropbox allows users to conve- share over its seemingly identical alternatives (such niently share their filesacrossplatforms;usersare as OneDrive). It is also pointed out in several studies likely to choose Dropbox over other service providers (Banerji and Dutta 2009, Bimpikis et al. 2016)that (e.g.,GoogleDriveandOneDrive)ifmanyoftheir the network structure is related to market segmen- colleagues have already adopted Dropbox. As an- tation: Product adoption may vary tremendously other example, an increasing amount of users on one from segment to segment in the network (e.g., side of a two-sided market (such as Airbnb) attracts WeChat in China, Facebook’s Messenger in North more users on the other side and, thus, generates America, and WhatsApp in other regions; see Sevitt network effects for the users on the same side. 2016). Such market dominance and market segmen- Because network effects largely enhance individ- tation do not arise if the market were monopolized uals’ willingness to pay through the cascade of by a single firm.

This paper attempts to gain deep understanding To understand the intuition, note that when the of the firms’ competitive strategies when customers’ network effects are not strong, customers’ purchas- purchasing decisions are influenced by network effects. ing decisions are primarily driven by their intrinsic We pay particular attention to the emergence of market consumption values. Thus, given that the firms are ex dominance and market segmentation in the form of ante symmetric, they equally split the market. When asymmetric equilibria. In pursuit of this goal, we con- the network effects are very strong, however, the sider two firms that sell their substitutable products network effects start to take over. This, in fact, ho- to a market of network-connected customers. The mogenizes the products, because now the firms pri- customers choose between the two products or leave marily compete on attracting the critical mass of the market without purchasing. They make their customers. In this vein, equal splitting becomes un- decisions based on price, quality,andthe anticipated stable, and the market structure is geared toward one network effect, in the sense that they are influenced by firm dominating the other. Notably, this asymmetric the choices of their neighbors in the network. We equilibrium is also more efficient in terms of boosting describethechoiceprocessbyamultinomiallogit customers’ willingness to pay. (MNL) model (Anderson et al. 1992, McFadden 2001). Under the same assumption of homogeneous cus- The MNL model has been very successful in cap- tomers and products, when the product quality is low turing discrete choices, due to its analytical tracta- and the network effects are neither too weak nor too bility and interpretability. We focus on the Cournot- strong, the resulting market equilibrium is never sym- type competition: The firms’ decision variables are metric,evenifthefirms are ex ante symmetric. This customers’ choice probabilities, or, equivalently, the result is somewhat paradoxical, but particularly ro- market shares, rather than the prices. We provide bust, because it implies the nonexistence of even practical and theoretical justifications for this mod- unstable symmetric equilibria. In this sense, our result eling choice in Section EC.3.1 of the online appendix. demonstrates a strong rebuttal to the conventional, Despite the parsimony of our model, the equilib- and perhaps na¨ıve, intuition that symmetric equi- rium analysis turns out to be highly intractable. The libria shall exist for symmetric firms. We are not complexity can be attributed to three sources: the aware of any prior work that proves this in the context network interactions, the nonconcave payoff function of network effects with competition. arising from the MNL model, and asymmetric equi- We next consider products with heterogeneous qual- libria. To the best of our knowledge, the first issue ities. When the network effects are strong enough, we has only been addressed under models with special establish the existence of two Nash equilibria corre- structures (such as Hotelling’s model and the linear- sponding to the respective market-dominance positions quadratic framework; see Section 2 for more details) of each firm, regardless of their quality difference.This in the literature, and the second and third issues have implies that, even though one product is far inferior to not been addressed in this context. To circumvent the its competitor, it may still retain market dominance difficulty and find the pure-strategy Nash equilibria due to the strong network effects. This could happen, of the duopoly competition, we adopt a novel ap- for example, if the product has a first-mover advan- proach that focuses on the inverses of the best- tage. Some classical examples include the battle be- response functions (see Section 4). It allows us to tween VHS and Beta in the videotape industry and obtain analytical results, whereas selecting best re- the victory of the QWERTY keyboard over others.1 sponses from the local maxima of a nonconcave Our second result justifies the investment in quality function is virtually impossible. improvement: If the quality difference is sufficiently We show that, depending on the products’ qualities large, the superior product can penetrate the network and the strength of the network effects, the Nash equi- effects and secure its dominating position, which is the libria exhibit highly distinct features. When the prod- only equilibrium outcome of the duopoly competition. ucts are symmetric and customers are homogeneous— When the network effects are heterogeneous among that is, the network that connects customers is a customers, we show that the firms are more likely to sell complete graph—a single symmetric Nash equilib- to customers with higher social influences and larger rium arises if the network effects are weak. At the price sensitivities in the symmetric equilibrium. We also other extreme, when the network effects are strong study networks consisting of two communities—that is, enough, there exist three Nash equilibria: two stable the connectivity inside each community is homoge- asymmetric Nash equilibria, in which one firm cap- neous, which is different from the externality between tures almost all the entire market and the other firm is communities. Such a network structure encompasses left with little market share, and an unstable sym- many special networks that are investigated in the metric Nash equilibrium. The stable equilibria exhibit literature, such as star graphs and complete bipartite some form of market dominance, and it emerges be- graphs. We show that market segmentation may arise cause of strengthened competition. under the network effects: One product is dominating Chen and Chen: Duopoly Competition with Network and Discrete Choices Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS 547 in one community, while the other product is popular market dynamics. For example, Laffont et al. (1998) in the other community. This market segmentation study duopoly competition for vertically differenti- emerges if the network effects inside communities are ated products in Hotelling’s model. They show the relatively strong in comparison with those between existence of two asymmetric customer equilibria, in communities. In the opposite case, wherein the net- which the market is cornered by one of the firms, in work effects between communities is stronger than addition to a symmetric equilibrium. See also Einhorn those within communities, the aforementioned seg- (1992), Grilo et al. (2001), and Katz and Shapiro (1992). mentation is no longer sustainable. See Farrell and Klemperer (2007) for a comprehensive Finally, the game is extended to a dynamic setting, review of earlier papers. when customers cannot coordinate perfectly and the In a more recent paper, Aoyagi (2018)alsodem- network effects build up over time. We show the onstrates the multiplicity of Nash equilibria in the existence of the pure-strategy, open-loop Nash equilib- customers’ buying subgame. His model uses a graph rium. The steady state of the dynamic game is closely to describe the network effects, and the two firms can related to the Nash equilibrium of a static game with price-discriminate. In contrast, Chen et al. (2018) transformed parameters. We also provide an expla- characterize the unique Nash equilibrium and the nation for the first-mover advantage using the dy- associated prices and consumptions in a closed form, namic game. using quadratic utility functions and linear network The rest of this paper is organized as follows. effects. Banerji and Dutta (2009)studymarketseg- Section 2 reviews some relevant literature. Section 3 mentation when two firms use uniform pricing for introduces our model setup. In Section 4,weconsider their network goods. In terms of model assumptions, the case with homogeneous customers—that is, the the setups in Calzada and Valletti (2008) and Tan and network is a complete graph. We discuss the possible Zhou (2021) are close to ours. They consider Bertrand- equilibria with both symmetric and heterogeneous type network competition under the MNL model firms. Section 5 discusses heterogeneous network (or more general choice models). However, they only effects. In Section 6, we investigate the dynamic study symmetric equilibria and cannot obtain the version of the game. Section 7 concludes. All proofs insights provided by our model, such as the nonex- are included in the online appendix. istence of symmetric equilibria when the network effects are neither too weak nor too strong. In a setting 2. Literature Review similar to ours, Feng and Hu (2017)observethat The study of the network effect and its influence on market concentration increases with the network economic activities has attracted many researchers in effects. Different from us, their model does not use the the last few decades. Because of the seminal paper by MNLmodelforcustomerchoice. Katz and Shapiro (1985), monopoly pricing for net- Our paper differs from the previous literature in the work goods has been studied extensively in various following ways. First, we introduce customers’ dis- settings; see Economides (1996)forareview.Re- crete choice modeling into competition. Customers cently, local network effects have become a central donotselectaproductsimplybecauseitgeneratesa research topic in this area (see, e.g., Ballester et al. higher utility than its competitor; instead, a choice 2006). As opposed to global network effects, local net- probability is assigned to each product, as well as the work effects capture the heterogeneous interaction be- no-purchase option. Second, in some previous pa- tween any pair of agents in the network and, thus, allow pers, the multiplicity of Nash equilibria arises from for the analysis of agents’“centrality” measure. For customers’ buying subgame, as customers have dif- example, Candogan et al. (2012), Bloch and Quérou ferent expectations of the network effects. In contrast, (2013), and Fainmesser and Galeotti (2015)investi- we focus on Cournot-type competition rather than gate the pricing problem when customers have local price competition, following papers such as Katz and network effects. Hu et al. (2015)incorporateso- Shapiro (1985), Economides (1996), and De Palma and cial influence into the newsvendor model and study Leruth (1996). As a result, there is no purchasing marketing strategies to leverage the network effect. subgame in our model and the source of the multi- Abeliuk et al. (2015) and Maldonado et al. (2018) plicity of Nash equilibria is the strategic interactions study a monopoly offering multiple products to cus- between the firms. Third, we identify a scenario (see tomers with social influence and position bias. Proposition 4)inwhichno symmetric equilibrium exists Our paper is related to a stream of literature that for symmetric firms. This happens because the net- investigates competition and network effects. Net- work effects make the firms’ best response discon- work effects create complementarity between cus- tinuous. To the best of our knowledge, this phe- tomers’ choices and may significantly change the nomenon has not been analyzed in the literature. Chen and Chen: Duopoly Competition with Network and Discrete Choices 548 Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS

As aforementioned, we use the multinomial logit customers form their choice probabilities by the model for customers’ discrete choices; see Anderson MNL model. More precisely, customer i’schoice et al. (1992) and McFadden (2001)forcomprehensive probability of product X and Y, denoted xi and yi, reviews. In this regard, our paper is related to the following satisfies2 fi {}∑ studies. Du et al. (2016) study the pro t-maximization X X n exp α − β p + g x problem of a monopoly offering a set of substitutable {}i i j 1 ij j xi ∑ , (1) X X n products when customers have network effects. The 1 + exp α − β p + g x + {}i i j 1 ij j customers make their choices according to the MNL ∑ αY − β Y + n model. Like us, they transform the decision variable exp ipi j1 gijyj {}∑ from pricing to market share. Wang and Wang (2016) Y Y n exp α − β p + g y show that assortment optimization under the MNL {}i i j 1 ij j yi ∑ . (2) model with network effects is NP-hard, but a simple + αX − β X + n + 1 exp ipi j1 gijxj heuristic can achieve near-optimal performance. {}∑ αY − β Y + n Cohen and Zhang (2017)usetheMNLmodelto exp ipi j1 gijyj capture the decision process of riders choosing between platforms. Anderson et al. (1992), Bernstein The MNL model has been very popular and successful and Federgruen (2004), Gallego et al. (2006), Aksoy- in modeling customer choices. It can be rationalized Pierson et al. (2013), and Gallego and Wang (2014) by a random utility model with extreme value distri- establish various conditions for the existence and butions. More precisely, the utility of product X (Y)to αX − uniqueness of Nash equilibria for the MNL model and customer∑ i is the sum of a deterministic∑ part β X + n αY − β Y + n its generalizations, such as the mixed multinomial ipi j1 gijxj (or ipi j1 gijyj)andarandom logit model. They do not consider network effects utility following a standard Gumbel distribution. among customers. Brock and Durlauf (2001)inves- The deterministic part features the quality of the tigate the equilibrium properties when individuals product itself, the price, and the network effects. make choices in the presence of social interactions, We refer the readers to Anderson et al. (1992)for according to the MNL model. They find that there more details. can be one or three choice levels in equilibrium, We investigate personalized pricing here—that is, depending on the parameter values. Although this ob- different prices are set for different customers. Given servation seems similar to our Proposition 2 and the unique positions of customers in a network, Proposition 3,ourmodelandfindings differ, in that personalized pricing is more natural within the the- there is no competition between firms involved in oretical framework and generates more profits. This their model, and, as mentioned above, symmetric is why it is the focus of many papers in this area equilibrium may not exist in our setup. (Candogan et al. 2012,BlochandQuérou2013). Personalized pricing is also made increasingly fea- 3. The Model sible for digital services and products. For example, Consider two firms, X and Y, selling their substitut- Dropbox used to provide free storage for customers able products to a market of n customers. We use making referrals, which can be translated to lower superscripts X and Y to denote the features of their prices. These are the reasons behind our modeling respective products. The utilities and choices of choice. Nevertheless, some of our results apply to customers are subject to the network effects—that uniform pricing; see Section 4 and Section EC.3.4 in is, customer j imposes network externality gij ≥ 0on the online appendix. customer i. The externality gij quantifies the level of influence customer j has on customer i,or,equiva- 3.1.2. Network Externality or Network Effects. From (1) ’ lently, customer i s susceptibility to the behavior of and (2), we observe that customer i is more∑ likely αX − β X + n customer j.Theroleofgij will become clear when we to choose product X than∑ Y if ipi j1 gijxj fi αY − β Y + n de ne the choice modeling of customers later in is larger than ipi j1 gijyj. This can happen the section. if product X has a better quality (αX >αY), a lower X < Y price∑ (pi ∑pi ), or stronger network externalities n > n 3.1. Customer Choices ( j1 gijxj j1 gijyj). The quantity gij captures the Let αX and αY be the quality parameters of the heterogeneous interactions between customers in the products and βi be the price sensitivity of customer i. network. For example, in the choice of competing social networks, if i and j are close friends, then the X Y 3.1.1. Choice Probabilities. Given prices pi and pi choice of j mayhaveastrongerimpactonthechoiceof for i ∈{1, ... , n}, the customers anticipate the choice i than mere acquaintances, and, thus, gij is large. In the probabilities of others and the resulting network case of collaboration tools such as Dropbox, the choice effects. It leads to a rational equilibrium, and the is mainly driven by the network effects, and we expect gij Chen and Chen: Duopoly Competition with Network and Discrete Choices Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS 549 to be large if i and j are collaborators. Moreover, we do revenue from customer i, while setting xi <1 − yi not assume gii 0. Notice that the choice probability for a sufficiently small always guarantees a positive of customer i depends on the choice probabilities of revenue from customer i and does not reduce the other customers, not the realization of their discrete revenues from other customers for firm X.Therefore, choices. This is because the choice decision is made any equilibrium must satisfy 0 < xi < 1 − yi.Itim- simultaneously, and customers do not communicate plies that the equilibria are fully characterized by or coordinate their decisions. the first-order conditions. That is, the best response { ∗}n fi xi i1 satis es () 3.2. Nash Equilibrium ∑n ∗ ∗ ∑ βigji ∗ x x fi ’ n X αX − 1 + g + x − log i − i 0, The ∑rms expected revenues are given by i1 xipi ij β j − ∗ − − ∗ − n Y j1 j 1 xi yi 1 xi yi and i1 yipi , respectively, which are simply the (expected) sales multiplied by the prices. They are the (5) payoff functions of both firms in the game. We use n i , ... , n fi the choice probabilities {x } , or, equivalently, the for all 1 . A similar rst-order condition is i i 1 fi fi { }n market shares, as the firms’ decision variables. That is, satis ed by the best response of rm Y given xi i1: () we consider Cournot competition. We refer to Section ∑n β ∗ ∗ Y igji ∗ yi yi EC.3.1 in the online appendix for the justification for α − 1 + gij + y − log ∗ − ∗ 0. β j 1 − x − y 1 − x − y this modeling choice. The key step is the following j1 j i i i i transformation (see, e.g., Du et al. 2016): (6) ()() ∗ n ∗ n ∑n If {x } and {y } is a Nash equilibrium in the X 1 αX + − xi . ( ) i i 1 i i 1 pi gijxj log 3 duopoly competition, then the two sets of first-order βi 1 − xi − yi j 1 conditions ((5)and(6)) must be satisfied simulta- The second term in the parentheses represents the neously. In the remainder of this paper, we will use total network externality customer i receives; the this necessary condition to identify pure-strategy third term is the logarithm of the relative choice Nash equilibria in different cases.

probability of choosing product X to that of the no- X purchase option. In general, pi can be negative for 4. Homogeneous Customers { }n given xi i1 and yi. This cannot happen in equilib- In this section, we assume that rium, however, when customers are homogeneous Assumption 1 (Homogeneous Customers). β ≡ β, ∀i (see Proposition EC.1 in the online appendix). For i 1, ..., n, and g ≡ γ/2n, ∀i, j 1, ..., n. heterogeneous customers (Section 5), negative prices ij may arise and can be interpreted as coupons used to Assumption 1 indicates that customers are equally 3 β ≡ β attract high-value customers. price-sensitive ( i ), and the network effects are ho- ≡ γ/ As Cournot-type competition, the best response of mogeneous (gij 2n). This is an assumption adopted fi { }n by many previous papers (e.g., Katz and Shapiro 1985, rm X is therefore to choose a vector xi i1 to max- { }n Cabral 2011).4 Becausecustomersarehomogeneous, imize its revenue for given yi i1: ( ) we also restrict the set of strategies in the assump- ∑n ∑n () xi X tion below. max α + gijxj − log()xi + log 1 − xi − yi {}x n β i i1 i1 i j1 Assumption 2. The strategies are symmetric for all customers: x ≡ x and y ≡ y. subject to 0 ≤ xi ≤ 1 − yi. (4) i i In general, even though customers are homoge- Having defined the game primitives, our next prop- neous and identical to the firms, there may exist osition establishes the existence of pure-strategy Nash Nash equilibria in which the choice probabilities of a 5 equilibria. We transform the strategy of firm X to product differ across customers. We restrict our at- obtain a supermodular game. The existence, thus, tention to symmetric strategies because: (1) Our goal follows from Milgrom and Roberts (1990). is to examine the asymmetric equilibria arising from the interaction between the firms, rather than those from fi Proposition 1. The duopoly game de ned above has at least customers. Therefore, focusing on symmetric strategies one pure-strategy Nash equilibrium. allows us to isolate the firms’ competitive behaviors To characterize the equilibrium, we first argue that that lead to market dominance and market segmen- the constraints 0 ≤ xi ≤ 1 − yi must not be binding in tation. (2) Some prior papers have examined the any best response of firm X,unlessyi 1. Note that asymmetry arising from consumers (such as Brock xi 1 − yi generates −∞ revenue and, thus, can never and Durlauf 2001 and Du et al. 2016). We deviate from be optimal. On the other hand, xi 0 garners zero these papers because asymmetry arising from firms’ Chen and Chen: Duopoly Competition with Network and Discrete Choices 550 Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS strategic interactions is orthogonal to their analysis Assumption 3 (Identical Qualities). αX αY α. and less explored. (3) In practice, it is often not fea- Assumption 3 suggests that the game is entirely sible to discriminate customers with identical fea- symmetric, and in the sequel, we characterize Nash tures. (4) In Section 5, we relax Assumptions 1 and 2 equilibria that may arise. ∗ ∗ and show that many insights carry over to the general Under Assumption 3, symmetric equilibria x y case based on the analysis in this section. satisfy the first-order condition ∗ ∗ 4.1. Preliminaries x + x α − + γ ∗, ( ) log − ∗ − ∗ 1 x 8 Given yi ≡ y, the best response of firm X solves the 1 2x 1 2x following univariate{ ( optimization problem: )} following (5). ∗() γx () x y ≡ arg max x αX + − log()x + log 1 − x − y . 0≤x≤1−y 2 4.2.1. Weak Network Effects. When the network ef- (7) fects are not strong, our firstresultfeaturesasingle α fi γ fi symmetric Nash equilibrium. For a given ,de ne 1 By Assumption 1 and the rst-order condition (5), x γ −1(α − + γ )+ ∗( ) fi x + x αX − + γ fi to be the unique solution to 1 h 1 1 1. x y satis es log 1−x−y 1−x−y 1 x.The rst- ∗( ) Proposition 2. Suppose Assumptions 1–3 hold. If γ<γ, order condition for y x can be obtained by symmetry. ∗ ∗1 fi then there exists one and only one equilibrium (x , x ), The dif culty in analyzing the Nash equilibrium ∗ lies in the fact that (7) may have multiple solutions where x is the only solution to (8). because of the nonconcavity of the objective function. ∗ The characteristics of symmetric equilibria for sym- To obtain x (y),onehastocompareseveralpotential metric firms have been documented in, for example, local maxima. This is virtually infeasible because Calzada and Valletti (2008), Chen et al. (2018), and those local maxima do not have tractable expressions. Laffont et al. (1998), where they use the linear demand To deal with this difficulty, we use the following model or Hotelling’s model to describe the network novel approach. In (7), if the best response of X is ∗ effects. To understand Proposition 2, we illustrate the given as x x (y),theny has a unique solution that best responses in Figure 1.Wenotethatasufficient can be expressed in closed form. More precisely, condition for the existence of a single symmetric fi fi equilibrium is the following: The derivative of the best De nition 1. De ne ∗ response, (x ) (y), is between −1and0inthedomain. ¯() − − ()x , yx 1 x − We can show that when γ<γ,thederivativeofy¯(·) is h 1 αX − 1 + γx 1

− where h 1(·) is the well-defined inverse of h(t)≡t+ log(t).6 Figure 1. (Color online) One Symmetric Nash Equilibrium Corresponding to Proposition 2 Clearly, a given x is the optimal solution to (7)only if y y¯(x). The function y¯(x) (and symmetrically x¯(y)) can be interpreted as follows: If x is the action of firm X, then y¯(x) is the action of firm Y that makes x the best response. We shall also elaborate on the mathematical ∗ connection between y¯(·) and x (·).Ify¯(·) has a well- defined inverse, for example, when y¯(·) is strictly deceasing, then the inverse y¯−1(·) coincides with the ∗ best response x (·).Ify¯(·) does not have an inverse, then there may be multiple solutions of x to y¯(x)y, ∗ and one of them is the best response x (y). ∗ In our approach, we focus on y¯(·) instead of x (·), thanks to the explicit expression of the former. The main technical challenge remains to be the nonmonotonicity of y¯(·). As we shall see in Figure 3,such nonmonotonicity may lead to discontinuities in the best response, a surprising outcome in symmetric games.

4.2. Products with Identical Qualities In this section, our analysis is based on the following assumption. Note. In this example, we set αX αY 1 and γ 5. Chen and Chen: Duopoly Competition with Network and Discrete Choices Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS 551 less than −1. Because y¯(x) is strictly decreasing, it Figure 2. (Color online) One Symmetric Nash Equilibrium ∗ follows that its inverse is exactly x (y).Bytheproperty and Two Asymmetric Nash Equilibria Corresponding to ∗ of function inverses, the derivative of x (·) is be- Proposition 3 tween −1 and 0. Symmetrically, the same property can ∗ be obtained for y (x),andtheremustbeasingle symmetric and stable Nash equilibrium. When the network effects are not strong, both ﬁrms take up equal market shares in equilibrium, according to Proposition 2. This result is consistent with our observations of traditional industries: If two products are similar, such as Pepsi and Coca-Cola, then the market is more or less evenly split, and both ﬁrms coexist. This may no longer be the case when the network effects are strong.

4.2.2. Strong Network Effects. Our next result features three Nash equilibria when the network effects are strong enough. Proposition 3. Suppose Assumptions 1–3 hold. For any δ ∈(0, 1/4), there is a constant Γ(α, δ) such that when γ>Γ, there exist three Nash equilibria: • ( ∗, ∗) Two stable asymmetric Nash equilibria x1 y1 and ( ∗, ∗) ∗ ∗ ∈( − δ, ) ∗ ∗ ∈( ,δ) x3 y3 , in which x1 y3 1 1 and y1 x3 0 ; Note. In the example, we set αX αY −1 and γ 10. and • ( ∗, ∗) An unstable symmetric Nash equilibrium x2 y2 . more likely to emerge than the unstable symmet- The rise of asymmetric equilibria is pertinent to ric equilibria in the following sense: If the two firms discrete-choice modeling. When customers have startfromatupleofactions(x0, y0) and play the best ( , )( ∗( ), ∗( )) linear utilities (Chen et al. 2018)ormakechoicesby response sequentially xt yt x yt−1 y xt for t Hotelling’s law (Laffont et al. 1998), only symmetric 1, 2, ..., then the actions converge to one of the asym- equilibria exist. metric Nash equilibria as t →∞. In other words, when The best responses and the Nash equilibria of the network effects are strong, one of the products Proposition 3 are illustrated in Figure 2.Thederiva- will eventually dominate the market, rather than ∗ tive of the best response y (x) changes more drasti- two products equally split the market. cally compared with Figure 1. In particular, there is an The emergence of market dominance arises from ∗ interval of x (when x is small), in which (y ) (x) < −1, strengthened competition. Specifically, recall from (1) ∗ whereas in Proposition 2,wealwayshave(y ) (x) > −1. and (2)thatcustomers’ choices are influenced by two ∗ This phenomenon suggests that y (x)+x may de- factors: the appeal of the product itself (measured by αX − β X αY − β Y crease in x. It, therefore, implies that when the net- ipi and ∑ ipi ) and∑ the network externality n n work effects strengthen, the total market share may (measured by j1 gijxj and j1 gijyj). In addition, the decrease as the dominated firm (firm X when x is small) choice model also embeds a random utility that tries to promote its product. Because of the dominated captures product differentiation (i.e., the two prod- firm’s aggressive behavior, the leading firm (firm Y) ucts are differentiated in nature). When the network holds back its market share substantially in order to effects are not strong, customers’ purchasing decisions maintain a high profit margin. This crowding-out effect are primarily driven by their intrinsic consumption is more pronounced than the usual strategic substitution, values. Thus, given that the firms are ex ante symmetric, which only indicates the direction rather than the they naturally equally split the market. magnitude of firm Y’sadjustment. When the network effects are very strong, however, An implication of the above difference is that when the network externality terms start to take over. This, the network effects are strong enough, there are in fact, homogenizes the products, because now the two asymmetric equilibria that can be arbitrarily firms primarily compete on attracting the critical close to a market dominance. Here, market dominance mass of customers. In this vein, equal splitting be- ∗ ∗ refers to a Nash equilibrium (x , y ),inwhichoneof comes unstable, and the market structure is geared the products takes almost all the market share—for toward one firm dominating the other. Notably, this ∗ ∗ example, x <δ and y > 1 − δ for some small δ. asymmetric equilibrium is also more efficient in Moreover, the two asymmetric Nash equilibria are terms of boosting customers’ willingness to pay. Chen and Chen: Duopoly Competition with Network and Discrete Choices 552 Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS

When customers “coordinate” on purchasing from the the best response among the three solutions. This same firm, their decisions collectively enhance others’ observation rulesouttheexistenceofanysymmetric utilities all together. This probably explains why market equilibria.Third,weanalyzethebehaviorofthe dominance can emerge among seemingly identical function y¯(x) when x is approaching 0 and 1 and, products, such as Dropbox and OneDrive. In addition, hence, establish the existence of two asymmet- the result also implies that when investors invest in ric equilibria. startup firms in industries with strong network effects, Having articulated the mathematical reasoning for the outcome tends to be binary and riskier. the nonexistence of symmetric equilibrium, we now The “asymmetry” in the equilibrium is regarding turn to the economic rationale for Proposition 4. the strategy of the firms—that is, the market share. It To decipher the counterintuitive phenomenon, it is does not directly imply that the dominating firm is crucial to understand why the firms’ strategies change earning more revenue, because it could be retaining dramatically in response to the competitor’saggres- the position by setting very low prices. Our next re- siveness (the discontinuity of the best-response func- sult demonstrates the power of network effects: The tion). It can only happen when the network effects are leading firm is able to charge a higher price while neither too strong nor too weak (γ is in a bounded gaining a majority of the market at the same time. interval) and the quality is low: • When the opponent acts passively—that is, Corollary 1. In the asymmetric Nash equilibrium of takes a small market share—then the best response is Proposition 3, the firm dominating in market share charges to take a large market share to capitalize the (not too higher prices. ∗ weak) network effects—that is, x (y)≈1 − y when y is small. 4.2.3. Nonexistence of Symmetric Equilibrium. Our • When the opponent acts aggressively—that is, third result features two asymmetric equilibria (and takes a substantial market share—then the best re- ∗ no symmetric equilibrium) when the quality is low sponse is to virtually quit the market—that is, x (y)≈0 and the network effects are neither too weak nor when y is above a threshold (in Figure 3, this happens too strong. when y ≥ 0.36). This is because the remaining market – share does not create strong enough network effects Proposition 4. Suppose Assumptions 1 3 hold. There exist to overcome the low quality. Therefore, customers constants α0, Γ(α), and Γ(α) such that if α<α0 and are unwilling to pay much for the product, and γ ∈(Γ(α), Γ(α)), then there are only two asymmetric equilibria. virtually exiting the market turns out to be a better Somewhat paradoxically, Proposition 4 documents option for the firm. the possibility that even if the firms are ex ante symmetric, the resulting market equilibrium is never Figure 3. (Color online) Two Asymmetric Nash Equilibria symmetric. This statement remains true, even if we Corresponding to Proposition 4 allow for unstable equilibria. In this sense, our result demonstrates a strong rebuttal to the conventional, and perhaps na¨ıve, intuition that symmetric equilibria shall exist for symmetric firms. To help visualize the nonexistence of symmetric equilibrium, the best responses and the Nash equilibria are illustrated in Figure 3. In the online appendix,weprovidetheexactexpressionsofα and the interval. We verify that for such α and γ in the interval, y¯(x) is not monotone. Becauseofthis,thebestresponse ∗ x (y) involves selecting the global maximum from the multiple solutions (local maxima) to y¯(x)y,whichis virtually intractable. To circumvent this difficulty, we directly deal with y¯(x). ¯( ) First, we show that y x x has only one solution x0, which implies that if a symmetric equilibrium exists, ¯ it must be (x0, x0). Second, we show that y(x)x0 has three solutions, including x x0.Then,thebestresponse of firm X when firm Y plays x0 is selected from the three solutions. Furthermore, we prove that x0 is X Y not a global maximum. In other words, x0 cannot be Note. In this example, we set α α −2.5 and γ 15. Chen and Chen: Duopoly Competition with Network and Discrete Choices Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS 553

As a result, only asymmetric equilibria arise due to victory could be partially explained by our theory, as the discontinuity. Our analysis, therefore, indicates a the strong network effects lead to the dominance of concrete operating regime in which such “anom- the QWERTY keyboard over the Dvorak typewriter aly” occurs. and other alternatives. Although Proposition 5 hints at the possibility that Remark 1. It is worth pointing out that our method- an inferior product may dominate the market, the ology can be generalized to oligopoly competitions. next result shows that the quality superior firm can When there are n symmetric firms competing with each secure its dominating position if the quality difference other in our model, there does not exist any symmetric is sufficiently large. equilibrium when the quality is low and the network effects are neither too weak nor too strong. Proposition 6. Suppose Assumptions 1 and 2 hold. For given αY and γ, and δ ∈(0, 1/4), there is a constant Remark 2. Propositions 2–4 do not exhaust all the α (αY,γ,δ) such that for αX >α, there exists one and only 0 ∗ ∗ 0 ∗ possible values of parameters—for example, when α one Nash equilibrium (x , y ), where x ∈(1 − δ,1). and γ are moderate. In theory, more complex equi- Proposition 6 does some justice to the firm that librium structures may emerge. However, based on the invests substantially in quality: It suggests that if the numerical study in Section 4.4, the three types of qualities are not “in the same league,” then the high- equilibria demonstrated from Propositions 2–4 cover quality firm will unambiguously become the market all the cases. leader. This may explain why Gmail was able to 4.3. Differentiated Products overtake Hotmail and Yahoo Mail as the largest In this section, we consider differentiated products—that global email service, despite the huge customer base is, αX αY.Onefirm’s product is inherently superior to the others have. the competitor’s. Our first result extends Proposition 3: There are two asymmetric Nash equilibria that are close 4.4. Parameter Ranges and Equilibrium Types to a market dominance when the network effects are To give a concrete view of the connection between pa- strong enough, regardless of their quality difference. rameters and equilibrium types, we conduct a numerical study assuming identical qualities (Assumption 3)to Proposition 5. Suppose Assumptions 1 and 2 hold. For given complement our theoretical studies. More precisely, αX αY δ ∈( , / ) Γ(αX,αY,δ) , , and 0 1 4 , there is a constant such we sample α and γ from the range [−6, 6]×[0, ∞). γ>Γ that for , there are three Nash equilibria: For a particular combination of α and γ,weinvesti- • ( ∗, ∗) ( ∗, ∗) Two stable Nash equilibria x1 y1 and x3 y3 , in gate the resulting equilibrium type after numerically ∗, ∗ ∈( − δ, ) ∗, ∗ ∈( ,δ) which x1 y3 1 1 and y1 x3 0 , and solving the best-response function. We do not sample • ( ∗, ∗) ∗ > ± An unstable Nash equilibrium x2 y2 such that x1 |α| > 6 because exp(±6)≈10 2 already has a different ∗ > ∗ ∗ < ∗ < ∗ x2 x3 and y1 y2 y3. order of magnitude from the utility of the outside It is worthwhile to elaborate on the procedure to option 1 in the Choice Probability (1). Thus, α ∈[−6, 6] prove Proposition 5.Becausethefirms (products) are seems to cover the practical range of parameters. heterogeneous, we cannot apply the same approach For all the combinations, the numerical results of as in the proof of Proposition 4 (which requires the equilibrium structure can always be categorized symmetry). In the first step, we show that when γ is into three types: type I (one symmetric Nash equi- sufficiently large, both y¯(x) and x¯(y) are decreasing librium; e.g., Figure 1), type II (two Nash equilibria; and have inverse functions. Moreover, there are, at e.g., Figure 3), and type III (three Nash equilibria; e.g., most, three Nash equilibria. Next, we compare the Figure 2). The ranges of the parameters correspond- ∗ functions y¯(x) and y (x) in a neighborhood of x 0. We ing to the equilibrium types are listed in Table 1. ∗ conclude that y¯(0)1 > y (0),buty¯(·) is decreasing For α ≥−1, a type-II equilibrium structure does not ∗ slightly faster than y (·), which leads to an intersection emerge. The types are not monotone in the parame- (a Nash equilibrium) in that neighborhood. By the ters: For example, a type-III equilibrium structure same token, a Nash equilibrium exists close to y 0. emerges for two disjoint intervals of γ, [8.5, 9.0) and Finally, a Nash equilibrium always exists between the [13.7, +∞),whenα −2. two equilibria mentioned because the best response is continuous in this case. 5. Heterogeneous Customers Proposition 5 states that, even though one product, In this section, we investigate the general setting and say, product X, is far inferior to its competitor, it analyze a market of customers who might be het- can still retain market dominance due to the strong erogeneous in their price sensitivities and network network effects. In the battle between the QWERTY effects. We first show that when the network effects keyboard and others, QWERTY is widely acknowl- are weak, then there exists a unique Nash equilib- edged to be inferior to the Dvorak typewriter. Its rium, extending Proposition 2. Chen and Chen: Duopoly Competition with Network and Discrete Choices 554 Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS

Table 1. Ranges of Parameters and the Equilibrium Types pay even a small amount for the products. Therefore, neither firmcanmakemuchrevenuefromhim. α Type I Type II Type III The only purpose for the firms to attract such a 6 γ ∈[0, 200.3) — γ ∈[200.3, +∞) customer is to benefit from his network effects on 4 γ ∈[0, 112.2) — γ ∈[112.2, +∞) other customers. The benefit from network effects 3 γ ∈[0, 43.2) — γ ∈[43.2, +∞) is always increasing in the choice probability. Thus, γ ∈[ , . ) — γ ∈[ . , +∞) 2 0 17 9 17 9 the firms are willing to set very low prices and 1 γ ∈[0, 8.8) — γ ∈[8.8, +∞) 0 γ ∈[0, 6.0) — γ ∈[6.0, +∞) forgo the revenue. For customers who are less price- −1 γ ∈[0, 6.5) — γ ∈[6.5, +∞) sensitive, the firms attempt to earn revenue and −2 γ ∈[0, 8.5) γ ∈[9.0, 13.7) γ ∈[8.5, 9.0)∪[13.7, +∞) utilize the network effects at the same time. The −3 γ ∈[0, 11.1) γ ∈[11.2, 19.4) γ ∈[11.1, 11.2)∪[19.4, +∞) former objective is not always increasing in the − γ ∈[ , . ) γ ∈[ . , . ) γ ∈[ . , . )∪[ . , +∞) 4 0 13 5 13 6 24 5 13 5 13 6 24 5 choice probability. Facing such trade-off, the choice −6 γ ∈[0, 18.2) γ ∈[18.2, 34.1) γ ∈[34.1, +∞) probabilities are relatively low for such customers Note. We use bisection search to find the cutoff values of γ, which are (so that higher prices can be charged) in equilibrium. This rounded to one digit. is common in the online game industry. For example, Hearthstone, a popular collectible card game, is F2P X Y Proposition 7. Given α , α and βi for all i, there exists a (free to play). For price-sensitive players, they can (αX,αY,β , ...,β ) constant G 1 n such that when the network spendmoretimetogrindandwaitfortheneeded effects gij < G for all i and j, there exists a unique pure- cards to drop, while VIP players can skip the strategy Nash equilibrium. grinding and directly buy their dream decks. In the proof, we are able to obtain an explicit If the customers are equally price-sensitive, then − condition. Let c (1 + h−1(αX)+h−1(αY)) 1.If Proposition 8 implies: (){} Corollary 3. Suppose Assumption 3 holds. If β ≡ β for all i ∑n g g c2 i ij + ji ≤ , , and g + g is decreasing in i for all j 1, 2, ..., n, then in β β min 1 β ()− for all j ij ji ∗ ∗ i1 i j j 1 c any symmetric Nash equilibrium, xi yi is decreasing in i. then the Nash equilibrium is unique. If the social influences of the customers can be ranked unanimously, then the firms tend to increase their market share for customers with higher network 5.1. Symmetric Nash Equilibrium effects in equilibrium. This is consistent with the in- By Proposition 7, if the network effects are weak and tuition: The market share for those customers is of qualities of the two products are identical (αX αY α; higher value because it significantly increases the i.e., Assumption 3), then the unique Nash equilibrium ∗ ∗ choice probabilities of all customers and, thus, the must be symmetric—that is, x y for all i in the i i total revenue. As a special case, for a network with equilibrium. Naturally, we are interested in the fol- asingle“star”—that is, g ≡ γ>0andg 0for lowing question: To which customer are the firms i1 ij j ≥ 2—the equilibrium market share for the star is eager to sell their products? The following proposi- higher than the rest of the market. This is usually tion gives a sufficient condition. implemented through the referral program, in which Proposition 8. Suppose Assumption 3 holds. If βjgij + βigji the influential customers earn bonuses and are happy is decreasing in i for all j 1, 2, ..., n, then in any sym- to consume more at a lower or even negative price. ∗ ∗ fi metric Nash equilibrium, xi yi is decreasing in i. The rm makes up for the revenue loss by monetizing fl When customers can be ranked in the same order by their in uences in the network. the quantity βjgij + βigji for all j,thefirms are vying for the customers who rank high—that is, their equilib- 5.2. Two Communities rium choice probabilities for both products are large. In this section, we consider a network with two To better interpret the quantity, consider a homo- communities. Suppose n n1 + n2,wheren1 and n2 geneous network. are the sizes of the two communities. The network effects are specified as follows: Corollary 2. Suppose Assumption 3 holds. If gij > 0 are β ≥ β ≥ ...≥ β equal for all i and j and 1 2 n, then in any Assumption 4 (Two Communities). The network matrix ∗ ∗ symmetric Nash equilibrium, xi yi is decreasing in i. {gij} satisfies: The corollary states that customers who are more ⎧⎪ γ , ∈ {}, ..., price-sensitive turn out to be more likely to pur- ⎪ 1 i j 1 n1 ⎨⎪ γ , ∈ {}+ , ..., chase either product in equilibrium. To understand 2 i j n1 1 n gij ⎪ γ ∈ {}, ..., , ∈ {}+ , ..., this seemingly counterintuitive result, note that an ⎩⎪ 12 i 1 n1 j n1 1 n γ ∈ {}, ..., , ∈ {}+ , ..., . extremely price-sensitive customer is unwilling to 21 j 1 n1 i n1 1 n Chen and Chen: Duopoly Competition with Network and Discrete Choices Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS 555

Under Assumption 4, the connectivity inside each Probabilities (1)and(2). In reality, network effects community is homogeneous, which is different usually build up over time: Customers cannot foresee from the externality between communities. This the potential network effects in the distant future. network structure encompasses many special net- Rather, they make consumption based on the current works that are investigated in the literature, such as network effects. This assumption is made in other star graphs and complete bipartite graphs. In prac- dynamic competitions (Mitchell and Skrzypacz 2006, tice, it can capture a segmented market structure Cabral 2011). It also better explains the first-mover whose customers in the two groups differ signifi- advantage, as early entrants are able to build stronger cantly in their locations, ages, or other characteris- current network effects and attach customers, which tics. We assume βi ≡ β for all customers and focus on potentially blocks later entrants that may have better firms’ strategies that induce the same market share quality. In this section, we extend the game to such a inside a community. dynamic setting.

6.1. Open-Loop Nash Equilibrium 5.2.1. Market Segmentation. The next result is con- Next, we introduce the dynamic game and the as- cerned with market segmentation under network sociated equilibrium concept. Suppose xt and yt are effects: Product X is dominating in one community, fi the market shares of rm X and Y∑in period t. whereas product Y is popular in the other community. n ∑The current network effects are, thus, j1 gijxj,t and n + Proposition 9. Suppose Assumption 4 holds and βi ≡ β, ∀i. j1 gijyj,t for customer i.Inperiodt 1, the customers • For any δ ∈(0, 1/4), there is a constant Γ that may observe and enjoy the network effects generated in X Y depend on α , α , β, n1, n2, γ12, γ21, and δ, such that if period t, and their market shares are given by the 7 γ1 > Γ and γ2 > Γ, then market segmentation arises in the following choice model: Nash equilibrium—that is, there exists a Nash equilibrium {}∑ ∗ ∗ ∗ ∗ ∗ ∗ X X n ( , , , ) > − δ > − δ exp α − βip , + + gijxj,t x1 y1 x2 y2 satisfying x1 1 and y2 1 . {}i t 1 j 1 xi,t+1 ∑ , (9) • Suppose the two communities are identical—that is, X X n 1 + exp α − βip , + + gijxj,t + n n , γ γ . If γ + γ ≥ γ + γ , then any Nash {}i t 1 j 1 1 2 1 ∗ 2 ∗ ∗ 12∗ 21 1 ∗ 2 ∗ ∗ ∗ ∑ ( , , , ) fi αY − β Y + n , equilibrium x1 y1 x2 y2 satis es x1 x2 and y1 y2. exp ipi,t+1 j1 gijyj t {}∑ This proposition gives a potential explanation for Y Y n exp α − β p , + + g y , {}i i t 1 j 1 ij j t , ( ) market segmentation. It can arise if the network yi,t+1 ∑ 10 X X n effects inside communities are relatively strong in 1 + exp α − β p , + + g x , + {}i i t 1 j 1 ij j t comparison with those between communities. This ∑ αY − β Y + n is parallel to the equilibrium behavior in Bimpikis exp ipi,t+1 j1 gijyj,t et al. (2016), where in the context of duopoly-targeted where the network effects are based on the market advertising, each of the two firms chooses to target a share in period t. Such dynamic setting with myopic “hub” in one of the two clusters in the graph. This expectation of other customers’ consumption was may explain why different messaging apps domi- originally proposed in the seminal paper of Brock and nated different regions and developed their own Durlauf (2001) to understand the limiting behavior ecosystems—that is, WeChat in China, Facebook’s of a dynamic game in the network context. When Messenger in North America, and WhatsApp in customers have perfect foresight and can anticipate other regions. the network effects in the next period (replacing x , On the other hand, when the connections between j t and y , by x , + and y , + ), it is equivalent to the static communities are stronger, we find that the afore- j t j t 1 j t 1 game (1)and(2). mentioned segmentation is no longer feasible. Under The dynamic game proposed in this section is more the symmetric-community assumption, each firm suitable to describe the market evolution of nondu- will obtain the same market share in the two com- rable services/products with network effects, such munities. Notably, market dominance may still exist, as Netflix subscriptions. The customers repeatedly in the sense that one firm dominates the other in make purchases in each period based on the network both communities. effects experienced in the last period. The model does not apply to durable network goods that customers 6. Dynamic Game only purchase once, such as video game consoles. The The game studied in the previous sections is static. It dynamic game of competing durable goods requires can be viewed as the market outcome when all cus- new formulations that depart from the static game tomers have perfect foresight: They can predict the and is, thus, out of the scope of this paper. consumption of connected customers and are willing In the static MNL model, we implicitly assume that to coordinate with each other to realize the Choice customers generate random utilities for both products, Chen and Chen: Duopoly Competition with Network and Discrete Choices 556 Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS as well as the outside option. In a dynamic model, one-to-one correspondence between the Nash equilib- however, it is not reasonable to assume that customers ria of the transformed game and the original game, we independently redraw their random utilities over time can establish the existence of the OLNE of the original for the same product. Although this setup has appeared game as well. in the literature of dynamic mechanism design and Proposition 10. There exists a pure-strategy OLNE. structural models for dynamic discrete choices (Aguirregabiria and Mira 2010, Pavan et al. 2014), we There are other methods commonly used to establish fi ’ acknowledge that this is a weakness of the model when existence, including xed-point theorems (Brouwer s ’ extended to the dynamic setting, and a dynamic model or Kakutani s). However, as we have seen, the best- with microfoundations remains our priority in fu- response functions exhibit discontinuity and non- ture research. concavity, even in the static case. Thus, they cannot be Again, we consider Cournot competition in the applied to the dynamic game. dynamic setting: Given initial market share x0 and y0, firm X is deciding x for t 1, 2, ...to maximize 6.3. Necessary Conditions t { }∞ { }∞ Suppose xt t1 and yt t1 form an OLNE. Given ∑∞ ∑n { }∞ { }∞ fi t X yt t1, xt ∑t1 maximizes∑ the payoff function of rm , ∞ max∞ r xi tpi,t — t n X {}xt X that is, r xi,tp , .Therefore,foragivent, xt t1 t1 i1 t 1 i 1 i t ( ())must maximize the terms in the payoff function that ∑n X 1 X xi,t . . α + , − − involve xt.Moreprecisely, s t pi,t gijxj t 1 log ∞ βi 1 − xi,t − yi,t ∑ ∑n j 1 s X xt arg max r xi,spi,s ≤ ≤ − , , ,..., ≤ ≤ −y 0 xt 1 yt t 1 2 0 x 1 t s1 i1 ( ( ) ∑n ∑n x rgjiβi where r ∈(0, 1) is the discount factor, and pi,t is solved i αX + + arg max β gijxj,t−1 β xj,t+1 from (9)and(10). The payoff function of firm Y can be 0≤x≤1−y i j t i 1 ()j 1 ) defined symmetrically. xi We investigate the equilibrium-concept open-loop − log . 1 − xi − yi,t

Nash equilibrium (OLNE) (see, e.g., Basar and Olsder Similarly, y satisfies 1999) arising from the dynamic game. Given the t ( ( ) x fi initial market share at t 0, 0,andy0, rm X (Y)is ∑n ∑n β ∞ ∞ yi Y rgji i { } { } α + , − + , + deciding xt t1 ( ∑yt t1)∑ to maximize∑ the∑ total dis- yt arg max gijyj t 1 yj t 1 ∞ ∞ ≤ ≤ − β β t n X t n Y 0 y 1 xt i 1 i j 1 ) j counted revenue t1 r i1 xi,tpi,t ( t1 r i1 yi,tpi,t) () subject to the constraint x + y ≤ 1. This is in contrast yi t t − log . to the feedback Nash equilibrium (FNE), in which the 1 − xi,t − yi firms are deciding xt+1 (yt+1) dynamically based on the fi market condition (xt, yt) at t. In general, the two The necessary conditions are satis ed by any OLNE. concepts do not lead to the same equilibrium strategy. From the necessary condition, it is clear that xt and In this paper, we adopt OLNE instead of FNE because yt form a Nash equilibrium of the following static of two reasons: (1) We focus on the industries where Cournot game: Firm X (Y) is choosing∑ the∑ market x x y y n X n Y the infrastructure and capacity have to be planned share t ( t) to maximize i1 xipi ( i1 yipi ), X Y ahead; and (2) technically, the Bellman equations where the relationship between xi and pi (yi and pi )is arising from FNE are intractable because of the lack of given by the vanilla MNL model: structure in the objective function. {} ˜ X X exp α − βip x {}i {}, i ˜ X X ˜ Y Y 1 + exp α − βip + exp α − βip 6.2. Existence {}i i α˜ Y − β Y fi exp ip We rstestablishtheexistenceofsuchanOLNE. y {}i {}, fi i + α˜ X − β X + α˜ Y − β Y Similar to Proposition 1,we rst transform the game 1 exp ipi exp ipi to a supermodular game and apply the lattice theory. ∑ β α˜ X ≜αX + n ( + rgji i ) α˜ Y ≜ αY + where j1 gijxj,t−1 β xj,t+1 and Note that the transformation is crucial: Firms X and Y ∑ β j n rgji i (g y , − + y , + ). In particular, if we treat x − , are clearly selling substitutable products, and the j 1 ij j t 1 βj j t 1 t 1 dynamic game is not supermodular/complementary. xt+1, yt−1,andyt+1 as exogenous variables, then xt fi After the transformation, we replace rm X by an- and yt are the Nash equilibrium of a classic MNL other player X ,whichturnsouttobecomplementary Cournot game, in which the qualities are en- to firm Y. This allows us to use the theory of supermodular hanced by the network effects in the neighbor- games and establish the existence. Because there is a ing periods. Chen and Chen: Duopoly Competition with Network and Discrete Choices Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS 557

6.4. Connection Between the Dynamic and given by (9)and(10).Thegamelastsfortwoperiods, Static Games and the payoff function of firm X is given by Dynamic games are generally intractable, especially ∑2 ∑n t X given the complexity of the static game. A crucial , max, r xi tpi,t x1 x2 question in this section is, thus, whether the equi- t 1 i 1( ()) librium structures and insights derived in the static ∑n . . X 1 αX + − xi,t case continue to hold in the dynamic game. We are s t pi,t gijxj,t−1 log βi 1 − xi,t − yi,t able to provide a positive answer to this question: j 1 0 ≤ x ≤ 1 − y , t 1,2. Proposition 11. If the dynamic game converges, then the t t x y fi limit ∞ and ∞ satisfy the Nash equilibrium rst-order The benefit of considering a two-period model is to conditions (5) and (6) of the following static game (denoted ∑ β˜ ˜ ∑ avoid the complexity and potential ill behavior (see ˜ X X ˜ Y Y n ˜ igji n by ˜·): α α , α α , and (g + ˜ ) (g + Remark 3) of a full dynamic game, while retaining the j 1 ij βj j 1 ij rβigji 8 β ) for all i. dynamic feature. j fi fi > If rm X has the rst-mover advantage, then x0 y0. Proposition 11 connects the static and the dynamic That is, at the beginning of the game (possibly the time of fi → games. When both rms are myopic (r 0), the dy- entry of firm Y), firm X gains more market share than namic game is essentially equivalent to the static game as firm Y. Then, a natural question is, under which con- the evolving market reaches a steady state. For r > 0, — > ∑ β˜ ˜ ∑ β ditions the advantage is retained that is, x2 y2 for n ˜ igji n r igji the relationship (g + ˜ ) (g + β ) implies ( , ) ( , ) fi j 1 ij βj j 1 ij j any OLNE x1 x2 and y1 y2 ?Forexample,if rm X that the static game discounts the network effects by r, has inferior quality and the network effects are weak, and the qualitative characteristics of the equilib- then firm Y may end up being the market leader. The rium still hold. For example, in the homogeneous next proposition provides such a sufficient condition. case (Assumptions 1–3) g γ/2n and g˜ γ/˜ 2n ij ij Proposition 12. For any , δ > 0, there are constants (1 + r)γ/4n, and Propositions 2–4 can be used to predict G(, δ, αX,αY, {β }n ) and R(, δ, αX,αY, {β }n , {g }n ) the equilibrium outcome of the dynamic game. This i i1 i i1 ij i,j1 such that if g > G for all i, j and r < R, then x , − y , >δ connection makes the theoretical results derived for the ij i 0 i 0 , − , > ( − )δ

for all i implies x y 1 for all i in any OLNE. static game applicable to the dynamic game. i 2 i 2 Moreover, firm X makes more revenues than firm Y. Remark 3. The proposition has two restrictions. First, Proposition 12 states that firm X is able to retain the we have to assume that the OLNE has a steady state. market-leader position over periods, regardless of This is not a merely technical simplification. In general, fi the quality differences, if (1) the network effects are because the dynamic game has an in nite number of strong, and (2) the discount factor is small. The first decision variables and the payoffs are highly non- point is straightforward, as strong network effects are concave, OLNEs may well be cyclic, in which case we necessary to persuade customers to keep using product X, cannot draw useful insights. It also a common as- even though its quality could be inferior. To see the sumption adopted in the literature (Mitchell and second point, consider firm Y’s strategy to break the Skrzypacz 2006). Second, the connection between fi fi rst-mover advantage of the opponent: It can subsi- the dynamic and static games is based on the rst- dize the customers heavily to counter the network order conditions, instead of the Nash equilibria. effects of firm X. In the long run, the subsidy may Again, the complexity arises from the nonconcavity fi fi secure the market-dominance position for rm Y and of the payoff functions, as the rst-order conditions makes the investment worthwhile. Therefore, first- do not necessarily lead to Nash equilibria. mover advantage remains valid when future revenues are greatly discounted and subsidizing customers is costly. 6.5. First-Mover Advantage One appealing feature of dynamic games is the ca- pability to capture the phenomena emerging from the 7. Conclusion market evolution, which cannot be explained by the In this paper, we consider two firms that sell their static game. For example, it is widely believed that an substitutable products to a market of network- earlier entrant to a market with strong network effects connected customers, who make their purchasing is able to enjoy the first-mover advantage and dom- decisions through the MNL model. We demon- inate the market eventually without having superior strate the different market structure when the quality. To simplify the analysis, we consider a two- strength of network effects varies. The results can period model in this section: The choice model is be used to predict the competitive outcome under Chen and Chen: Duopoly Competition with Network and Discrete Choices 558 Operations Research, 2021, vol. 69, no. 2, pp. 545–559, © 2021 INFORMS various market conditions. They can also consis- Aoyagi M (2018) Bertrand competition under network externalities. – tently explain real-world market phenomena. J. Econom. Theory 178:517 550. Ballester C, Calvo-Armengol´ A, Zenou Y (2006) Who’s who in net- Our analysis hopefully opens up potential research – fi ’ works. Wanted: The key player. Econometrica 74(5):1403 1417. directions related to rms competitive strategies Banerji A, Dutta B (2009) Local network externalities and market when confronted with network-connected customers. segmentation. Internat. J. Indust. Organ. 27(5):605–614. In particular, we believe that (1) investigating the Basar T, Olsder GJ (1999) Dynamic Noncooperative Game Theory, strategic interaction between firms and customers in a Classics in Applied Mathematics, vol. 23 (Society for Industrial dynamic setting can generate valuable insights and and Applied Mathematics, Philadelphia). Bernstein F, Federgruen A (2004) A general equilibrium model for in- explain market phenomena; and (2) analyzing com- dustries with price and service competition. Oper. Res. 52(6):868–886. plicated network structures may reveal the link be- Bimpikis K, Ozdaglar A, Yildiz E (2016) Competitive targeted ad- tween competitive outcomes and social interactions. vertising over networks. Oper. Res. 64(3):705–720. Bloch F, Quérou N (2013) Pricing in social networks. Games Econom. Behav. 80:243–261. Endnotes Brock WA, Durlauf SN (2001) Discrete choice with social interactions. 1 The QWERTY keyboard is widely regarded to be inferior to the Rev. Econom. Stud. 68(2):235–260. Dvorak typewriter (David 1985). 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