## Introduction

Consumers’ purchasing decisions and perceived utility are significantly influenced by their social interactions with other consumers (Ameri, Honka, Xie, 2019, Hu, Milner, Wu, 2015). Social interactions are typically structured as a network of participants (i.e., nodes) and relationships (i.e., links), and their impact on consumers’ behavior is referred to as network externalities, which are manifested either globally or locally. While the global network externalities are derived from the collective behavior of all consumers of a good (Katz & Shapiro, 1985), local externalities are derived from a consumer’s interaction with their immediate peers (Banerji & Dutta, 2009). The local network effects have particularly paved the way for viral and word-of-mouth (WOM) marketing, which exploit the interconnected nature of consumer networks to facilitate the diffusion of information and provide a fertile ground for promoting products and services to groups of potential consumers (Bapna & Umyarov, 2015). This influence is extensively studied, and it is estimated to be the driving force behind 25% to 50% of all purchasing decisions (Meyners, Barrot, Becker, & Bodapati, 2017), and it can increase the likelihood of purchasing a product up to 60% (Bapna & Umyarov, 2015). Marketers accordingly use this propagation of information to execute viral marketing campaigns and generate sales through peer-to-peer influence (Kahr, Leitner, Ruthmair, Sinnl, 2021, Samadi, Nagi, Semenov, Nikolaev, 2018). However, since pricing decisions can excite or inhibit the effects of WOM marketing and network externalities (Ajorlou, Jadbabaie, & Kakhbod, 2016), the pricing strategies are needed to be carefully synthesized to maximize the network effect’s positive impact.

In addition to the network effects, competition can also add to the pricing complexities. Therefore, a firm operating in a networked market needs to adjust its pricing strategy to account for network effects and competition. In such a market, consumers are connected through different interactions. For example, an online social medium can act as the framework on which consumers interact, and each consumer might utilize several social media. Similarly, a firm might concentrate its WOM activities on a particular platform or group of outlets. Therefore, while the consumers are the same, the topology of their connection might be different in each network’s layer. In its general form, competition might occur through various networks, where each consumer has a different set of peers in each network. In other words, a multilayer networked model is a more realistic representation of a market when it comes to competition. In this case, network layers represent the distinct transmission routes of the information by which the network effect is imposed. Such network models have already been studied when multiple rival information propagation processes are considered (DarabiSahneh & Scoglio, 2014) and can propose a novel framework for modeling competition in networked markets. Using multilayer networks in the pricing literature is unprecedented to the best of the authors’ knowledge.

WOM marketing relies on information diffusion in networks, a stochastic process that in its general form is of complicated nature, even for simple scenarios (Kiss, Miller, Simon, etal., 2017, Newman, 2018). Additionally, analyzing such stochastic processes becomes intractable in realistic problems (e.g., pricing in large consumer networks) where the network size is large (Sahneh, Chowdhury, Brase, Scoglio, 2014, Van Mieghem, Omic, Kooij, 2009). Deterministic approximations such as the mean field method are proposed to overcome the intractability of modeling stochastic processes effectively (Sahneh, Scoglio, Van Mieghem, 2013, Tavasoli, Shakeri, Ardjmand, Young, 2021, Van Mieghem, Omic, Kooij, 2009). Following the successful application of the mean field method, this study utilizes the same methodology.

This study contributes to the pricing literature by considering a duopoly in a multilayered networked market where consumers are exposed to local externalities. The competing products are functionally identical, and each customer’s purchasing decision is based on the number of their peers who have purchased the product and its price. The objective is to maximize the long-term profit of each firm and both firms collectively. Each firm is assumed to utilize a separate network to promote its product where the nodes (i.e., consumers) have different neighbors in each network. It is noteworthy that by considering each network’s layer to have the same topology, the problem will be reduced to the case where both firms use the same connection network. Consumers’ purchasing decision occurs in two stages. First, they will be exposed to a product based on the number of neighbors who possess it in the network layer chosen by the product’s firm to promote it. After being exposed, a consumer will decide to buy the product based on its price. Section3 elaborates on this idea and formally outlines the model.

A key modeling aspect of the presented study, borrowed from the epidemiology field, is the application of multilayer networks in formulating the diffusion process and local externalities (DarabiSahneh & Scoglio, 2014). Specifically, a two-layer network will be used to model a duopoly competition. In this context, each layer represents a distinct network by which consumers are connected (e.g., two social media platforms). If both firms utilize the same platform, then both layers will be the same. A two-layer setting offers a more flexible modeling tool for scenarios where different firms target different information mediums to reach consumers. The architecture of network layers and their correlation plays a critical role in determining sales propagation among consumers. When the two network layers hold similar structural properties (e.g., a significant overlap among their central nodes), they correlate positively. Conversely, when the central nodes of layers do not typically match, the layers correlate negatively. As will be demonstrated later, a positive correlation of network layers makes it difficult for products to survive the competition, while survival is easier in a competition that is characterized by a network with negatively correlated layers. From a technical standpoint, since the interconnections in multilayer networks with negative correlation do not contribute much to the interlayer diffusion process (Tavasoli, Ardjmand, & Shakeri, 2020), different layers work distinctly, which makes the presence of products over different layers more probable. On the other side, the interlayer diffusion in networks with positive correlation is facilitated by the interconnections, and different layers can not operate distinctly. In this scenario, one product will have a higher chance of dominating both layers. In the context of competitive marketing, this work identifies two different settings of positive and negative correlation as the conditions where the influential agents of one product may or may not be at the reach of the other product’s network, respectively, and studies the impact of each setting on the network equilibrium and pricing strategies.

Firms may set their prices differently for each consumer (heterogeneous or differential pricing) to further exploit the network effects and increase their revenues. For this purpose, companies can identify influential agents and incentivize them to promote their products by offering lower prices to them (Cohen & Harsha, 2019). An optimal pricing strategy can be characterized as a function of the underlying social interactions over a network and consumers’ centrality measures (Candogan, Bimpikis, & Ozdaglar, 2012). In extreme instances, it might even be optimal to offer below-cost or free products to some influential consumers to exploit the network effect (Cohen & Harsha, 2019). However, this does not mean that influential consumers always get discounted prices. Pricing under externality boils down to finding the right trade-off between influence and exploitation, and in some scenarios, this trade-off might lead to higher prices for more central nodes compared to the less central ones (Bloch, Quérou, 2013, Tavasoli, Shakeri, Ardjmand, Young, 2021). Therefore, carefully synthesized prices are needed based on the marketing objectives and network conditions.

In a competitive market, firms might decide to set their prices independently (competitive strategy) or price coordinate (cooperative or collaborative strategy). A cooperative strategy is sought, for example, by firms that form a strategic alliance and develop a cooperative alliance brand (Yan, 2009). Collaborative pricing has a high potential of increasing the prices for consumers. For example, a symmetric demand model tested in a semiconductor industry showed that the equilibrium prices in the collusive case are always higher than the prices in the Nash equilibrium (Coughlan, 1985). Price coordination can occur in three manners: price competition without channel coordination, price competition with channel coordination, and global coordination (Coughlan, 1985, Sinha, Sarmah, 2010). In general, while competition enhances market efficiency in terms of social welfare, a cooperative pricing strategy enhances overall market profitability (Sinha & Sarmah, 2010).

The key contributions of this study can be summarized as follows:

- •
Pricing strategies in a duopoly networked market with local network effects are investigated.

- •
A mean field representation of a stochastic model formulated based on epidemic spread theory over networks is used to examine the implications of the local network effects on competitive pricing.

- •
Several types of equilibrium and optimal pricing policies based on multiplex structural properties are characterized.

- •
Different optimal pricing strategies and their properties under cooperating and competitive profit maximization policies, as well as homogeneous and heterogeneous pricing are investigated.

- •
Existing epidemic models over complex networks are extended to cover a duopoly under local network externalities and tools from nonlinear analysis are borrowed to analyze a complex competition problem.

- •
Differential pricing, which has not been considered in previous studies on competition with local externalities is investigated.

The remainder of this research is organized as follows. First, Section2 briefly reviews the literature of pricing under network externalities and modeling methodologies. Section3 formulates the proposed model, while the properties of the model are investigated in section4. Sections5 and 6 investigate optimal pricing for homogeneous and heterogeneous cases, respectively. A discussion of the results and the overall conclusion are stated in sections7 and 8.

## Section snippets

## Related Literature

This paper aims to study pricing strategies in a duopoly networked market whose consumers’ behavior is impacted by price and local network effects. Local externalities for each firm’s product are exerted through a separate network whose nodes represent consumers and links symbolize different social settings under which the consumers are connected. The nodes of the networks correspond to the same set of consumers, while each network can have a different set of edges. Each firm might utilize

## Model development

The contact topology (i.e., the consumers’ network) in this paper is modeled by two undirected generic graphs ${\mathcal{G}}_{1}({\mathcal{V}}_{1},{\mathcal{E}}_{1})$ and ${\mathcal{G}}_{2}({\mathcal{V}}_{2},{\mathcal{E}}_{2})$ of individuals, where ${\mathcal{V}}_{j}$ denotes the set of vertices and ${\mathcal{E}}_{j}$ denotes the set of edges ($j\in \{1,2\}$). This study considers the competition of two goods 1 and 2, which spread through two different networks ${\mathcal{G}}_{1}$ and ${\mathcal{G}}_{2}$, having adjacency matrices $A={\left[{a}_{ij}\right]}_{N\times N}$ and $B={\left[{b}_{ij}\right]}_{N\times N}$, respectively, where $N$ denotes the number of individuals in the consumers’ society. Therefore, the

## Equilibrium, stability, and model properties

This sectioninvestigates the dynamical properties and different kinds of equilibrium for the proposed model (3). Since some results in this sectionare based on a single-product model, such a model is studied in A.1. Henceforth, the vector $x={[{{r}^{1}}^{T},{{s}^{1}}^{T},{{r}^{2}}^{T},{{s}^{2}}^{T}]}^{T}\in {\mathbb{R}}^{4N}$ will denote the overall system state, and $\lambda (.)$ will denote a member of the eigenvalue set of the corresponding matrix argument where ${\lambda}_{1}(.)$ denotes the largest member. The state vector may also be considered as $x={\left[{{x}^{1}}^{T}{{x}^{2}}^{T}\right]}^{T}$ where ${x}^{j}={[{{r}^{j}}^{}}^{}$

## Homogeneous pricing

This sectionexamines the optimal homogeneous pricing strategy, where the same price is offered to every consumer. Every time an individual transitions to one of ${O}_{1}$ or ${O}_{2}$ states, the corresponding product generates a certain amount of profit. The pricing strategies can be designed to maximize the profit of a single firm or both together.

## Heterogeneous prices

In the heterogeneous pricing strategy, each consumer $i$ ($i=1,\cdots ,N$) is assigned a price ${{p}_{j}}_{i}$ for the product $j$ ($j=1,2$) (see (Fainmesser & Galeotti, 2020b) for several examples of heterogeneous pricing under network effect). Defining the scalars ${{\alpha}_{j}}_{i}=f\left({{p}_{j}}_{i}\right)$ and ${{\gamma}_{j}}_{i}=f\left(\frac{1}{{{p}_{j}}_{i}}\right)$, as well as the diagonal matrices ${\Omega}_{j}=\text{diag}\left[{{\alpha}_{j}}_{i}\right]$ and ${\Gamma}_{j}=\text{diag}\left[{{\gamma}_{j}}_{i}\right]$, the mean field approximation of the underlying stochastic process is obtained as follows.$\begin{array}{c}{\dot{r}}^{1}={\beta}_{1}(I-{R}^{1}-{R}^{2}-{S}^{1}-{S}^{2})A{s}^{1}-({\Omega}_{1}+{\Gamma}_{1}){r}^{1}\\ {\dot{s}}^{1}={\Omega}_{1}{r}^{1}-{\delta}_{1}{s}^{1}\\ {\dot{r}}^{2}={\beta}_{2}(I-{R}^{1}-{R}^{2}-{S}^{1}-{S}^{2})B{s}^{2}-({\Omega}_{2}+{\Gamma}_{2}\end{array}$

## Managerial implications

Several parameters affect the network equilibrium and different pricing strategies when different firms compete in the presence of network externalities. A factor determining the network equilibrium is the reproduction number. When the reproduction numbers of the two products are largely different, the product with a smaller reproduction number will become extinct while the other with a larger reproduction number pervades the network. Investigation of the reproduction number’s role reveals

## Conclusion

In this article, a new competitive pricing model was developed under network externalities. A mean field model was introduced by constructing a stochastic model based on different transition probabilities, considering the local network effects and product prices. Dynamical properties in terms of equilibrium conditions of the model are studied, and several corollaries are presented. In addition, different optimization problems were considered to maximize profits in possible circumstances. A

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