Competition for networked agents in the lottery Blotto game
Introduction
It is widely acknowledged that agents are embedded in a social network, they make consumption decisions based on whether their close friends, neighbors, and celebrities also adopt the same products. Adolescents’ consumption of alcohol and tobacco is affected by their friends’ consumption. When deciding whether to use a new web conference platform, enterprise staffs rely on information from colleagues and peers. Such peer influence can also be found in school choice, job searching and criminal offense.1
In the meanwhile, firms in the market make marketing effort to attract the attention from agents, increase the consumption of its products. Consider, for example, in the public cloud service market, enterprises can choose among AWS, Microsoft Azure and Google Cloud Platform. In the market of social media messaging apps, Facebook Messenger, Snapchat, and WhatsApp are fighting for market shares. Considering agents are more inclined to choose products that are used by more peers, service providers need to make an effort to lobby “important” agents and strive to induce agents to choose their own products among similar products.
In this paper, we study marketing competition between firms when products are differentiated and exhibit local (positive) network effects. Two products, provided by two firms, are perfect substitutable and agents can freely access to them. As in Chen et al. (2018b), we assume that an agent’s utility consists of two parts: one part corresponds to the utility from her own usage level, and the other part describes the positive externality of her peers.2 In contrast, to capture the competition between firms, we assume that each agent’s total consumption level for the two products will not be affected by firms’ marketing effort.3 The duopoly firms make efforts such as advertising to target their marketing budgets to agents in the market.
How can the duopoly firms exploit the above network externality, and allocate their marketing budgets to agents so as to maximize the total consumption towards their products? To answer this question, we model the competition between firms as a lottery Blotto game adopted from Xu and Zhou (2018). Viewing each consumer’s consumption level as a prize in the competition between firms, we characterize the equilibrium marketing strategies and study how the strength of network effect and network structure affect firms’ budget allocation decision.
We show that the equilibrium consumption level for each agent is the agent’s Bonacich centrality. The fraction of a firm’s marketing budget targeted to an individual is just the proportion of individual consumption in total market consumption. In other words, firms’ equilibrium marketing strategies only depend on the underlying social network structure and the strength of network effect. Specifically, when the strength of network effect is relatively small, the network externality mainly comes from the number of neighbors. While when the strength of network effect from peers is large enough, firms’ tend to allocate more budget to those agents with higher eigenvector centralities, who are more influential.
This paper belongs to the literature on firms’ competition in social networks. The Bonacich centrality measure always appears in social and economic networks literature. Following the seminal work of Ballester et al. (2006), a bunch of papers expand research on the above topics.4 The social network-based pricing decision has been well studied by Candogan et al. (2012), Bloch and Querou (2013), Fainmesser and Galeotti (2016) and many others. While early papers primarily focus on the optimal pricing strategies of a monopolist, Chen et al. (2018b) study the price competition between competing firms who sell heterogeneous products. They show that firms’ discriminatory pricing based on network structure is related to the Bonacich centrality measures. While all of the literature assumes that consumers buy product from a given manufacturer, our paper differs from these by allowing agents to choose which firm to buy from, and we study competition between firms on agents’ consumption while pricing is no longer firms’ strategy in our model.
Our study is related to the literature on social network-based marketing decision, including Hartline et al. (2008), Carroni et al. (2020), and Manshadi et al. (2020).5 While these studies focus on the optimal marketing or seeding strategies of a monopolist, competition between firms on targeted advertising has received relatively little attention. Using the Blotto game (e.g., Friedman, 1958, Roberson, 2011), Bimpikis et al. (2016) study consumers’ awareness levels for firms in the word-of-mouth process and highlight their dependence on the underlying social network structure. Two firms make advertising efforts to compete for agents’ awareness according to contest success functions. Goyal et al. (2019) consider a stochastic dynamics of local adoption model. Two firms choose an allocation of budget to “seed” the initial adoption of their products, so as to maximize the total number of eventual product adoptions. They use the linear selection function to specify the probability of infection by each firm in terms of the local relative market share split. Similar with these two papers, our paper also uses contest success functions to model the competition between firms with marketing budget. While differ from the literature considering the dynamic process on social networks, we assume that when firms make marketing efforts to compete for agents’ consumptions, firms’ marketing strategies will not affect the amount of consumption of each agent. Our main focus is to study the optimal budget allocation of firms and how network externality influences it.
Section snippets
Model setup
Consider a market with a set of agents (consumers) and two firms (producers). Agents are embedded in a connected social network represented by an matrix . We assume if and are connected for all , otherwise . Each agent cannot connect with himself, i.e., . The network structure is symmetric in the sense that . Let denote the set of agent ’s neighbors: . Let denote the number of agent ’s neighbors.
Assume the prices
Equilibrium analysis
Let denote the -dimensional identity matrix, and denote the -dimensional column vector of ones. Note that is a real symmetric matrix, Spectral Theorem implies that each eigenvalue of is real. We assume that the distinct eigenvalues of are (), then the largest eigenvalues of is . Let denote the vector of Bonacich centralities of parameter .7
Examples
To better understand the effect of network structure on the consumption of agents and the budget allocation of firms, we provide three examples.
Example 1 A Regular Network A network is regular of degree if each agent has exactly neighbors. Using Lemma 1, we obtain the following equilibrium consumptions: From Lemma 2, the equilibrium allocation for firm in battle is given by Given the regular network structure, the consumption level of each agent is the same. Thus, firms distribute
Conclusion
In this paper, we consider firms’ competition for attention of networked agents. Two firms, targeting their marketing budgets to individuals embedded in a social network, compete for agents’ consumption. The competition between firms is modeled as a lottery Blotto game. We characterize the equilibrium marketing strategies and highlight their dependence on the underlying network externality.
Our main focus of this paper is to study the effect of network externality on each agent’s consumption and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We thank an anonymous referee for helpful comments. We are grateful to Junjie Zhou and Xiang Sun for insightful and constructive suggestions. Jiao gratefully acknowledges the financial support from Humanities and Social Science Youth Foundation of Ministry of Education of China (No. 20YJC790052). Xu gratefully acknowledges the financial support from China Postdoctoral Science Foundation (No. 2019M662380) and the National Natural Science Foundation of China (No. 72073083). The usual disclaimer
References (19)
- et al.
Pricing in social networks
Games Econom. Behav.
(2013) - et al.
Competitive contagion in networks
Games Econom. Behav.
(2019) - et al.
Bipartite conflict networks with returns to scale technology
J. Econ. Behav. Organ.
(2019) - et al.
Discriminatory power and pure strategy Nash equilibrium in the lottery Blotto game
Oper. Res. Lett.
(2018) - et al.
Who’s who in networks. Wanted: The key player
Econometrica
(2006) - et al.
Competitive targeted advertising over networks
Oper. Res.
(2016) Power and centrality: a family of measures
Am. J. Sociol.
(1987)- et al.
Optimal pricing in networks with externalities
Oper. Res.
(2012) - et al.
Bring a friend! privately or publicly?
Manage. Sci.
(2020)