1 Introduction
Over the past few years, the problem of influence and spread in networks has been subject to intense study (Ballester et al., 2006; Bharathi et al., 2007; Galeotti and Goyal, 2009; Kempe et al., 2003, 2005; Chasparis and Shamma, 2010; Vetta, 2002). Furthermore, modeling and analysis of the spread of new strategies and behaviors via local coordination games has been an ongoing field of research (Ellison, 1993; Kandori et al., 1993; Harsanyi and Selten, 1988; Young, 1993, 2001, 2002; Montanari and Saberi, 2010; Kleinberg, 2007). For example, in LópezPintado (2006) the authors show that the contagion of an action in a random network depends on the distribution of the connectivities. In Amini et al. (2009), authors provide an upper bound on the proportion of agents adopting a new product assuming a threshold model for product adoption. Montanari and Saberi (2010) studies the diffusion of innovation in social networks based on the dynamics of coordination games and shows that innovation spreads much more slowly on wellconnected network structures dominated by longrange links than in lowdimensional ones dominated by geographic proximity, contrasting some earlier works on epidemic models (e.g., (Ganesh et al., 2005; Draief et al., 2006)).
A game theoretic model of competition and product adoption has been proposed in Goyal and Kearns (2012). The authors use the proposed model to come up with upper bounds on the price of anarchy and show how network structure may increase the gap between the initial budgets. Similarly, in Bimpikis et al. (2013), the authors propose a game theoretic model for competition between firms, where firms can target their marketing budgets toward attracting individual consumers embedded in a social network. They subsequently provide conditions under which it is optimal for the firms to asymmetrically target a subset of the individuals. As another relevant work, (Chasparis and Shamma, 2010) considers a dynamical model of preferences in a duopoly setting and characterize optimal policies for both finite and infinite time horizons, studying the effect of endogenous network influences as well as network uncertainties. The equilibria of network games with linear best response dynamics have been completely characterized in Bramoullé et al. (2014). Considering a monopoly setting, optimal pricing policies are derived in Candogan et al. (2012) assuming quadratic utility functions for the agents, which is a common theme in gametheoretic social network analysis as also previously used in Ballester et al. (2006); Corbo et al. (2007) for instance.
Despite the tremendous development made in the past decade (see e.g., (Seeman and Singer, 2013; Fazeli et al., 2017; Bharathi et al., 2007; Montanari and Saberi, 2010; Chasparis and Shamma, 2010)), influence maximization algorithms are typically devised under idealized assumptions regarding accessibility of individual consumers for targeted advertisement (e.g., via seeding) as well as the linearity of the cost function with respect to the size of the influence on individual’s consumption behavior. In practice, however, firms often have direct access only to a small subset of consumers. In addition, the extent to which firms can influence consumption behavior of consumers is limited, no matter how much they spend on seeding/advertisement. This can be more formally stated as diminishing returns on changes in individual consumption levels, which we model by assuming a convex seeding cost function in our work. It seems also quite compelling to investigate approaches that can support sparse seeding as (at least) nearoptimal marketing strategies, in order to account for the limitations in directly accessing individuals especially in large populations.^{1}^{1}1By sparse seeding we mean only seeding a subset of consumers with an infinitesimal size compared to the size of the whole population.
In this paper, we study strategic competition between two firms seeking to maximize their product consumption in a network. The consumption of each product by each agent is the result of her myopic best response to the previous consumptions of her peers. A firm can thus improve its market share by targeting its advertising budget toward seeding key individuals in the network, whose consumption of the product can in turn incentivize their peers to consume more of the same product, subsequently affecting the consumption behavior of the individuals all over the network via interagent influences. We model the problem above as a fixedsum game between the two firms, where each firm tries to maximize a utility function which is a discounted sum of its product consumption over time less a seeding cost which is appropriately chosen to account for the diminishing return in seeding/advertisement budget as well. We characterize the unique Nash equilibrium of the resulting duopoly game in terms of the network structure and the market price. The resultant seeding strategy typically prescribes seeding all the agents (in an amount proportional to their influence), which is rather an infeasible task. As a remedy, we propose studying the equilibria of the game and derive conditions under which such equilibria can asymptotically lead to sparse seeding strategies in large populations.
The rest of this paper is organized as follows: in Section 2, we present basic mathematical notations. In Section 3, we introduce our model and update dynamics for agents by applying the myopic best response. We then study the game played between the firms and how they decide to seed key individuals in Section 4. Next, we define a nearNash equilibrium concept as a relaxation of the standard Nash equilibrium with the aim of expanding the equilibrium set to include sparse seeding strategies in Section 5. In Section 6, we consider the case of a large population and characterize network structures for which a pair of sparse seeding strategies can be asymptotically realized. The results are illustrated via an example in Section 7. Finally, in Section 8, we conclude the paper.
2 Mathematical Notations
Throughout the paper, the discrete time index is denoted by . The sets of real (integer), positive real (integer), and strictly positive real (integer) numbers are represented by (), () and (), respectively. The set of natural numbers is denoted by . Bold letters, such as or
, stand for realvalued vectors. Capital letters, such as
or , stand for realvalued matrices. We use to denote the norm of vector . We denote the number of nonzero elements in vector by . The byidentity matrix is denoted by . Also, we represent the by vector of ones by and the by and by matrices of zeros by and , respectively. The transpose of matrix is denoted by . For two matrices and , we denote the Hadamard product by and the Kronecker product by .3 Spread Dynamics
We consider a social network consisting of a group of consumers (agents) denoted by . The relationship among agents is given by a weighted directed graph . The weighted adjacency matrix of is denoted by where its th entry denoted by if otherwise . Link weight presents the strength of the influence of agent on .
We assume there are two competing firms producing product and , respectively. Let and denote the indegree and outdegree of node , respectively. Assume and denote agent ’s consumption of product and at time , respectively. The initial consumption of agent from each product is determined by the effort made by each firm in seeding/marketing its product to agent at time , and are denoted by and . We refer to and as the control/seeding of firm and on agent , respectively. The relation between the controls and the states is motivated by (Chasparis and Shamma, 2010) on social networks. We consider a convex cost function for seeding to reflect the diminishing return of seeding/advertisement budget on changing the consumption behavior in the population. The cost reflects the monetary value required to increase the consumption of the agents in the population by amount . For the sake of simplicity, we develop our results for a quadratic cost function of the form .
The total utility of agent from taking action is given by
(1)  
(2)  
(3) 
where . In the above equation denotes an action vector of all agents but agent . The first part of the utility () represents the normalized second order approximation of a concave selfutility function, as is commonly used in the literature. The fact that externality is weaker for consuming different products is captured by . Finally, the price represents the common market price for the products.
3.1 Myopic Best Response Dynamics
We assume agents repeatedly apply myopic best response to the consumption levels of their neighbors in the previous stage to update their consumption of each product, that is,
and
Consumption levels for agent at time thus find the following linear dynamics:
and
This results in the following closedform update dynamics:
(4) 
where and .
Assumption 1.
We assume to guarantee that and are nonnegative for all feasible initial seedings.
In the next section, we show how firms can exploit the structure of the network to maximize their product consumption, and we then characterize the unique Nash equilibrium of the game played between these two firms.
We first recall the definition of the KatzBonacich centrality measure.
Definition 1.
For a given attenuation factor
less than the reciprocal of the absolute value of the largest eigenvalue of the adjacency matrix
, the KatzBonacich centrality is given byDefinition 2.
For a given subset , we define an indicator binary vector as follows:
(5) 
We then use these definitions in the next section to characterize equilibrium strategies in a duopoly game between the firms.
4 Optimal Seeding Strategies
This section describes the game between two firms where each firm aims to maximize the consumption of its product over an infinite horizon. Each firm can invest in promoting its product by seeding some of the agents. Initial seeding could be viewed as free offers to promote the product in the social networks. We thus consider the problem of deriving optimal advertising policies for the spread of innovations/consumption in a network. In Chasparis and Shamma (2010), an analytical solution to the optimal advertising problem in the absence of a competing firm is provided, and it is shown that the solution can be related to previously introduced centrality measures in sociology.
We define the utility of each firm as the discounted sum of its product consumption over time minus the squared norm of its seeding, which are defined formally as follows:
(6) 
and
(7) 
Definition 3.
A pair of seeding strategies is said to be a Nash equilibrium of the duopoly game described above if none of the players can improve her payoff by unilaterally deviating from her strategy. That is,
and
Assumption 2.
We assume that the absolute value of the largest eigenvalue of is less than or equal to where
We then use Assumption 2 to get welldefined centrality measures for the graph with adjacency , and make the matrix pencil in Definition 1 invertible.
In the following lemma, an equilibrium strategy for the duopoly game between the firms is characterized based on the node centrality measures.
Lemma 1.
Proof 1.
Let us define the following Jacobian matrices for firm as follows:
(11) 
and
(12) 
Similarly, we define the following Jacobian matrices for firm :
(13) 
and
(14) 
Next, we write the update dynamics for and according to their definitions and update dynamics (4):
(15) 
Then, based on update dynamics (15), (12) and (14) we get
(16) 
where , and . We can rewrite (17) in the following compact form
(17) 
From this, it follows that
(18) 
We then use the following property to simplify (18)
(19) 
Let us define
(20) 
based on (19) and (20), we get
(21) 
With a simple calculation we get:
(22) 
Next by expanding and then applying (19), (21), and (22), it follows that
By substituting (18) in (24), we have
(23) 
From (6), we get
(24) 
Finally, by substituting (23) in (24), we get the desired result.
Theorem 1.
Proof 2.
The proof is a direct consequence of Lemma 1.
5 Sparse Seeding
In what follows, we first define a nearNash equilibrium concept as a relaxation of the standard Nash equilibrium defined in Definition 3, with the aim of expanding the equilibrium set to include sparse seeding strategies (cf. (Daskalakis et al., 2006)). In what follows, we will make this statement formal.
Definition 4.
Given , a pair of seeding strategies is said to be equilibrium of the duopoly game described in Section 4 if none of the players can improve her payoff by an amount more than fraction of her current payoff, by unilaterally deviating from her strategy. That is,
and
Every Nash Equilibrium is equivalent to a equilibrium where .
The next lemma provides a closedform expression for the utility functions of the firms when there is not any seeding.
Lemma 2.
Proof 3.
Let us assume , then
(26) 
where is given by (20). With a simple calculation we get:
(27) 
Then, based on the definition of the utility, we have
Theorem 2.
6 Asymptotically Realizable Sparse Equilibria
In this section, we consider the case of a large population for which . We characterize network structures for which a pair of sparse seeding strategies can be realized as the limit of a sequence of equilibria with . In what follows, we will make this statement formal.
Definition 5.
We call a pair of equilibrium seeding strategies asymptotically sparserealizable (ASR) if and only if , and .^{2}^{2}2Given functions and , the asymptotic notations and mean and , respectively.
We begin our analysis by a lemma that presents a necessary and sufficient condition for a pair of strategies be asymptotically sparserealizable in terms of the biproduct centrality (10).
Lemma 3.
A pair of strategies is asymptotically sparserealizable if and only if
(34) 
where is given by (10), , , , and .
Proof 5.
We first start with the fact that if and only if . Therefore, the pair of equilibrium seeding strategies is asymptotically sparserealizable if and only if
(35) 
Let us recall
(36)  
(37) 
where , and . It can be seen that
where in the last inequality, we use the fact that .
(38) 
where .
As the next result, we derive a necessary condition for the existence of asymptotically sparserealizable equilibrium strategies.
Proposition 1.
Let , and suppose that
Then, there exists no pair of equilibrium seeding strategies that is asymptotically sparserealizable.
Proof 6.
This result implies that networks with bounded outdegree (i.e., ) are not asymptotically sparserealizable.
We conclude this section by stating a sufficient condition for the existence of asymptotically sparserealizable equilibrium strategies.
Proposition 2.
Proof 7.
The proof is a direct consequence of Lemma 3.
7 Discussion
We illustrate our results using a coreperiphery network structure, that is a network with few highly interconnected and many sparsely connected nodes.
Consider a network consisting of communities of size denoted by for . Consumption levels of the agents within each community are influenced by an agent called a role model that induces an externality effect of magnitude on each consumer. Consumption levels of role models themselves are each influenced by the consumption of a role model in another community whose structure is assumed to form a cycle. Let agent be the role model in community for . The corresponding entries of the adjacency matrix are given by
(39) 
The case is depicted in Fig. 1.
Characterizing the equilibrium seeding strategies (both Nash equilibrium and equilibria) requires finding centrality vectors
and
where . Let and be the corresponding centralities of a role model and a periphery consumer in , respectively. A periphery consumer does not induce externality on any other consumer, resulting in a centrality of . For a role model, on the other hand, from the definition of KatzBonacich centrality we can obtain
(40) 
resulting in . Similarly, we can find and .
Applying Theorem 1, the symmetric Nash equilibrium strategy thus involves seeding a periphery consumer by , and every role model by an amount of
(41) 
Keeping the number of communities fixed and shifting , we can use Lemma 3 to verify that seeding only the role models according to (7) is an asymptotically sparserealizable equilibrium strategy. Finally, Assumption 2 requires which can be satisfied if noting that (this follows from for all ).
8 Conclusion And Future Work
We proposed and studied a strategic model of marketing and product consumption in social networks. Two firms offer substitutable products and compete to maximize the consumption of their products in a social network. Consumers are myopic and update their consumption level as the best response to the consumption of their neighbors in the previous period. This results in linear update dynamics for the product consumption. Moreover, each consumer receives externality from the consumption of each neighbor; the externality is stronger for consumption of the same product. Firms can improve their market share by seeding the key consumers in the market, as their consumption will incentivize the consumption of the same product by their peers given the interagent externalities, which in turn can affect the consumption behavior all over the network. We represented the above setting as a duopoly game between the firms and introduced a novel framework that allows for sparse seeding to asymptotically emerge as an equilibrium strategy. We then studied the effect of the network structure on the optimal seeding strategies and the extent to which these strategies could be sparsified, under the proposed equilibrium concept. In particular, we derive necessary and sufficient conditions under which Nash equilibrium strategies can asymptotically lead to sparse seeding in large populations. The results were demonstrated using a large coreperiphery network structure with few highly connected and many sparsely connected nodes. Extending our analysis to timevarying seeding strategies (recurring seeding), networks with uncertainties (e.g., in valuations of consumers about the products), and oligopoly setting are some of the potential venues for future research.
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