Optimal Budget Allocation in Social Networks: Quality or Seeding

In this paper, we study a strategic model of marketing and product consumption in social networks. We consider two competing firms in a market providing two substitutable products with preset qualities. Agents choose their consumptions following a myopic best response dynamics which results in a local, linear update for the consumptions. At some point in time, firms receive a limited budget which they can use to trigger a larger consumption of their products in the network. Firms have to decide between marginally improving the quality of their products and giving free offers to a chosen set of agents in the network in order to better facilitate spreading their products. We derive a simple threshold rule for the optimal allocation of the budget and describe the resulting Nash equilibrium. It is shown that the optimal allocation of the budget depends on the entire distribution of centralities in the network, quality of products and the model parameters. In particular, we show that in a graph with a higher number of agents with centralities above a certain threshold, firms spend more budget on seeding in the optimal allocation. Furthermore, if seeding budget is nonzero for a balanced graph, it will also be nonzero for any other graph, and if seeding budget is zero for a star graph, it will be zero for any other graph too. We also show that firms allocate more budget to quality improvement when their qualities are close, in order to distance themselves from the rival firm. However, as the gap between qualities widens, competition in qualities becomes less effective and firms spend more budget on seeding.Comment: 7 page

Targeted Marketing and Seeding Products with Positive Externality

We study a strategic model of marketing in social networks in which two firms compete for the spread of their products. Firms initially determine the production cost of their product, which results in the payoff of the product for consumers, and the number and the location of the consumers in a network who receive the product as a free offer. Consumers play a local coordination game over a fixed network which determines the dynamics of the spreading of products. Assuming myopic best response dynamics, consumers choose a product based on the payoff received by actions of their neighbors. This local update dynamics results in a game-theoretic diffusion process in the network. Utilizing earlier results in the literature, we derive a lower and an upper bound on the proportion of product adoptions which not only depend on the number of initial seeds but also incorporate their locations as well. Using these bounds, we then study which consumers should be chosen initially in a network in order to maximize product adoptions for firms. We show consumers should be seeded based on their eigenvector centrality in the network. We then consider a game between two firms aiming to optimize their products adoptions while considering their fixed budgets. We describe the Nash equilibrium of the game between firms in star and k-regular networks and compare the equilibrium with our previous results

Duopoly Pricing Game in Networks With Local Coordination Effects

In this paper, we study a duopoly pricing problem in which two firms compete for selling two products in a network. Our proposed model consists of two stages. In the first stage, firms set the price they charge agents for their product and the quality of the product they offer. For agents, the quality of the product can be interpreted as the payoff of a local coordination game played among them in the network. In the second stage, agents in the network decide what fraction of these two products to purchase. We first characterize the Nash equilibrium of the game played among agents in the network. We show that agents’ actions in the Nash equilibrium consist of two terms, one of which is proportional to the agents’ centrality in the network. Conditioned on agents playing the equilibrium policy, we find the Nash equilibrium of the pricing game played between firms. We show that even when firms are similar and offer a uniform price for agents, their Nash equilibrium price depends on the network structure.We then analyze sensitivity of the agents’ consumption with respect to the price and quality of the product. We finally show that depending on a firm’s opponent’s price and quality, the optimal price of a firm can be higher, equal or less than the monopoly optimal price

Consensus Over Martingale Graph Processes

In this paper, we consider a consensus seeking process based on repeated averaging in a randomly changing network. The underlying graph of such a network at each time is generated by a martingale random process. We prove that consensus is reached almost surely if and only if the expected graph of the network contains a directed spanning tree. We then provide an example of a consensus seeking process based on local averaging of opinions in a dynamic model of social network formation which is a martingale. At each time step, individual agents randomly choose some other agents to interact with according to some arbitrary probabilities. The interaction is one-sided and results in the agent averaging her opinion with those of her randomly chosen neighbors based on the weights she assigns to them. Once an agent chooses a neighbor, the weights are updated in such a way that the expected values of the weights are preserved. We show that agents reach consensus in this random dynamical network almost surely. Finally, we demonstrate that a Polya Urn process is a martingale process, and our prior results in [1] is a special case of the model proposed in this paper

Game Theoretic Analysis of a Strategic Model of Competitive Contagion and Product Adoption in Social Networks

In this paper we propose and study a strategic model of marketing and product adoption in social networks. Two firms compete for the spread of their products in a social network. Considering their fixed budgets, they initially determine the payoff of their products and the number of their initial seeds in a network. Afterwards, neighboring agents play a local coordination game over a fixed network which determines the dynamics of the spreading. Assuming myopic best response dynamics, agents choose a product based on the payoff received by actions of their neighbors. This local update dynamics results in a game-theoretic diffusion process in the network. Utilizing earlier results in the literature, we find a lower and an upper bound on the proportion of product adoptions. We derive an explicit characterization of these bounds based on the payoff of products offered by firms, the initial number of adoptions and the underlying structure of the network. We then consider a case in which after switching to the new product, agents might later switch back to the old product with some fixed rate. We show that depending on the rate of switching back to the old product, the new product might always die out in the network eventually. Finally, we consider a game between two firms aiming to optimize their products adoptions while considering their fixed budgets. We describe the Nash equilibrium of this game and show how the optimal payoffs offered by firms and the initial number of seeds depend on the relative budgets of firms

Multi-agent Flocking With Random Communication Radius

In this paper, we consider a multi-agent system consisting of mobile agents with second-order dynamics. The communication network is determined by a metric rule based on a random interaction range. The goal of this paper is to determine a bound on the probability that the agents asymptotically agree on a common velocity (i.e. a flocking behavior is achieved). This bound should depend on practical conditions (on the initial positions and velocities of agents) only. For this purpose, we exhibit an i.i.d. process bounding the original system’s dynamics. We build upon previous work on multi-agent systems with switching communication networks. Though conservative, our approach provide conditions that can be verified a priori