Competitive Diffusion 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 firms in a market competing to maximize the consumption of their products. Firms have a limited budget which can be either invested on the quality of the product or spent on initial seeding in the network in order to better facilitate spread of the product. After the decision of firms, agents choose their consumptions following a myopic best response dynamics which results in a local, linear update for their consumption decision. We characterize the unique Nash equilibrium of the game between firms and study the effect of the budgets as well as the network structure on the optimal allocation. We show that at the equilibrium, firms invest more budget on quality when their budgets are close to each other. However, as the gap between budgets widens, competition in qualities becomes less effective and firms spend more of their budget on seeding. We also show that given equal budget of firms, 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 as well. As a practical extension, we then consider a case where products have some preset qualities that can be only improved marginally. At some point in time, firms learn about the network structure and decide to utilize a limited budget to mount their market share by either improving the quality or new seeding some agents to incline consumers towards their products. We show that the optimal budget allocation in this case simplifies to a threshold strategy. Interestingly, we derive similar results to that of the original problem, in which preset qualities simulate the role that budgets had in the original setup.

💡 Research Summary

The paper develops a game‑theoretic model of competition between two firms that sell substitutable products in a social network. Each firm has a fixed budget that can be allocated between improving product quality and “seeding” a subset of consumers (i.e., giving them an initial consumption advantage). Consumers have unit demand and decide each period how much of product a versus product b to consume. Their payoff consists of an isolation term (a concave quadratic function of their own consumption) and a network externality term that rewards them when neighbors choose the same product.

The authors assume agents apply a myopic best‑response rule: given neighbors’ current consumptions, each agent chooses the consumption that maximizes its instantaneous payoff. This leads to a linear update dynamics for the vector of deviations from a 50‑50 split:

 y(t + 1) = W y(t) + u_a,

where W = (1⁄2β) G (G is the row‑stochastic adjacency matrix) and u_a is a constant vector proportional to the quality difference (q_a − q_b). The solution can be written in closed form as

 y(t) = W^t y(0) + ∑_{k=0}^{t‑1} W^k u_a.

Thus the entire consumption trajectory depends on the initial seed vector y(0), the quality pair (q_a, q_b), and the network structure.

Each firm’s objective is the discounted sum of its product’s consumption over an infinite horizon, with discount factor δ. After algebraic manipulation, the utility of firm a becomes

 U_a = constant + vᵀ S_a − vᵀ S_b + λ · (q_a − q_b)/(q_a + q_b),

where v = (I − δWᵀ)⁻¹ 1 is a centrality vector that captures how much a unit of seed at a node propagates through the network, and λ is a positive scalar that aggregates model parameters (α, β, δ, n). The utility of firm b is symmetric.

Seeding costs c_s per unit, quality improvement costs c_q per unit, and each firm’s budget constraint is

 c_s ‖S_i‖₁ + c_q q_i = K_i (i = a,b).

Because the term vᵀ S_i is linear in the ℓ₁‑norm of the seed vector, the optimal seeding policy for any firm is to allocate seeds to the highest‑centrality agents first, filling each up to the maximum feasible amount (0.5) until the budget is exhausted. Consequently, the strategic decision reduces to choosing the quality level q_i; the seed allocation follows automatically.

The interaction is a fixed‑sum game (U_a + U_b is constant), which can be transformed into a zero‑sum game. The authors prove that the utility function is strictly concave in a firm’s own quality and strictly convex in the rival’s quality, satisfying the conditions of Sion’s minimax theorem. Hence a unique pure‑strategy Nash equilibrium exists.

The equilibrium quantities are derived analytically:

 q_a* = (2λ / (c_s c_q)) ·