Competitive Contagion in Networks (1110.6372v3)

Abstract: We develop a game-theoretic framework for the study of competition between firms who have budgets to "seed" the initial adoption of their products by consumers located in a social network. The payoffs to the firms are the eventual number of adoptions of their product through a competitive stochastic diffusion process in the network. This framework yields a rich class of competitive strategies, which depend in subtle ways on the stochastic dynamics of adoption, the relative budgets of the players, and the underlying structure of the social network. We identify a general property of the adoption dynamics --- namely, decreasing returns to local adoption --- for which the inefficiency of resource use at equilibrium (the Price of Anarchy) is uniformly bounded above, across all networks. We also show that if this property is violated the Price of Anarchy can be unbounded, thus yielding sharp threshold behavior for a broad class of dynamics. We also introduce a new notion, the Budget Multiplier, that measures the extent that imbalances in player budgets can be amplified at equilibrium. We again identify a general property of the adoption dynamics --- namely, proportional local adoption between competitors --- for which the (pure strategy) Budget Multiplier is uniformly bounded above, across all networks. We show that a violation of this property can lead to unbounded Budget Multiplier, again yielding sharp threshold behavior for a broad class of dynamics.

Competitive Contagion in Networks: An Analytical Overview

The paper "Competitive Contagion in Networks" by Sanjeev Goyal and Michael Kearns explores a game-theoretic framework to analyze how firms with budget constraints compete to maximize the adoption of their products within a social network. This paper leverages stochastic diffusion processes to simulate how the influence of seeded initial adopters spreads through a network. Central to the research are the large-scale dynamics of strategic seed allocation by two competing firms, referred to as Red and Blue, which are analyzed through various properties of contagion processes such as the Price of Anarchy and the Budget Multiplier.

Key Contributions

• Game-Theoretic Framework: The paper develops a framework that models the allocation and influence of seed budgets within a social network. Each firm strategically allocates resources to maximize the spread of its product under dynamic contagion settings.
• Adoption Dynamics and Strategy: The analysis identifies two significant properties influencing equilibrium contagion strategies:
  1. Decreasing Returns to Local Adoption: The inefficiency in resource usage at equilibrium, termed as the Price of Anarchy (PoA), is shown to be uniformly bounded when there is decreasing returns to local adoption.
  2. Proportional Local Adoption: Defined through the concept of the Budget Multiplier, where budget imbalances can be amplified, there is a bound on this amplification when proportional local adoption dynamics are present.

• Theoretical Results: The work provides rigorous proofs for bounding PoA and Budget Multiplier. Notably, for concave switching functions and linear selection functions, the PoA is uniformly bounded by 4, emphasizing how strategic inefficiency can be controlled under certain dynamic properties.

Results and Implications

• Price of Anarchy (PoA): The PoA illustrates the inefficiency impacts due to decentralized strategies. With concave switching functions, the paper shows a PoA bound, highlighting that strategic non-cooperation does not lead to drastic inefficiencies when local adoption follows a particular concave pattern.
• Budget Multiplier: The concept measures amplification of budget advantages, finding that under certain conditions beyond linear selection, significant inequalities can result. The paper highlights how even minor deviations from linear selection can result in unbounded Budget Multipliers.
• Nonlinearity and Network Effects: Findings suggest that both the structure of the network and the nonlinear properties of switching and selection functions critically impact contagion dynamics. This illuminates the intricacies of strategic influence in networked environments, as slight deviations in dynamics can dramatically alter competitive outcomes.
• Threshold Behavior and Dynamics Amplification: The research identifies sharp threshold behaviors in dynamic classes—where crossing a threshold in parameters flips the outcome from bounded to unbounded inefficiency or budget amplification.

Speculation and Future Work

While the current research is substantive, several avenues remain for further exploration. Future studies could investigate the dynamics in multi-firm settings or under dynamic budget adjustments influenced by past performance and network feedback. Additionally, exploring the computational challenges in tracking optimal strategies in expansive networks or varying consumer behaviors offers fertile ground for further theory and algorithm development.

This research outlines the multilayered complexity of competitive diffusion models, offering insights into how industrial strategies can be optimized or need rethinking based on network topology and adoption dynamics. As networks grow and evolve, such frameworks will be pivotal in strategizing and optimizing social influence, competitive marketing, and informational campaigns within interconnected populations.