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New Models for Competitive Contagion

Last modified: 2014-06-21

#### Abstract

In this paper, we introduce and examine two new models for competitive contagion in networks, a game-theoretic generalization of the viral marketing problem. In our setting, firms compete to maximize their market share in a network of consumers whose adoption decisions are stochastically determined by the choices of their neighbors. Building on the switching-selecting framework introduced by Goyal and Kearns, we first introduce a new model in which the payoff to firms comprises not only the number of vertices who adopt their (competing) technologies, but also the network connectivity among those nodes. For a general class of stochastic dynamics driving the local adoption process, we derive upper bounds on (1) the (pure strategy) Price of Anarchy (PoA), which measures the inefficiency of resource use at equilibrium, and (2) the Budget Multiplier, which captures the extent to which the network amplifies the imbalances in the firms' initial budgets. These bounds depend on the firm budgets and the maximum degree of the network, but no other structural properties. In addition, we give general conditions under which the PoA and the Budget Multiplier can be unbounded. We also introduce a model in which budgeting decisions are endogenous, rather than externally given as is typical in the viral marketing problem. In this setting, the firms are allowed to choose the number of seeds to initially infect (at a fixed cost per seed), as well as which nodes to select as seeds. In sharp contrast to the results of Goyal and Kearns, we show that for almost any local adoption dynamics, there exists a family of graphs for which the PoA and Budget Multiplier are unbounded.

#### Keywords

Competitive Contagion, Price of Anarchy, Connectivity, Endogenous Budget Constraints

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