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We introduce a new approach to the study of influence in strategic settings where the action of an individual depends on that of others in a network-structured way. We propose network influence games as a game-theoretic model of the behavior of a large but finite networked population. In particular, we study an instance we call linear-influence games that allows both positive and negative influence factors, permitting reversals in behavioral choices. We embrace pure-strategy Nash equilibrium, an important solution concept in non-cooperative game theory, to formally define the stable outcomes of a network influence game and to predict potential outcomes without explicitly considering intricate dynamics. We address an important problem in network influence, the identification of the most influential individuals, and approach it algorithmically using pure-strategy Nash-equilibria computation. Computationally, we provide (a) complexity characterizations of various problems on linear-influence games; (b) efficient algorithms for several special cases and heuristics for hard cases; and (c) approximation algorithms, with provable guarantees, for the problem of identifying the most influential individuals. Experimentally, we evaluate our approach using both synthetic network influence games and real-world settings of general interest, each corresponding to a separate branch of the U.S. Government. Mathematically, we connect linear-influence games to important models in game theory: potential and polymatrix games. © 2014 Published by Elsevier B.V.