  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 \emph{influence games} as a
\emph{game-theoretic} model of the behavior of a large but finite networked
population. Influence games allow \emph{both} positive and negative
\emph{influence factors}, permitting reversals in behavioral choices. We
embrace \emph{pure-strategy Nash equilibrium (PSNE)}, an important solution
concept in non-cooperative game theory, to formally define the \emph{stable
outcomes} of an influence game and to predict potential outcomes without
explicitly considering intricate dynamics. We address an important problem in
network influence, the identification of the \emph{most influential
individuals}, and approach it algorithmically using PSNE computation.
\emph{Computationally}, we provide (a) complexity characterizations of various
problems on 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.
\emph{Experimentally}, we evaluate our approach using both synthetic influence
games as well as several real-world settings of general interest, each
corresponding to a separate branch of the U.S. Government.
\emph{Mathematically,} we connect influence games to important game-theoretic
models: \emph{potential and polymatrix games}.
