Social networks are the substrate upon which we make and evaluate
many of our daily decisions: our costs and benefits depend on
whether---or how many of, or which of---our friends are willing to go
to that restaurant, choose that cellular provider, already own that
gaming platform. Much of the research on the ``diffusion of
innovation,'' for example, takes a game-theoretic perspective on
strategic decisions made by people embedded in a social context.
Indeed, multiplayer games played on social networks, where the
network's nodes correspond to the game's players, have proven to be
fruitful models of many natural scenarios involving strategic interaction.
In this paper, we embark on a mathematical and general
exploration of the relationship between 2-person strategic
interactions (a ``base game'') and a ``networked'' version of that
same game. We formulate a generic mechanism for superimposing a
symmetric 2-player base game $M$ on a social network $G$: each node
of $G$ chooses a single strategy from $M$ and simultaneously plays
that strategy against each of its neighbors in $G$, receiving as its
payoff the sum of the payoffs from playing~$M$ against each
neighbor. We denote the \emph{networked game} that results
by~$\gamenet{M}{G}$. We are broadly interested in the relationship
between properties of~$M$ and of~$\gamenet{M}{G}$: how does the
character of strategic interaction change when it is embedded in a
social network? We focus on two particular properties: the (pure)
price of anarchy and the existence of pure Nash equilibria. We show
tight results on the relationship between the price of anarchy
in~$M$ and $\gamenet{M}{G}$ in coordination games. We also show
that, with some exceptions when $G$ is bipartite, the existence or
absence of pure Nash equilibria (and even the guaranteed convergence
of best-response dynamics) in $M$ and $\gamenet{M}{G}$ are not
entailed in either direction. Taken together, these results suggest
that the process of superimposing $M$ on a graph is a nontrivial
operation that can have rich, but bounded, effects on the strategic
environment.