  There has been significant interest in studying security games for modeling
the interplay of attacks and defenses on various systems involving critical
infrastructure, financial system security, political campaigns, and civil
safeguarding. However, existing security game models typically either assume
additive utility functions, or that the attacker can attack only one target.
Such assumptions lead to tractable analysis, but miss key inherent dependencies
that exist among different targets in current complex networks. In this paper,
we generalize the classical security game models to allow for non-additive
utility functions. We also allow attackers to be able to attack multiple
targets. We examine such a general security game from a theoretical perspective
and provide a unified view. In particular, we show that each security game is
equivalent to a combinatorial optimization problem over a set system
$\varepsilon$, which consists of defender's pure strategy space. The key
technique we use is based on the transformation, projection of a polytope, and
the elipsoid method. This work settles several open questions in security game
domain and significantly extends the state of-the-art of both the polynomial
solvable and NP-hard class of the security game.
