  In this work, we provide a structural characterization of the possible Nash
equilibria in the well-studied class of security games with additive utility.
Our analysis yields a classification of possible equilibria into seven types
and we provide closed-form feasibility conditions for each type as well as
closed-form expressions for the expected outcomes to the players at
equilibrium. We provide uniqueness and multiplicity results for each type and
utilize our structural approach to propose a novel algorithm to compute
equilibria of each type when they exist. We then consider the special cases of
security games with fully protective resources and zero-sum games. Under the
assumption that the defender can perturb the payoffs to the attacker, we study
the problem of optimizing the defender expected outcome at equilibrium. We show
that this problem is weakly NP- hard in the case of Stackelberg equilibria and
multiple attacker resources and present a pseudopolynomial time procedure to
solve this problem for the case of Nash equilibria under mild assumptions.
Finally, to address non-additive security games, we propose a notion of nearest
additive game and demonstrate the existence and uniqueness of a such a nearest
additive game for any non-additive game.

Computer Science and Game Theory