  In this work, we investigate a security game between an attacker and a
defender, originally proposed in \cite{emadi2019security}. As is well known,
the combinatorial nature of security games leads to a large cost matrix.
Therefore, computing the value and optimal strategy for the players becomes
computationally expensive. In this work, we analyze a special class of zero-sum
games in which the payoff matrix has a special structure which results from the
{\it additive property} of the utility function. Based on variational
principles, we present structural properties of optimal attacker as well as
defender's strategy. We propose a linear-time algorithm to compute the value
based on the structural properties, which is an improvement from our previous
result in \cite{emadi2019security}, especially in the context of large-scale
zero-sum games.
