  Many real-world systems are composed of interdependent networks that rely on
one another. Such networks are typically designed and operated by different
entities, who aim at maximizing their own payoffs. There exists a game among
these entities when designing their own networks. In this paper, we study the
game investigating how the rational behaviors of entities impact the system
robustness. We first introduce a mathematical model to quantify the interacting
payoffs among varying entities. Then we study the Nash equilibrium of the game
and compare it with the optimal social welfare. We reveal that the cooperation
among different entities can be reached to maximize the social welfare in
continuous game only when the average degree of each network is constant.
Therefore, the huge gap between Nash equilibrium and optimal social welfare
generally exists. The rationality of entities makes the system inherently
deficient and even renders it extremely vulnerable in some cases. We analyze
our model for two concrete systems with continuous strategy space and discrete
strategy space, respectively. Furthermore, we uncover some factors (such as
weakening coupled strength of interdependent networks, designing suitable
topology dependency of the system) that help reduce the gap and the system
vulnerability.
