Game-theoretic strategies for systems of components using product-form utilities

Many critical infrastructures are composed of multiple systems of components which are correlated so that disruptions to one may propagate to others. We consider such infrastructures with correlations characterized in two ways: (i) an aggregate failure correlation function specifies the conditional failure probability of the infrastructure given the failure of an individual system, and (ii) a pairwise correlation function between two systems specifies the failure probability of one system given the failure of the other. We formulate a game for ensuring the resilience of the infrastructure, wherein the utility functions of the provider and attacker are products of an infrastructure survival probability term and a cost term, both expressed in terms of the numbers of system components attacked and reinforced. The survival probabilities of individual systems satisfy first-order differential conditions that lead to simple Nash Equilibrium conditions. We then derive sensitivity functions that highlight the dependence of infrastructure resilience on the cost terms, correlation functions, and individual system survival probabilities. We apply these results to simplified models of distributed cloud computing and energy grid infrastructures.