Decomposition of Probability Marginals for Security Games in Abstract Networks
Given a set system (E, š«), let Ļā [0,1]^š« be a vector of requirement values on the sets and let Ļā [0, 1]^E be a vector of probability marginals with ā_e ā PĻ_e ā„Ļ_P for all P āš«. We study the question under which conditions the marginals Ļ can be decomposed into a probability distribution on the subsets of E such that the resulting random set intersects each P āš« with probability at least Ļ_P. Extending a result by Dahan, Amin, and Jaillet (MOR 2022) motivated by a network security game in directed acyclic graphs, we show that such a distribution exists if š« is an abstract network and the requirements are of the form Ļ_P = 1 - ā_e ā PĪ¼_e for some Ī¼ā [0, 1]^E. Our proof yields an explicit description of a feasible distribution that can be computed efficiently. As a consequence, equilibria for the security game studied by Dahan et al. can be efficiently computed even when the underlying digraph contains cycles. As a subroutine of our algorithm, we provide a combinatorial algorithm for computing shortest paths in abstract networks, answering an open question by McCormick (SODA 1996). We further show that a conservation law proposed by Dahan et al. for requirement functions in partially ordered sets can be reduced to the setting of affine requirements described above.
READ FULL TEXT