Probability Distributions on Partially Ordered Sets and Network Security Games
share
We consider the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset's elements and maximal chains is satisfied? In this article, we present a combinatorial algorithm to positively resolve this question. We show that this result plays a crucial role in the equilibrium analysis of a generic security game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs, and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. The first (resp. second) player seeks to maximize the value of effective (resp. interdicted) flow net the total transportation (resp. interdiction) cost. Using our existence result on posets and strict complementary slackness in linear programming, we show that the equilibrium properties of this game can be described using primal and dual solutions of a minimum cost circulation problem. Our analysis provides a new characterization of the critical network components.
READ FULL TEXT