Learning an Unknown Network State in Routing Games

We study learning dynamics induced by myopic travelers who repeatedly play a routing game on a transportation network with an unknown state. The state impacts cost functions of one or more edges of the network. In each stage, travelers choose their routes according to Wardrop equilibrium based on public belief of the state. This belief is broadcast by an information system that observes the edge loads and realized costs on the used edges, and performs a Bayesian update to the prior stage's belief. We show that the sequence of public beliefs and edge load vectors generated by the repeated play converge almost surely. In any rest point, travelers have no incentive to deviate from the chosen routes and accurately learn the true costs on the used edges. However, the costs on edges that are not used may not be accurately learned. Thus, learning can be incomplete in that the edge load vectors at rest point and complete information equilibrium can be different. We present some conditions for complete learning and illustrate situations when such an outcome is not guaranteed

Network Inspection for Detecting Strategic Attacks

This article studies a problem of strategic network inspection, in which a defender (agency) is tasked with detecting the presence of multiple attacks in the network. An inspection strategy entails monitoring the network components, possibly in a randomized manner, using a given number of detectors. We formulate the network inspection problem $(\mathcal{P})$ as a large-scale bilevel optimization problem, in which the defender seeks to determine an inspection strategy with minimum number of detectors that ensures a target expected detection rate under worst-case attacks. We show that optimal solutions of $(\mathcal{P})$ can be obtained from the equilibria of a large-scale zero-sum game. Our equilibrium analysis involves both game-theoretic and combinatorial arguments, and leads to a computationally tractable approach to solve $(\mathcal{P})$. Firstly, we construct an approximate solution by utilizing solutions of minimum set cover (MSC) and maximum set packing (MSP) problems, and evaluate its detection performance. In fact, this construction generalizes some of the known results in network security games. Secondly, we leverage properties of the optimal detection rate to iteratively refine our MSC/MSP-based solution through a column generation procedure. Computational results on benchmark water networks demonstrate the scalability, performance, and operational feasibility of our approach. The results indicate that utilities can achieve a high level of protection in large-scale networks by strategically positioning a small number of detectors

Evaluating Resilience of Electricity Distribution Networks via A Modification of Generalized Benders Decomposition Method

This paper presents a computational approach to evaluate the resilience of electricity Distribution Networks (DNs) to cyber-physical failures. In our model, we consider an attacker who targets multiple DN components to maximize the loss of the DN operator. We consider two types of operator response: (i) Coordinated emergency response; (ii) Uncoordinated autonomous disconnects, which may lead to cascading failures. To evaluate resilience under response (i), we solve a Bilevel Mixed-Integer Second-Order Cone Program which is computationally challenging due to mixed-integer variables in the inner problem and non-convex constraints. Our solution approach is based on the Generalized Benders Decomposition method, which achieves a reasonable tradeoff between computational time and solution accuracy. Our approach involves modifying the Benders cut based on structural insights on power flow over radial DNs. We evaluate DN resilience under response (ii) by sequentially computing autonomous component disconnects due to operating bound violations resulting from the initial attack and the potential cascading failures. Our approach helps estimate the gain in resilience under response (i), relative to (ii)

Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses 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? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction 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. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation