Multi-agent teamwork and defender-attacker security games are two areas that are currently receiving significant attention within multi-agent systems research. Unfortunately, despite the need for effective teamwork among multiple defenders, little has been done to harness the teamwork research in s ecurity games. The problem that this paper seeks to solve is the coordination of decentralized defender agents in the presence of uncertainty while securing targets against an observing adversary. To address this problem, we offer the following novel contributions in this paper: (i) New model of security games with defender teams that coordinate under uncertainty; (ii) New algorithm based on column generation that utilizes Decentralized Markov Decision Processes (Dec-MDPs) to generate defender strategies that incorporate uncertainty; (iii) New techniques to handle global events (when one or more agents may leave the system) during defender execution; (iv) Heuristics that help scale up in the number of targets and agents to handle real-world scenarios; (v) Exploration of the robustness of randomized pure strategies. The paper opens the door to a potentially new area combining computational game theory and multi-agent teamwork.

In recent years, several security agencies have been deploying scheduling systems based on algorithmic advances in Stackelberg security games (SSGs). Unfortunately, none of the existing algorithms can scale up to domains where benefits are accrued from multiple defender resources performing jointly coordinated activities. Yet in many domains, including port patrolling where SSGs are in use, enabling multiple defender resources to perform jointly coordinated activities would significantly enhance the effectiveness of the patrols. To address this challenge, this paper presents four contributions. First, we present Smart (Security games with Multiple coordinated Activities and Resources that are Time-dependent), a novel SSG model that explicitly represents jointly coordinated activities between defender’s resources. Second, we present two branch-and-price algorithms, \(S\textsc {mart}_{\textsc {O}}\,\)—an optimal algorithm, and \(S\textsc {mart}_{\textsc {H}}\,\)—a heuristic approach, to solve Smart instances. The two algorithms present three novel features: (i) a novel approach to generate individual defender strategies by ordering the search space during column generation using insights from the Traveling Salesman Problem(TSP); (ii) exploitation of iterative modification of rewards of multiple defender resources to generate coordinated strategies and (iii) generation of tight upper bounds for pruning using the structure of the problem. Third, we present an extensive empirical and theoretical analysis of both \(S\textsc {mart}_{\textsc {O}}\,\)and \(S\textsc {mart}_{\textsc {H}}\,\). Fourth, we describe a large scale real-world experiment whereby we run the first head-to-head comparison between game-theoretic schedules generated using \(S\textsc {mart}_{\textsc {H}}\,\)against schedules generated by humans on a one-day patrol exercise over one train line of the Los Angeles Metro System. Our results show that game-theoretic schedules were evaluated to be superior to ones generated by humans.

This paper proposes the Multi-Operation Patrol Scheduling System (MOPSS), a new system to generate patrols for transit system. MOPSS is based on five contributions. First, MOPSS is the first system to use three fundamentally different adversary models for the threats of fare evasion, terrorism and crime, generating three significantly different types of patrol schedule. Second, to handle uncertain interruptions in the execution of patrol schedules, MOPSS uses Markov decision processes (MDPs) in its scheduling. Third, MOPSS is the first system to account for joint activities between multiple resources, by employing the well known SMART security game model that tackles coordination between defender's resources. Fourth, we are also the first to deploy a new Opportunistic Security Game model, where the adversary, a criminal, makes opportunistic decisions on when and where to commit crimes. Our fifth, and most important, contribution is the evaluation of MOPSS via real-world deployments, providing data from security games in the field.

This paper presents actual results from the field using a novel deployed system referred to as the Multi-Operation Patrol Scheduling System (MOPSS). MOPSS generates patrols for a transit system considering three different threats: fare evasion (FE), terrorism (CT) and crime (CR). In so doing, this paper present four contributions: (i) we propose the first multi-operation patrolling system; (ii) MOPSS uses Markov decision processes (MDPs) to handle uncertain interruptions in the execution of patrol schedules; (iii) we are the first to deploy a new Opportunistic Security Game model, where the adversary, a criminal, makes opportunistic decisions on when and where to commit crimes and, finally (iv) we evaluate MOPSS in the real-world, providing some of the first actual data from security games in the field.

This paper presents actual results from the field using a novel deployed system referred to as the Multi-Operation Patrol Scheduling System (MOPSS). MOPSS generates patrols for a transit system considering three different threats: fare evasion (FE), terrorism (CT) and crime (CR). In so doing, this paper present four contributions: (i) we propose the first multi-operation patrolling system; (ii) MOPSS uses Markov decision processes (MDPs) to handle uncertain interruptions in the execution of patrol schedules; (iii) we are the first to deploy a new Opportunistic Security Game model, where the adversary, a criminal, makes opportunistic decisions on when and where to commit crimes and, finally (iv) we evaluate MOPSS in the real-world, providing some of the first actual data from security games in the field.

Multiagent teamwork and defender-attacker security games are two areas that are currently receiving significant attention within multiagent systems research. Unfortunately, despite the need for effective teamwork among multiple defenders, little has been done to harness the teamwork research in security games. This paper is the first to remedy this situation by integrating the powerful teamwork mechanisms offered by Dec-MDPs into security games. We offer the following novel contributions in this paper: (i) New models of security games where a defender team's pure strategy is defined as a Dec-MDP policy for addressing coordination under uncertainty; (ii) New algorithms based on column generation that enable efficient generation of mixed strategies given this new model; (iii) Handling global events during defender execution for effective teamwork; (iv) Exploration of the robustness of randomized pure strategies. The paper opens the door to a potentially new area combining computational game theory and multiagent teamwork.

Despite recent successful real-world deployments of Stackelberg Security Games (SSGs), scale-up remains a fundamental challenge in this field. The latest techniques do not scale-up to domains where multiple defenders must coordinate time-dependent joint activities. To address this challenge, this paper presents two branch-and-price algorithms for solving SSGs, SMARTO and SMARTH, with three novel features: (i) a column-generation approach that uses an ordered network of nodes (determined by solving the traveling salesman problem) to generate individual defender strategies; (ii) exploitation of iterative reward shaping of multiple coordinating defender units to generate coordinated strategies; (iii) generation of tighter upper-bounds for pruning by solving security games that only abide by key scheduling constraints. We provide extensive experimental results and formal analyses.

Despite recent successful real-world deployments of Stackelberg Security Games (SSGs), scale-up remains a fundamental challenge in this field. The latest techniques do not scale-up to domains where multiple defenders must coordinate time-dependent joint activities. To address this challenge, this paper presents two branch-and-price algorithms for solving SSGs, SMARTO and SMARTH , with three novel features: (i) a column-generation approach that uses an ordered network of nodes (determined by solving the traveling salesman problem) to generate individual defender strategies; (ii) exploitation of iterative reward shaping of multiple coordinating defender units to generate coordinated strategies; (iii) generation of tighter upper-bounds for pruning by solving security games that only abide by key scheduling constraints. We provide extensive experimental results and formal analyses.

In this paper, we describe the model, theory developed, and deployment of PROTECT, a game-theoretic system that the United States Coast Guard USCG uses to schedule patrols in the Port of Boston. The USCG evaluated PROTECT's deployment in the Port of Boston as a success and is currently evaluating the system in the Port of New York, with the potential for nationwide deployment. PROTECT is premised on an attacker-defender Stackelberg game model; however, its development and implementation required both theoretical contributions and detailed evaluations. We describe the work required in the deployment, which we group into five key innovations. First, we propose a compact representation of the defender's strategy space by exploiting equivalence and dominance, to make PROTECT efficient enough to solve real-world sized problems. Second, this system does not assume that adversaries are perfectly rational, a typical assumption in previous game-theoretic models for security. Instead, PROTECT relies on a quantal response QR model of the adversary's behavior. We believe this is the first real-world deployment of a QR model. Third, we develop specialized solution algorithms that can solve this problem for real-world instances and give theoretical guarantees. Fourth, our experimental results illustrate that PROTECT's QR model handles real-world uncertainties more robustly than a perfect-rationality model. Finally, we present 1 a comparison of human-generated and PROTECT security schedules, and 2 results of an evaluation of PROTECT from an analysis by human mock attackers.

While three deployed applications of game theory for security have recently been reported at AAMAS [12], we as a community remain in the early stages of these deployments; there is a continuing need to understand the core principles for innovative security applications of game theory. Towards that end, this paper presents PROTECT, a game-theoretic system deployed by the United States Coast Guard (USCG) in the port of Boston for scheduling their patrols. USCG has termed the deployment of PROTECT in Boston a success, and efforts are underway to test it in the port of New York, with the potential for nationwide deployment. PROTECT is premised on an attacker-defender Stackelberg game model and offers five key innovations. First, this system is a departure from the assumption of perfect adversary rationality noted in previous work, relying instead on a quantal response (QR) model of the adversary's behavior --- to the best of our knowledge, this is the first real-world deployment of the QR model. Second, to improve PROTECT's efficiency, we generate a compact representation of the defender's strategy space, exploiting equivalence and dominance. Third, we show how to practically model a real maritime patrolling problem as a Stackelberg game. Fourth, our experimental results illustrate that PROTECT's QR model more robustly handles real-world uncertainties than a perfect rationality model. Finally, in evaluating PROTECT, this paper for the first time provides real-world data: (i) comparison of human-generated vs PROTECT security schedules, and (ii) results from an Adversarial Perspective Team's (human mock attackers) analysis.

Building upon previous security applications of computational game theory, this paper presents PROTECT, a game-theoretic system deployed by the United States Coast Guard (USCG) in the port of Boston for scheduling their patrols. USCG has termed the deployment of PROTECT in Boston a success, and efforts are underway to test it in the port of New York, with the potential for nationwide deployment. PROTECT is premised on an attacker-defender Stackelberg game model and offers five key innovations. First, this system is a departure from the assumption of perfect adversary rationality noted in previous work, relying instead on a quantal response (QR) model of the adversary's behavior - to the best of our knowledge, this is the first real-world deployment of the QR model. Second, to improve PROTECT's efficiency, we generate a compact representation of the defender's strategy space, exploiting equivalence and dominance. Third, we show how to practically model a real maritime patrolling problem as a Stackelberg game. Fourth, our experimental results illustrate that PROTECT's QR model more robustly handles real-world uncertainties than a perfect rationality model. Finally, in evaluating PROTECT, this paper provides real-world data: (i) comparison of human-generated vs PROTECT security schedules, and (ii) results from an Adversarial Perspective Team's (human mock attackers) analysis.

Randomized first-mover strategies of Stackelberg games are used in several deployed applications to allocate limited resources for the protection of critical infrastructure. Stackelberg games model the fact that a strategic attacker can surveil and exploit the defender's strategy, and randomization guards against the worst effects by making the defender less predictable. In accordance with the standard game-theoretic model of Stackelberg games, past work has typically assumed that the attacker has perfect knowledge of the defender's randomized strategy and will react correspondingly. In light of the fact that surveillance is costly, risky, and delays an attack, this assumption is clearly simplistic: attackers will usually act on partial knowledge of the defender's strategies. The attacker's imperfect estimate could present opportunities and possibly also threats to a strategic defender.In this paper, we therefore begin a systematic study of security games with limited surveillance. We propose a natural model wherein an attacker forms or updates a belief based on observed actions, and chooses an optimal response. We investigate the model both theoretically and experimentally. In particular, we give mathematical programs to compute optimal attacker and defender strategies for a fixed observation duration, and show how to use them to estimate the attacker's observation durations. Our experimental results show that the defender can achieve significant improvement in expected utility by taking the attacker's limited surveillance into account, validating the motivation of our work.

k- and t-optimality algorithms [9, 6] provide solutions to DCOPs that are optimal in regions characterized by its size and distance respectively. Moreover, they provide quality guarantees on their solutions. Here we generalise the k- and t-optimal framework to introduce C-optimality, a flexible framework that provides reward-independent quality guarantees for optima in regions characterised by any arbitrary criterion. Therefore, C-optimality allows us to explore the space of criteria (beyond size and distance) looking for those that lead to better solution qualities. We benefit from this larger space of criteria to propose a new criterion, the so-called size-bounded-distance criterion, which outperforms k-and t-optimality.

Given the real-world deployments of attacker-defender Stackelberg security games, robustness to deviations from expected attacker behaviors has now emerged as a critically important issue. This paper provides four key contributions in this context. First, it identifies a fundamentally problematic aspect of current algorithms for security games. It shows that there are many situations where these algorithms face multiple equilibria, and they arbitrarily select one that may hand the defender a significant disadvantage, particularly if the attacker deviates from its equilibrium strategies due to unknown constraints. Second, for important subclasses of security games, it identifies situations where we will face such multiple equilibria. Third, to address these problematic situations, it presents two equilibrium refinement algorithms that can optimize the defender's utility if the attacker deviates from equilibrium strategies. Finally, it experimentally illustrates that the refinement approach achieved significant robustness in consideration of attackers' deviation due to unknown constraints.

We provide an overview of two recent applications of security games. We describe new features and challenges introduced in the new applications.

Distributed constraint optimization (DCOP) is a promising approach to coordination, scheduling and task allocation in multi-agent networks. DCOP is NP-hard [6], so an important line of work focuses on developing fast incomplete solution algorithms that can provide guarantees on the quality of their local optimal solutions. Region optimality [11] is a promising approach along this line: it provides quality guarantees for region optimal solutions, namely solutions that are optimal in a specific region of the DCOP. Region optimality generalises kand t-optimality [7, 4] by allowing to explore the space of criteria that define regions to look for solutions with better quality guarantees. Unfortunately, previous work in region-optimal quality guarantees fail to exploit any a-priori knowledge of the reward structure of the problem. This paper addresses this shortcoming by defining reward-dependent region optimal quality guarantees that exploit two different levels of knowledge about rewards, namely: (i) a ratio between the least minimum reward to the maximum reward among relations; and (ii) the minimum and maximum rewards per relation.

This technical report is written as a support material for the reader in [1], detailing the transformations to simplify an initial program to compute a tight bound for a C-optimal assignment into a linear program (LP). In this technical report we show how the initial program to compute a tight bound for a C-optimal assignment, can be transformed into a linear program. Our departure point is the following program. Find R, xC and x∗ that minimize R(x C) R(x∗) subject to xC being a C-optimal for R We start by analyzing what exactly means saying that xC is C-optimal. The condition can be expressed as: for each x inside region C of xC we have that R(x) ≥ R(x). However, instead of considering all the assignments for which x is guaranteed to be optimal, we consider only the subset of assignments such that the set of variables that deviate with respect to x take the same value than in the optimal assignment. If we restrict to this subset of assignments, then each neighborhood covers a 2|C | assignments, one for each subset of variables in the neighborhood. Let 2 α stand for the set of all subsets of the neighborhood C. Then for each A ∈ 2Cα we can define an assignment xk such that for every variable xi in a relation completely covered by A we have that xk i = x ∗ i , and

This technical report is written as a support material for the reader in [1], detailing the transformations to simplify an initial program to compute a tight bound for a C-optimal assignment into a linear program (LP).