Introduction

Game theory's popularity continues to increase in a whole variety of disciplines, including economics, biology, political science, computer science, electrical engineering, business, law, and public policy. In the arena of security, where game theory has always been popular, there now seems to be an exponential increase in interest. This increase is in part due to the new set of problems our societies face, from terrorism to drugs to crime. These problems are ubiquitous. Yet, limited security resources cannot be everywhere all the time, raising a crucial question of how to best utilize them.

Game theory provides a sound mathematical approach for deploying limited security resources to maximize their effectiveness. While the connection between game theory and security has been studied for the last several decades, there has been a fundamental shift in the relationship due to the emergence of computational game theory. More specifically, with the development of new computational approaches to game theory over the past two decades, very large-scale problems can be cast in game-theoretic contexts, thus providing us computational tools to address problems of security allocations.

My research group has been at the forefront of this effort to apply computational game theory techniques to security problems. We have led a wide range of actual deployed applications of game theory for security. Our first application, Assistant for Randomized Monitoring Over Routes (ARMOR), successfully deployed game-theoretic algorithms at the Los Angeles International Airport (LAX) in 2007 and has been in use there ever since.

Introduction

As discussed in other chapters of this book, an increasing number of technically sophisticated tools are available to support decisionmaking for security resource allocation in many different domains. In this chapter we discuss the question of evaluating these deployed security systems, using examples from our own research to illustrate some of the key challenges in doing evaluation for security systems. Two of the primary difficulties are (1) that we cannot rely on adversaries to cooperate in evaluation, which makes it difficult to validate models, and (2) there is (thankfully) very little data available about real-world terrorist attacks.

Despite the difficulties of comprehensive evaluation in security domains, it is only by asking the question, how well does a system work? that policy makers can decide how to allocate finite resources to to different security measures. We discuss the goals of security systems, the elements that comprise these systems, and different approaches for evaluation. Every approach has drawbacks, so in lieu of an ideal test, we advocate a comprehensive style of evaluation that uses diverse metrics and data to perform cost-benefit analysis for the complete system. We also emphasize that the focus of the evaluation is not, is system X the perfect security system? which is an impossible standard. Rather, the relevant question is which of the available alternatives should be used? Providing strong evidence that one alternative is superior to other approaches is often feasible, even when providing exact quantitative measures of value is not.

Introduction

There has been significant recent research interest in game-theoretic approaches to security at airports, ports, transportation, shipping and other infrastructure (Basilico, Gatti, and Amigoni, 2009; Conitzer and Sandholm, 2006; Kiekintveld et al., 2009; Pita et al., 2008). Much of this work has used a Stackelberg game framework to model interactions between the security forces and attackers. That is, the defender (i.e., the security forces) acts first by committing to a patrolling or inspection strategy, and the attacker chooses where to attack after observing the defender's choice. The typical solution concept applied to these games is strong Stackelberg equilibrium (SSE), which assumes that the defender will choose an optimal mixed (randomized) strategy based on the assumption that the attacker will observe this strategy and choose an optimal response. This leader-follower paradigm appears to fit many real world security situations. Indeed, Stackelberg games are at the heart two major decision-support applications: the ARMOR program in use at the Los Angeles International Airport since 2007 to randomize allocation of checkpoints and canine patrols (Pita et al., 2008), and the IRIS program in use by the U.S. Federal Air Marshals to randomize assignments of air marshals to flights (Tsai et al., 2009).

However, there are legitimate concerns about whether the Stackelberg model is appropriate in all cases.

Introduction

Algorithms for attacker-defender Stackelberg games, resulting in randomized schedules for deploying limited security resources at airports, subways, ports, and other critical infrastructure have garnered significant research interest (Parachuri et al. 2008; Kiekintveld et al. 2009). Indeed, two important deployed security applications are using such algorithms: ARMOR and IRIS. ARMOR has been in use for over two years by Los Angeles International Airport police to generate canine-patrol and vehicle-checkpoint schedules (Pita et al., 2009). IRIS was recently deployed by the Federal Air Marshals Service (FAMS) to create flight schedules for air marshals (Tsai et al., 2009). These applications use efficient algorithms that solve large-scale games (Parachuri et al., 2008; Conitzer and Sandholm, 2006; Basilico, Gatti, and Amigoni, 2009), the latest being ERASER-C, the algorithm used in IRIS.

Unfortunately, current state-of-the art algorithms for Stackelberg games are inadequate for many applications. For example, U.S. carriers fly over 27,000 domestic and 2,000 international flights daily, presenting a massive scheduling challenge for FAMS. IRIS addresses an important part of this space – the international sector – but only considers schedules with a single departure and return flight. The ERASER-C algorithm used in this application does not provide correct solutions for longer and more complex tours (which are common in the domestic sector). In fact, recent complexity results show that the problem of finding Stackelberg equibria with general scheduling constraints is NP-hard (Korzhyk, Conitzer, and Parr, 2010) and can be solved in polynomial time only for restricted cases.

Since the days of Sparta and Athens the use of the world's Maritime Transportation System (MTS) to move goods and services has been a critical facet of a nation's economic well-being. The MTS served as a “center of gravity” with nations trading as far away as distant continents or as close as two ports located in the same country or region. Corbett and Winebrake (2008: 6) summarized that the MTS “is an integral, if sometimes less publicly visible, part of the global economy” and that the MTS consists of “a network of specialized vessels, the ports they visit, and transportation infrastructure from factories to terminals to distribution centers to market.” The security of this system is imperative as goods move through the ports and waterways within national boundaries, into the littorals, and out into the world-wide Global Maritime Commons.

The issue of security within this global system is complicated because the number of attack vectors and methods an adversary can take are endless. The attackers also hold an advantage in their ability to select the time, place, and method of an attack … and to abort an attack if counterdetection occurs. The introduction of suicide attackers has made the security challenge even more daunting as bombers are willing to give their own lives for their cause. Pape (2003: 344) noted that “suicide terrorism is strategic. The vast majority of suicide attacks are not isolated or random acts by individual fanatics.”

Introduction

In Stackelberg games, one player, the leader, commits to a strategy publicly before the remaining players, the followers, make their decisions (Fudenberg and Tirole, 1991). There are many multi-agent security domains, such as attacker-defender scenarios and patrolling, for which these types of commitments by the security agent are necessary (Agmon et al., 2008; Brown et al., 2006; Kiekintveld et al., 2009; Paruchuri et al., 2006), and it has been shown that Stackelberg games appropriately model these commitments (Paruchuri et al., 2008; Pita et al., 2008). For example, security personnel patrolling an infrastructure decide on a patrolling strategy first, before their adversaries act taking this committed strategy into account. Indeed, Stackelberg games are at the heart of theARMOR system, deployed at LAX since 2007 to schedule security personnel (Paruchuri et al., 2008; Pita et al., 2008), and they have recently been applied to federal air marshals (Kiekintveld et al., 2009). Moreover, these games have potential applications for network routing and pricing in transportation systems, among many others possibilities (Cardinal et al., 2005; Korilis, Lazar, and Orda, 1997).

Existing algorithms for Bayesian Stackelberg games find optimal solutions considering an a priori probability distribution over possible follower types (Conitzer and Sandholm, 2006; Paruchuri et al., 2008). Unfortunately, to guarantee optimality, these algorithms make strict assumptions on the underlying games; namely, that players are perfectly rational and that followers perfectly observe the leader's strategy. However, these assumptions rarely hold in real-world domains, particularly those involving human actors.

Enumerative combinatorics has undergone enormous development since the publication of the first edition of this book in 1986. It has become more clear what the essential topics are, and many interesting new ancillary results have been discovered. This second edition is an attempt to bring the coverage of the first edition more up to date and to impart a wide variety of additional applications and examples.

The main difference between this edition and the first is the addition of ten new sections (six in Chapter 1 and four in Chapter 3) and more than 350 new exercises. In response to complaints about the difficulty of assigning homework problems whose solutions are included, I have added some relatively easy exercises without solutions, marked by an asterisk. There are also a few organizational changes, the most notable being the transfer of the section on P-partitions from Chapter 4 to Chapter 3, and extending this section to the theory of (P, ω)-partitions for any labeling ω. In addition, the old Section 4.6 has been split into Sections 4.5 and 4.6.

There will be no second edition of volume 2 nor a volume 3. Since the references in volume 2 to information in volume 1 are no longer valid for this second edition, I have included a table entitled “First Edition Numbering,” which gives the conversion between the two editions for all numbered results (theorems, examples, exercises, etc., but not equations).