A branch and price algorithm for a Stackelberg Security Game

Highlights

• A column generation method for tight formulation of Stackelberg Security game model.

• Investigated speedups with use of greedy algorithm and dual variable smoothing.

• Evaluation of its use in a branch and price algorithm.

Abstract

Mixed integer optimization formulations are an attractive alternative to solve Stackelberg Game problems thanks to the efficiency of state of the art mixed integer algorithms. In particular, decomposition algorithms, such as branch and price methods, make it possible to tackle instances large enough to represent games inspired in real world domians.

In this work we focus on Stackelberg Games that arise from a security application and investigate the use of a new branch and price method to solve its mixed integer optimization formulation. We prove that the algorithm provides upper and lower bounds on the optimal solution at every iteration and investigate the use of stabilization heuristics. Our preliminary computational results compare this solution approach with previous decomposition methods obtained from alternative integer programming formulations of Stackelberg games.

Introduction

Stackelberg games model the strategic interaction between players, where one participant – the leader – is able to commit to a strategy first, knowing that the remaining players – the followers – will take this strategy into account and respond in an optimal manner. These games have been used to represent markets in which a participant has significant market share and can commit to a strategy (von Stackelberg, Bazin, Hill, & Urch, 2010), where government decides tolls or capacities in a transportation network (Labbé, Marcotte, & Savard, 1998), and of late have been used to represent the attacker-defender interaction in security domains (Jain et al., 2010). These games are examples of bilevel optimization problems, which are in general non convex optimization problems that are difficult to solve.

In this work we focus on a specific class of Stackelberg games which we refer to as Stackelberg Security Games (SSG) that arise in security domains and have a particular payoff structure (Yin, Korzhyk, Kiekintveld, Conitzer, & Tambe, 2010). In a SSG, the security (or defender) behaves as the leader selecting a patrolling strategy first and then, possibly many attackers act as the follower, observing the defender’s patrolling strategy and deciding where to attack. Such Stackelberg Security Game models have been used in the deployment of decision support systems with specialized algorithms in real security domain applications (Jain et al., 2010, Pita et al., 2011, Shieh et al., 2012).

Recent work has developed efficient integer optimization solution algorithms for different variants of the SSGs (Hochbaum et al., 2014, Jain et al., 2010, Jain et al., 2011, Jain et al., 2011, Kiekintveld et al., 2009). In general terms these optimization problems are formulated with the defender committing to a mixed (randomized) strategy and the attacker(s) responding with a pure strategy after conducting surveillance of the defender’s mixed strategy. A mixed strategy refers to a probability distribution over the possible actions while a pure strategy corresponds to selecting one of the possible actions. In this SSG, the defender mixed strategies are probability distributions over possible patrolling strategies, while the attacker’s pure strategy corresponds to selecting a specific target to attack. In addition, the number of actions of the defender can be exponential in size, with respect to the targets and defense resources, due to the combinatorics of using N resources to patrol m targets. This illustrates that to solve SSGs we have to address mixed integer optimization problems with exponential number of variables. Addressing the combinatorial size of defender strategies has led to both development of branch and price methods (Kiekintveld et al., 2009) and constraint generation methods (Yang, Jiang, Tambe, & Ordóñez, 2013). There are, however, problem instances that arise from real security applications that still challenge existing solution methods. Here we investigate a new branch and price method developed for a novel formulation of Stackelberg games (MIPSG), introduced in Casorrán-Amilburu, Fortz, Labbé, and Ordóñez (2017). This new formulation has been shown to provide tighter linear relaxations than other existing mixed integer formulations and to give the convex hull of the feasible integer solutions when there is only one follower.

We begin by introducing notation and describing the integer optimization formulations that have been considered previously in the next section. We also introduce the equivalent MIPSG formulation. In Section 3 we present the column generation algorithm for the solution of the linear relaxation of MIPSG, along with a speed up that can be obtained by aggregating subproblems, and the existence of upper and lower bounds at every iteration. We also describe the branching strategies used in adapting this column generation to a Branch and Price method and how to apply dual stabilization techniques. We present our preliminary computational results in Section 4 and provide concluding remarks in Section 5.