A study of general and security Stackelberg game formulations

Abstract

In this paper, we analyze different mathematical formulations for general Stackelberg games (GSGs) and Stackelberg security games (SSGs). We consider GSGs in which a single leader commits to a utility maximizing strategy knowing that p possible followers optimize their own utility taking the leader's strategy into account. SSGs are a type of GSG that arise in security applications where the strategies of the leader consist of protecting a subset of targets and the strategies of the p followers consist of attacking a single target. We compare existing mixed integer linear programming (MILP) formulations for GSGs, ranking them according to the tightness of their linear programming (LP) relaxations. We show that SSG formulations are projections of GSG formulations and exploit this link to derive a new SSG MILP formulation that i) has the tightest LP relaxation known among SSG MILP formulations and ii) has an LP relaxation that coincides with the convex hull of feasible solutions in the case of a single follower. We present computational experiments empirically comparing the difficulty of solving the formulations in the general and security settings. The new SSG MILP formulation remains computationally efficient as problem size increases.

Coordinating resources in Stackelberg Security Games

“…First, we present formulations for general Stackelberg games where a complete enumeration of the pure strategy set of the leader is needed. These formulations were introduced in [25,6]. Then, we present compact formulations of SSG with coordinating resources which consider fewer variables but more constraints in order to improve the computational performance.…”

“…The (MIP-G) equivalent formulation, introduced in [6], is constructed by using the linearizing variable h k ij = x i q k j and dropping out variables f and s through Fourier-Motzkin Elimination. The formulation is as follows:…”

“…While (MIP-G) has more variables and constraints, [6] shows that it provides a tighter linear optimization relaxation. We point the reader to that work for a detailed study of the strength and efficiency of these formulations.…”

Optimal k Cut-Sets in Attack/Defense Strategies on Networks

“…The model considers a non-cooperative two-period game. Casorran et al [26] consider multi attackers and one defender in general and security Stackelberg games. Li [27] studies Stackelberg games with two attackers and one defender that fight over the hub nodes by considering cost-sensitive parameters.…”

Discussion of Fairness and Implementability in Stackelberg Security Games

“…Otherwise, it is called with heterogeneous resources. Several mixed integer formulations, for both, general Bayesian Stackelberg games and Bayesian Stackelberg Security games are presented in [9], [4].…”