Résumé : In this paper we analyze general Stackelberg games (GSGs) and Stackelberg security games (SSGs). Stackelberg games are hierarchical adversarial games where players select strategies to optimize their payoffs in a sequential manner. SSGs are a type of GSGs that arise in security applications where the strategies of the player that acts first consist in protecting subsets of targets and the strategies of the followers consist in attacking a target. We present a comparative study of existing mixed integer linear programming (MILP) formulations for GSGs, where we rank them according to the tightness of their linear programming (LP) relaxations. We establish a theoretical link between GSG and SSG formulations through projections of variables and exploit this link to derive a new tight SSG MILP formulation. We extend our comparison of GSG formulations to the security setting, showing that the new SSG formulation we derive i) has the tightest LP relaxation known among SSG MILP formulations and ii) its LP relaxation coincides with the convex hull of feasible solutions in the case of a single follower. We run computational experiments in both the general and the security setting to measure the performance of the corresponding formulations. Our new SSG formulation outperforms competing formulations and is better suited to tackle scaling up of the instances.