In this paper we analyze general Stackelberg games (SGs) and Stackelberg security games (SSGs). SGs are hierarchical adversarial games where players select actions or strategies to optimize their payoffs in a sequential manner. SSGs are a type of SGs 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 one of the targets. We review existing mixed integer optimization formulations in both the general and the security setting and present new formulations for both settings. We compare the SG formulations and the SSG formulations both from a theoretical and a computational point of view. Our theoretical results show that the new formulations provide tighter linear relaxations. Our computational experiments show that the new formulations give better solution times.