  There has been significant recent interest in leader-follower security games,
where the leader dominates the decision process with the Stackelberg
equilibrium (SE) strategy. However, such a leader-follower scheme may become
invalid in practice due to subjective or objective factors, and then the Nash
equilibrium (NE) strategy may be an alternative option. In this case, the
leader may face a dilemma of choosing an SE strategy or an NE strategy. In this
paper, we focus on a unified three-player leader-follower security game and
study the coincidence between SE and NE. We first explore a necessary and
sufficient condition for the case that each SE is an NE, which can be further
presented concisely when the SE is unique. This condition not only provides
access to seek a satisfactory SE strategy but also makes a criterion to verify
an obtained SE strategy. Then we provide another appropriate condition for the
case that at least one SE is an NE. Moreover, since the coincidence condition
may not always be satisfied, we describe the closeness between SE and NE, and
give an upper bound of their deviation. Finally, we show the applicability of
the obtained theoretical results in several practical security cases, including
the secure transmission problem and the cybersecurity defense.
