  Secure equilibrium is a refinement of Nash equilibrium, which provides some
security to the players against deviations when a player changes his strategy
to another best response strategy. The concept of secure equilibrium is
specifically developed for assume-guarantee synthesis and has already been
applied in this context. Yet, not much is known about its existence in games
with more than two players. In this paper, we establish the existence of secure
equilibrium in two classes of multi-player perfect information turn-based
games: (1) in games with possibly probabilistic transitions, having countable
state and finite action spaces and bounded and continuous payoff functions, and
(2) in games with only deterministic transitions, having arbitrary state and
action spaces and Borel payoff functions with a finite range (in particular,
qualitative Borel payoff functions). We show that these results apply to
several types of games studied in the literature.
