  This paper studies a language-based opacity enforcement in a two-player,
zero-sum game on a graph. In this game, player 1 (P1) wins if it can achieve a
secret temporal goal described by the language of a finite automaton, no matter
what strategy the opponent player 2 (P2) selects. In addition, P1 aims to win
while making its goal opaque to a passive observer with imperfect information.
However, P2 colludes with the observer to reveal P1's secret whenever P2 cannot
prevent P1 from achieving its goal, and therefore, opacity must be enforced
against P2. We show that a winning and opacity-enforcing strategy for P1 can be
computed by reducing the problem to solving a reachability game augmented with
observer's belief states. Furthermore, if such a strategy does not exist,
winning for P1 must entail the price of revealing his secret to the observer.
We demonstrate our game-theoretic solution of opacity-enforcement control
through a small illustrative example and in a robot motion planning problem.
