  We introduce Boolean Observation Games, a subclass of multi-player finite
strategic games with incomplete information and qualitative objectives. In
Boolean observation games, each player is associated with a finite set of
propositional variables of which only it can observe the value, and it controls
whether and to whom it can reveal that value. It does not control the given,
fixed, value of variables. Boolean observation games are a generalization of
Boolean games, a well-studied subclass of strategic games but with complete
information, and wherein each player controls the value of its variables.
  In Boolean observation games, player goals describe multi-agent knowledge of
variables. As in classical strategic games, players choose their strategies
simultaneously and therefore observation games capture aspects of both
imperfect and incomplete information. They require reasoning about sets of
outcomes given sets of indistinguishable valuations of variables. An outcome
relation between such sets determines what the Nash equilibria are. We present
various outcome relations, including a qualitative variant of ex-post
equilibrium. We identify conditions under which, given an outcome relation,
Nash equilibria are guaranteed to exist. We also study the complexity of
checking for the existence of Nash equilibria and of verifying if a strategy
profile is a Nash equilibrium. We further study the subclass of Boolean
observation games with `knowing whether' goal formulas, for which the
satisfaction does not depend on the value of variables. We show that each such
Boolean observation game corresponds to a Boolean game and vice versa, by a
different correspondence, and that both correspondences are precise in terms of
existence of Nash equilibria.
