Universal games of incomplete information

We consider two-person games of incomplete information in which certain portions of positions are private to each player and cannot be viewed by the opponent. We present various games of incomplete information which are universal for all reasonable games. The problem of determining the outcome of these universal games from a given initial position is shown to be complete in doubly-exponential time. We also define “private alternating Turing machines” which are alternating Turing machines with certain tapes and portions of states private to universal states. The time and space complexity of these machines is characterized in terms of the time complexity of deterministic Turing machines, with single and double exponential jumps.