Universal games of incomplete information

Abstract

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. We also consider blindfold games, which are restricted games in which the second player is not allowed to modify the common position. We show various blindfold games to have exponential space complete outcome problems and to be universal for reasonable blindfold games. We define “blind alternating Turing machines” which are private alternating Turing machines with the restriction that the universal states cannot modify the public tapes and public portion of states. A single exponential jump characterizes the relation between the space complexity of deterministic Turing machines.