This paper is both a survey and a research announcement. The author considers a class of two-person games with ``slightly imperfect'' information called Blackwell games. These games belong to the broader class of stochastic games. Two-person games with perfect information have played an important role in set theory via the Axiom of Determinacy (AD) . Blackwell games give rise to corresponding ``axioms'', which the author calls pBl-AD and Bl-AD. It is known that AD\(\implies\)Bl-AD\(\implies\)pBl-AD and that all three axioms have the same consistency strength. However, the question of equivalence remains open. From the author's introduction: ``In this survey, we describe how to get the strong partition property of \(\aleph_1\) and further consequences for the infinitary combinatorics of projective ordinals from Bl-AD and an interesting definability hierarchy of sets of reals under the assumption of pBl-AD.'' Details and proofs are to appear in a later paper.