  It is straightforward to see that very simple sets are determined. For example, if A is the set of all reals then clearly I has a winning strategy; if A is empty then clearly II has a winning strategy; and if A is countable then II has a winning strategy (by “diagonalizing”). This might lead one to expect that all sets of reals are determined. However, it is straightforward to use the Axiom of Choice (AC) to construct a non-determined set (by listing all winning strategies and “diagonalizing” across them). For this reason AD was never really considered as a serious candidate for a new axiom. However, there is an interesting class of related axioms that are consistent with AC, namely, the axioms of definable determinacy. These axioms extend the above pattern by asserting that all sets of reals at a given level of complexity are determined, notable examples being Δ̰11-determinacy (all Borel sets of reals are determined), PD (all projective sets of reals are determined) and ADL(ℝ) (all sets of reals in L(ℝ) are determined). 