When doing descriptive set theory, it is conventional to use either $\omega^\omega$ or $2^\omega$ as your space of ``reals'' (these spaces are homeomorphic to the irrationals and the Cantor set, respectively). Throughout this article, I will use the term ``reals'' to refer to $\omega^\omega$

Let $X \subseteq \omega^\omega$ be given and consider the following game on $X$ played between two players, I and II: I starts by saying a natural number; II hears this number and replies with another (or possibly the same one); I hears this and replies with another; etc. The sequence of numbers said (in the order they were said) is a point in $\omega^\omega$ I wins if this point is in $X$ otherwise II wins.

A map $\sigma: \omega^{<\omega} \to \omega$ is said to be a winning strategy for I if it has the following property: if, after the play has gone $n_0 n_1 \dotsc n_M$ I plays $\sigma(n_0 \dotsc n_M)$ for each move, then I wins. A winning strategy for II is defined analogously.

The axiom of determinacy (AD) states that every such game is determined, that is either I or II has a winning strategy.

Using choice, a non-determined game can be constructed directly: for $\alpha< \cc$ enumerate the uncountable closed subsets of the reals $F_\alpha$ Now construct two sequences $\gen{x_\alpha : \alpha < \cc}$ and $\gen{y_\alpha: \alpha < \cc}$ by choosing $x_\alpha, y_\alpha$ as distinct points from $F_\alpha$ which are not in $\set{x_\gamma, y_\gamma : \gamma < \alpha}$ (this is possible as each uncountable closed set has cardinality $\cc$ . Then the game on the set of all $x_\alpha$ is non-determined.

From ZF+AD, one may prove many nice facts about the reals, such as: any subset is Lebesgue measurable, any subset has a perfect subset and the continuum hypothesis. ZF+AD also proves the axiom of countable choice.

AD itself is not taken seriously by many set theorists as a genuine alternative to choice. However, there is a weakening of AD (QPD, which states that all games in $\LL[\RR]$ are determined) which is consistent with ZFC (in fact, it's equiconsistent to a large cardinal axiom) which is a serious axiom candidate.