Axiom of determinacy

The axiom of determinacy asserts that every infinite two-player game of perfect information on natural numbers, with a specified set of winning plays, has a winning strategy for one player.

Gale–Stewart game: An infinite game in which two players alternately choose natural numbers, producing an infinite sequence. The axiom asserts determinacy for this class of games.
Axiom of dependent choice: A choice principle allowing successive choices from a sequence of nonempty sets. Determinacy is often studied alongside this weaker choice principle.
David Gale: An American mathematician whose work included game theory and mathematical economics. Gale co-authored the foundational study that introduced these infinite games.
Axiom of determinacy for finite games: The principle that finite two-player games with perfect information have optimal strategies. Finite-game determinacy is provable by backward induction; the axiom addresses infinite play.
Winning strategy: A rule that ensures a player wins regardless of the opponent’s choices. Determinacy means one player has such a rule.
Lebesgue measurability: A property allowing a set to have a consistent Lebesgue measure. Strong determinacy hypotheses imply measurability for broad classes of sets of reals.
Frank M. Stewart: An American mathematician known for work on infinite games and set theory. Stewart co-authored the paper establishing the framework for these games.
Martin’s theorem: The theorem that every game with a Borel payoff set is determined. Unlike the unrestricted axiom, it establishes determinacy for Borel payoffs in ZFC.
Game tree: A tree whose branches encode the possible sequences of moves in a game. Strategies select moves at nodes, while infinite branches represent plays.
Perfect set property: The property that every uncountable set of reals contains a nonempty perfect subset. Determinacy provides this regularity property for many definable sets.

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