---
tag: QXECDPAYTN
kind: concept
name: Determinacy
msc: ["03E60", "91A44", "03E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Determinacy
same_as:
  wikidata: Q1201165
  wikipedia: Determinacy
---

## Summary

Determinacy is the property of an infinite two-player game of perfect information that one of the two players has a winning strategy.
In the games of descriptive set theory, the players alternately choose natural numbers forever.
The first player wins when the resulting [[sequence|0G9EKM0E15]] lies in a fixed payoff set.
Gale and Stewart proved that games with an open or a closed payoff set are determined.
The [[Borel determinacy theorem|8FVWYEZWTK]] extends this to all Borel payoff sets.
Stronger determinacy hypotheses, such as [[projective determinacy|2GY3131ZPS]] and the [[axiom of determinacy|PEXTM3MJ5Z]], imply strong regularity properties of sets of reals.

## Explanations

### L1 human

Two players take turns writing down numbers, and they never stop.
Before the game starts, a rule decides which infinite sequences count as a win for the first player.
A strategy is a recipe that tells a player which number to write in each position, and it is winning if it wins against every way the opponent can play.
In a finite game where every play ends with a win for one of the players, one of them always has a winning strategy.
For infinite games this is no longer automatic, and determinacy is the statement that some player has a winning strategy.