# Banach game

In mathematics, the Banach game is a topological game introduced by Stefan Banach in 1935 in the second addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.[1]

Given a subset ${\displaystyle X}$ of real numbers, two players alternatively write down arbitrary (not necessarily in ${\displaystyle X}$) positive real numbers ${\displaystyle x_{0},x_{1},x_{2},\ldots }$ such that ${\displaystyle x_{0}>x_{1}>x_{2}>\cdots }$ Player one wins if and only if ${\displaystyle \sum _{i=0}^{\infty }x_{i}}$ exists and is in ${\displaystyle X}$.[2]

One observation about the game is that if ${\displaystyle X}$ is a countable set, then either of the players can cause the final sum to avoid the set.[3] Thus in this situation the second player can win.

## References

1. Mauldin, R. Daniel (April 1981). The Scottish Book: Mathematics from the Scottish Cafe (PDF) (1 ed.). Birkhäuser. p. 113. ISBN 978-3-7643-3045-3.
2. Telgársky, Rastislav (Spring 1987). "Topological Games: On the 50th Anniversary of the Banach–Mazur Game" (PDF). Rocky Mountain Journal of Mathematics. 17 (2): 227–276. at 242.
3. Mauldin 1981, p. 116.
• Moran, Gadi (September 1971). "Existence of nondetermined sets for some two person games over reals". Israel Journal of Mathematics. 9 (3): 316–329. doi:10.1007/BF02771682.