Banach game explained

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

X

of real numbers, two players alternatively write down arbitrary (not necessarily in

X

) positive real numbers

x0,x1,x2,\ldots

such that

x0>x1>x2>

Player one wins if and only if
infty
\sum
i=0

xi

exists and is in

X

.[2]

One observation about the game is that if

X

is a countable set, then either of the players can cause the final sum to avoid the set. Thus in this situation the second player has a winning strategy.

Further reading

Notes and References

  1. Book: Mauldin. R. Daniel. The Scottish Book: Mathematics from the Scottish Cafe. April 1981. Birkhäuser. 978-3-7643-3045-3. 113. 1.
  2. Telgársky. Rastislav. Topological Games: On the 50th Anniversary of the Banach–Mazur Game. Rocky Mountain Journal of Mathematics. Spring 1987. 17. 2. 227–276. at 242.