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
of
real numbers, two players alternatively write down arbitrary (not necessarily in
) positive real numbers
such that
Player one wins if and only if
exists and is in
.
[2] One observation about the game is that if
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
- Moran. Gadi. Existence of nondetermined sets for some two person games over reals. Israel Journal of Mathematics. September 1971. 9. 3. 316–329. 10.1007/BF02771682. free.
Notes and References
- Book: Mauldin. R. Daniel. The Scottish Book: Mathematics from the Scottish Cafe. April 1981. Birkhäuser. 978-3-7643-3045-3. 113. 1.
- 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.