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.
Given a subset
X
of real numbers, two players alternatively write down arbitrary (not necessarily in
X
) positive real numbers
x
0
,
x
1
,
x
2
,
…
such that
x
0
>
x
1
>
x
2
>
⋯
Player one wins if and only if
∑
i
=
0
∞
x
i
exists and is in
X
.
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 can win.
