Search Results for “banach game”

Source: 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.
Given a subset

X

{\displaystyle X}

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

X

{\displaystyle X}

) positive real numbers

x

0

,

x

1

,

x

2

,
…

{\displaystyle x_{0},x_{1},x_{2},\ldots }

such that

x

0

>

x

1

>

x

2

>
⋯

{\displaystyle x_{0}>x_{1}>x_{2}>\cdots }

Player one wins if and only if

∑

i
=
0

∞

x

i

{\displaystyle \sum _{i=0}^{\infty }x_{i}}

exists and is in

X

{\displaystyle X}

.
One observation about the game is that if

X

{\displaystyle 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.