Banach–Mazur Game

Around 1928¹, the Polish mathematician Stanisław Mazur invented the following mathematical game.

Setup of the Game

We start with a complete metric space $X$ and a particular "playground" $M \subseteq X$. There are 2 players.

Player 1's goal is to land on $M$, and Player 2's goal is to avoid $M$.

Player 1 starts the game by picking a closed ball $B_1 \subseteq X$, then Player 2 picks a closed ball $B_2 \subseteq B_1$, then Player 1 picks a closed ball $B_{3} \subseteq B_{2}$, and so on. These closed balls must be non-degenerate, of course.

We then consider the countable intersection $\bigcap_{n=1}^\infty B_n$ and check whether or not this intersects our playground $M$.

If $\left( \bigcap_{n=1}^\infty B_n \right) \cap M \neq \emptyset$, then Player 1 wins. If $\left( \bigcap_{n=1}^\infty B_n \right) \cap M = \emptyset$, then Player 2 wins.

(simulation of the game - allow the reader to play)

Winning Strategies

It is natural to wonder if either player has a "winning strategy", i.e., a set of moves that guarantees their victory no matter what the opponent does. Let us investigate this.

Intuitively, Player 1 has an upper hand, since she picks the first ball and all subsequent moves are subsets of that first one. So if the choice of $X$ and $M$ were random, we would expect Player 1 to win more often than not. (simulation?)

Footnotes

1. See page 20 of these notes. ↩