# Banach-Mazur game

A game that appeared in the famous Scottish Book [a11], [a6], where its initial version was formulated as Problem 43 by the Polish mathematician S. Mazur: Given the space of real numbers and a non-empty subset of it, two players and play a game in the following way: starts by choosing a non-empty interval of and then responds by choosing a non-empty subinterval of . Then player in turn selects a non-empty interval and continues by taking a non-empty subinterval of . This procedure is iterated infinitely many times. The resulting infinite sequence of nested intervals is called a play. By definition, the player wins this play if the intersection has a common point with . Otherwise wins. Mazur had observed two facts:

a) if the complement of in some interval of is of the first Baire category in this interval (equivalently, if is residual in some interval of , cf. also Category of a set; Baire classes), then player has a winning strategy (see below for the definition); and

b) if itself is of the first Baire category in , then has a winning strategy. The question originally posed by Mazur in Problem 43 of the Scottish Book (with as prize a bottle of wine!) was whether the inverse implications in the above two assertions hold. On August 4, 1935, S. Banach wrote in the same book that "Mazur's conjecture is true" . The proof of this statement of Banach however has never been published. The game subsequently became known as the Banach–Mazur game.

More than 20 years later, in 1957, J. Oxtoby [a8] published a proof for the validity of Mazur's conjecture. Oxtoby considered a much more general setting. The game was played in a general topological space with and the two players and were choosing alternatively sets from an a priori prescribed family of sets which has the property that every element of contains a non-empty open subset of and every non-empty open subset of contains an element of . As above, wins if , otherwise wins. Oxtoby's theorem says that has a winning strategy if and only if is of the first Baire category in ; also, if is a complete metric space, then has a winning strategy exactly when is residual in some non-empty open subset of .

Later, the game was subjected to different generalizations and modifications.

## Generalizations.

Only the most popular modification of this game will be considered. It has turned out to be useful not only in set-theoretic topology, but also in the geometry of Banach spaces, non-linear analysis, number theory, descriptive set theory, well-posedness in optimization, etc. This modification is the following: Given a topological space , two players (usually called and ) alternatively choose non-empty open sets (in this sequence the are the choices of and the are the choices of ; thus, it is player who starts this game). Player wins the play if , otherwise wins. To be completely consistent with the general scheme described above, one may think that and starts by always choosing the whole space . This game is often denoted by .

A strategy for the player is a mapping which assigns to every finite sequence of legal moves in a non-empty open subset of included in the last move of (i.e. ). A stationary strategy (called also a tactics) for is a strategy for this player which depends only on the last move of the opponent. A winning strategy (a stationary winning strategy) for is a strategy such that wins every play in which his/her moves are obtained by . Similarly one defines the (winning) strategies for .

A topological space is called weakly -favourable if has a winning strategy in , while it is termed -favourable if there is a stationary winning strategy for in . It can be derived from the work of Oxtoby [a8] (see also [a4], [a7] and [a9]) that the space is a Baire space exactly when player does not have a winning strategy in . Hence, every weakly -favourable space is a Baire space. In the class of metric spaces , a metric space is weakly -favourable if and only if it contains a dense and completely metrizable subspace. One can use these two results to see that the Banach–Mazur game is "not determined" . I.e. it could happen for some space that neither nor has a winning strategy. For instance, the Bernstein set in the real line (cf. also Non-measurable set) is a Baire space which does not contain a dense completely metrizable subspace (consequently does not admit a winning strategy for either or ).

The above characterization of weak -favourability for metric spaces has been extended for some non-metrizable spaces in [a10].

A characterization of -favourability of a given completely-regular space can be obtained by means of the space of all continuous and bounded real-valued functions on equipped with the usual sup-norm . The following statement holds [a5]: The space is weakly -favourable if and only if the set

is residual in . In other words, is weakly -favourable if and only if almost-all (from the Baire category point of view) of the functions in attain their maximum in . The rich interplay between , and is excellently presented in [a3].

The class of -favourable spaces (spaces which admit -winning tactics) is strictly narrower than the class of weakly -favourable spaces. G. Debs [a2] has exhibited a completely-regular topological space which admits a winning strategy for in , but does not admit any -winning tactics in . Under the name "espaces tamisables" , the -favourable spaces were introduced and studied also by G. Choquet [a1].

[a10] is an excellent survey paper about topological games (including ).

#### References

[a1] | G. Choquet, "Une classe régulières d'espaces de Baire" C.R. Acad. Sci. Paris Sér. I , 246 (1958) pp. 218–220 |

[a2] | G. Debs, "Strategies gagnantes dans certain jeux topologique" Fundam. Math. , 126 (1985) pp. 93–105 |

[a3] | G. Debs, J. Saint-Raymond, "Topological games and optimization problems" Mathematika , 41 (1994) pp. 117–132 |

[a4] | M.R. Krom, "Infinite games and special Baire space extensions" Pacific J. Math. , 55 : 2 (1974) pp. 483–487 |

[a5] | P.S. Kenderov, J.P. Revalski, "The Banach–Mazur game and generic existence of solutions to optimization problems" Proc. Amer. Math. Soc. , 118 (1993) pp. 911–917 |

[a6] | "The Scottish Book: Mathematics from the Scottish Café" R.D. Mauldin (ed.) , Birkhäuser (1981) |

[a7] | R.A. McCoy, "A Baire space extension" Proc. Amer. Math. Soc. , 33 (1972) pp. 199–202 |

[a8] | J. Oxtoby, "The Banach–Mazur game and the Banach category theorem" , Contributions to the Theory of Games III , Ann. of Math. Stud. , 39 , Princeton Univ. Press (1957) pp. 159–163 |

[a9] | J. Saint-Raymond, "Jeux topologiques et espaces de Namioka" Proc. Amer. Math. Soc. , 87 (1983) pp. 499–504 Zbl 0511.54007 |

[a10] | R. Telgárski, "Topological games: On the fifth anniversary of the Banach–Mazur game" Rocky Mount. J. Math. , 17 (1987) pp. 227–276 |

[a11] | S.M. Ulam, "The Scottish Book" , A LASL monograph , Los Alamos Sci. Lab. (1977) (Edition: Second) |

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Banach-Mazur game.

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