Banach–Mazur game

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In general topology, set theory and game theory, a BanachMazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spaces. This game was the first infinite positional game of perfect information to be studied. It was introduced by Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach.

Definition[edit]

Let be topological space, a fixed subset of and a family of subsets of that have the following properties:

  • Each member of has non-empty interior.
  • Each non-empty open subset of contains a member of .

Players, and alternatively choose elements from to form a sequence

wins if and only if

Otherwise, wins. This is called a general Banach–Mazur game and denoted by

Properties[edit]

  • has a winning strategy if and only if is of the first category in (a set is of the first category or meagre if it is the countable union of nowhere-dense sets).
  • If is a complete metric space, has a winning strategy if and only if is comeager in some non-empty open subset of
  • If has the Baire property in , then is determined.
  • Any winning strategy of can be reduced to a stationary winning strategy.
  • The siftable and strongly-siftable spaces introduced by Choquet can be defined in terms of stationary strategies in suitable modifications of the game. Let denote a modification of where is the family of all non-empty open sets in and wins a play if and only if
Then is siftable if and only if has a stationary winning strategy in
  • A Markov winning strategy for in can be reduced to a stationary winning strategy. Furthermore, if has a winning strategy in , then has a winning strategy depending only on two preceding moves. It is still an unsettled question whether a winning strategy for can be reduced to a winning strategy that depends only on the last two moves of .
  • is called weakly -favorable if has a winning strategy in . Then, is a Baire space if and only if has no winning strategy in . It follows that each weakly -favorable space is a Baire space.

Many other modifications and specializations of the basic game have been proposed: for a thorough account of these, refer to [1987].

The most common special case arises when and consist of all closed intervals in the unit interval. Then wins if and only if and wins if and only if . This game is denoted by

A simple proof: winning strategies[edit]

It is natural to ask for what sets does have a winning strategy. Clearly, if is empty, has a winning strategy, therefore the question can be informally rephrased as how "small" (respectively, "big") does (respectively, the complement of in ) have to be to ensure that has a winning strategy. The following result gives a flavor of how the proofs used to derive the properties in the previous section work:

Proposition. has a winning strategy if is countable, is T1, and has no isolated points.
Proof. Index the elements of X as a sequence: Suppose has chosen if is the non-empty interior of then is a non-empty open set in so can choose Then chooses and, in a similar fashion, can choose that excludes . Continuing in this way, each point will be excluded by the set so that the intersection of all will not intersect .

The assumptions on are key to the proof: for instance, if is equipped with the discrete topology and consists of all non-empty subsets of , then has no winning strategy if (as a matter of fact, her opponent has a winning strategy). Similar effects happen if is equipped with indiscrete topology and

A stronger result relates to first-order sets.

Proposition. has a winning strategy if and only if is meagre.

This does not imply that has a winning strategy if is not meagre. In fact, has a winning strategy if and only if there is some such that is a comeagre subset of It may be the case that neither player has a winning strategy: let be the unit interval and be the family of closed intervals in the unit interval. The game is determined if the target set has the property of Baire, i.e. if it differs from an open set by a meagre set (but the converse is not true). Assuming the axiom of choice, there are subsets of the unit interval for which the Banach–Mazur game is not determined.

References[edit]