Alan Watts and the Infinite Game

Author(s):  
Nathan L. Hulsey
Keyword(s):  

1985 ◽  
Vol 50 (1) ◽  
pp. 110-122
Author(s):  
Howard Becker

For any A ⊂ R, the Banach game B(A) is the following infinite game on reals: Players I and II alternately play positive real numbers a1; a2, a3, a4,… such that for n > 1, an < an−1. Player I wins iff ai exists and is in A.This type of game was introduced by Banach in 1935 in the Scottish Book [15], Problem 43. The (rather vague) problem which Banach posed was to characterize those sets A for which I (II) has a winning strategy in B(A). (There are three parts to Problem 43. In the first, Mazur defined a game G**(A) for every set A ⊂ R and conjectured that II has a winning strategy in G**(A) iff A is meager and I has a winning strategy in G**(A) iff A is comeager in some neighborhood; this conjecture was proved by Banach. Presumably Banach had this result in mind when he asked the question about B(A), and hoped for a similar type of characterization.) Incidentally, Problem 43 of the Scottish Book appears to be the first time infinite games of any sort were studied by mathematicians.This paper will not provide the reader with any answer to Banach's question. I know of no nontrivial way to characterize when player I (or II) wins, and I suspect there is none. This paper is concerned with a different (also rather vague) question: For which sets A is the Banach game B(A) determined? To say that B(A) is determined means, of course, that one of the players has a winning strategy for B(A).



2021 ◽  
Author(s):  
Zahra Gambarova ◽  
Dionysius Glycopantis
Keyword(s):  


Author(s):  
William M. Donaldson
Keyword(s):  


2016 ◽  
Vol 103 (2) ◽  
pp. 15-16
Author(s):  
Katherine Strand
Keyword(s):  


2006 ◽  
Vol 113 (1-2) ◽  
pp. 145-158 ◽  
Author(s):  
Jan Malý ◽  
Miroslav Zelený
Keyword(s):  


2017 ◽  
Vol 103 (4) ◽  
pp. 13-14
Author(s):  
Katherine Strand
Keyword(s):  


2013 ◽  
Vol 13 (01) ◽  
pp. 1350002
Author(s):  
JINDŘICH ZAPLETAL

Certain separation problems in descriptive set theory correspond to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle 𝕋 can be covered by ℵ1 many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it.



1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling

Abstract.Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.



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