scholarly journals A TWO-PERSON INFINITE GAME SUGGESTED FROM A QUIZ OF GUESSING A NUMBER

10.5109/13342 ◽  
1983 ◽  
Vol 20 (3/4) ◽  
pp. 23-32
Author(s):  
Yoshinobu Teraoka
Keyword(s):  
Infinite Game

Determinacy of Banach games

1985 ◽  
Vol 50 (1) ◽  
pp. 110-122
Author(s):  
Howard Becker
Keyword(s):  
Winning Strategy ◽  
Similar Type ◽  
Infinite Games ◽  
Real Numbers ◽  
Positive Real ◽  
Infinite Game ◽  
First Time

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).

scholarly journals On the Analytics of Infinite Game Theory Problems

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

The Infinite Game

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

The Infinite Game: Part Three

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

SEPARATION PROBLEMS AND FORCING

2013 ◽  
Vol 13 (01) ◽  
pp. 1350002
Author(s):  
JINDŘICH ZAPLETAL
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Closed Sets ◽  
Infinite Game ◽  
Preservation Property ◽  
Sets Of Uniqueness ◽  
Descriptive Set ◽  
Fusion Type ◽  
Separation Problems

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.

Banach games

1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Perfect Information ◽  
Real Numbers ◽  
The Real ◽  
Infinite Game ◽  
Image Position ◽  
The Relationship

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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