Gale-Stew a rt games and Blakw ell games Daisuk e Ik egami - - PowerPoint PPT Presentation

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Gale-Stew a rt games and Blakw ell games Daisuk e Ik egami - - PowerPoint PPT Presentation

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Gale-Stew a rt games and Blakw ell games Daisuk e Ik egami (Universit y of Califo rnia, Berk eley) 1st of Ma y , 2012 P erfet & imp erfet info rmation fo r games P erfet info rmation: Pla y ers kno

slide-1
SLIDE 1 Gale-Stew a rt games and Bla kw ell games Daisuk e Ik egami (Universit y
  • f
Califo rnia, Berk eley) 1st
  • f
Ma y , 2012
slide-2
SLIDE 2 P erfe t & imp erfe t info rmation fo r games P erfe t info rmation: Pla y ers kno w ab
  • ut
the p revious moves
  • f
  • pp
  • nents.
E.g., Gale-Stew a rt games. Imp erfe t info rmation: Pla y ers do not kno w ab
  • ut
what
  • pp
  • nents
did p reviously . E.g., Bla kw ell games.
slide-3
SLIDE 3 F rom no w
  • n:
: : W
  • rk
in ZF+DC R . F
  • r
a set X , DC X : F
  • r
any relation R
  • X
  • X
su h that (8x 2 X ) (9y 2 X ) (x ; y ) 2 R , there is a fun tion f : ! ! X su h that (f (n ); f (n + 1)) 2 R fo r any natural numb er n .
slide-4
SLIDE 4 F rom no w
  • n:
: : W
  • rk
in ZF+DC R . F
  • r
a set X , DC X : F
  • r
any relation R
  • X
  • X
su h that (8x 2 X ) (9y 2 X ) (x ; y ) 2 R , there is a fun tion f : ! ! X su h that (f (n ); f (n + 1)) 2 R fo r any natural numb er n . DC: DC X holds fo r any set X .
slide-5
SLIDE 5 Gale-Stew a rt games Fix a pa y
  • set
A
  • 2
! . I's turn. ; h0i h1i h00i h01i h10i h11i
slide-6
SLIDE 6 Gale-Stew a rt games td. I has pla y ed. ; h0i h1i h00i h01i h10i h11i
slide-7
SLIDE 7 Gale-Stew a rt games td.. I I's turn. ; h0i h1i h00i h01i h10i h11i
slide-8
SLIDE 8 Gale-Stew a rt games td... I I has pla y ed. ; h0i h1i h00i h01i h10i h11i
slide-9
SLIDE 9 Gale-Stew a rt games td.... I's turn again. ; h0i h1i h00i h01i h10i h11i
slide-10
SLIDE 10 Gale-Stew a rt games td..... After innitely many times: : : ; h0i h1i h00i h01i h10i h11i x 2 2 ! Pla y er I wins if x is in the pa y
  • set
A and
  • therwise
Pla y er I I wins.
slide-11
SLIDE 11 The Axiom
  • f
Determina y A subset A
  • f
2 ! is determined if
  • ne
  • f
the pla y ers has a winning strategy in the Gale-Stew a rt game with the pa y
  • set
A. Denition (My ielski-Steinhaus) The Axiom
  • f
Determina y (AD) asserts the follo wing: Every subset A
  • f
2 ! is determined.
slide-12
SLIDE 12 The Axiom
  • f
Determina y A subset A
  • f
2 ! is determined if
  • ne
  • f
the pla y ers has a winning strategy in the Gale-Stew a rt game with the pa y
  • set
A. Denition (My ielski-Steinhaus) The Axiom
  • f
Determina y (AD) asserts the follo wing: Every subset A
  • f
2 ! is determined. Rema rk 1 AD
  • ntradi ts
the Axiom
  • f
Choi e (A C). 2 AD has many b eautiful
  • nsequen es,
e.g., every set
  • f
reals is Leb esgue measurable. 3 Mo dels
  • f
AD (o r AD + ) a re losely
  • nne ted
to mo dels with W
  • din
a rdinals.
slide-13
SLIDE 13 Extensions
  • f
AD One an dene AD X fo r any nonempt y set X . (Note: AD = AD 2 ). Denition AD X : Every A
  • X
! is determined.
slide-14
SLIDE 14 Extensions
  • f
AD One an dene AD X fo r any nonempt y set X . (Note: AD = AD 2 ). Denition AD X : Every A
  • X
! is determined. Rema rk AD X is in onsistent if there is an inje tion from ! 1 to X . Out interest: AD and AD R .
slide-15
SLIDE 15 Bla kw ell games Fix a pa y
  • set
A
  • 2
! . I's turn. ; h0i h1i h00i h01i h10i h11i
slide-16
SLIDE 16 Bla kw ell games td. I has pla y ed. ; h0i 1=2 h1i 1=2 h00i h01i h10i h11i
slide-17
SLIDE 17 Bla kw ell games td.. I I's turn. ; h0i ? h1i ? h00i h01i h10i h11i
slide-18
SLIDE 18 Bla kw ell games td... I I has pla y ed. ; h0i ? h1i ? h00i 2=3 h01i 1=3 h10i 1=4 h11i 3=4
slide-19
SLIDE 19 Bla kw ell games td.... I's turn again. ; h0i 1=2 h1i 1=2 h00i ? h01i ? h10i ? h11i ?
slide-20
SLIDE 20 Bla kw ell games td..... After innitely many times: : : ; h0i 1=2 h1i 1=2 h00i 2=3 h01i 1=3 h10i 1=4 h11i 3=4
slide-21
SLIDE 21 Bla kw ell games td...... Cal ulate the p robabilit y as b elo w. ; h0i 1=2 h1i 1=2 h00i 2=3 h01i 1=3 h10i 1=4 h11i 3=4 |{z} |{z} |{z} |{z} 1=2
  • 2=3
= 1=3 1=2
  • 1=3
= 1=6 1=2
  • 1=4
= 1=8 1=2
  • 3=4
= 3=8
slide-22
SLIDE 22 Bla kw ell games td....... Pla y er I wins if the p robabilit y
  • f
the pa y
  • set
is 1. Pla y er I I wins if the p robabilit y
  • f
the pa y
  • set
is 0.
slide-23
SLIDE 23 F
  • rmal
denitions; Bla kw ell games
  • is
a mixed strategy fo r I if
  • :
2 Even ! Prob(2).
  • is
a mixed strategy fo r I I if
  • :
2 Odd ! Prob(2).
slide-24
SLIDE 24 F
  • rmal
denitions; Bla kw ell games
  • is
a mixed strategy fo r I if
  • :
2 Even ! Prob(2).
  • is
a mixed strategy fo r I I if
  • :
2 Odd ! Prob(2). F
  • r
a mixed strategy
  • fo
r I and a mixed strategy
  • fo
r I I, dene
  • :
2 <! ! Prob(2) as follo ws:
  • (s
) = (
  • (s
) if lh (s ) is even,
  • (s
) if lh (s ) is
  • dd.
slide-25
SLIDE 25 F
  • rmal
denitions; Bla kw ell games
  • is
a mixed strategy fo r I if
  • :
2 Even ! Prob(2).
  • is
a mixed strategy fo r I I if
  • :
2 Odd ! Prob(2). F
  • r
a mixed strategy
  • fo
r I and a mixed strategy
  • fo
r I I, dene
  • :
2 <! ! Prob(2) as follo ws:
  • (s
) = (
  • (s
) if lh (s ) is even,
  • (s
) if lh (s ) is
  • dd.
Then dene
  • ;
: 2 <! ! [0; 1℄ as follo ws:
  • ;
(s ) = Y i <lh(s )
  • (s
  • i
)(s (i )): With the help
  • f
DC R ,
  • ne
an uniquely extend
  • ;
to a Bo rel p robabilit y measure
  • n
the Canto r spa e.
slide-26
SLIDE 26 F
  • rmal
denitions; Bla kw ell games td. Let A
  • 2
! . A mixed strategy
  • fo
r I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I I,
  • ;
(A) = 1. A mixed strategy
  • fo
r I I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I,
  • ;
(A) = 0.
slide-27
SLIDE 27 F
  • rmal
denitions; Bla kw ell games td. Let A
  • 2
! . A mixed strategy
  • fo
r I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I I,
  • ;
(A) = 1. A mixed strategy
  • fo
r I I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I,
  • ;
(A) = 0. A is Bla kw ell determined if either I
  • r
I I has an
  • ptimal
strategy in A. Bl-AD: Every A
  • 2
! is Bla kw ell determined. Note: There is another fo rmulation
  • f
Bla kw ell games
  • ming
from game theo ry .
slide-28
SLIDE 28 F
  • rmal
denitions; Bla kw ell games td.. Let X b e a non-empt y set.
  • is
a mixed strategy fo r I if
  • :
X Even ! Prob ! (X ).
  • is
a mixed strategy fo r I I if
  • :
X Odd ! Prob ! (X ). F
  • r
a mixed strategy
  • fo
r I and a mixed strategy
  • fo
r I I, dene
  • :
X <! ! Prob (X ) as follo ws:
  • (s
) = (
  • (s
) if lh (s ) is even,
  • (s
) if lh (s ) is
  • dd.
Then dene
  • ;
: X <! ! [0; 1℄ as follo ws:
  • ;
(s ) = Y i <lh(s )
  • (s
  • i
)(s (i )): If w e have DC (RX ! ) , w e an uniquely extend
  • ;
to a Bo rel p robabilit y measure
  • n
X ! . Note: DC (RR ! ) follo ws from DC R
slide-29
SLIDE 29 F
  • rmal
denitions; Bla kw ell games td... Let X b e a non-empt y set and A
  • X
! . Assume w e have DC (RX ! ) . A mixed strategy
  • fo
r I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I I,
  • ;
(A) = 1. A mixed strategy
  • fo
r I I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I,
  • ;
(A) = 0. A is Bla kw ell determined if either I
  • r
I I has an
  • ptimal
strategy in A. Bl-AD X : every A
  • X
! is Bla kw ell determined.
slide-30
SLIDE 30 F
  • rmal
denitions; Bla kw ell games td... Let X b e a non-empt y set and A
  • X
! . Assume w e have DC (RX ! ) . A mixed strategy
  • fo
r I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I I,
  • ;
(A) = 1. A mixed strategy
  • fo
r I I is
  • ptimal
in A if fo r any mixed strategy
  • fo
r I,
  • ;
(A) = 0. A is Bla kw ell determined if either I
  • r
I I has an
  • ptimal
strategy in A. Bl-AD X : every A
  • X
! is Bla kw ell determined. Rema rk Bl-AD X is in onsistent if there is an inje tion from ! 1 to X Our interest: Bl-AD and Bl-AD R .
slide-31
SLIDE 31 Observation 1 Observation Let X b e a nonempt y set and A b e a subset
  • f
X ! . If A is determined, then A is Bla kw ell determined.
slide-32
SLIDE 32 Observation 1 Observation Let X b e a nonempt y set and A b e a subset
  • f
X ! . If A is determined, then A is Bla kw ell determined. P
  • int:
Given a strategy
  • ,
  • ne
an naturally transfo rm it to a mixed strategy ^
  • .
If
  • is
winning, then ^
  • is
  • ptimal.
slide-33
SLIDE 33 Observation 1 Observation Let X b e a nonempt y set and A b e a subset
  • f
X ! . If A is determined, then A is Bla kw ell determined. P
  • int:
Given a strategy
  • ,
  • ne
an naturally transfo rm it to a mixed strategy ^
  • .
If
  • is
winning, then ^
  • is
  • ptimal.
Co rolla ry (Ma rtin) AD implies Bl-AD. AD R implies Bl-AD R .
slide-34
SLIDE 34 Observation 1 Observation Let X b e a nonempt y set and A b e a subset
  • f
X ! . If A is determined, then A is Bla kw ell determined. P
  • int:
Given a strategy
  • ,
  • ne
an naturally transfo rm it to a mixed strategy ^
  • .
If
  • is
winning, then ^
  • is
  • ptimal.
Co rolla ry (Ma rtin) AD implies Bl-AD. AD R implies Bl-AD R . Conje ture (Ma rtin) Bl-AD implies AD.
slide-35
SLIDE 35 Observation 2 Observation If a nite game is Bla kw ell determined, then it is determined when X is totally
  • rdered.
Finite games = games ending at some xed round n < ! . Sk et h
  • f
p ro
  • f.
In bla kb
  • a
rds.
slide-36
SLIDE 36 Observation 2 Observation If a nite game is Bla kw ell determined, then it is determined when X is totally
  • rdered.
Finite games = games ending at some xed round n < ! . Sk et h
  • f
p ro
  • f.
In bla kb
  • a
rds. Co rolla ry (L
  • w
e) Assume Bl-AD R . Then Unifo rmization holds, i.e., every relation
  • n
the reals an b e unifo rmized b y a fun tion.
slide-37
SLIDE 37 Observation 2 td. By the same a rgument: : : Prop
  • sition
If a lop en set is Bla kw ell determined, then it is determined.
slide-38
SLIDE 38 Observation 2 td. By the same a rgument: : : Prop
  • sition
If a lop en set is Bla kw ell determined, then it is determined. Theo rem (Neeman) Assume Bl-AD. Then every Suslin &
  • -Suslin
subset
  • f
the Canto r spa e is determined. Denition 1 A subset A
  • f
the Canto r spa e is Suslin if there is an
  • rdinal
  • and
a tree T
  • n
2
  • su h
that A = p[T ℄. 2 A subset A
  • f
the Canto r spa e is
  • -Suslin
if the
  • mplement
  • f
A is Suslin.
slide-39
SLIDE 39 Observation 2 td.. Theo rem (Ke hris & W
  • din)
If every Suslin &
  • -Suslin
set is determined, then AD L(R) holds.
slide-40
SLIDE 40 Observation 2 td.. Theo rem (Ke hris & W
  • din)
If every Suslin &
  • -Suslin
set is determined, then AD L(R) holds. Co rolla ry (Ma rtin, Neeman & V ervo
  • rt)
L(R )
  • \AD
( ) Bl-AD ". In pa rti ula r, AD and Bl-AD a re equi onsistent.
slide-41
SLIDE 41 Observation 3 Observation Assume Bl-AD R . Let A
  • R
! . If A is range-inva riant, then A is determined. Denition A set A
  • R
! is range-inva riant if fo r any ~ x ; ~ y 2 R ! with the same range, ~ x 2 A ( ) ~ y 2 A.
slide-42
SLIDE 42 Observation 3 td. Theo rem (de Klo et, L
  • w
e, I.) Assume Bl-AD R . Then there is a ne, no rmal,
  • omplete
ultralter
  • n
P ! 1 (R ). Denition Let U b e a lter
  • n
P ! 1 (R ). 1 U is ne if fo r any x 2 R , fa 2 P ! 1 (R ) j x 2 a g 2 U . 2 U is no rmal if fo r any family fA x 2 U j x 2 R g , 4 x 2R A x = fa 2 P ! 1 (R ) j (8x 2 a ) a 2 A x g 2 U .
slide-43
SLIDE 43 Observation 3 td.. Theo rem (Solova y) If there is a ne, no rmal
  • omplete
ultralter
  • n
P ! 1 (R ), then R # exists. P
  • ints:
1 By assumption, ! 1 is measurable and hen e a # exists fo r all a 2 P ! 1 (R ) . 2 Letting U b e a ne no rmal measure
  • n
P ! 1 (R ),
  • 2
R # ( ) fa 2 P ! 1 (R ) j
  • 2
a # g 2 U :
slide-44
SLIDE 44 Observation 3 td.. Theo rem (Solova y) If there is a ne, no rmal
  • omplete
ultralter
  • n
P ! 1 (R ), then R # exists. P
  • ints:
1 By assumption, ! 1 is measurable and hen e a # exists fo r all a 2 P ! 1 (R ) . 2 Letting U b e a ne no rmal measure
  • n
P ! 1 (R ),
  • 2
R # ( ) fa 2 P ! 1 (R ) j
  • 2
a # g 2 U : Co rolla ry (de Klo et, L
  • w
e & I.) Assume Bl-AD R . Then R # exists. Hen e Bl-AD R ` Con (AD).
slide-45
SLIDE 45 AD R vs Bl-AD R Question Do es Bl-AD R imply AD R ?
slide-46
SLIDE 46 AD R vs Bl-AD R Question Do es Bl-AD R imply AD R ? Theo rem (W
  • din
& I.) Under ZF+DC, AD R and Bl-AD R a re equivalent.
slide-47
SLIDE 47 AD R vs Bl-AD R Question Do es Bl-AD R imply AD R ? Theo rem (W
  • din
& I.) Under ZF+DC, AD R and Bl-AD R a re equivalent. But! Rema rk (Solova y) AD R +DC implies the
  • nsisten y
  • f
AD R . So assuming DC is not
  • ptimal
fo r the ab
  • ve
theo rem.
slide-48
SLIDE 48 AD R vs. Bl-AD R td. Question Are AD R a re Bl-AD R equi onsistent?
slide-49
SLIDE 49 AD R vs. Bl-AD R td. Question Are AD R a re Bl-AD R equi onsistent? Conje ture (W
  • din)
AD R and Bl-AD R a re equi onsistent.
slide-50
SLIDE 50 Bl-AD R and generi emb eddings Theo rem (Sa rgsy an) Assume CH and that there is a generi emb edding j : V ! M su h that 1 M is transitive and M ! \ V [G ℄
  • M
, 2 G is a generi lter
  • f
a homogeneous fo r ing, and 3 j
  • Ord
is denable in V . Then there is a mo del
  • f
ZF+AD R +\ is regula r". The metho d: Co re Mo del Indu tion
slide-51
SLIDE 51 Bl-AD R and generi emb eddings Theo rem (Sa rgsy an) Assume CH and that there is a generi emb edding j : V ! M su h that 1 M is transitive and M ! \ V [G ℄
  • M
, 2 G is a generi lter
  • f
a homogeneous fo r ing, and 3 j
  • Ord
is denable in V . Then there is a mo del
  • f
ZF+AD R +\ is regula r". The metho d: Co re Mo del Indu tion Theo rem Assume Bl-AD R . Then fo r any
  • <
  • and
A
  • R
, there is a generi emb edding j : L(A; R ) ! M su h that 1 M is transitive, R V [G ℄
  • M
, and
  • is
  • untable
in M , 2 G is a generi lter
  • f
a homogeneous fo r ing, and 3 j
  • Ord
is denable in V .
slide-52
SLIDE 52 Gratitude Thank y
  • u
very mu h fo r y
  • ur
attention!