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

Gale-Stew a rt games and Bla kw ell games Daisuk e Ik egami (Universit y of Califo rnia, Berk eley) 1st of Ma y , 2012

P erfe t & imp erfe t info rmation fo r games P erfe t info rmation: Pla y ers kno w ab out the p revious moves of opp onents. E.g., Gale-Stew a rt games. Imp erfe t info rmation: Pla y ers do not kno w ab out what opp onents did p reviously . E.g., Bla kw ell games.

F rom no w on : : : W o rk in ZF+DC . R F o r a set X , DC : F o r any relation R � X � X su h that X ( 8 x 2 X ) ( 9 y 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 .

F rom no w on : : : W o rk in ZF+DC . R F o r a set X , DC : F o r any relation R � X � X su h that X ( 8 x 2 X ) ( 9 y 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 holds fo r any set X . X

Gale-Stew a rt games ! Fix a pa y o� set A � 2 . I's turn. ; h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Gale-Stew a rt games td. I has pla y ed. ; h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Gale-Stew a rt games td.. I I's turn. ; h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Gale-Stew a rt games td... I I has pla y ed. ; h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Gale-Stew a rt games td.... I's turn again. ; h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Gale-Stew a rt games td..... After in�nitely many times : : : ; h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i ! x 2 2 Pla y er I wins if x is in the pa y o� set A and otherwise Pla y er I I wins.

The Axiom of Determina y ! A subset A of 2 is determined if one of the pla y ers has a winning strategy in the Gale-Stew a rt game with the pa y o� set A . De�nition (My ielski-Steinhaus) The Axiom of Determina y (AD) asserts the follo wing: ! Every subset A of 2 is determined.

The Axiom of Determina y ! A subset A of 2 is determined if one of the pla y ers has a winning strategy in the Gale-Stew a rt game with the pa y o� set A . De�nition (My ielski-Steinhaus) The Axiom of Determina y (AD) asserts the follo wing: ! Every subset A of 2 is determined. Rema rk 1 AD ontradi ts the Axiom of Choi e (A C). 2 AD has many b eautiful onsequen es, e.g., every set of reals is Leb esgue measurable. + 3 Mo dels of AD (o r AD ) a re losely onne ted to mo dels with W o o din a rdinals.

Extensions of AD One an de�ne AD fo r any nonempt y set X . (Note: AD = AD ). X 2 De�nition ! AD : Every A � X is determined. X

Extensions of AD One an de�ne AD fo r any nonempt y set X . (Note: AD = AD ). X 2 De�nition ! AD : Every A � X is determined. X Rema rk AD is in onsistent if there is an inje tion from ! to X . X 1 Out interest: AD and AD . R

Bla kw ell games ! Fix a pa y o� set A � 2 . I's turn. ; h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Bla kw ell games td. I has pla y ed. ; 1 = 2 1 = 2 h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Bla kw ell games td.. I I's turn. ; ? ? h 0 i h 1 i h 00 i h 01 i h 10 i h 11 i

Bla kw ell games td... I I has pla y ed. ; ? ? h 0 i h 1 i 2 = 3 1 = 3 1 = 4 3 = 4 h 00 i h 01 i h 10 i h 11 i

Bla kw ell games td.... I's turn again. ; 1 = 2 1 = 2 h 0 i h 1 i ? ? ? ? h 00 i h 01 i h 10 i h 11 i

Bla kw ell games td..... After in�nitely many times : : : ; 1 = 2 1 = 2 h 0 i h 1 i 2 = 3 1 = 3 1 = 4 3 = 4 h 00 i h 01 i h 10 i h 11 i

Bla kw ell games td...... Cal ulate the p robabilit y as b elo w. ; 1 = 2 1 = 2 h 0 i h 1 i 2 = 3 1 = 3 1 = 4 3 = 4 h 00 i h 01 i h 10 i h 11 i |{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

Bla kw ell games td....... Pla y er I wins if the p robabilit y of the pa y o� set is 1. Pla y er I I wins if the p robabilit y of the pa y o� set is 0.

F o rmal de�nitions; Bla kw ell games Even � is a mixed strategy fo r I if � : 2 ! Prob(2). Odd � is a mixed strategy fo r I I if � : 2 ! Prob(2).

F o rmal de�nitions; Bla kw ell games Even � is a mixed strategy fo r I if � : 2 ! Prob(2). Odd � is a mixed strategy fo r I I if � : 2 ! Prob(2). F o r a mixed strategy � fo r I and a mixed strategy � fo r I I, de�ne <! � � � : 2 ! Prob(2) as follo ws: ( � ( s ) if lh ( s ) is even, � � � ( s ) = � ( s ) if lh ( s ) is o dd.

F o rmal de�nitions; Bla kw ell games Even � is a mixed strategy fo r I if � : 2 ! Prob(2). Odd � is a mixed strategy fo r I I if � : 2 ! Prob(2). F o r a mixed strategy � fo r I and a mixed strategy � fo r I I, de�ne <! � � � : 2 ! Prob(2) as follo ws: ( � ( s ) if lh ( s ) is even, � � � ( s ) = � ( s ) if lh ( s ) is o dd. <! Then de�ne � : 2 ! [0 ; 1℄ as follo ws: � ;� Y � ( s ) = � � � ( s � i )( s ( i )) : � ;� i < lh( s ) With the help of DC , one an uniquely extend � to a Bo rel R � ;� p robabilit y measure on the Canto r spa e.

F o rmal de�nitions; Bla kw ell games td. ! Let A � 2 . A mixed strategy � fo r I is optimal in A if fo r any mixed strategy � fo r I I, � ( A ) = 1. � ;� A mixed strategy � fo r I I is optimal in A if fo r any mixed strategy � fo r I, � ( A ) = 0. � ;�

F o rmal de�nitions; Bla kw ell games td. ! Let A � 2 . A mixed strategy � fo r I is optimal in A if fo r any mixed strategy � fo r I I, � ( A ) = 1. � ;� A mixed strategy � fo r I I is optimal in A if fo r any mixed strategy � fo r I, � ( A ) = 0. � ;� A is Bla kw ell determined if either I o r I I has an optimal strategy in A . ! Bl-AD: Every A � 2 is Bla kw ell determined. Note: There is another fo rmulation of Bla kw ell games oming from game theo ry .

F o rmal de�nitions; Bla kw ell games td.. Let X b e a non-empt y set. Even � is a mixed strategy fo r I if � : X ! Prob ( X ). ! Odd � is a mixed strategy fo r I I if � : X ! Prob ( X ). ! F o r a mixed strategy � fo r I and a mixed strategy � fo r I I, de�ne <! � � � : X ! Prob ( X ) as follo ws: ( � ( s ) if lh ( s ) is even, � � � ( s ) = � ( s ) if lh ( s ) is o dd. <! Then de�ne � : X ! [0 ; 1℄ as follo ws: � ;� Y � ( s ) = � � � ( s � i )( s ( i )) : � ;� i < lh( s ) If w e have DC ! , w e an uniquely extend � to a Bo rel � ;� ( R � X ) ! p robabilit y measure on X . Note: DC ! follo ws from DC R ( R � R )

F o rmal de�nitions; Bla kw ell games td... ! Let X b e a non-empt y set and A � X . Assume w e have DC ! . ( R � X ) A mixed strategy � fo r I is optimal in A if fo r any mixed strategy � fo r I I, � ( A ) = 1. � ;� A mixed strategy � fo r I I is optimal in A if fo r any mixed strategy � fo r I, � ( A ) = 0. � ;� A is Bla kw ell determined if either I o r I I has an optimal strategy in A . ! Bl-AD : every A � X is Bla kw ell determined. X

F o rmal de�nitions; Bla kw ell games td... ! Let X b e a non-empt y set and A � X . Assume w e have DC ! . ( R � X ) A mixed strategy � fo r I is optimal in A if fo r any mixed strategy � fo r I I, � ( A ) = 1. � ;� A mixed strategy � fo r I I is optimal in A if fo r any mixed strategy � fo r I, � ( A ) = 0. � ;� A is Bla kw ell determined if either I o r I I has an optimal strategy in A . ! Bl-AD : every A � X is Bla kw ell determined. X Rema rk Bl-AD is in onsistent if there is an inje tion from ! to X X 1 Our interest: Bl-AD and Bl-AD . R

Observation 1 Observation ! Let X b e a nonempt y set and A b e a subset of X . If A is determined, then A is Bla kw ell determined.

Observation 1 Observation ! Let X b e a nonempt y set and A b e a subset of X . If A is determined, then A is Bla kw ell determined. P oint: Given a strategy � , one an naturally transfo rm it to a mixed strategy � ^ . If � is winning, then � ^ is optimal.