2013 04 14_game_theory_lecture_07

Ⱥɪɛɢɬɪɚɠɧɵɟ ɫɯɟɦɵ ɢ ɤɨɨɩɟɪɚɬɢɜɧɵɟ ɢɝɪɵ

ɂɥɶɹ Ʉɚɰɟɜ1

1ɋɚɧɤɬ�ɉɟɬɟɪɛɭɪɝɫɤɢɣ ɷɤɨɧɨɦɢɤɨ�ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɢɧɫɬɢɬɭɬ ɊȺɇ

��

ɂ�ȼ�Ʉɚɰɟɜ �ɋɉɛ ɗɆɂ� Ʉɨɨɩɟɪɚɬɢɜɧɚɹ ɬɟɨɪɢɹ ��

Ⱥɪɛɢɬɪɚɠɧɵɟ ɫɯɟɦɵȺɪɛɢɬɪɚɠɧɨɣ ɫɯɟɦɨɣ ɧɚɡɵɜɚɟɬɫɹ ɩɚɪɚ (;, G)� ɝɞɟ ; ! R2 � ɩɟɪɟɝɨɜɨɪɧɨɟɦɧɨɠɟɫɬɜɨ� ɚ G " ; � ɬɨɱɤɚ ɧɟɫɨɝɥɚɫɢɹ� ,5(;) � ɦɧɨɠɟɫɬɜɨ ɢɧɞɢɜɢɞɭɚɥɶɧɨɪɚɰɢɨɧɚɥɶɧɵɯ ɜɟɤɬɨɪɨɜ ɩɟɪɟɝɨɜɨɪɧɨɝɨ ɦɧɨɠɟɫɬɜɚ� ,5(;) := {[ " ; | [ # G}.

Ɋɟɲɟɧɢɟɦ ɞɥɹ ɤɥɚɫɫɚ ɚɪɛɢɬɪɚɠɧɵɯ ɫɯɟɦ B ɧɚɡɵɜɚɟɬɫɹ ɨɬɨɛɪɚɠɟɧɢɟ! : B $ R2�


Ⱥɤɫɢɨɦɵ

�� ɉɚɪɟɬɨ�ɨɩɬɢɦɚɥɶɧɨɫɬɶ� !(;, G) " ";�

�� ɂɧɞɢɜɢɞɭɚɥɶɧɚɹ ɪɚɰɢɨɧɚɥɶɧɨɫɬɶ� !(;, G) # G�

�� ɇɟɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɚɮɮɢɧɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ� ɞɥɹ D > 0, E " R2

!(D;+ E, DG+ E) = D!(;, G) + E.

�� Ⱥɧɨɧɢɦɧɨɫɬɶ� ɟɫɥɢ # : R2 $ R2 � ɫɢɦɦɟɬɪɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɹɦɨɣ \ = [�ɬɨ !(#;,#G) = #!(;, G)�

�� ɇɟɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɧɟɫɭɳɟɫɬɜɟɧɧɵɯ ɚɥɶɬɟɪɧɚɬɢɜ� ɟɫɥɢ ;! ! ; ɢ!(;, G) " ;!� ɬɨ !(;!, G) = !(;, G)�

�� Ɉɝɪɚɧɢɱɟɧɧɚɹ ɦɨɧɨɬɨɧɧɨɫɬɶ� ɉɭɫɬɶ ɞɜɟ Ⱥɋ %1 = (;1, G), %2 = (;2, G),ɬɚɤɨɜɵ� ɱɬɨ ;1 ! ;2 ɢ PD[[",5(;1,G) [L = PD[[",5(;2,G) [L ɞɥɹ L = 1, 2� Ɍɨɝɞɚ

!(;1, G) % !(;2, G)


Ⱥɪɛɢɬɪɚɠɧɨɟ ɪɟɲɟɧɢɟ ɇɷɲɚ

�Ɍɟɨɪɟɦɚ �ɇɷɲ� ��

��

ɋɭɳɟɫɬɜɭɟɬ ɬɨɥɶɤɨ ɨɞɧɨ ɪɟɲɟɧɢɟ� ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɚɤɫɢɨɦɚɦ �� Ɉɧɨɡɚɞɚɟɬɫɹ ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɭɥɨɣ ɞɥɹ % " B :

!(%) = DUJ PD[[",5(;)

Q!

L=1

([L & GL).


Ⱥɪɛɢɬɪɚɠɧɨɟ ɪɟɲɟɧɢɟ Ʉɚɥɚɢ�ɋɦɨɪɨɞɢɧɫɤɨɝɨɂɞɟɚɥɶɧɨɣ ɬɨɱɤɨɣ ɚɪɛɢɬɪɚɠɧɨɣ ɫɯɟɦɵ % = ';, G( ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪ,(;, G) " R2�

(,(;, G))L = PD[[",5(;,G)

[L.

!

"

G

,(;, G)

•[.&


Ⱥɪɛɢɬɪɚɠɧɨɟ ɪɟɲɟɧɢɟ Ʉɚɥɚɢ�ɋɦɨɪɨɞɢɧɫɤɨɝɨ

�7KHRUHP��

��

ɋɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɞɥɹ ɤɥɚɫɫɚ B2, ɭɞɨɜɥɟɬɜɨɪɹɸɲɟɟɚɤɫɢɨɦɚɦ �� ɗɬɨ ɪɟɲɟɧɢɟ Ʉɚɥɚɢ±ɋɦɨɪɨɞɢɧɫɬɤɨɝɨ�


Ʉɨɨɩɟɪɚɬɢɜɧɵɟ ɢɝɪɵ

ɉɟɪɟɝɨɜɨɪɧɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨ� ɱɬɨ ɜɫɟ ɢɝɪɨɤɢ ɦɨɝɭɬ ɩɨɥɭɱɢɬɶ ɜɦɟɫɬɟ

Ɍɨɱɤɚ ɧɟɫɨɝɥɚɫɢɹ ɬɨ� ɱɬɨ ɢɝɪɨɤɢ ɦɨɝɭɬ ɩɨɥɭɱɢɬɶ ɩɨ ɨɬɞɟɥɶɧɨɫɬɢ�

ɇɚɞɨ ɭɱɢɬɵɜɚɬɶ ɢ ´ɩɪɨɦɟɠɭɬɨɱɧɵɟ´ ɜɚɪɢɚɧɬɵ � ɤɨɚɥɢɰɢɢ ɢɝɪɨɤɨɜ�



ɉɟɪɟɝɨɜɨɪɧɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨ� ɱɬɨ ɜɫɟ ɢɝɪɨɤɢ ɦɨɝɭɬ ɩɨɥɭɱɢɬɶ ɜɦɟɫɬɟ

Ɍɨɱɤɚ ɧɟɫɨɝɥɚɫɢɹ ɬɨ� ɱɬɨ ɢɝɪɨɤɢ ɦɨɝɭɬ ɩɨɥɭɱɢɬɶ ɩɨ ɨɬɞɟɥɶɧɨɫɬɢ�

ɇɚɞɨ ɭɱɢɬɵɜɚɬɶ ɢ ´ɩɪɨɦɟɠɭɬɨɱɧɵɟ´ ɜɚɪɢɚɧɬɵ � ɤɨɚɥɢɰɢɢ ɢɝɪɨɤɨɜ�



Ɋɚɫɫɦɨɬɪɢɦ ɛɟɫɤɨɚɥɢɰɢɨɧɧɭɸ ɢɝɪɭ

! = '1, {;L}L"1, {.L}L"1(.

ɂɝɪɨɤɢ$ ɤɨɚɥɢɰɢɹ$ ɜɵɢɝɪɵɲ ɤɨɚɥɢɰɢɢ�

ȿɫɥɢ ɢɝɪɨɤɢ ɤɨɚɥɢɰɢɢ ɦɨɝɭɬ ɩɟɪɟɪɚɫɩɪɟɞɟɥɹɬɶ ɜɵɢɝɪɵɲ ɦɟɠɞɭ ɫɨɛɨɣ� ɬɨ ɷɬɨɢɝɪɚ ɫ ɬɪɚɧɫɮɟɪɚɛɟɥɶɧɵɦɢ ɩɨɥɟɡɧɨɫɬɹɦɢ �78 JDPH�� ɢɧɚɱɟ � ɫɧɟɬɪɚɧɫɮɟɪɚɛɟɥɶɧɵɦɢ �178 JDPH��




! = '1, {;L}L"1, {.L}L"1(.






! = '1, {;L}L"1, {.L}L"1(.





Ɋɚɫɫɦɨɬɪɢɦ ɢɝɪɭ ! ɢ ɤɨɚɥɢɰɢɸ 6� Ɇɧɨɠɟɫɬɜɨ ɟɟ ɫɬɪɚɬɟɝɢɣ ɹɜɥɹɟɬɫɹ ɦɧɨɠɟɫɬɜɨ;6 =

"L"6 ;L.

Ɋɚɫɫɦɨɬɪɢɦ ɚɧɬɚɝɨɧɢɫɬɢɱɟɫɤɭɸ ɢɝɪɭ !6�

!6 = ';6, ;1\6,#

L"6.L([)(.

ȼ ɢɝɪɟ !6 ɢɦɟɸɬɫɹ ɞɜɚ ɢɝɪɨɤɚ � ɤɨɚɥɢɰɢɹ 6 ɢ ɟɟ ɞɨɩɨɥɧɟɧɢɟ 1 \ 6. Ɏɭɧɤɰɢɹɜɵɢɝɪɵɲɚ ɢɝɪɨɤɚ 6 ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɜɵɢɝɪɵɲɟɣ ɜɫɟɯ ɟɟ ɱɥɟɧɨɜ�Ɋɚɫɫɦɨɬɪɢɦ ɨɫɬɨɪɨɠɧɭɸ ɫɬɪɚɬɟɝɢɸ ɢɝɪɨɤɚ 6�

Y!(6) = PD[[6";6

PLQ[1\6";1\6

#

L"6.L([6, [1\6).

ȼɟɤɬɨɪ ɡɧɚɱɟɧɢɣ {Y!(6)}6#1 ɧɚɡɵɜɚɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɢɝɪɵ !.




"L"6 ;L.


!6 = ';6, ;1\6,#

L"6.L([)(.

ȼ ɢɝɪɟ !6 ɢɦɟɸɬɫɹ ɞɜɚ ɢɝɪɨɤɚ � ɤɨɚɥɢɰɢɹ 6 ɢ ɟɟ ɞɨɩɨɥɧɟɧɢɟ 1 \ 6. Ɏɭɧɤɰɢɹɜɵɢɝɪɵɲɚ ɢɝɪɨɤɚ 6 ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɜɵɢɝɪɵɲɟɣ ɜɫɟɯ ɟɟ ɱɥɟɧɨɜ�

Ɋɚɫɫɦɨɬɪɢɦ ɨɫɬɨɪɨɠɧɭɸ ɫɬɪɚɬɟɝɢɸ ɢɝɪɨɤɚ 6�

Y!(6) = PD[[6";6

PLQ[1\6";1\6

#

L"6.L([6, [1\6).





"L"6 ;L.


!6 = ';6, ;1\6,#

L"6.L([)(.


Y!(6) = PD[[6";6

PLQ[1\6";1\6

#

L"6.L([6, [1\6).





"L"6 ;L.


!6 = ';6, ;1\6,#

L"6.L([)(.


Y!(6) = PD[[6";6

PLQ[1\6";1\6

#

L"6.L([6, [1\6).



ɋɜɨɣɫɬɜɚ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ

�� Y!()) = 0 ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ�

�� ɋɭɩɟɪɚɞɞɢɬɢɜɧɨɫɬɶ� Y!(. * /) # Y!(.) + Y!(/) ɞɥɹ ɥɸɛɵɯ ., / ! 1�. + / = )�

�� Ⱦɥɹ ɢɝɪ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɭɦɦɨɣ �$

L"1 .L([) = F ,[ " ;�Y!(6) + Y!(1 \ 6) = Y!(1) = F ɞɥɹ ɜɫɟɯ 6 ! 1;

�� ɂɧɜɚɪɢɚɧɬɧɨɫɬɶ ɨɬɧɨɫɢɬɟɥɶɧɨ ɚɮɮɢɧɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ� ɟɫɥɢ ɢɝɪɵ ! ɢ!! ɨɬɥɢɱɚɸɬɫɹ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɬɨɥɶɤɨ ɮɭɧɤɰɢɹɦɢ ɜɵɢɝɪɵɲɟɣ�

.!L ([) = D · .L([) + EL, D > 0,

ɬɨ ɞɥɹ ɥɸɛɨɣ ɤɨɚɥɢɰɢɢ 6 Y!!(6) = DY!(6) +$

L"6 EL;

�� Ⱦɥɹ ɥɸɛɨɝɨ ɢɡɨɦɨɪɮɢɡɦɚ # : ! $ !!

Y!!(#6) = Y!(6).



Ʉɨɨɩɟɪɚɬɢɜɧɨɣ ɢɝɪɨɣ ɫ ɬɪɚɧɫɮɟɪɚɛɟɥɶɧɵɦɢ ɩɨɥɟɡɧɨɫɬɹɦɢ ɢɥɢ ɢɝɪɨɣ ɜ ɮɨɪɦɟɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ ɧɚɡɵɜɚɟɬɫɹ ɩɚɪɚ (1, Y), ɝɞɟ 1& ɦɧɨɠɟɫɬɜɨ ɢɝɪɨɤɨɜ�Y : 21 $ R& ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ�

Ɇɵ ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɛɟɫɤɨɚɥɢɰɢɨɧɧɭɸ ɢɝɪɭ� ɧɚ ɨɫɧɨɜɟ ɤɨɬɨɪɨɣ ɩɨɫɬɪɨɟɧɚɞɚɧɧɚɹ ɤɨɨɩɟɪɚɬɢɜɧɚɹ ɢɝɪɚ�


ɉɪɢɦɟɪ� ɚɞɞɢɬɢɜɧɚɹ ɢɝɪɚ

ȿɫɬɶ Q ɢɝɪɨɤɨɜ ɫ ɜɟɫɚɦɢ D1, D2, ..., DQ > 0� Ɋɚɫɫɦɨɬɪɢɦ ɢɝɪɭ (1, Y)� ɝɞɟ

1 = {1, 2, ..., Q},

Y(6) =#

L"6DL.


ɉɪɢɦɟɪ� ɝɨɥɨɫɨɜɚɧɢɹ

ȿɫɬɶ Q ɢɝɪɨɤɨɜ ɫ ɜɟɫɚɦɢ �ɤɨɥɢɱɟɫɬɜɚ ɝɨɥɨɫɨɜ� D1, D2, ..., DQ > 0� Ɂɚɤɨɧɩɪɢɧɢɦɚɟɬɫɹ� ɟɫɥɢ ɡɚ ɧɟɝɨ ɨɬɞɚɧɨ ɤɚɤ ɦɢɧɢɦɭɦ T > 0 ɝɨɥɨɫɨɜ�Ɉɩɪɟɞɟɥɹɟɦ ɤɨɨɩɟɪɚɬɢɜɧɭɸ ɢɝɪɭ (1, Y) ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ�

1 = {1, 2, ..., Q},

Y(6) =

%1 ɟɫɥɢ

$L"6

DL # T

0 ɢɧɚɱɟ

ɗɬɨ ɜɡɜɟɲɟɧɧɚɹ ɦɚɠɨɪɢɬɚɪɧɚɹ ɢɝɪɚ�

ɂɝɪɨɤ L " 1 ɹɜɥɹɬɫɹ ɛɨɥɜɚɧɨɦ� ɟɫɥɢ ɞɥɹ ɥɸɛɨɣ ɤɨɚɥɢɰɢɢ 6 ɜɵɩɨɥɧɟɧɨY(6) = Y(6 * {L})�


ɉɪɢɦɟɪ� ɝɨɥɨɫɨɜɚɧɢɹ

ȿɫɬɶ Q ɢɝɪɨɤɨɜ ɫ ɜɟɫɚɦɢ �ɤɨɥɢɱɟɫɬɜɚ ɝɨɥɨɫɨɜ� D1, D2, ..., DQ > 0� Ɂɚɤɨɧɩɪɢɧɢɦɚɟɬɫɹ� ɟɫɥɢ ɡɚ ɧɟɝɨ ɨɬɞɚɧɨ ɤɚɤ ɦɢɧɢɦɭɦ T > 0 ɝɨɥɨɫɨɜ�Ɉɩɪɟɞɟɥɹɟɦ ɤɨɨɩɟɪɚɬɢɜɧɭɸ ɢɝɪɭ (1, Y) ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ�

1 = {1, 2, ..., Q},

Y(6) =

%1 ɟɫɥɢ

$L"6

DL # T

0 ɢɧɚɱɟ

ɗɬɨ ɜɡɜɟɲɟɧɧɚɹ ɦɚɠɨɪɢɬɚɪɧɚɹ ɢɝɪɚ�

ɂɝɪɨɤ L " 1 ɹɜɥɹɬɫɹ ɛɨɥɜɚɧɨɦ� ɟɫɥɢ ɞɥɹ ɥɸɛɨɣ ɤɨɚɥɢɰɢɢ 6 ɜɵɩɨɥɧɟɧɨY(6) = Y(6 * {L})�


ɉɪɢɦɟɪ� ɩɪɨɫɬɚɹ ɢɝɪɚ

ɂɝɪɚ (1, Y) ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɫɬɨɣ� ɟɫɥɢ�� Ⱦɥɹ ɥɸɛɨɝɨ 6 ! 1 Y(6) " {0, 1}�� Y(1) = 1�� ȿɫɥɢ Y(6) = 1 ɢ 6 ! 7� ɬɨ Y(7) = 1�

ȼɟɪɧɨ ɥɢ� ɱɬɨ ɥɸɛɚɹ ɩɪɨɫɬɚɹ ɢɝɪɚ ɹɜɥɹɟɬɫɹ ɜɡɜɟɲɟɧɧɨ ɦɚɠɨɪɢɬɚɪɧɨɣ"


ɉɪɢɦɟɪ� ɢɝɪɚ ɛɚɧɤɪɨɬɫɬɜɚ

ɉɭɫɬɶ ɟɫɬɶ ɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɢɝɪɨɤɨɜ 1 ɫ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɜɟɫɚɦɢ {DL}L"1�Ɋɚɫɫɦɨɬɪɢɦ ɬɚɤɠɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ $ <

$L"1

DL� Ɍɪɨɣɤɚ (1, {DL}, $)ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɟɣ ɛɚɧɤɪɨɬɫɬɜɚ�

ɇɚ ɟɟ ɨɫɧɨɜɟ ɟɫɬɟɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɨɩɟɪɚɬɢɜɧɚɹ ɢɝɪɚ (1, Y)�

Y(6) = PD[{0,#

L"1\6

DL}.


ɉɪɢɦɟɪ� ɢɝɪɚ ɛɚɧɤɪɨɬɫɬɜɚ

ɉɭɫɬɶ ɟɫɬɶ ɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɢɝɪɨɤɨɜ 1 ɫ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɜɟɫɚɦɢ {DL}L"1�Ɋɚɫɫɦɨɬɪɢɦ ɬɚɤɠɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ $ <

$L"1

DL� Ɍɪɨɣɤɚ (1, {DL}, $)ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɟɣ ɛɚɧɤɪɨɬɫɬɜɚ�

ɇɚ ɟɟ ɨɫɧɨɜɟ ɟɫɬɟɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɨɩɟɪɚɬɢɜɧɚɹ ɢɝɪɚ (1, Y)�

Y(6) = PD[{0,#

L"1\6

DL}.


ɉɪɢɦɟɪ� ɚɞɞɢɬɢɜɧɚɹ ɢɝɪɚ ɫ ɜɟɬɨ�ɢɝɪɨɤɨɦ

ȿɫɬɶ Q ɢɝɪɨɤɨɜ ɫ ɜɟɫɚɦɢ D1, D2, ..., DQ > 0� Ɋɚɫɫɦɨɬɪɢɦ ɢɝɪɭ (1, Y)� ɝɞɟ

1 = {1, 2, ..., Q},

Y(6) =

% $L"6

DL ɟɫɥɢ 1 " $

0 ɢɧɚɱɟ


ɉɪɢɦɟɪ� ɢɝɪɚ ɫ ɢɟɪɚɪɯɢɱɟɫɤɨɣ ɫɬɪɭɤɬɭɪɨɣ

ȿɫɬɶ Q ɢɝɪɨɤɨɜ ɫ ɜɟɫɚɦɢ D1, D2, ..., DQ > 0� ɂɝɪɨɤɢ ɹɜɥɹɸɬɫɹ ɜɟɪɲɢɧɚɦɢɨɪɢɟɧɬɢɪɨɜɚɧɧɨɝɨ ɞɟɪɟɜɚ ɫ ɨɞɧɨɣ ɜɟɪɲɢɧɨɣ ɛɟɡ ɩɪɟɞɲɟɫɬɜɟɧɧɢɤɨɜ�Ɋɚɫɫɦɨɬɪɢɦ ɢɝɪɭ (1, Y)� ɝɞɟ

1 = {1, 2, ..., Q},

Y(6) =#

L"!(6)

DL

ɝɞɟ $(6) � ɦɚɤɫɢɦɚɥɶɧɨɟ ɞɨɩɭɫɬɢɦɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ 6�


Algorithms for computing the nucleolus of disjunctive games with permission structure

Restricted games

Peer-group game

Peer-group game. Example.

Coalition ABC is feasible.Coalition ABFE is not feasible.r(ABC)=v(ABC)=9.v(ABFE)=9,r(ABFE)=v(ABF)=7.


Restricted games

Peer-group game


Coalition ABC is feasible.

Coalition ABFE is not feasible.r(ABC)=v(ABC)=9.v(ABFE)=9,r(ABFE)=v(ABF)=7.


Restricted games

Peer-group game


Coalition ABC is feasible.Coalition ABFE is not feasible.

r(ABC)=v(ABC)=9.v(ABFE)=9,r(ABFE)=v(ABF)=7.


Restricted games

Peer-group game


Coalition ABC is feasible.Coalition ABFE is not feasible.r(ABC)=v(ABC)=9.

v(ABFE)=9,r(ABFE)=v(ABF)=7.


Restricted games

Peer-group game


Coalition ABC is feasible.Coalition ABFE is not feasible.r(ABC)=v(ABC)=9.v(ABFE)=9,

r(ABFE)=v(ABF)=7.


Restricted games

Peer-group game


Coalition ABC is feasible.Coalition ABFE is not feasible.r(ABC)=v(ABC)=9.v(ABFE)=9,r(ABFE)=v(ABF)=7.


Restricted games

Game with permission structure

Game with permission structure. Disjunctive approach.

Coalition ABCDG is feasible.Coalition ACDFG is not feasible.r(ABCDG)=v(ABCDG)=17.v(ACDFG)=18,r(ACDFG)=v(ACF)=9.


Restricted games



Coalition ABCDG is feasible.

Coalition ACDFG is not feasible.r(ABCDG)=v(ABCDG)=17.v(ACDFG)=18,r(ACDFG)=v(ACF)=9.


Restricted games



Coalition ABCDG is feasible.Coalition ACDFG is not feasible.

r(ABCDG)=v(ABCDG)=17.v(ACDFG)=18,r(ACDFG)=v(ACF)=9.


Restricted games



Coalition ABCDG is feasible.Coalition ACDFG is not feasible.r(ABCDG)=v(ABCDG)=17.

v(ACDFG)=18,r(ACDFG)=v(ACF)=9.


Restricted games



Coalition ABCDG is feasible.Coalition ACDFG is not feasible.r(ABCDG)=v(ABCDG)=17.v(ACDFG)=18,

r(ACDFG)=v(ACF)=9.


Restricted games



Coalition ABCDG is feasible.Coalition ACDFG is not feasible.r(ABCDG)=v(ABCDG)=17.v(ACDFG)=18,r(ACDFG)=v(ACF)=9.


Restricted games


Game with permission structure. Conjunctive approach.

Coalition ABCDE is feasible.Coalition ACDFG is not feasible.r(ABCDE)=v(ABCDE)=12.v(ABCDG)=17,r(ABCDG)=v(ABCD)=11.


Restricted games



Coalition ABCDE is feasible.

Coalition ACDFG is not feasible.r(ABCDE)=v(ABCDE)=12.v(ABCDG)=17,r(ABCDG)=v(ABCD)=11.


Restricted games



Coalition ABCDE is feasible.Coalition ACDFG is not feasible.

r(ABCDE)=v(ABCDE)=12.v(ABCDG)=17,r(ABCDG)=v(ABCD)=11.


Restricted games



Coalition ABCDE is feasible.Coalition ACDFG is not feasible.r(ABCDE)=v(ABCDE)=12.

v(ABCDG)=17,r(ABCDG)=v(ABCD)=11.


Restricted games



Coalition ABCDE is feasible.Coalition ACDFG is not feasible.r(ABCDE)=v(ABCDE)=12.v(ABCDG)=17,

r(ABCDG)=v(ABCD)=11.


Restricted games



Coalition ABCDE is feasible.Coalition ACDFG is not feasible.r(ABCDE)=v(ABCDE)=12.v(ABCDG)=17,r(ABCDG)=v(ABCD)=11.


Restricted games


Case of multiple top-players, disjunctive approach

Coalition ABCEFI is feasible.Coalition ABCEIJ is not feasible.r(ABCEFI)=v(ABCEFI)=19.v(ABCEIJ)=19,r(ABCEIJ)=v(ABCE)=10.


Restricted games



Coalition ABCEFI is feasible.

Coalition ABCEIJ is not feasible.r(ABCEFI)=v(ABCEFI)=19.v(ABCEIJ)=19,r(ABCEIJ)=v(ABCE)=10.


Restricted games



Coalition ABCEFI is feasible.Coalition ABCEIJ is not feasible.

r(ABCEFI)=v(ABCEFI)=19.v(ABCEIJ)=19,r(ABCEIJ)=v(ABCE)=10.


Restricted games



Coalition ABCEFI is feasible.Coalition ABCEIJ is not feasible.r(ABCEFI)=v(ABCEFI)=19.

v(ABCEIJ)=19,r(ABCEIJ)=v(ABCE)=10.


Restricted games



Coalition ABCEFI is feasible.Coalition ABCEIJ is not feasible.r(ABCEFI)=v(ABCEFI)=19.v(ABCEIJ)=19,

r(ABCEIJ)=v(ABCE)=10.


Restricted games



Coalition ABCEFI is feasible.Coalition ABCEIJ is not feasible.r(ABCEFI)=v(ABCEFI)=19.v(ABCEIJ)=19,r(ABCEIJ)=v(ABCE)=10.


Restricted games


Case of multiple top-players, conjunctive approach

Coalition ABCEF is feasible.Coalition ABCEFI is not feasible.r(ABCEF)=v(ABCEF)=13.v(ABCEFI)=19,r(ABCEFI)=v(ABCEF)=13.


Restricted games



Coalition ABCEF is feasible.

Coalition ABCEFI is not feasible.r(ABCEF)=v(ABCEF)=13.v(ABCEFI)=19,r(ABCEFI)=v(ABCEF)=13.


Restricted games



Coalition ABCEF is feasible.Coalition ABCEFI is not feasible.

r(ABCEF)=v(ABCEF)=13.v(ABCEFI)=19,r(ABCEFI)=v(ABCEF)=13.


Restricted games



Coalition ABCEF is feasible.Coalition ABCEFI is not feasible.r(ABCEF)=v(ABCEF)=13.

v(ABCEFI)=19,r(ABCEFI)=v(ABCEF)=13.


Restricted games



Coalition ABCEF is feasible.Coalition ABCEFI is not feasible.r(ABCEF)=v(ABCEF)=13.v(ABCEFI)=19,

r(ABCEFI)=v(ABCEF)=13.


Restricted games



Coalition ABCEF is feasible.Coalition ABCEFI is not feasible.r(ABCEF)=v(ABCEF)=13.v(ABCEFI)=19,r(ABCEFI)=v(ABCEF)=13.

ɉɪɢɦɟɪ� DVVLJQPHQW PDUNHW

Ɋɚɫɫɦɨɬɪɢɦ ɞɜɭɞɨɥɶɧɵɣ ɝɪɚɮ ɫ ɦɧɨɠɟɫɬɜɚɦɢ ɜɟɪɲɢɧ 11,12� Ⱦɥɹ 6 ! 11 * 12

ɨɩɪɟɞɟɥɢɦ Y(6) ɤɚɤ ɦɚɤɫɢɦɭɦ ɬɨɝɨ� ɱɬɨ ɦɨɠɟɬ ɩɨɥɭɱɢɬɶ ɩɨɞɝɪɚɮ ɧɚ ɦɧɨɠɟɫɬɜɟɜɟɪɲɢɧ 6�


Ⱦɨɦɚɲɧɟɟ ɡɚɞɚɧɢɟ

�� Ⱦɨɤɚɠɢɬɟ� ɱɬɨ ɚɪɛɢɬɪɚɠɧɨɟ ɪɟɲɟɧɢɟ ɇɷɲɚ ɧɟ ɹɜɥɹɟɬɫɹ ɦɨɧɨɬɨɧɧɵɦ �ɬɨ ɟɫɬɶɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɚɤɫɢɨɦɟ ��



�� Ɋɚɫɫɦɨɬɪɢɦ ɭɫɢɥɟɧɧɭɸ ɚɤɫɢɨɦɭ ɦɨɧɨɬɨɧɧɨɫɬɢ �ɨɬɛɪɚɫɵɜɚɟɦ ɭɫɥɨɜɢɟ,(;1, G) = ,(;2, G)��

�� Ɉɝɪɚɧɢɱɟɧɧɚɹ ɦɨɧɨɬɨɧɧɨɫɬɶ� ɉɭɫɬɶ ɞɜɟ Ⱥɋ %1 = (;1, G), %2 = (;2, G),ɬɚɤɨɜɵ� ɱɬɨ ;1 ! ;2� Ɍɨɝɞɚ

!(;1, G) % !(;2, G)

Ⱦɨɤɚɠɢɬɟ� ɱɬɨ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɚɪɛɢɬɪɚɠɧɨɝɨ ɪɟɲɟɧɢɹ� ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨɚɤɫɢɨɦɚɦ �� ɢ ɧɨɜɨɣ ɚɤɫɢɨɦɟ ɦɨɧɨɬɨɧɧɨɫɬɢ ��



��ɚ� ȼɟɪɧɨ ɥɢ� ɱɬɨ ɥɸɛɚɹ ɢɝɪɚ ɛɚɧɤɪɨɬɫɬɜɚ ɹɜɥɹɟɬɫɹ ɫɭɩɟɪɚɞɞɢɬɢɜɧɨɣ"ɛ� ȼɟɪɧɨ ɥɢ� ɱɬɨ ɜ ɢɝɪɟ ɛɚɧɤɪɨɬɫɬɜɚ ɢɝɪɨɤ ɹɜɥɹɟɬɫɹ ɛɨɥɜɚɧɨɦ ɜ ɬɨɦ ɢ ɬɨɥɶɤɨ ɜɬɨɦ ɫɥɭɱɚɟ� ɟɫɥɢ ɟɝɨ ɜɟɫ ɪɚɜɟɧ �"



�� ȼɟɪɧɨ ɥɢ� ɱɬɨ ɥɸɛɚɹ ɩɪɨɫɬɚɹ �ɤɨɨɩɟɪɚɬɢɜɧɚɹ� ɢɝɪɚ ɹɜɥɹɟɬɫɹ ɜɡɜɟɲɟɧɧɨɦɚɠɨɪɢɬɚɪɧɨɣ"


Economy & Finance

2013 04 14_game_theory_lecture_07