KL mMoGgAhGaZ - Brown Universitycs.brown.edu/people/amy/papers/cuny.pdf · 2004. 1. 7. · x9 n b a...
Transcript of KL mMoGgAhGaZ - Brown Universitycs.brown.edu/people/amy/papers/cuny.pdf · 2004. 1. 7. · x9 n b a...
� � ����� ��� � � �� � � ��� ��� �
� � ��� � � ���� �� � � ��� ��� �
� � � � ����� ��! "$#&%
'�(*),+- "/.102% � 3 �4%$�65
7 �80:9<; = ">#?#
@ ACBED F GHAJILKHMONPACBED QHR S$K�T U6QVMXW
Y Q[Z\A^] _[Ga` b _dc KeN fKgc AhGi_jM
k KHlmMoni_jMpD q[rstqVuduwv
� " �X9 �
� n�� B A _aZdKHG B � � Y KL_mMXGHAhGaZ
� ��� ������� ) ������� ��������� ( �� ! " #�%$����� &�(' )*��+ , ( � ( " �&�-�/.10 ) ��243 ( � (65 � ( 3&+ ,7#� ( " ( �8'
� 9:�%' + ��� ;=<>3 �?��� @ ���A�-+ �?�CBD�E&E?F1G! " #�%$����� &�(' )*��+ , ( � ( " �&�-�/.10HB 5 3 ) �I ) ��� " �8'J'K��� ( �/. )
�%243 ( � (L5 � ( 3M+ ,7#� ( " ( �('
� N ( � ( + �8O%��� ;P� ( ) ) + �?��BQD�E&E�R?G! " #�%$����� &�(' )S�?�1�&�T. )E( " �?�A�T.�0 ) �%243 ( � (L5 � ( 3&+ ,?#� ( " ( �('( � " #��' ) �?� ) �*'83&+ U�?+ �('
V W V ZdKHGa`V_ Y KL_jMXG � �YX _��&nLKeN
Z
� " �X9 �K�
V ���jM Q��dAhc _ B K � � Y KL_mMoGgAhGaZ
� 9�1� � �� 4� � ) �>� � �I��� � ( �� ! 9� � O ) �����1�?� � �� 4� � ) � �I��� � ( �� � " #��$����� &�(' ) + ( � ( + �8O ' ) � � ) �% ( �('( � " ��' ) �&� ) � '(3&+ ��?+ �('
! 9� ( � ) �����1�?� ���% 4��� ) � ������� ( �� � " #��$����� &�(' ) " ���� � � � ) �%� �%2#3 ( � (65 � ( 3&+( � &� ����� �?� � '(3&+ ��?+ �('
b � V ZdKHGa`V_ Y KL_jMXG ��� ngA � AhlmM A _
� " � 5�3 . - ���80��V0 3 � � �,3 �������m��� � � - � ��
� " ( ' ( #� � � " �('J'
� � ( ' � ' � ) � ' ) � ) �8' )���� � 0
� � ( ' � ' � ) � � " ) ( ��' )�� � � 0
� � ��� � � � � ( ' � � � ' ��� � �3M� " ) ( #�
� � ;����! "$#%� �'& � �'&)()()()& �+* & ��* G ( ' � ,�� 5 � 5 ( � ( ' ) ( " ) � �&��' ( )E( #� �3M� " ) ( #� ) + � ) �&�(' " � (65 �8' ) � �?��' (*) ( #��' 5 � )�' �%��� ' ) � ) �('IB" #��� (*) ( ���%� #� ,1�%' ) ' ) � ) �(' �?��� � " ) ( ��'
N � , � " ( ' ( #� � � " �('J' - N ����. $ � � ,?��� ) . �� ;����! "/#%� � & � � &)()()()& �+* & �0* G�, � ;��1�2 3"4#%� � & � � G
576 B 5 � * &)()()()& � � � � B 5 � * &)()()(8& � � � �
9
� � #?#?� "$� � � � � �8" 9d0 3 � �
��� ) � & ��0�, � ) � & ��0 - ��
� ;���� # � & �%G� �� ) ���L0
�� )��K0�, + �8O�����������
��� )�� & ��0
� ">#��� � 9L� � "�9d0 3 �
� � �"!$# %& #(' � &)%+* 9 ) N � & ��0% �&,&3 ) ' � ( ' " #3&� ) �� " ) ��,�* 3 ) ,&3 ) #, )E( + �?�U' ) � ) �Y� $��?�A3�� �3&� " )E( #� �
#, )E( + �?�#� " )E( #�8�*$1�?�A3�� 3&� " )E( #� � �% � ( )E( �?� (.- � , � , /'0#��1# � &
�� �?�A� � � � U� �?�-� � � �
� )�� & ��0�, � ) � & ��0 - �32 � � ;�� � #%� & �HG� ) � � 0 ) � 0 , + �8O � � )�� & ��0
4 * '0# � #('
5
� � � " � ��0 ���
� �"# � ' 9 % 9�� ) N � & � &��& 0% �M,&3 ) ' � ( ' " #3&� ) � " ) �� �
� � ) � � �%$���� �I ( �� �� � ) � � � OM,&� �� � )E( #�
* 3 ) ,&3 ) #, )E( + �&��' ) � ) �Y�*$1�?�A3�� 3&� " )E( #� �#, )E( + �&�4� " ) ( �8� $��?�-3�� �3&� " )E( #� � �
% � (*) ( �?� (.- � , � , /'0#��1# � &
( � (*) ( �&� (.- � � & �@ < % �"# � ( ' �I#�1� 5 ' �� 5 ( �� *
' ( + 3M� � ) � � " ) ( � � ( � ' ) � ) � � 5 ' � � $�� ��� ' ����� � �?��� ��� O ) ' ) � ) � � �" #+ ,&3 ) � ) � � 0 , + �8O � ��������� � )�� & �103&,?��� ) � � )�� & �10 , ) D�� � 0 � ) � & ��0 - � ;�� - � )�� � 0�G
" + 47' � � " ) ( � � � ) #�H� �� � � ,?#� ( " .�0��, � � B � , � ��&� " � . �
4 * '0# � #('
� �LK Q M Kgc ��� _ B�WaAhG N s����gr����� � �JKa_jMXGHA GLZ ] Q/G IaKHM ZwK N B Q � � _[GL` � �
�
� " � 5�3 . � "$� ���
� ) " + �%' ) ( " � �?+ �
� � ( ' � ' � ) � �� ,&��� .8� � ' )�� � �%0
� � ( ' � ' � ) � ' ) � ) �8' )���� � 0
� ��� )��K0 ( ' ) + � � ),+ ,M� � .H���� ' ' � ) � � " ) ( #��' � ) ' ) � ) � �� � ) � )��K0�, � " )��K0 � ()()( � ���)��K0 )��� � � ) � 0 0
� � ;�� �! " #%� � & �� � &)()()()& � * & �� * G ( ' � ,�� 5 � 5 ( � ( ' ) ( " ) � �&��' ( )E( #� �3M� " ) ( #� ) + � ) �&�(' " � (65 �8' ) � �?��' (*) ( #��' 5 � )�' �%��� ' ) � ) �('IB" #��� (*) ( ���%� #� ,1�%' ) ' ) � ) �(' �?��� � " ) ( ��'
� � � )�� & ���0 ( ' ),+ � � ) + ,&��� .8� � � ' � � ' ��� � � ) ' ) � ) � � ��� " ) ( � $�� " ) �����
N ����.K $ � �?+ � , � ) " + ��' )E( " � �&+ � - N ��� . $ �:� #,?� � ) . �� ;����! "/#%� � & �� � &)()()()& �+* & ��0* G�, � ;��1�2 3"4#%� � & �� � G
576 B 5 � * &)()()()& � � � � B 5 �� * &)()()(8& �� � � �
�
� � #?#?� "$� � � � �8"$# 3�� �4�
� �� ) � & ���0�, � ��) � & ���0 - ��
� ;�� � #%� & ��8G� �� ) � � 0
k Q\KM�H� �" ) � 0 , + �HO ��� ��� � � ��� + ( � ��������� ��� � � " ) � & " & ���K0 , � �� ) � 0k M A^KHGa` �H� �� ) � 0 , + �HO �� ����� ��� � �� )�� & ���0S _ N � �H� �� ) � 0 � 9 �%' + � ) � � " ) � 0 &1()()( & � � ) � 0 0@ � �H� �� ) � 0 � � # � )*� � " )��K0 &�()()()& � � )��K0 0
�
� �>#^9d0 " �>� ��9 � � � " � ��0 ���
N ! � &)% � ) N � �?+ � & � &��& 0'0#��1# � &
( � (*) ( �&� (.- � � & � " &)()()()& � @ < % �"# � ( ' �I#�1� 5 ' �� 5 ( �� *
' ( + 3M� � ) � � " ) ( ��' � " &)()()()& � ( � ' ) � ) � � 5 ' � � $�� ��� ' ����� ' � " &)()()()& � �&��� ��� O ) ' ) � ) � � � U� �?�-� � � �
" + ,M3 ) � � )�� � 03&,?�U� ) � � ��) � & � " &)()()()& � 0
) ' ( + 3&� ) �?���%#3�'(�/.10 " + �7' � � " )E( #��' � � " &)()(8()& � ���, � � B � " , � � " &)()()()& � , � ��&� " � . �
4 * '0# � #('
9:�%' + ��� " #�%$����� &�(' )*��+ , ( � ( " �&�-�/.10�I ) ��� " �('J' ��� ( �T. ) ��243 ( � (65 � ( 3&+ ,7#� ( " ( �('
4 4 � � " #�%$����� &�(' )S�?�1�&�T. )E( " �?�A�T.�0) �%243 ( � (L5 � ( 3M+ ,7 � ( " ( �(' ( � ���(' ) � ( " ) ��� " � ��'J' �(' � ��&+ �8'
� # ��� " �%$�� � &�8' )*� + , ( � ( " �?�A�T.�0) �%243 ( � (L5 � ( 3M+ ,7 � ( " ( �('
�
;/� � � �
� ���%' ( �/. " #+ ,&3 ) � 5 � � $ ( � � ( ������� ,1� M 4� �?+ + ( �� B3&�&� ( . � 9:�%' + �%2#3 ( � (65 � ( 3&+
� ,&��� .8� � ' " �?� � " + ( �($�� ,1� .8 � ' #3 ) ' ( �&� ),+ � " #��$�� O + 3&�A�� 9:��' + ,�� .( � ' ; � 3M+ �&�&�CB�� R7G
� ,&��� .8� � ' � �I��� � " �� ������� ) ��� ��243 ( � (65 � ( 3&+ $ ( � �I1� ���% 4��� )�?� ?�� ( ) + + ' ; 4 7' ) � ��� � + � � B EME�G
� " #��' ( ' ) ��� ) '�( ) + ),+ � 3�'(31�?� � % $ ( � ' � ( ��� ( $ ( �431�?�A�T.� � ) ( �1�?� 5 � + �H$ ( U�
;/� � � � �� " � ; 3 � � � � �
� �%243 ( � (L5 � ( 3&+ ' � � � " ) ( #� ,�� 5 � ��+
� 3 � � � #h" 9L��% � � ��0 # 0�� � 0 �>�
@ �gA^]�W KHG� �
� � s � q s��� � s q u/shu
@ �� �
� �Hq �� �� � � u
+ �8O D���������- E������ - E������ - /������
'(3 5�� � " ) ) ,�� 5 � 5 ( � ( ) . " #��' ) � � ( � ) '� ��� - � ��� - � ��� - � ��� , D
� ��� & � ��� & � ��� & � ��� � /
� ( ��� ( $ ( �43��?� � � )E( #�1�?� (*) . " #��' ) � � ( � ) '� ����� - ������� � ������� - /��������� ��� - /�� ��� � � � � � - ��� ���� � ��� - ��� ��� � ��� ��� - /�� ���������� - /��!��� � � ����� - ���!���
Z#"
� # ��)*� " )��K0 &)()()()& � )��K0 0 , � 2 �� ��� � ) ���0 � ��)�� & ���0 # � 'K� ) ( '�� �(' # 2��DMB�� B�� B U� R��
! )E( � (*) ��� ( �?� + �8O ( + ( - � ) + � '83&+ � ��� ' ����� '
��� ����������������� � ! �#" $% �'&)(+*,�-'. ! (0/21+*,3- ( Z -
# ��&� ( ) ��� ( �&� + �8O ( + (.- � ) + � + ( � ( + 3&+ � � ' ��� �
� ������4�5�6��7��� � �98;:! �#" $% �#&)(+*,�-'. ! (0/<1<*,3- ( -
'� ,&3 5 � ( " �?� + �8O ( + (.- � ),+ � + �8O ( + 3&+ ��� ' �����
� ����=���5�6��7���>� �5�6�! �#" $% �#&)(?*,�-'. ! (0/<1<*,3- ( 9 -
� (65 � � ) ��� ( �?� � + �8O ( + ( - �(' #�&�/. � � ' ��� ' ����� '� � ) � , @ �
� '>+ � � �
! ������A�5�6��7���>� $% �#&)(+*,�-'. ! (0/<1<*,3- ( 5 -
Z Z
� � 02% � "$� ���
b b � b b q b b v
A B
A
A B A B
B A,B A,B
0 1 2 3 4 5 6 7 8 9 10
x 105
0
0.5
1
1.5
2
2.5
3Grid Game 1
Simulation Iteration
Mea
n Q
−va
lue
Diff
eren
ce
Q Foe−Q Friend−Q Correlated−Q
0 1 2 3 4 5 6 7 8 9 10
x 105
0
0.5
1
1.5
2
2.5
3Grid Game 2
Simulation Iteration
Mea
n Q
−va
lue
Diff
eren
ce
Q Foe−Q Friend−Q Correlated−Q
0 1 2 3 4 5 6 7 8 9 10
x 105
0
0.5
1
1.5
2
2.5
3Grid Game 3
Simulation Iteration
Mea
n Q
−va
lue
Diff
eren
ce
Q Foe−Q Friend−Q Correlated−Q
Z
��� � �� �� � � � � � � � � � � �
��� �� ��� � �� � �� � �� � ��
� � � !� �" # � $ % !� � ��� � �� $ % !� � ��� � �� $ % !� � ��� � � �
& �' ' ( �' ' �) ' ' * + ( � ' ' � � � � �' ' ( � �) � � � �
, ! � - & ' (' ' ./ ( .0 � ' ' � � �' ( � �' � � � �
,� � �1 � - & 2 �' 354 2 �' 3 ' 2 �' 34 2 �' 3 ' 2 �' 354 2� ' 3 '
6 78 - & �' ' ( �' ' �) ' ' ) ' ( � ' ' � � � � � � . ( � � . � � � �
9 78 - & �' ' ( �' ' �) ' ' ) � ( � ' ' � � � � � � / ( � � / � � � �
: 78 - & �' ' ( �' ' �) ' ' � ' ' ( * + � � � � � �) ( �' ' � � � �
; 78 - & �' ' ( �' ' �) ' ' � ' ' () � � � � � 2 �' 3<4 2� ' 3 '
Z89
� ����� �
������� ������������ �������� �!��"
# $&%'%)( *,+.-0/1 2 %3*546+879$:+./ ;<+8= >@?A7B? 2DC&EFEHGJILK % C * +5M)N@? E ?PO67Q?PNR= >@% E ? 2SG
# O C ( *,+S-0/1 +5M)N@? E ?PO67Q?PNR= >@% E ? 2SG ?T/ > C�KVU (:+8>:+8*,(:+W* KXZY\[,Y!] ( G * C = ? 2 / C 79+ *�%3*,+W7^$&%)('? 2
_ ����� �`�
a �b�W��3"c<�d� � �&�e������ �!��"
# fg%Jhi+kj K +87l* C&E hm79+5$)79+ K1 2 %3*546+879$:+ K % = ?A*@?A= C j / K 7 CnK +o$�?p+n/?P* 2 %3*D/ KmC * K hq/nN:= $ C = +./
# fg%Jhr?P* K +87l* C&E hm79+5$)79+ K1 2 %3*546+879$:+ K % 2 %s7t79+ EuCnK +o( +5M)N@? E ?PO67Q?PNR=?P* $:+8*6+87 C:E h</.N:= $ C = +./
vxw
� ����� � ����� � � ���
$ C = + ? / C K N:> E + � � ;����m;���������� I ����� I - U +W7^+
# � ? / C / + K %! > EuC G +W7!/ ;#"%$ � I# � � ? / C / + K %! >:N:79+ C�2kK ? %3*D/ ;�& � $ � � I# ����'(� ) * ? / C 79+S- C 79( N:* 2kK ?p%3* ;+& $ � � , � ��� I
79+8>&+ CnK +o( $ C = + ? / C /�+5M)N,+W* 2 + %! K N:> E +./ �.- %s7/�10
v�2
� ��� � ��� � �� � ��� � �� � �
� +5$37^+ K ?T/ KVU + ('?��g+W7^+8* 2 + ?A* 79+S- C 79(W/ Q%s7�> EuC G ?P*,$ C�2kK ? %�* &���7 CnKkU +87 KkUJC * &(� CnK K ?P= + � '���� ;+& � ��& �� I � � �� ;�& �� I�� � � ;+& � ��& � � � I�� � � ;+& �� ��& � � � I��
E + C 7l*@?A*,$ C&E $&%s7B? KVU = +Sj U ?AO@? K / *�%Jh +Sj K +87t* C&E h 7^+o$)7^+ K ?��? K $:+W*,+87 CnK +n/ - +J?p$ U�K / � � ��"! # Y%$ Y Q%s7 C&EFE %3>:>&%@/k?A*,$ >@% E ? 2 ?p+n/ ]KVU +W7^+ +kj:?T/ K /'& # Y%$ Y Q%s7 C&EFE & ( &*) ]
= C j+-, �/. ,0&
-
�2143 +65, �/. ,� �� ;�& �� I � �� ;�&(����& �� I�7 8 ��� ;"& I
- U +879+ 8 ��� ;"& I ) 9 C /�& ) : ;
E + C 7l*@?A*,$ C&E $&%s7B? KVU = +Sj U ?AO@? K / *�%Jh ?A* K +87t* C&E h 7^+o$)7^+ K ?<�? K $:+W*,+87 CnK +n/ - +J?p$ U�K / � � ��"! # Y%$ Y Q%s7 C&EFE %3>:>&%@/k?A*,$ >@% E ? 2 ?p+n/ ]KVU +W7^+ +kj:?T/ K /'& # Y%$ Y Q%s7 C&EFE & ( & ) ]
= C j+-, � . ,0& +65, �/. ,
-
�2143� �� ;�& �� I � �� ;+& � ��& �� I
=7 8 �>� ;?& I
- U +879+ 8 ��� ;"& I ) 9 C /�& ) : C *6( @ = � = C jA�B@ ��9 !DCv E
� ��� � ��� � ��� � � � � � ���� �
� �� ;�& �� ��& � I � 0�
� 1 3
� � ;+& �� ��& � I
�� ;+& � I �+ 5, �/. ,
� �� ;+& �� � & � I � �� ;+& � I �+ 5, �/. ,
� �� ;+& �� I � �� ;�& �� ��& � I
� C 7 K C *,( C /xh�� % E + EFE ; � � I
�*� = 3� ;+& � I � � �� ;+& � I�� =� +-, �/. , � �� ;+& � I�� =
f 8 � E + C 7t*@?A*,$ C >:>J7 %oj:?P= CnK +./ = ?P*@?A= C j +5M3N@? E ?AO67B? C��� 7^+WN:*,( C *,( � 2WUJC >@?A7^+ ] 0������ �
� � � +J?p$ U5K +5( 46+87 C $:+ � +5$37^+ K ; � � I
� � = 3� ;+& � I � � � �� ;+& � I�� =� +6, �/. , � � �� ;+& � I�� =
f�� � E + C 7t*&?P*,$ C >R>67 %oj:?P= CnK +./ 2 %s7t79+ EuCnK +o( +oM)N@? E ?AO67Q? C��� %@/ K +87 C *,( � % U 7 C ] 0���� � �
v"!
� � �! � � � � � � � ���
� � ����� �!��" ���g���� �
12 H T
H
T
1,0 0,1
0,1 1,0
� � ����� �� � ���
T 0,1
1,00,0
CL12
M
B 0,1 0,0 1,0
0,1
0,0
R
1,0
v��
� � ��� �� � _ � � ���
����� � � ���� �"�� �� � � �
0 100 200 300 400 500 600 700 800 900 1000−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Ext
erna
l Reg
ret
Time
Hart and Mas−Colell: Matching Pennies
Player 1: HPlayer 1: TPlayer 2: HPlayer 2: T
����� ��� ���� 3"�� �� � �
0 100 200 300 400 500 600 700 800 900 1000−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Ext
erna
l Reg
ret
Time
WAR: Matching Pennies
Player 1: HPlayer 1: TPlayer 2: HPlayer 2: T
v��
� � ��� �� � _ � � ���
� � � ��� ���� 3"�� �� � � �
0 100 200 300 400 500 600 700 800 900 1000−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Reg
ret M
atrix
Time
Hart and Mas−Colell: Matching Pennies
� � � ��� ���� �"�� �� � �
0 100 200 300 400 500 600 700 800 900 1000−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Reg
ret M
atrix
Time
WAR: Matching Pennies
���
� � ��� �� � _ � � ���
� � ���� �� ���� � � � �
0 100 200 300 400 500 600 700 800 900 1000
0
0.2
0.4
0.6
0.8
1
Em
piric
al F
requ
ency
Time
Hart and Mas−Colell: Matching Pennies
Player 1: HPlayer 1: TPlayer 2: HPlayer 2: T
� � ���� �� ��� � � �
0 100 200 300 400 500 600 700 800 900 1000
0
0.2
0.4
0.6
0.8
1
Em
piric
al F
requ
ency
Time
WAR: Matching Pennies
Player 1: HPlayer 1: TPlayer 2: HPlayer 2: T
��v
� � � � � ��� ���
����� � � ���� �"�� �� � � �
0 100 200 300 400 500 600 700 800 900 1000−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Ext
erna
l Reg
ret
Time
Hart and Mas−Colell: Rock, Paper, Scissors
Player 1: RPlayer 1: PPlayer 1: SPlayer 2: RPlayer 2: PPlayer 2: S
����� ��� ���� 3"�� �� � �
0 100 200 300 400 500 600 700 800 900 1000−2
−1.5
−1
−0.5
0
0.5
1
Ext
erna
l Reg
ret
Time
WAR: Rock, Paper, Scissors
Player 1: RPlayer 1: PPlayer 1: SPlayer 2: RPlayer 2: PPlayer 2: S
� �
� � � � � ��� ���
� � � ��� ���� 3"�� �� � � �
0 100 200 300 400 500 600 700 800 900 1000−1.5
−1
−0.5
0
0.5
1
1.5
Reg
ret M
atrix
Time
Hart and Mas−Colell: Rock, Paper, Scissors
� � � ��� ���� �"�� �� � �
0 100 200 300 400 500 600 700 800 900 1000−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Reg
ret M
atrix
Time
WAR: Rock, Paper, Scissors
���
� � � � � ��� ���
� � ���� �� ���� � � � �
0 100 200 300 400 500 600 700 800 900 1000
0
0.2
0.4
0.6
0.8
1
Em
piric
al F
requ
ency
Time
Hart and Mas−Colell: Rock, Paper, Scissors
Player 1: RPlayer 1: PPlayer 1: SPlayer 2: RPlayer 2: PPlayer 2: S
� � ���� �� ��� � � �
0 100 200 300 400 500 600 700 800 900 1000
0
0.2
0.4
0.6
0.8
1
Em
piric
al F
requ
ency
Time
WAR: Rock, Paper, Scissors
Player 1: RPlayer 1: PPlayer 1: SPlayer 2: RPlayer 2: PPlayer 2: S
�`w
� � �� � ��� � � �
����� � � ���� �"�� �� � � �
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1.5
−1
−0.5
0
0.5
1
Ext
erna
l Reg
ret
Time
Hart and Mas−Colell: Shapley Game
Player 1: TPlayer 1: MPlayer 1: BPlayer 2: LPlayer 2: CPlayer 2: R
����� ��� ���� 3"�� �� � �
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Ext
erna
l Reg
ret
Time
WAR: Shapley Game
Player 1: TPlayer 1: MPlayer 1: BPlayer 2: LPlayer 2: CPlayer 2: R
� 2
� � �� � ��� � � �
� � � ��� ���� 3"�� �� � � �
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Reg
ret M
atrix
Time
Hart and Mas−Colell: Shapley Game
� � � ��� ���� �"�� �� � �
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Reg
ret M
atrix
Time
WAR: Shapley Game
� E
� � �� � ��� � � �
� � ���� �� ���� � � � �
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0
0.2
0.4
0.6
0.8
1
Em
piric
al F
requ
ency
Time
Hart and Mas−Colell: Shapley Game
Player 1: TPlayer 1: MPlayer 1: BPlayer 2: LPlayer 2: CPlayer 2: R
� � ���� �� ��� � � �
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0
0.2
0.4
0.6
0.8
1
Em
piric
al F
requ
ency
Time
WAR: Shapley Game
Player 1: TPlayer 1: MPlayer 1: BPlayer 2: LPlayer 2: CPlayer 2: R
� !
� � �� � � ��� � ��� � ��� � � � �D� �
�� � ;�& � ��& � � � I � � ,�� + ,�� +� ,�� � �, � + , � ? & �� � & �9 % KVU +W7 - ? / +
�� �� � ; 0 � ��I � ���� �� � � �
� U +5%s7^+W=� C * ?P* B%s7l= +5( E + C 7t*@?A*,$ C&E $&%s7B? KVU = � �e+kj U ?PO@? K / *�%Jhm79+5$)79+ K ]KVU +W* KkU + * C ? 46+ E + C 7t*@?A*,$ C&E $&%s7Q? KkU = �� � +kj U ?AO@? K / � h *�%Jhm79+5$)79+ K ;
� �
� ��� � ��� � ��� � � � � �A� �
f � � ; � C = + ��� ��� � � I� *R>:N K / (�? / 2 %3N:* K C�2kK %s7��7 CnK + %! C 4,+87 C $�?A*,$ �7 CnK + %! +SjR> E %s7 CnK ?p%3* �
� N K >:N K +oM)N@? E ?AO67B?AN:= / KmC�K +Vhq4 C&E N,+ iN:* 2kK ? %�* �+oM)N@? E ?AO67B?AN:= C�2kK ? %�*nh 4 C&E N6+ N:* 2kK ?p%3* �
� *&? K ? C&E ? ��+ � � � 9� 8�� 8 �
?A*@? K ? C:E ? �5+ � ��& 3 ������� ��&��� � ��� 8 � ? / *�%3* K +87l= ?A* C&E�� �/k?A= N EpCnK + C�2kK ? %�*o/ & 3 ������� ��&�� ?A* / KmCnK + �%3Oo/ +W7 46+ 79+S- C 79(W/ � 3 ������� ��� � C *,( *,+kj K / K CnK + � � B%d7 C&E E " $ �E + K ���;�� � ���& I � � � �� �9;�� � ��& � I2 %�= >RN K + � ;�� � I � � �+ �/. ���;�� � ���& I � � ;�� � ��& IN:>&( CnK + � � ;�� ���& I � ; 0 � � I � � ;�� ���& I � �! � ;�� ��& � I- ? KVU � ;�� ��& � I � � � � �" � ;�� � I
N:>&( CnK + >@% E ? 2 ?u+./?A* B%d7t= +5(* C ?T4,+
;t/k?A= N E KmC *,+5%3No/ EHGJI 2WU %'%@/�+ C�2SK ?p%3*o/ & � 3 �����#� ��& ����� � � ] & 3 � & � 3 ������� � & � � &���(:+ 2�C G �
� � � 8 � 8 �� �
_ � � � �
��� � � � � � ������ � � ��� �!��"
# $&%'%)( *,+.-0/1 2 %3*546+879$:+./ ;<+8= >@?A7B? 2DC&EFEHGJILK % C * +5M)N@? E ?PO67Q?PNR= >@% E ? 2SG
# O C ( *,+S-0/1 +5M)N@? E ?PO67Q?PNR= >@% E ? 2SG ?T/ > C�KVU (:+8>:+8*,(:+W* K
_ � � � ���
a�� � 3"�� � ��� � � ��� �!��"
# 2 %3* + 2kK NR7^+./1 � � C *,( � 8 � 8 +kj U ?AO@? K *�%8h ?A* K +W7t* C&E 7^+o$)7^+ K1 f 8 � � h���+ C 7t*@?A*,$ 2 %3*54,+W7^$:+n/ K % = ?A*@?A= C j/ K 7 CnK +o$�?p+n/ ?A* 2 %3*D/ KmC * K hq/nN:= C 7���%W4 $ C = +./
1 f�� � � h��'+ C 7l*@?P*6$ 2 %3*546+879$:+./ K % 2 %s7l7^+ EuCnK +5(+5M)N@? E ?PO67Q?PNR= ?A* $R+8*,+W7 C&E hq/.NR= C 7 ��%W4 $ C = +n/
���
� � � � � � � � � �
�� � �� �
AL
1,0
R
0,0.25D
U
0,0
0.25,1
B
� ��� ��� � �
A
D
U
B
1,0.75
1,2L R
1.75,1 1,1
� *@?uM)N,+ ? j6+o( � K 7 CnK +5$ G 8 M)N@? E ?AO67B?AN:=���r;�� I � � 0�� C *,( �� ;�� I � � �
��v