A Novel Method for Automatic Strategy Acquisition in N

A Novel Method for Automatic Strategy Acquisition in
N-player Non-zero-sum Games
S. Phelps
M. Marcinkiewicz
S. Parsons
Dept. of Computer Science
University of Liverpool
Liverpool L69 3BX
United Kingdom
Dept. of Computer Science
Graduate Center
City University of New York
365, 5th Avenue
New York, NY 10016
USA
Dept. of Computer Science
City University of New York
365, 5th Avenue
New York, NY 10016, USA
[email protected]
[email protected]
[email protected]
!"#$%&')(*&))+,(*-"'.')/01
23
/')456%"7)8"(')$97+:($47')')/;<"#')$=
+>3
')')?*'<9$/$@BAC'.DE#')('))*F/@3G@'<(8&=.3
+#
')HJI#')"1"=#')K')L')%')D=*8"
*D12&2
(3G<"#')$=
+J)/$&')MCNJAC'.J;/$$,
&(23
')/4 &0POQ7R1S6
2')/B1
2&/+HTU*AV*MAW&/$"U=
"(@>)/$&')K8&59')XY(') &4')BZ+R6/&
7R(*"
[+8\')667R+]&E$#'<&[/&3^&')(
MV=:'.4')["()*&*MjAC=*Mk'.C+')/#M+
_@`$,a b$c\d^#`&e='<fZg2')`&?eR')g2hZ/i=Hmc lQ%')0=&R*M#A[$#$#4W#$#'N
(@')@'NB "((
5$n$)"#'<=
+%/@3G
+%7
*%$
V)')o.+B$#$#'<3^2&VRC/$')41
2/+B')6<&/$VR$"<&3
');!
2M]&((2#')/$.+8A,[7<W61
2
&$"#C<$*]1
2&/+PR
&A)\H
ABSTRACT
&"(')!&=W)(@
$')(m%&o@M*6")'.3Z/$)"')jM&$#3
2')$4=,)*')/#M$/$&')(p "&')!&R/!1+#13
Keywords
q rINTRODUCTION
"#$%'<(]')
(
+L&W/&3G<+#'</F1
2/$')%
)/:7RD($
'<*LE(
2#$7<s')Lt
'.u=('<&0lQ.3
)')/$($Hvqr12=&2D(@'<-";')J$)"#'<=
+w("#
3
')/] &B')
(')/]Ax1
2/$')B') 3Z$)"')jM!'):AC')(@
u
4&y(2L')')'N#"=&V')D9g2$` $)'</rR$"<&')$D&
1
2/$')]'<P
*z<&')r9]')')'N#"=&<%')z=
R"N')$;7+("#'</BU*+XY0$7#2')8AC8W
3
)(@*r'<')'<"=&)y')2(@*Hy{V3G<"#')$r(*&:
')03
".!')%|}2~ _2f }1| g2h@_ M'),AC')(@W($<@'.Z+W&=U$7"1
&
1
2/$')C')yR$"<&')$8')(*&
C+K($"
3Z#
K$&2'<')P')R$H
1.
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission and/or a fee.
Copyright 200X ACM X-XXXXX-XX-X/XX/XX ... 5.00.
€
 5jM‚*A*M‚(3G$$)"')$&
+E)*')/P(*E\&'<8($#3
/y87"11
2&/$')HmlQ')p=R@A#)y
$
)')'.2&')$#("#
(3G<"#')$=
+,&</'.M&=UA
')
"(p0&$$y &k&"$%*W1
2&/+3Z(*-"')
'.')$
7+z(67'<')/D9/&3^&')(&=.+#
')P/$;AC'.ƒ
)"')$&
+8
*&(28'<)#')/K[
'</$),3G($$)'</6R"3
<&')$‚HWq[
R(Um$"#W]')y&2&@y
"
')/B+XY['<')'<"=&=1
2&/$')0&V$"#Vu
"(')$‚M
AU'<1*$8"
U
')?,&'.C7=
')8
2(')]"0
)"')$&
+K/$3^')(C&=.+#
')Hk>C&.+4$"#V
KB#$"7)4&"(')r%&oB„ …&†^‡C#$%&'<;AC')(@r</
]y47R(2%&o#$7<ˆ'<]"=#12')/41
2&/')(3
')2(')'<P6")'.3Z/C1+#1H
q[")')[')C=&Ry'<UC <)*ACHlQ:‰(')$WŠK&=
‹ AC/'<$yW7#')Y')
#"(')6,/&&
+6)"')$#3
&
+P/&W&
+PAC'.'<P,(@#C&m"#$%&P1
2&/+3
(-"')
')')jHklZB‰#(')BŒ0AVm')A
$#kAo
!
1226(3G$$)"')$&
+K)/$&')mN$0R@
R(')C
/&3^&
+H!lZ‰#(')$8ŽAV/')U7')Y')AJ‚y(@#3
'<-"](*))*z"')1')(3G1
2&/&+*#')%&')$‚MC)
9o*AC
'.')(*C/&P
+Mp$DAC')(2J"9'<7
*YM
#'<
("
,*Ax'.[')B7)6P(K
$K&VK)'<'.23
')$6C(3G<"#')$=
+D)/'.[ACD&=&)+#
')/E1
23
/$')(%')2(')B/$1 h
d _c\d e‘ AV))3Go*AC>1
2/$')B &
/$')r/$$HClZ:‰#(@'<E…KA[R
&’"')1')(3G1
2&/&+
.+#
')8w:$'<%y;#$"7);&"(')$J%&o@(*))*
E()*&')/&3G$"
$H9lZw‰(')$J“;A]
(')7R8$"#V($#3
')7"#'<]'<[#$%&'<‚”k')?$Mk*AxA%%7<K]($47')
"')1')(3G1
2&/&+r.+#
')W])"')$&
+;
&(2r'<]2@
&"$%')(*).+r&(*-"'. eYh@• 1
2&/$')y5[#$"7)["(@3
')$L/&$M0&=9u==&))+D')L
(')$%–&=L—EAVr')
("
$"
")H
qy\'))",&k(
2&')PQ+#RVk(&3G$)"#'<=
+]&)/$'.
%($@/$B$;$7"1U1
2&/'<W'):(
2')E
(&')$W')A)
o*ACjM6=&p%&+[R
')7)y("
” p)$M0R$#3
"<'<r%+%y6)'<'.(+#(),'.1
2&/')y)*y');&)')
/$2'<B&K7)K8#)$'.[("#
,$R$[=E("#
3
%$R$8*zOZ5/
S9*A˜>7R*&%]')*
)')#')/> $
')\H'<)18%+LXY(@'<$;(@'<-"%$r7R
L$)$R*wD(:
E7)Mr%&'<M
*AV*MF#R#$7<™AC')(2J').+D7R/$')')/E7R
$#
D
"((
15").+HlZ9
$%/&Mm
"(2DK{V
MA
2. NASH EQUILIBRIUM
(E&5.+E7RU&y')p<+($
')1)+;7R*&y<+M
&=E<+ ˜($
')1.+E7R*W<+ 4MRE<+ ')
)')o.+%67R&C<+ 4H‰')(W#$')=&(<&')$
');')
2&
'.')$MkAVP(*&97"'<<')/ "y}1| c\d e‘F
_ &_cZh ~ _ &K73
Q(').+r2&o')/8<+U')EU&&7'))'.Q+Mjr4"
6&
"(@:2o#')/]1+#1,(*&7RK"
*:]
[>OZ=&)S
u
1
2/$')U$)]"
')/%K(3G<"#')$=
+;#(
&=w
";&rR$"<&')$z')8<')/FA&2w7R@
&=>7R
@41
2/$')HElZ>%+:B/&M*A*M!
#$')=&(p/2&6')k'</.+B')
2&
'.')$M%o')/'.'<R
')7)
[2oK1
2&/$')0$P,
')/$)W
()$HmlQ%
"(2%/&M'.0%o
)'.
)4
,62)o]7R$"#]OZ7R1SMywOQ/$#SMY1
2/$')
')(P>'.y;/$')91
2/+7R*&4;<&/]"47R,3
RV1
2&/'<CyAC'))‚.A0+#07R%+$R$p=
7)0,7R*p'.*HqC7R1m1
2&/&+6,<+4'<6
"(2W/&
')U.A0+#,R=#yr[1
2&/'),$#$#*;7+;$ U3
RH
&
+6'<p"AC')8[R*A
5"=((V5&3
')/J7R";E7R1r1
2&/+zJ$#$]')ƒ
"(@ ('.("K3
12&(‡W4')F&0 ,| _ >h bd\a d } d b ~6H4t @[1
23
/')W &BP/$')F/$B'<[ W&
F-"')<')7#'<"’'.QM:.+E'.QM
4<*+0(*&P')C')=+X]7+"'<<2).+81AC')(2')/
Kr&)@=&'),1
2/+H
lG@K')4;')=[1
2/+ , &4K/&$MFAV
$"N8<+y1
2&/&+8&0/')C"VU7R1C+Xr7
*
AC=[A7R)')K"B$R$WAC'<)p<+HlZAV%
"
"8$R$Pr=+&X %&#')'<
@MLAro*A˜=
+zAC')),<+ W&
ƒ1
2/+ @;7+ } h &bgc\d\` | | _b } f
+ W
F:7+:u'.')$>B<1
&b ~]”k'.+F#'NE&B<
y‚ ($"N%#47R
@7+(2=/$')/[')1
2&/&+M#=
(%+:A$"<>B7R%%&#')')
')/]'.6*+XkHrq')4')
$
+R*AV
"($(#*Mk
')(8.$"/9&6$
+/$
4'<1
2/+M
+u='.U/&UR
C&C)*&1C$
} ~ˆ
<"#')$9„ –&†^HptU'.')$=&))+M'.!A,oA W'.
h @Wbd\a dp 1
d b2&/$'),&=;=+&XYMRAB(*:#')')1')(*).+r(K3
"U W&
]1
2&/$')HlGk$.+;6
')/$)[*-"'))'<7#')" #')1
5PE/$')J/Mp'<4=&*Mp')J
+L)*1*MAV
(:.A0+#W("#[EO1'<&SP1
2&/&+; &[/$')
/$$H
y$Mj*AV*M‚=W
U
:1
2&/&+'),[&)A+,
2h@_c 1
2&/&+6,N+6'<%&)=('.("12&(H Š3GN+@V?@3
"ˆ/&M,(*&;
*Aƒ=0 W&
]1
2/+]'<C3
)$'.27<HpTy*A@*M')j$"#R$#BV<+K'.! W
1
2/+M][%+]7R,&y"3 W&
;1
2&/')U=
]7R
@&6#)$'.')/E&XR3Z-"')<')7#'<"™<*+H9ty'.')$#3
&))+M&%&+y*-"'))')7'<C%*+U#')1kW')B#3G<+j$#3G($12
3
"˜/y'.y%*+87R[(
&
+P &W&/$y g2`*` } d e | cZh $
]&]*-"'))')7')"™'.U')K1
2&/&+J')6:%')D
("#
&/&'<1V@)$'.2&')$‚”'.=AAVWN+4 U
61
2/+B5
K-"')<')7#'<" AC'<)1B$"#,$R$W<+:P1
2/+E5
&r.&')[*-"'))')7')" A[%+A)ju=%=&C"=+&X
')
')/$')u(*.+4<*AVp4')RAV0=4(&2')=*6$4&
-"')<')7#'<"˜C$"#$RH
‰ 2BEYOND
&2/$CNASH
&
+8EQUILIBRIUM
p)‚"pAC'<(28‚%+
R
')7)W
w1
2&/'<]$"#8$R8&r)')o.+L><+H
#%$
|} /$K
+L'„ &&@†0F'.[$'<B&
2(2oE')
#$`$7a b<c\ d^`e 7+; R
'.'</=&*M2&@U=&r($"')/K W
1
2/$')j k/"
'</C7"@3G5(m&=U,
)(')/$
tJ1
2&/+6AC')(@K')m.A*+#07R1m[$#$#pW%&
@
AC&y+8$R$H
3.
WK<*+M#"C$R$y,W)')o.+%K/&2$").+
$( 1"16')61
2&/&+9B')8')9
R
P;rR&*
$7
&')$p&R'.m*AC%=4@) =+&XYH  #&(@
9)"')$&
+z/$3^
+w"
%>} hZiRa d^g | cG` } *&e |~ d^g_
*-"=')$]K
R('.5+]W5*-"(+%AC'.]AC'<(2E'.XYC"
1
2&/$')r
"<z7R:<+* R=')/9$ $"#;$R$+ 1
2&/+Y‡
2& 9
- , .0 „ 31 24 6. 587- 93: ;1 2 <7- 5 =7- 9 † -/.
/
AC@ ')6'.#*#3G1
2&/+%(@*M>132
')0U
=+&XEAC7- E&<<+<+ 7- MR?1324 .@58<7- 7- 589 7- ')94*@2/$
=+&X>]"41
2&/&+B;A ACF)<+,<+ 7- Mj&=C- , .
')6%u1K')&')8 - . AC'.D
R(Kr')$H9‰
2&@3
/$')B=6/$')L7R3Z*2&/$8=+XJ7R(%)')o.+
7Rm<+*‚M&=W')-"&')$[#)!C
'<) g`$fQh $ `$a bc\d^`e |} (
&')')(
+y)*&'</M&')WAC')(2[/$j1AC'.(@WC1
2&@3
/$')C=R*&CK7R,&W
"((
1 "^H
&%+D')'.'<C'.#*#3G1
2&/+>AV;(*Du=>]"=&
$"#($kN$ ')j(3G<"#')$=
+W#(
!7+y
$)'<D
/ , E
&8)G AFEu==Pu=V')#*3G1
2&/&+9P($- . 0F
/*
R$"<&')jH4q'),!=&W4&
2(@'<$KR
')W=*‡
2\' ) W&
]*-"'))')7'<4&!y/&W&,12')$=
+8R$')0"3
@9 U[<')(*B+')(”‚&=
2\')' )! (*&R$')Wm
<')(*W#+#=&'<(yH U
;-"')<')7#9 'N&m,<"#')$=
+
/&$H
q"mI W
%*-"'<)')7')" <"#')$Cy67R**K')
12&')=&
+6R$')p&RC#')(')$Ku=N4RC+#=')(
R(@3
'.u=*r7+r*-"&')$
&$H,ty."/$E[&<k12')$=
+ER$'),&
W&
r*-"'))')7'<#M7+P<+#'</%#+#=&')(W‚m)*')/
$PW*-"'<)')7'<4AV,(*P
WAC')(2;
$)"#'<y&,&W)'<o$.+
7RB')
(;7+ `bJe h #a $f }
| c\d\`e | a /Hq
H U
*-"'))')7'<B&&U12')$=
+8R$')0&AC')(@PB</$02&/$
B')'.'<U12&%AC')<U=w"jMy&E*-"'<)')7'<&]&E
)')o.+47R*(2 2^&
"')/48')'.'<‚')1
')7"#')$%=&V')
"'. 9 H
q')E'):)4AV)6&=v/ ')v
+M[7"#r>#[')
4)')')z2(@'<(,"
:
')(F%+J'<1'</9*.3^A&N
/&&W~ b#a c\d\fG_c | cZ(h K Hm‰"(24/$k(*&67RV
2&15*6')
&%.3^ &ˆ/M7"C$.+P7+%')
"(')/&;')
2&(27.+
<&/8"67R[&C"61
2&/'<Mp%&o'</P=+XD%
'.
')R$
')7)WK($"#$H
LV"#!AC=')AmAVU*#'<%mp)'<(&p+K3
')(7+"
'</6&P$)"')=&
+%
&(2Ppy1
2&/&+%
=((M
Ny&@P ($
'<')/L&ƒ')#u='.ER$"<&')ƒ($
')1')/
0P'."BC&<R
'<7)"41
2&/')M!A6"
8
%)
u='.0R$"<&')$K&=2)+B&)*61
2/$'),*#'.3
%&p0/&$H Lp+B')
"(')/W4"2&')K4(
13G&$*MA
(*&6
*(@4'.
"#)[/')$k&=p1
2&/&+B
=($H
O]')/$,
"(@E%(
W($@/$[8
4o')=;&&##')%&3
')$J,r
"B W&
J*-"'<)')7')"P M’lQ=#*YMp'<6'<]A+
'<#')/P#'<1')/](&3Z$)"')=&
+F&</'.”u
13
R
')$&P
)(')$J<+6r
'<')<&6$)Pr%<')(*3
0#+#=&'<(V*-"=')$%')P
"')/,&0
"((
15"Y1
2&/')
=&/&MR]/$')([$R2y))*A nK
*(@r
@W1
2/$')HƒqrrF"47RK&W#$7)
AC'. "(@ #&(@;5 J/&3^&')(>R
R(@'<$$M
*AV*M#AC')(2]A,
)#')
("
')P"‚H
q6*+X5[/'<$FB,+r12&/$4mB/4@3
RK 0$PW'<1&
+]&k<+H
4. CO-EVOLUTION
!'.1.+MR
')$8jR$"<&')$8<+#'</6#'.XY
1
2/$')[
$,8"=&!$)K'):8(3G<"#')$=
+:)/$&'.
„ Œ&†GH  B0=YM$R
')$K&RR"N')$KN+3
')/D>/$')z1
2/+J
%r7=7')<'.Z+z[<*+#')/
&P"#r1
2&/&+z') >'.#3G1
2&/&+ W
w*-"'))'<7#')";H
 LP@%=‚M0$)"')=&
+L
*(@L*-"'.%#'<$
'.Q+
')zrR$"<&')$')z&2897RrXY(')Hq')8
"//$1
&p'.jAVyy
&(2')/[ &% W&
*-"'))')7'<,'<)#')/B'.#*3
1
2/$')KAC]P8"%1
2/+D(R$4
4')/$85*-"(+M(&
R$')/44(&3G$)"#'<=
+P
&(2
AC%]'</R(2/$&6R$"<&')$>')B$#$#'</P
&1
2&/&+M&UAVp%+y7Rm')[/j#
3G($1
2')')/
"
*(@r&0AB&(@]6
$)"#'<jH
‰(=.+64<&*#.+M.$"/Ku=#
pY1
2/$')
');B($/*]R$"<&'):%+87R[7R1
R$
y*&(@
&*M[')y/$"&2[=C,u='.8m1
2/$')
(%7RP'<$#*97+>@+
3^3G7R3G("*>1
2/$')
')E
*(@ ($MU]1
2&/'<]=;7R(*&E'<(P'<
&)')%/$2&')7R(*&"
8+R
&*LR&.+L/&')1
&6*)'<@p1
2/+B'.W&m'.XY[N$ pu=(#$/$*
1
2/+;')YH @'<([R2&&W
"(2r6"#2&')$]y($
13
@pAC'))R7RCR0
*(@')/, &1
2&/$')V&p($"N
R&'<).+['<$#mpAC.+,127)'<
*[*-"'))'<7#')"ƒ7R(*"
&7Rp7);H!lG
(=#')')k<‚M[mu=
'.r&1
2/$')['),')<"
'<7)K[
"H W&
F*-"'<)')7')"
& V‰‰YM#
')(WUAC'))7RW"
*(@*P1
2/$')C=C($"N
R&'<).+[7*&oUm*-"'))')7')"ƒ7+W$7#2')')/y7R
@!=+&XY
5m(
2&')6<+H>V')/*M&
)
M7Rp&')1u=*BAC')
;u=C'.J&,1
2&/$');&8&*#')%&')$L9F
"
U
*-"'<)')7')" KC/"=#m&p'.j"V(3G)"')$&
+
&)/$'.’')U"=7)[u=]*-"'<)')7')"K3Z7#*o')/1
2/$')M
w>8)/'. AC'))U7RE7);>
HƒTU*A*M
&')
("
;7R$M#A,@R(B| i } d\` } d =(3G)"')$&
+
&)/$'.4AC'))7R%=&
')("<&.+ };&4
*&(2')/r5K
1
2/$')0(C@+K=*y')
(@i#`**` %2\&
'< 9 *-"'))'<7#')";H
!')).+M(&3Z$)"')=&
+L&)/$'.K)*+DE"67R@6&
#'.XYk
)(')K#M&m)AC')(@[+#'<N[R$"<&')
+#=')(&C(/$,;/&3^&')(,*-"'))')7'<]„ ކ^H
q
:$7)P*r)*z
*&(2@]') (3G)"')$&
+
($"#')/V
'</,A>)/$&'.)*+')//&3^&')(
)"')$ ((#F„ ‹ †GHxlZz=
')("<&*MB„ Œ†,
(')7R:>/&3
@'<(8
*(@>(2'<-"K &K&(*-"'.')/E*#')%&')$4&
U
1
2/$')P'<wN/$:1+#
')(EŠ3G<+8(12&
3G
"
/$PAC')ƒQ+#R3G')=#R=#8=+XYH˜lZ');=&R*MUAV
4#3G<+B3G($12
3G
" 4".'.3G12&/&[AC')
Z+#R3ZRB=+&XYH:lQD
"(29/&Mm<+')/:" W
1
2/+ 2\&Ev*#'<%')$ #r&r/$"&2&3
8"4
("'.Q+>/$')16#)$'.2&')jM"9 ['.08&%4".3
')<4*-"'))')7'<#M‚'.,%+;7RB4$#'<&4%N+E 2h@_c^f
} h@_^i#`e_h 6W1
2&/$')C=&CAV,'<# y,'<]<+H
5.
NEMPIRICAL
$p\H0„'&†Y&=[9GAME-THEORY
)
K@V&\HC„ &*ІR7#'<0%+[R
#$7<!R12&=&2[(3G)"')$&
+4)/$&'.k7+,1
')(
3
')/E&
')$]
%)p#
2&')&)%OZ"#')1'<(S
1
2/$')&&UoAC[7R($.+KN+K'<8[/$')
4".'.3G12&B/H &,%+P/Mj"
"#')
')/$.+r$4&
U1
2/$')0($)+6')P"
U')0')=py&H
')w;<&(@oDBF'<%1
2&/+M0'.8')%L=&"#2
P
or'.p4&K'."y&V
OQ"S81
2&/$')W=
($1'."#,/@3G@'<(,-"')<')7#'NH
y
%)"47R0mN+@U&=;"')1')(B1
2/$')MRAV
(:($1
"([<&'))+E
%&)k%&.3G5 =+X>%&
'.
AC')(24')&=&7)p/&3^&')(.+#
')Hmq') h@b } d _c\d^g
=+&X%&
'.,')k(*&)'<7#2&*B7+,"')/C%+,'.2'<
/&$”$'N')$C')]+XYC#"yK'.XY0N+@
3GZ+#R2\/
7<&(@oPAC'.$M=7"+@y&U
<) y1(@1')(B#'.2
\&( 2\/I NI * 9 &[*9 2&/$*P&$C%+&)C
Q+#RK')# %')$>
")')/;')9;
')/$)%*&=+&X9;*&(@
<+4 &K*&(@D
+>')9%*+XJ%&
'.YHI<+) kQ+#R
&B&
"*%K7R[#2AC;')=#R.+5˜W[#')13
')7"#'<jMp9L/+ K(@$')(8y1
2&/&+9')K&
"*
7R6')RU'.WZ+RMRAC')(@))*AC[6*+X>%
'.
r7R "#
4($#
*YM
'<(A8
').+*r
R(')N+
E"47R%B&/$P<+')/9*(21
2&/&+wL@')
[#R(@*]=+&XE8(2E/$*H0q#"5WK/,AC'. 1
2&/$')MAVV
'<k'<Bp"#'<1')(+X%
'.
C(CkU &
P7 0 2 5
9
AC@ R(')uVy"47Rpk&/$VAC6UN+#')/[
;1
2&/+Hp&(@r
+ ')y%&R*8$&;$"#($
(@ &!y 7 0 2 5 9
AC@ R('.u=j#R(*W+XB0 B1
2/+H;&
]/$6AC'.J/M!K"67RUC
'),')6=+X
%&
'.8')C/$');7+
2 &
2GŠ 9
0 2 : : & 9 9
&
%) &=9
%) >')B
".K')D=+&Xz%&
')(K
%/*7<%
')?$”5! 0 ‹ &="
0 …#M–M & E AV=*
I0 Š&–#MŒŽM=]…$…B
R(').+H
 (6*+XD%&
'.:=&[7R($"#*rA%(*F
"7#3
1(V'.m[U')/$"p/@3G@'<(&=.+#
')M
*&(2K &% U
*-"'))')7'<E
$)"')$%L&)+>'.XY6)K&U)*')/
w)"')$‚MC
"(2%;)'<(&;#+#=&')(\MC')
2#m[=&.+
0y+#=')(k&j$ Q"1k,*-"'<)')7')";H
qr-"')<')7#'NF
$)"')$8&P&;"$72&')*zE
')/$&$"I W&
]*-"'<)')7'<[ CU "<‚6".')3G12U/&$”#
')m.A0+#m0R$
')7'))'.Q+B=V&K"($
'<*B1
2&/&+6($"<
')$kk*-"'<)')7')";;H y
)
M"#')1'<(@3Z1
2/+W*-"'.3
)')7'<6U<&"
')7),&)&*&.3GAV<P/$<+3
')/L$
r72&'<*J"
')/912&=&2J(3G<"#')$=
+
*(@%(')
.+7R(*&"
C+8} } ')$K[1
2&/')
=y&,($.+%o*AC]=]h@&_c ,d^g')c;($8"
$H>U(*
@ &7RV($#uRk&W($.+[o*ACK1
2/+B &
B/&6,=&=]AC'))7*&o;$"#W*-"'))')7')";MR=;"
*-"'))')7')"n12=#&C)*1y
$U(@(WkR@
'<1')/'<8
&
C "#"$H
 k("
$M$(B;*-"'<)')7')"ˆ'<C12&7)')
*YM=[#
')/$#3
,&8=&
')("<&[1
2&/+E%+r&,7R6')1u=*rAC'.F')
1
2&/+ [$')$#3^2&K')EB/@3Z<*+#')/8R$"<&')
<&/$HtyB'„ &І!
"/$/1*M=4#
')/$y&QMR5y)$M=K=
3
')("<k
2$')/U1
2&/+B')6%&o/&V%+,=*mu=('N&
')(')4
"(@9&4&[')/$[r'<(*&
8'.OZ%&o
3
=$S #&0')MU#$R&
'<;!<+0"
')/4')1
2
3
/&+M*M&')B/$3^')(kM*m#$77'))'.Q+,&'."
1
2&/+r7R')/<+*;')E')#*3G1
2&/&+]*-"'<)')7')"˜AC'.
6<&/[7=
');&m&
2&(')$jHpq+8/$$P6R$
B6
'))
$)/+, &kR
5')/
"(2[$')')?*'<4"
')/%"=&
'</E#HCt '<
')/3G)$o')/%(*'<[1
2&/&+r')
(2$
; UR
"7&')$E&=&)+#
')”‚K@A,MRR
"7R‚M
')$
U&')/$')=&"')1')(W=+&Xr%&
'.%')0($"*%')PAC'<(2
m=+&XY!#m(*&='<&V1
2/+BV')(@*
*B7+,C
%)
u#*zR(2&/$$M"8))')/L9+#R&')(*&UQAV*ow
41
2/+]=&y+#'<N#U'):
%)k')(*&
6')r=+&XYH,q
)')(*&&
3Q+#=')('.(')[u<,'<‚W)
*WC12&73
)')
6ACk+#R@'<().+#3G')')?*[1
2&/&+6')m7<
K(2'),4')/$]$')$P2&,')PWR$"<&')$‚HV‰
2&/+
#
')/$](*&wz($(
2&;'.]@XY
%$')')/
$
[1
2&/$')C=&7R($,1
$/%&
2&(CAC'.r6
%)
')(*
[');*+XYH
lZB')mR*M&A0@#=,')!(2'<-"p7+["
')/UU/$')(3
&)/$'. 2 t 9 ]| bcG` ~K| c\d^g | a a $#'<')?(*&='<&U1
23
/')67+F
*&(2')/r5;'.
&3Z"o*AC>7R1
3G
R$
&*M4"
UU$#'<&U$(<&"#$M } }
™W&6#')1')/W'.[&="')1')(01
2&/'<H NU2h@c\cZh kf h@_Gi#B`&e"_1h 3
')/[W12=&26(3G<"#')$=
+&</'.x,R
503
')')?*'<jMAU"
W[
')/<@3ZR"N')$ tAC@yu
&Bz')')'N#"=&y1
2/+w'<8("DN$ r"')1')(3
1
2/+:*+X9%
'):&((&2')/8P47=&
'<F
')?K'.,+#'<N#
"=#0U)')(*&&W+')(H
6THE
&)+;DOUBLE-AUCTION
$"U]%4(-"')
')')F&V1
2/$')W5
`&b a<h | bgc\d^`&e „ …&†^H4q6$"7)K&"(')F'),%/$2)')&3
')KY&($.+3Go#*ACB
')/$)3G
'<*K&"(')M
"(@6
m/$)')
&
(=#')/8"(')$jMAC')(@r')<$4
')/)6
))
2')/rAC'.96".'<)87"+@HrlQDr#$"7)P&"(')jMAP.3
)*Av6".'))U
2C]7R]
'N#C!W%&o$*”0A)
)'<(')')/P&XY07"+P6/]5 7"+@M&')y7'N#MAV
&)
]
$)')('.BXYyP
)m%/$r5$’
)<@M
P(*))*L| _ _ H
p&'<&C&k,#$"7)["(@'<r&,($$.+%"
*]')P%+
.3^A<;%o<(C
"(@]01(2o8@(2=&/C'<P
(&
3
')[AC
".+:#%F')/$)+F+')($HB'))1
')/$)3G
'<*:"(')$U&BA).3G"=#1;5 /&B3
&')(WR
R(@'<$$M$"7)3G
'<*%&"(')$%')%')
2(2&7)
E8 ")p/@3G@'<(8&=.+#
')4
R('<).+>AC%
N').+[ A9
2k4*&(@4
'<V&V%&o$*Hkq"6"(2
&=&)+#
')m='<k/$k ("
*66"
'</W"')1')(0#
6#),'<7<,7'<')/61
2&/$')H
lZ,')$"jAVo6'„ & †^MAVp"
*["#')1'<(@3Z1
2/+B&=&)+#
')
W=&.+
ZAW&'<&E&=0#$"7)C"(')$K%ok(2=3
')
R$"<&*%AC'.;6'.&k"#')1'<(U1
2&/'<MY%AV
&7)]:u==D&*#')%&P/@3G@'<(P*-"'))')7')" )"3
')HClQP')yR*MA["
,B&,7=&
'<([52&A&oRMR7"#
AV0 ("V$K ga<h |} d5e‘$f `b_h $"7)C"(')$ 2G{VT 9 „ …&†AC')
"'. & ')(')/M')[AC')(26&</&R)<B &k'.k&XY
7R@ 2(')$2o<&($M#%)=
2(')$C&C
@("*:&U6&6%o
3G()*&')/P#'<($HBlQr')WR*M
AV(
'<m.+,V5$))*AC'</WV"#'<1')(3G1
2/$')GH "#3
"#WA&oPAC'))"
B6&W
2&')/ 2^]<&/ 9 @U&
"')1')(3G1
2&/$')y6$')')?,&/&'<1*H
qB
"#3G<)')/%1
2/+ 2^q0q 9 MRAC7+E/W
"7#3
'.0XY@-"=6'.0)"='<P5CU
$"(W7R3
')/]
2* 2\'<]1
2&/+3GC%o*Mq0qxAC'))V7RK
')=C1
2/+ 9 ”
Ny&3 k1
2/+ 2 N #>:1
2/+97
*D$
qB
')5(0)*&'</M
(')7R9 *8'):„ І‚8()')72&
AC'.]U=&2&C
R('.u*]')„ —&†^”==
+ 2 „ “*†^M&AC7+U/$
q Q12$3 U'<(2o=&"j1
2&/&/
1')%&99$77'<)'.Q+vP&+v7'< 9 7R'</ ((*
7=&
*]]')1&'<(%oC&2K&=%P7'<84%&3
')')?,#R(*8&u*H
‰#')(;&<'.#*#3G1
2&/+9()');')J]"'.
3Z
')<@‚M
& 0 ‹ 1
2&/')UAB(*&r(1(U["')
3G
'))P$%
QAVK#'<
')$‚
=(,8P<&0W1AC'.(2')/67RQAV
1
2&/$')]&;(("P"=#8:#+#=&')(P&B*-"=&') ŠH
!')/$"#
&
*ACm0'.(@'<3^u=N4R)')(*&&
3Q+#=')(
*-"=')$[ mp"#'<1')(p1
2&/'<M
*AC')/=A
=*BQAV;*-"'))')7')" )"')$H !'.1)+MjAK
K=& ')
K7R1
3^
R
[')
)ZM‚]([')WK"#3G1
2&/&+]-"'.3
)')7')";Hv>E)
L
E'.]P>
+JN/$ | _d ev` | c\c }
| gf
”j5[&+]2=#$.+3G)*;')')'<&k($#u=/$"#2&')$Fm
Rc\d^$`e "<&')J1
=*ACKL"L')9P7R
K3G')/
3
=&=3Z(@*HtU'.')$=&))+MU')0B
($8'.#*3Z1
2/+
*-"'))')7'<r&[(2#'<&B 2 E H –$–M E H &ŠM E 9 ')>K
')<@
(&
R$')/K%&r–$– ˆ'.8mq0qv;B &Š ')8& N0M
*AV[%
2(&45B')B*-"'))'<7#')" ')44"(2F
%&)<@
=&PU"#3G1
2&/&+ *-"'))')7')";”.+P… nj2$
12&
m')=CV—&Œ V"# 4HTU($M((&2')/
W')V=&.+
')MAVy#R(@p1V&YCR$"<&')$&Y
2$@
$#C 1
2/+H
6.
RE
TT
GD
"!#$%&'(*)('++,&.-/#1023%&45,67089,-:;#%
< 9 '+0>=?#@&A9B9C&D9%E-F,'*&D$+%DCG!#86/H&D8,-:#%JIK-/!L-:!3
#$%&'8%3#)8-:M4NO90QPK#-:!RCTS *UV6/-/&D-/*2XW'&.YZ.''90
P[S]\^
TU*A 6"(@:(uR#((*AV/$')K]')6&=.+#
')4/')
=r*+XY"
L>(1
"(%r#')(')$3^u<J)&
A@1')%&*E7=&
*F$.+>Š EE*E ),&*&(@F/&( M
 E&#(2w9&1AV')/9')8-"1')$w'<9(=#"(8
')')'.Q+D&=&)+#
')”0APR
"79P*&9=+&XY4 &*&(@
1
2&/+>')>8%
'.7+;
%)VR(2/$r
8'.$"
*-"'))')7'<W&=.+#
')p')k$7"1kW@
')6+X%1')%H
!')/$"#rŠ;
*AC6P#'.(')$#3Gu<9)K&5@KA8R
& R
"#7=&')PAC@UAU&$,ŠH Ž nW=+&XYV5 q0qn&= ˜1
2&/$')&=>&
'</ ,Ž =+XY[; N
1
2&/+Hmq')V
".')84-"=&)')2').+%'.XYV
&-"'.3
)')7'<#” N ;1
2/+[7R($jC7R1
3^
R$
mC'.
.AC'.
W<&/y7
')K‚
2(') 2^@… & M=B#"kA0(()"=#0=&
$"#,*-"'))')7')" .+#
'),'<U
9 '.')48
%)k
&W')F+#3
XP1')%&M#=6&m$"#&'</')=R#*')(')YAC'<
#*$
$#$#'<9 ˜%+:B(("4')CAV8*"1')%*
U=+XY0 N0H
lZCAV]$7
Pr')"#8&W&)81
2&/')')J&("=&
<+M‚*A*M‚6R@
"7=')$F=&.+
')B)
;
"/$/1,=&
lQ')'<"=&)0'<8')0
*(@;
(WAVW#
*]&KŽ E 3
7'.1
')/#MAC$‡
7'. &3Z–6(#*85y=&2& - '<PU2&/$2& 5 & E9 ”
7'.C—&
3 &*…K(*% CW=2 0 ;')8U2&/$
2 E>5 & 9 ”
7'. &*“3QŠ&ŒB(5V=&2& ;')%02/ 2GŠ 5 Š$Ž&– 9 ”
E')JP2&/$
7'.8Š$Ž*3 ‹ Š:(#*L5%&2@ 9!
2 E>5 & 9 ”
7'. ‹‹ 3GŒ E (#*W &!=2@ &m')W!2&/$I 2& 5 &Ž E*EE9 ”
7'.4>Œ &3GŒŠ](*5B(2$')(%&y)*')/E&)/$'.
$/$1I NC>M WImqUM=‰ N[”=
2 E@5 & 9 H
LV'.Œ ‹ 3QŽ E (*P5=2@ '<8W2&/$/
RE’
Equilibrium 1
Equilibrium 2
TT
GD
BR PK*)('++,&D-/# 023%&45,6 089,-:#% < 9'+0Q=?#
& 9B9CG&.9%R- ,';9&.$+%DCT!3#86/ &D8,-:#% )Z*-:YZ90XIK;-:!
)8&*2# 6 - #X-/! PK#-:!RCTS *U 6/-/&D-/*2 W '&DYZ9'90
P[S \
AV;("<J7#'</>&7R$"6AC'N#
$z# (@'<J<N '.UAV
&7)4ZA*&o][1
2&/+r7+;')#&')/'.W+XF
)'</)+Y”
c h%ih } c\b } | c\d^`e | e | a _d _Ki#`&d ec\_;cG` #’| _ | g | e d | cZh ` }
i#`&cZh@ec\d | a!`@i=c\d ~ d | c\d^`e H
Zl USTRATEGY
#'<"‚
(')$ACQUISITION
,AVV#
(')7R*W*AFA"
*W"')1')(3
1
2/+,#*')%'<W0'<'.5+WR'<(=#'<&1
2
3
/+K5')')?*&')jHk>&<
[')
"(*')
')/$"'</[
3
')(C &2o#')/,1
2&/'<0$%,
')/$) ").+3G2@*K
(*)$‡m')?$M
y
')?y‚C1
2&/+ 07=
')%&
2&(')$8"mC)'.3
(&k#+#=&'<(HlZ[')
(')BAVV"
')!@
'<(kR
&
%"')1')(6
*(@E
=(6p1
2&/$'),()$
)+E<*E
= N 1
2&/+H lQL]5$))*AC'</9AE@u=P;
(r&
1
2/$')&pAU
&(2jM#Ky2&'<)&jy
&(2]&</3
'.;H
7.
q N ]1
2/+W"
Y')5(j)*&')/2 N 0(2
N$ >R$
')7)K%o"Cy4/+ U<')'.y')9 (67
*
;,A02P
')/$‚($"*%&CB "(')P&k&u**
')#')"p"%&7'<')/Ht/$V7'<%&C
oK&0#'<(
2‹ 9
P0-
AC ')WK/+ B)')'.,')(M ')W6""WN$’
)*')/K)/$&')˜ - '<CB
(*)')/6=&2&*Hmty'.')$#3
&))+M N#3 4)*&'</K&</'. '.
.!=&V
$2=N
&2&‡W4((+:=2@ #M‚6#R')2&')$
&2& M$=4K')')'<&<')'<%=&2& &$HklZK'.')$
4W&'</')= Ny&3 k 2 N 9 &</'.;M# @,&,
$2
&6)*&'</3Z)/'.B=&4=6$
"((
1 ").+D7R
"
*r N91
2&/$')[')t{ 0HR>B
*&(2F@W4$'.3
')=YR
'<7')<'.')‡p12&)
y3G)*&')P/ 2G‰ 9 M'.u(*&')$
N 9"
P7+E„ —&† 2 WImq 9 =P6(
‚)/$&')ˆAC')(@P
3
)(@U6"'. &.+82&=#$˜(@'<;/2)
kA&2]
')/=
2MN HR‰ =5B=2@‡0B')
("
3^2&
RM=#R'.3
29 ')$ M]6)*')/&3G2 H
7.1 Strategy space
>y"
*84/@'<(@3Q&)/$'. 2 t B
*&(2P')0
=(W
1 2&/$')MmAC%%u
4&*(2D9 '<')'<"=&1
2&/+9')
,
&(2:
(,AU($"#*P7+P1')%&')/'.7=&
'<r
')?
"6])'<(&]#+#')(K"=@')@2(')$JAC')w$"
#')1')/]#1
2&/')‡ 6Mq0q = N 0;H L
')9
')?A0&
1')%&;"
')/[&B7#"3^5(B#
(')7R*r')
‰#(')$6ŽM7"
')(VAVp($"#V&)#
')'<[p"#')1'<(@3
=+&X%%&
'.B')K
"R
mR(2K(*'<01
2&/&+MA0"
*
)*A,&)[
')?W')r2#U ('))'.2&K)"&')$Em%&+
1
2&/$')”!4&)6
')?65,4"47Ry/$U<+*
&m*(26
+[')["')1')(=+&X%%&
'.,A0&k')(*&
*K&
, "(')$%‚/$2'<]"47R*G‡ & E " 2& EE < 2#& 99
))*AC')/,0
*(@#3Z)/$&') W-"')(@o.+[u==B')/$#3^u
@3
/$')V&YC
*&(23G
(')*&)')/@2&')$06*"(')/
')
y&=K&)<*AC')/B@u=k‚
$)"#'<m')Np/$#3
2'<HkV"
6y($12&#"67R=)')(*&&
3Q+#=')(
2+ 1(&') Ž E ')K2#m,1')%&C7
')K
')?C5$
y=+&Xr%&
'.%$(W'.0$%7RP($"*K &C$"#C(*#3
'<&,1
2&/+H
>8(2$
; t K"
&(2z)/'. #')(')=&))+D7R@3
(*&"
CR'.p7'<)'.Q+BW(RAC'.6C'.')$$')
&
)*AB)K
')?')
#"(:')PK$@7 1(@'<$K "(')$% $H
 " tƒA0(u=/"*8AC');6R$"<&')$]
'<?B& & MAC'.
')/<@3ZR')K($
13G*Mpr($
13G62&8 &$Mpr6E*"E 2')$3
2&PH & E&(' 9u
13G#$R
')=&]
)(')$‚Hzq t
A0&"P5 ‹ Š6/$2'<M=AC')(@]o]&##')%&)+ &– EE
{*I ) "H
7.2 Search algorithm
!')/$"#Ž:
*AC8r*Lu
%, t R$"<&')$
&*&(@D/$2'<jHLt6(*&L7R8
‚Mp8'<41')<r<&/
"P4$'N&(F') u
P$&)"P'< <&@r/$2&')H
TU*AV*MY')
R(')$:V2&=#$’&)Bp1
2&/'<W5$
*&(@B/$2&')4)*B&
'<(#$/$(V&=Q+#R$M
7"#jAC'.,
')/$'.u=(*&3 ="("='<‚')Wu
)"j"k0
%)
)V
')?% 2\
07R 9 8H O]$1k2&7.+Mu
1k')')'N#"=&
&]/$2&') ‹ Š>=$&)
J&R**')@')
.+J8
u
1,')=')'<"ku=B')W')F6#')" & E /$2&')M
8"AVW$o%')0,""N$ˆW
&(2jH
 "m')')
*61
2&/+K"
012&<
+ y3G)*&'</[)/$&3
'. AC'.]U <)*AC')/%=2@‡
$ y$MY*A@*MR=&yA6&B g` ~6~ d c\cZh 4"
4&
t &y4
*&(2r)/'.K3p
W
"
0A&oP
(@'<
8. RESULTS
0
0
…
& Š6& E
—‹“
TT
0 E &– ‹ Ž&— ‹ “$Ž
Equilibrium A
0 E >Œ &Œ E …ŠŽ
0
E &–“$Ž
q &27<p *"0')k1
2&/&+B'<!
%)"47R!&
R
')7)W%&o"I2 9 M#&=KC&
*Az2/$y‚%&o"
'„ & †#!($=*WAC'.B')1
')7"#')$[$&<"&')$4')1
'.3
7"5 ')$- KAC'<Hq'<m *"CA0&p
&*7+K&)YCKu=
1
2/$'),')r[<1Ur/$2&')M=;'),&&y\(@
&')=#'<(&*]($/(UkU
*&(2jH
K(*:;&=&)+#?6$"[
R(')1
2/+:",
5")V"')1')(3G1
2&/+9&=&)+#
')6"
')/ & E M EE*E &)[0
/$U5*&(@;&!yŒŽŽ6
'<0'<8W+XE%
')YH!z')
("
j
')$,&#$"j
')6"N ' M*AVVp&7)0($<@
')%=&.+
')%')J)
DQAZ+> "%$""
')/F".3
#(
& ‹ H … ? U$]AVo12&')$‚H
!')/"p…
*ACjZAQ+
2+ 1(&')kk)'<(&
3Z+')(
)
*P4')3G
') &*(2]1
2/+M8
ACCy')3
@2(')$7R@QA60A,M')')
*61
2&/+M  ‰RM/$
AC'.],#')1'</K1
2&/$')‡ 6M=q0q &=? NCH
qo')/ & 2&=.+J&)*L'<'.'N&U'.#*#3G1
2/$')M
AVp()("<mEEE &‚m&QAVC&
2&(3‡ 0 2 E E>5 E E@5 & E>5 E E 9
&= 2 …$“ ‹ Š
? 2
Ht
3
2&( 0 (*E# "5 E y$5 E .+ E@5 ‹ E E 9 &m
2+ 1(5')Mj5AC
?5*&,H&9 2&(3
& ("#m#'.
").+KC'.0 $"#
3Z')
')=
'<)
2^—$“ H ty."/ ')]7=&
')ƒ');$
+z<&/$MW"]$')')?*
1
29 /+%
&m')V-"')<')7#'<" AC'.0
"#3^))'</[1
2&
3
//+ 2^q0q 9 M/'<')/C"u=$&2%oj
=m E …$“ E —“ 0
…$Ž 8Hpq'<p($&\*"2&7.+KAC'.8W%&o
3G
&C ‹ Š 5[
"#3^)<')/; ‹ & 6HjqK&'</')= N 1
2/+
')%#$')=*L7+9"%')')
*L1
2/+H !')/$"# ‹ &=DŒ
*A#')(')$Fu<E5,ZA]& ‹ 3G1
2&/+:($47')=3
')W')$)')/8$"#,$#'<')
*r1
2/+YH‡ 2  ‰YMYq0qUM 9 =
2  ‰RM 4M N 9 R(').+H
Gl ,')DISCUSSION
,
$AC&W%&o7)B=&W$"W
$AC&,
'))')1')(
')')
*61
2/+4')p&7)W/$')%5(&N$ U'</.+4
3
')1')(*&81
2/+)')o 6MAC'<)1CV&')0
"#3
<)')/L');&7)rD@2')ƒ>
=&:[5$))AVr') 9R$"<&3
')z*#$')=*J7
+ j 2^q0q R&>7R |}1|
 ‰ 9 Hk=&(($"6 &P7'))'.Q+Ly
%)  i ‰9_'.dc\d^g
B')$C')/$#3G$77'<)'.Q+K')#m,
$')1')(*&*8$#'<$
1
2/+ 2 9 M$AC'))1%&')')/[")@27)C['<
')$%7+6
)*A03G7=7')<'.Z+;'.%m6$#3Q&')U1
2/+ 2^q0q 9 M
 "#m("
m+#R
'<m')m&  ‰4')V&7)0,#)$'.V RA
')wPA*+L=A;($1
"(8$&)"=&')z')1
')7"#')$Hƒt
#')
("
**)'<@*M#AVy"
0CC6
')/$')/B$&<"#3
')$pm');'„ &Š*†G”&m')M &V*&(@4"6&R0/$$M0)*A
3
7R"‚M M#jW)"&')$]')1
')7"')]')0
)(*8"')5.+
y2&=#$ 5ˆW2/$%„ >… & &… †kP,"R
3G7R"= ')
')')<&.+L#2ACD5$ „ … 5 E@5 E vŠ E —†GH &6=K"D&
P/$$Mp*&(@L/+ K$&)"=&')J')6D#2ACD"')5.+
N$ „ 5 †^H4TU*A*M‚A+#R
'<
B=,')W
".,')F
12')1'<(‚(
<'<]7RZA8y23G7R$"pU*3
@2/$U
<RU&
"# "=
".+&=8%&=K
(2*#") =
9.
'!""$#%&'(*)+&,*.-0/1&"*%123$&14*'"3,',
Equilibrium B
Equilibrium C
OS
GD
65R@P[*)('+,&D- # 023%&45+,608;$9,-/#% < 9'+0 =?# &
9B9C&D9%E- ,';9&.$+%DCT!3#86/&,-:#% 6:!#9IK+% +%E-/*&,-:;#%
YZ*- I"*.% #)8-/M45+6/90 6/-/&.-/*2 W879\ U9$6/96 " &%0;:=<
')Mm/$')
8')1
')7"#'<9=2[@8')4'<
"?>('<
&'<(y')y'.XY(7RQAV%)"='<&j
2mAC
&C')/$7R$"#m6V
".+4&p%&=4("$Hk‰#')(AC&
"
')/>F"'. &K3G')(]o 0E H ŽE()*&')/")$Mp](@')
')4"<27)K;')(3G%&')"N')$N$ )*&16 >K(')
2$#”$V7"+jAC'.4p)AV1m%&(2*,7'NYM=,p
))
AC'.],')/$1U%(@*P&
o](*&;R'<&j%&')"NW
u=()*'</,')(3 i } ` $ d h [c | c!c h H`,eR`&ck` $ h } _c | cZhUc h@d }
$&| a bh4ga | d ~ cG`c h6h
cZh@ecc | cd5cpd ~ i=d e‘h@_,`eEc A
~| cGg h d $B[` } c =h @Je $f d<(‘ h@_2c ~K| cGg h | _ H h @')Je $9fGa)`
•V"?>6h@_3c
(')m$'<(')K)"&')$p7RZA60)*A1V4Š=3
)*A1C%(@*KXY 2\(*)R')m2=#$ $'N&7)C8OQ%&(2
)2&S MR'<y")2&7'))'.Q+r')UU*
').+r#)')‚HyTU*A$*M
')D]%9 &o$[AC')>;
%)V"47R[0
24>;=
*A
')1
')7"#')$E UB%&(2F#.2%B'<W:$R&
"')Z+]
2$#C&p
%)#%&/$')&7R0
"6'.R+"6u=B+"
.RAC'.
)"='<6()
V*-"'))')7')" ')($Hq')k'<k#(')
.+[
7R')$"#W&p41
2/$')W&UAV67
[87R4#*')3
=&V')C<&V/$2&')&" t3+&)R"
,
%)
"47RV&R
')7)W%&o"M(2P&j %&)R'<%(=
3
')
$>]KR$
')7<%)"=')$>7R"=#HPqK')5(
3
)*&')/6($R$&y1
2&/&+%')87)UBu3^"
C%o#"K#R=#')/,$KACC0
2$#pu=#
)
$;[
".+;W%&=8("B5U$&<"&')$W62*AC‚H
lZ'.y') &U*A+]5 ,*-"'))')7')"K3G')(,'.(*&E$ Q"1y'.
%&/')z()$
E>?#MCAC'))1]'.['.]')8*%r*-"'))')7')"K3
')(C'.p(*Ku=KW
%&<%&/'<K&ppp'<')/$0'.
&1
3G'</7R$"*H;q')6+#R
')['<4)
E(
')14AC'.
=2
'.')(6
"#3^)<')/”j')U'),*1+r8
4&W
"#3^))')/%')
[7R1
3^
R$
y &C4Š&=3G<*AV10%&(2K7'<#V4,<*AV1
%&(2;7'<#WN+#')/  ‰YJH "
yAVor*%7R4(*
3
')*r$"#U8'.5+;')U+#R
'<MY7"y*&.+E')=#')(*&')$,&
=0 >((+4Y"m$#'<')
*41
2&/&+6')m')/$)+4
')')
K(2=&/$')]W)"='<;23Z7R"H
q"j')kR&&!"k$)*,1
2/+[
3^uj0AC=
')kR'<).+46&
'.\&(*(H W!')k'<)'<$"#&(@ M
 4(
2
+M$AV07R)'<=')p#$1
2&!&m')/$.+
$')1')(*&:1
2/$'),(*&ER$
[&Ft(@')<)yk'<E(@
3
2')4
')"&')$M[&"[51
2&/+6(*-"')
'.')$
(*&;u==]=P#)$'.
[")@27'))'.')H xR
&&3
$"#W&#(2;'); "#"WAVoRH  ,&k,o@+;&(@o*AC)*#/$*
Ao
Um'.'<(!/$3^
+P'<,
AC=,&7'.3
2&
+E
<(')$>&')'.'<"')1')(3G1
2&/'<HKTU*AV*Mj'.,')
$"#0RU=
*(@P$"#)'<*%'<8')RAC'))
7R(667=
')[0'.2')%&(@:;')/]
')'.'<m
[%"=&<.+E(2
"')1')(3G1
2&/')47+ER"3
<&')/PK'.;AC'.9'.')$m"#')1'<(@3Z1
2/$'),&B&
')
(@*P"/]"')1')(3G
*&(2jH
tym/27 M
'</')/B
"(28,
*&(2]&)/$'.n')V "#"
A&oRH LV"U[9 = h@_^i=d cZh 4<&(@or&%("1K3Z
')/$*
*(@:)/'.;MRAV441'))7)B%"
B/$2&)3G"R$
"')1')(3G
*&(2F@P8
"((
1 "<.+:&(*-"'.6&E'<1
3
')/K‚1
2&/&+P5$"#y7R(2%&o#$7<;H
GD
Equilibrium A
Equilibrium B
OS
RE
3 PK9)8'++,&.-/#N02%3&45+,6089,-:#% < .'0 =#D &
*B9CG&DD.%E-1,'*&D$+%.CT!#86/&D8,-:;#% 6:!#9IK+%35+%E-/*&,-/#%
YZ*- I1*9% #D)8-:M456/.056/-/&D-/*2 W 7 98\ U*$6/96A:=< &%80N-:!
#D;+%&D' PK#-/!3CTS *UA6/- $&.-/*2JWP[S]\
7"1.+4/$')%W<&/y&'<&(')K%&(2)2M7"#m/$')
&
'<C#'<&Ko*AC)*#/$;&7R$"#$&)"=&')J#')1
'<7"')$K')L
%&)y%o*MVJ;%+D#')18]XY(')$M0=
')<@,1
2&/&+H6lZr6&
')("N[(*
4=&W$&<"&')$')13
')7"')$K&8(1
"(@*967R$Mk>%1
2&/+  ‰
AC'))R@
ˆ%o$&7.+A<\H
lZ')=&RAVW=*y
(@'<7RPB=@K &C1
2&
3
/+ &(*-"')
'.'<ƒ5r#3G<+P$#3G?3G
" /$MW=wAV
*y
"((
15").+P)'<8"@K,W&(*-"')
'.')$8
2')/U1
2&/&+450#$"7)3Z"(')$6%o*”$y(/$'<
7R(2%&o47)x 0')%')jH  "@K(67')
ZAF#'<1')/F</$')')JF
+>
'<)8A*+Y‡]'.'.3
(V/$3^
+:&=F"')1')(
*(@ 2\'):')[(*
$M!"
')/]
/@'<(@3Q&)/$'. H  "yo+]($
')7"#')$r')W#$%')r')
0"
CR07=&
'<9 #3G
'<?y
')(U2&oB1
2&/')Hpty."/$
AV%=*&B'N#*>]$"/$9&=&)+')(*&'.u=(&')$9&
"yYMAVB7R<')$[&U&')12/$B'.yj5*&"#
@97R%
P&y'.')(*/$3^
+>&8
'<)
$"/$]&6
"((
15")'<(&')$;&$"#C6A<.3
oACE7R(@%&oP$7)’
" >(]#$1
2&[BR3
'<&‚5 ""#U
*(@‚H
10. SUMMARY
qW%')%A*&o
0&k$"#0("#
C&(@P1nN$;‡
9 R&'<VAV*o
6')>"=#.+')/;'.')(*&V/&3
+%@)$/+M&=%7 9 U<(2o%kB("1')
*8
&(2
&)/$'.˜2&')<&*;
R('.u(*).+]5C')%')jH
t6/$&2% MVAV](/$')
]&'.')(*C/&3^&
+
')] /'</>9 u=<YHntP
"(@‚MW(P=w@u=
E($')"=).+J7R')/>%]D#
8
r['.%*&)')
AV*o
H &4@&)$M!>)
>6\H„ & ‹ †C#
(')7R8]3
uC[]AC')(2E))ACW5[&B#')('))*
&#(2r &,@')')/')%!&)[
')?$M‚>'„ &Œ†m#3
(@'<7R8P5"
4#*')%'<F(2'<-"6=B))*AC4 &4
/&*&.+J*"(*w"47R&[1
2&/&+z&u=)%97Rr($
'<#3
@*‚Hmq
W&=%@u=AC'<)Y7R'<(R&2&*8')
11. FURTHER WORK
12.
„'&@†HOFREFERENCES
H H.6H OFH NyMH KHJO;&(,')3 O;
$:&=
tBH  =+#
'<)'\H #)&'</7'<#')/61
2/$')C &
%&o
3G7
*%
(")')/#H h
gd _d^`e bii#` } c _cZh ~ _ M
Š EE ŽH
„ Š*†[l@H 8&=]t[H 0H Ny&jHI*#')(')/KAƒR),<+
/‡ N')5(C<&')/K');#R')2Y/&
AC')]"'<-"M'.#*%1
2&/&+]-"')<')7#'NH~ h } d^g | e
# g`e=` ~ d^g h $ d^h@• M–$–>2\Œ 9 ‡ –Œ$– #–$–>&M &*—$—–#H
„ ‹ †4‰YH !'<('<('^H `$a bc\d^`&e
m`&eRg2hZi=c\_,d e
m`h $ `$a bc\d^`&e |} } ~ HRID v
')M Lp2') )U')@
')Z+M O;+
Š Ea ‘E Œ` H d5c _
„ Œ&†4‰YH !'<('<('&=HI!$)<(2oRHRtƒ/@3G@'<(,
+
(@'<
y($$)"')$‚HlZ; @( @ M=/$
Š–$… ##Š&—“Hj‰')/
3?p@N&/M=Š E*E ‹ H
„ Ž*†4‰YH H !')(')('=H LyHRI!<<&(@oRHRtƒ/&3^&')(
#&(@]4W
'))W($)"#'<=
+r)/'.;HlQ
>2H‰#')/$ m</M &*…3QŠ E Š E*EE H
„ …† 6H #'<%PH Ny"1*H h `b a<h bgc\d^`e
F|} h@c H>1#')A,M &*—>— &H
„ “*†4‰YH 112;.H W')(@o=&"*HIk'<([5%'<;')
$"7),&"(')$HC|&~ h@_ | e # g`eR` ~ d^g!Uh
*| $ d^`&b }2M
Š$Ї & ##Š—M8&*——$–#H
„ –†"H U
‚H m-"'))'<7#')"ˆR$')C')]3GR
;/&HlQ
0}
[| | | ~
M=<" ‹ …M
=/`*g$C#m`ŒW– #c Œ—h 3M &*—$c\Ž d^E `He a g h 8` gd^h@eRg2h@_
„ —†"H y')($<')
‚M 4HI!
RM= mHq1\
')$jJH O;&o@
RAVyP >('<(+P')r6("2')$=&Y)(
')('.Q+
%&oVAC');')
(')')=&&
+;#$"7)3Z"(')$]#')(')/H
# #G#%}1|
# $
~ | MY@Ž 2GŽ ‡ Ž Œ ##ŽŠ ‹ M
 ($7RUŠ eEE_&#&H `e `$a'#
m` i=bc c\d^`e 9 E
„'& E †4‰YHIm)M=‰YHIk&
MY&=]I!H O]+( LV"#+Hty
$)"#'<=
+]/$3^')(W(='<
')r!ZA
$"7),&"(')$]%&o#
')/$H=lQ ‘h@ec^f h d | cZh # a<h
gc } `&ed^g
m` ~6~ h } g2h 2HR‰')/
3?p<&/M=Š EE ŒH
„'&&@†"HR‰'.jH # $ `a b#c\d\`e | Je c h[c h` } ]` ‘ |~ h@_ H
{67#'</$ )U')@
')Z+PI
M &*—–ŠH
„'&Š*†, H>)
jM N[.H WM Hq&"#MRH,=&
*H
ty=&.+?'</K()%1
2&/$')(['<2&(')$y')
6".'.3Z/C/HlZ()*) f,(+ @.-]` } _ #`@i>`&e 0|~ h
h
` } h@c\d^g | Je h
gd _d^`e h
` } h@c\d^g ‘h@ec\_ MYŠ EE ŠH
„'& ‹ †, H>)
j.M 6HIk&o$MR&=B N[.H ,H={V
'</<
K($"y"#'<1')(3G1
2/+P=&
;*-"'))')7')";H#lQ
‘h@ec^f h d | cZ
h # a<h
gc } `&ed^g
m` ~6~ h } g2h ‚MRŠ EE ‹ H
„'&Œ&H† OFH=I!H#>))%‚.M 4JH OFH NyMKJH OFH j(2o*M
NBH‰#"#'\HR‰*&(2')/4 CA)')H=lQ0} `*g2h2h d5e‘_4` Wc h
0/ * f1+32-;` } _ `@i`e }
| d e‘ ‘h@ec h@_d<‘e | e e | a &_d _ MRŠ EE ŽH
1
0.9
0.8
Mean fitness
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
Generation
25
30
35
; *&% < -:%96:6 #D= -:!3 : )Z#)88'&D-/#% IK-:! #% 6/-/&%0&D$0A*$#D
1
OS
TT
GD
RE
0.9
0.8
P(strategy)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
Time
E PK9)8'++,&.-/# 02%3&45+,6 -:M4N 6/*$96 )('#-5=# & 9B9CG&.9%R- ,'*&D$+%.CT!#86/J&8,-:#% 6:!#9IK+%3 +%E-/*&,-:;#%
YZ*- I1*9% #)-:M45+6/90A6/- $&.-/*2JW 7 98\1U*6/96 :=<Z" &%80 -:! #$%&'PK#-:!3CGS *UA6/-/&D- 9D2 WP[S7\