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¾ ] ¾ 68!eCS=(:t;=(*m"$E/%$<C%2R3:37*!(MR3:"$'&37:<+:E-hRW X'@"$O:j32:t%$GN*&=&a%$ U]\!G39%$*-RW X'@"$29%$*&=&%$E9]7<QL;- J#'@"$:j3$B:-%2%$"$*C<Q!*#/g!;H@%232L9T@Ä=@%$*'&Eh2<C*#/R'1]\#;=CORW Q'&"$33E"$;!jQ/<Q!*J: R3+_ JU'@"$r33E"$;!j:9T ¾ ] ÇOQP!RLS(TU(VWS(Xbg
i ∈ NR3:"$'&
si ∈ Si33E"$;!j:R3r/<Q!*J: R3_
si33E"$;!j:9]
\Jui(si, s−i) ≥ ui(si, s−i) ∀ s−i
A^E%"2
ui(si, s−i) > ui(si, s−i)G%$;&j: 6%$;=
s−iA^E%T
¾ ] Ã OQP!RLS(TU(VWS(Xb(i ∈ N
R3:"$'@si ∈ Si
33E"$;!j:R3r%$ENh"$E3%$G%$<H 1%$*D/<Q!*J: R3
si33E"B;!j:9]#\Jgb
ui(si, s−i) > ui(si, s−i) ∀ s−iA^E%T
¾ ] ¾ 68!S=(:É<+:3E3!PML;=Mj'(
A \ B J RW
RW (2,−2) (3,−3)
\!G39%$*AR3:"$'&-C J33E"$;!j:\JC*#%$<S;7R3C:j32*!^])L;=
BR3:"$'@-37:7A
<+:E+K2'iQ&%232%2"$;+<Q!*!<+!7::+<+E/o|oC J33E"$;!j:2¿XSE3!*(9]J<QL;_RW J Á SE3!*&r@%232L9To 7+I JPAÄ33E"$;!jM/<Q!*J: R3m_RW '@"$$AÄ33E"$;!j:9T 7H-R3:"$'D%$;=@%$*#3u=(01*&SR3CÂ RW U]g J(^AÄ33E"$;!j0J:E9T ¾ ] Æ ¤OQP!RLS(TU(VWS(XbO>?;=
s = (s1, . . . , sn)33E"B;!j9&%B'(E3Dú-3\gAÄ%$;=&%$*#3u=(*J'
*#%$&%B22w'1]\Jui(s) ≥ ui(si, s−i) ∀ i ∈ I
"2 ∀ si ∈ SiA^E%T
¾ ] Ê^OQP!RLS(TU(VWS(Xb)>?;=s = (s1, . . . , sn)
33E"$;!j2&%$'(E3-32!;@E3u"$E3%$G%$<_ )%$*&%$3ú-3\gAÄ%$;=&%$*#3u=(*J'I*#%$@%$22w'1]\J
ui(s) > ui(si, s−i) ∀ i ∈ I"2 ∀ si ∈ Si
A^E%Z0)"$G/#i: j:7: )?RWOj:3\J]\#;=Yä RW U] J(^AÄ33E"$;!j0J:E3 1 %$;=(!'
S"$*#%$'d%$< "BE/%$'@%D%$<C9/g*!^]?\!G39%$*vK2'sE37 Y\#%$=(9%$3 )%D'&%$E3w\#%$9T>?2+<C%$;@G/#:3<C&/ \LR'MX/<Q!*J:t33E"$;!j:'D%$!<Q!*J:K$!GR3:*J'1T
¾ Ç ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌ ¾ ] Á 68!Zå)c#;@=M/g!G%$<Q<+"$- 5#*#a9Nr3u=@t 5*("$*(=i%$'&a9&%$"2"$@%$:/*J'1]1"2-'(wGa*gA^'(wGa*E3SR3:'
N'&%BS;gT,.!9*(=(LP"B'M%$G%$*(w'MK2'DK$%$'&"$=I<C"$E3"$'( )%$*:oE%$*#/%$'&%$9"23E%T46!*#/g'&"$rS;@=37:<+:Em'&"$X%$E3*J3Li:.E%$*#/%$'@%$9"23E%bkj*!^]n;=7A
;&/g*!^|\JYj'(!^]'('&E6d 5*(:E:m:/ R3Y"2mR3:M<+;&:E33j*#:;&<C%$G%$3D"$E3@%$^TZ£Ä;='(! y 9%BN<+:3E3!PæS w*&%$"2I<C"$E3"$'@"$E%+'&a9&%$3'@%$9N'&"$00)%$*j'(^b
A \ B *#%$<Z
ÇC"$1]ÇC"$! SÈX"$1] Ã "$!*#%$<Z Ã "$1]JÈX"$! ¾ "$1] ¾ "$!
x: j:7r37:<+! 1(!j:;@*'&a9&%$3'@%$2!'1])\#;=i+33E"$;!jD/<Q!*J: R3I*#%$<ö33E"$;!j:9]U\!G39%$*
4 < 5"20 < 1
C%$E%2/g<C"$*(=@T£Ä;=i<Q!*#/sI'&"$ 5gA*#a9Nrj*!JS;_"2
(4, 4)"$Q w*(%$"23h'0*J'1](<C%$=C*J;=@ U]&<Q!*&
(1, 1)"$1T
7@]J\JDrR3:"$'@'m*#%$<'&&01%$E:*J'1]);=m*#%$<\!\#%$3*#%$'C<+:3!'m'&&01%$E3!PA(!:: J*U]J'('@E %$;=I!=@%$*m!*gS%$E3!GEt:j0)-:Pn%$GNTk%$'(!*&3w'6:j:*#:; J*U])\J+XS%$*(3!n01"$G/#: JC*#%$<"$&%$'@%$9]#\)*#%$<\J32*#WA
:;@'LE3*'1]J'('&E9b
A \ B*#%$<'&&01%$E: '@(0)%$E:
*#%$<Z'&&01%$E:(p, p) (t, s)'&&01%$E:(s, t) (r, r)
\#s < p < r < t
]g%$'('@EOG"$3E%WRWa*mrS;@=I/g!G%$<Q<+g]#7r R3:"$'@'+/#<Q!P*):*#33E"$;!j:R3CX/%$9%$E3:j:QS_*#%$<Z'&&01%$E:j:BT j: J*Ma D39%$E%$0GNhS;@=C/g!G%$<Q<+çkR3:"$'&'.39':h'@a3RW37:;#]@'&a$AG%$;@%$GNQ;=I:E/#<Q!kK$01/#-R3:"$'@'(*J'I*#%$&%$2*!^T ¾ T Ç68!scJ;@=D/g!G%$<Q<+:EgT"$k'(wGa*( )a9N-/gE;;&gw2G%$%$GN-/gE;'&%$E%23'@%2/Nt<C%$;&:j0)(/g!'1]\#;=r0E y )NA
(L%$*#%$'Q"2h-E%$*#/%$'&%$9"23w'(E%t:Gr/gE;_%$;=_E"239"$h%$GK2%$E"$!'C%$;=(<+:^T@ E: J**#%$<x!G3<C%$E3"$'X%$;=(<+:3?"2'&"2N Q%$< S;*J'Hj:'&2*!^TK2%$E"$?E%$*#/g'(L(w;=&EWA*m'@%$UG%$ )*(=@L*!^]17r0E J:EQ*!*#K2!G/N]'@"$E/"29]\#;=M"$E/%$<C%2WAÄ%rK2*!'1]7rG&/g!n/gE;C\#%$=@%$3\J<Q!Gt/g*!kQK2%BE"B*J"B(M/gE;C0!jK$!n"$E3"$'@%G%$;=@%$*mÇ+%B;=g"$;#]#a*'@a"$;@% ¾ %$;=g"$;#T
A \ B!;&2! \J<Q!G
!;&2!(3, 3) (−1, 4)\J<Q!G(4,−1) (0, 0)èéÂêìëíëî ïðñ î ò¡óó6ôõxò¡ömòø÷&é&î ù^òøúo÷&ûüxý`þmòøúCôð÷`ô8ÿrù^ò¡õôpôoëõxî G÷ xüB÷ ¡þmëþõpô@þ
ÇmÈ ÎÈ Òp@áßgß@Þ)â<Ë ¾9Ã
-M!;&2!_K2%$E"$*#%$'!;&2!E(4 − 1) ]k7 Á %$;=g"$;*&=(!h\J39*s'&%$G%$3'@%$2!'1]U\J\J<Q!Gt!;&2!Eg]1'('@E-Q\J<Q!Gt/E;@-/#CÇI%$;=g"$;+\J32*#9]J<QL;DQ<+:3!'DS"$ ¾
%$;=g"$;*&=(!n@%232%2"$;@%$E%9!7:^T): j:73 ))RWUj:32!'1]\#;=mX\)<Q!t33E"$;!j/<Q!*J: R3Cr!;&2!n33E"$;!j:9T ¾ ] Ã 68!Zå)út%$<C%$'I\J: 1E3uR3['&a2!G3<C%$E3 R3:"$'a gS"$G%+aE3"$*#%$3%$.!G3<C%$E39Tn>?;=0J:E_%$;=(w3_39%$E%$3*#"+a%$*!X%23"$9]1XS%$G%2"$;D32L*\J:2 Jg]oQS"$ESR<C%2K2K23E%_39%$E%$3*#%H<C%$**!^]1/%_%$GN*&=( 1%$*E"232%BA3L%$*#"$'MH%$;=(w3aa3 %23"$9T
S"$E y \ *#N 32L*\J: <C%2K2K232L*\J:
(1, 2) (0, 0)<C%2K2K2(0, 0) (2, 1)
Ö6×øØ× =<>B¥G7:H¨ £ÄESR*'rS%$):PBG:*J*+%$;=_'&"$39%$<C"$=@%2oR3:"$'&9]\J |S1| = |S2| = 2 T46!'&E?G%23%$;=IEa;2L%$3t33E"$;!j7A^0J:Et/<Q!*J:*#MÏ 4i%$=(!'_R3:"$'@:G32*):g]\J *mG%$**#%HrBTR3:"$'&M
_b BT \ BT l zÍ
(10, 1) (2,−1)(8, 5) (3, 4)
,-b BT \ BT l zÍ
(7, 3) (4, 4)(2, 5) (8, 8)
Ô! /3'&"$hOÍ AÄK2E3*Jg]&<Q!*#/g'@%$3NH'@"$W^R<H5#E3.3;&:E2\JO%$;=C/3h!G/NA0)*(3 J*U])_*#"$9%$3"$;!kE:*(=@'t_j: !U<+:3E3!PM<HSR3gb
1. \ 2. ck!< f&01E3ck!<
(55%, 45%) (52%, 48%)f&01E3(50%, 50%) (45%, 55%)
>?G%$<C%$22w'6tR3:"$'&946!oaE3"$*!'1]\J(F,F)
"BE"$'@%(66, 34)
A^E%:2!'!MÝ Ô! S%$j/#3 J*i/3?R3:"$'&'@*(@%$E3: R'M:9"$EAÄa39%$;5CR3:"$'(':g]g"2%$**#%$'%$;L"$;@"$@%$U<+;=(E:22'mHS%$*('013%$E%2/g<C"$*(=(9ì" "$?R3:"$'&!GK$!:\J
(s1, s2)¾$# A^EgT!-
s1 + s2 ≤ 1]#'('&E<Q!*#/g'@%$3%B*6<C%$;A
'0R3:'M_!GK$!SRw'&%$9T$-s1 + s2 > 1
]J'('@E%$;=(!'(w'D%$<'06%$<Q<Q!9TO46!'DQ32!;@E3uJ*6/<Q!*J:t33E"$;!j:'!M :O46!'DX;=@%$*;@"$*6/<Q!*J:t33E"$;!j:'!MKO,.!9*&=(L3'D )%]\#;=I*!*#K2 /<Q!*J:*# 33E"$;!jg
¾9Æ ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌÖ6×&%µ× ' 7:=IÙWF6>¡7(¥8?A«B5¥65¥ßC«B8t=('(E*U])-R3:"$'M<C%$;@G/#:+E:*XG%$;S*( i0E G"$<+g]J\#;=m<Q!=@%$*6S%B%BA&"$%$1"$\#%$3w*#'DH<+:3!'rR3:"$'&3E^Tg>?;=@%$GNE%tS%$"$%$G%$22w'1]g\#;=C<Q!*#/%$*'(!oEK$!GA*J:!G*6K2%$G%$'g32!'1T ¾ ] ËOQP!RLS(TU(VWS(Xb%$;=(w'CS%$^](\#;=
iR3:"$'&
ui%$;@*#/j:UE%$*#/%$'@%$2!'D%$G%$*gS%BA
G"$EN^T( -%$;@*#/j:2kRW%$G%$*&3!^]&\#;=Ir<+:3!'R3:"$'&O-%$;=&%2.33E"$;!j:!O<Q!=@%$*G32L*&5#"$;;@%$UR3:37I<C%$;#]U7
µi%$;=%$G32j:9To>?'('@E
iR3:"$'&-EK$!G*J:!G
µi%$;@*#/j:^]#\J
si<+7g!<+!7: R3+ ∑
s−i
ui(si, s−i)µ(s−i)'(!PS%WRW%$9"239T
`S;@=C/g!G%$<Q<+:*J:*) ¾ ] Á,+ 01"BG/#j:33'1]@\#;=IEK$!G*J:!GkR3:"$'&.\Jr%$<'&&0)%$E:^TJn%$'(!*(3w'I<C3t_'@a7@%$3'&%$9N-R3:"$'&9T
¾ \ ¿ s12 s22 s32s11 (2, 2) (1, 1) (4, 0)
s21 (1, 2) (4, 1) (3, 5)
¾ T-Q¿gTEK$!G*J:!Gs22A^*#%$<xR3:37gT
¿gT ¾ T#R3:"$'&r3#/R3m%$29]U\JDS%$%232!^]U\#;=¿gTUEK$!G*J:!G9]nL;=NM*#%$<R3:376s21A^9T
Á T¿gT?32!*("$*YS%$%232!^]\#;= ¾ TkEK$!G*J:!G9T-%$2XS%$%232!^]'('@EQ3#/R3g]\#;=*#%$<S;RW
s21A^?R3:37*!^T!£Ä;=+*#%$'(!U*#%$<"$E3%$<C%2
s32A^9T
Ç#T£Ä;=DK2'(s11, s
12)33E"$;!jX0J:E<+E/oT
Y%$*(3!(0)"$G/#: 1g(!j:;@::!'1]9\#;=r*#%$<CK2'XOS*(9]7\#;=H"$*_EK$!G*J:!G;=@'1]&\J*#%$<t!G9]&\#;=C%$G%$*gS%$G%$<Z@Tf(N9]g-%$G%$*gS%$G%$<Q*#%$'+3#/g*!jr'&%$^]&\#;="$*DEK$!G*J:!G;=@'I"22!G9](\#;=+3#/<m]g\#;=DN_EK$!G*J:!G9T)f(N9](3#/<m]\#;=NX3#/R3g]\#;=D"$*mEK$!G*J:!G;=&'1]#7__EK$!G*J!:-_'&a2a3#/#:E"239%T ¾ ] Ì6OQP!RLS(TU(VWS(Xb#Ä%$E: /<Q!*J*#K$!j¾ TgG"$01"29]G%$;=@%$*
S0i = Si¿gTgG"$01"29]G%$;=@%$*S1i =
si ∈ S0i | 6 ∃ s′i ∈ S0i : ui(s
′i, s−i) > ui(si, s−i)∀ s−i ∈ S0−i
kTgG"$01"29]G%B;=&%$*
Sk+1i =
si ∈ Sk
i | 6 ∃ s′i ∈ Ski : ui(s
′i, s−i) > ui(si, s−i)∀ s−i ∈ Sk
−i
ÇmÈ -È Ñ Þ@à^ÊUáÄÞ.)âã6Ñ ÌgßÌÓ!Ñ ß ¾ Ê
G%$;=@%$*UbS∞i =
∞⋂
i=1
SkiSH%$ R3:E:&"B;@%2G"B01"23 )%$*6 )%BS%WRW%$9N&/!P'1]\J |Si| <∞ ∀ i
- |S∞| = 1]1'('@E-rR3:"$'& !%$E:/<Q!*J*#K$!j:9<C%$;@G/g\J*J'6*#%$@%$$A
2w'1T ¾ ] ¿Í bo|fb|pt_@qn\J*G+%$ R3:E:U'@*#33E3J:\JDr!%$E: ;=@%$*;@%H/<Q!*J*#K$!PA:9UGX<C%$;@G/g\J:;#T ¾ ] Á Í bo|fb|pt_@q1(!;=&:79]r;=@%$*;@%-/<Q!*J*#K$!j:91G_!%$E:j:.*#%$< uhw;;@%$WAG%$*U ¾ ] Æ 68!T#k%$'(!*&3w'mHj: !#R3:"$'&9b1\2 s1
2s2
2
s1
1 /1032 465 /7482 05s2
1 /:92 05 /1032 9,5s3
1 /:92 05 /7482 05s3
1≥ s2
1=⇒
1\2 s1
2s2
2
s1
1 /1032 465 /7482 05s3
1 /:92 05 /7482 05 s1
2≥ s2
2=⇒
1\2 s1
2
s1
1 /1032 465s3
1 /:92 05s3
1≥ s1
1=⇒ 1\2 s1
2
s3
1 /:92 051\2 s1
2s2
2
s1
1 /1032 465 /7482 05s2
1 /:92 05 /1032 9,5s3
1 /:92 05 /7482 05s3
1≥ s1
1=⇒
1\2 s1
2s2
2
s2
1 /:92 05 /1032 9,5s3
1 /:92 05 /7482 05 s2
2≥ s1
2=⇒
1\2 s2
2
s2
1 /1032 9,5s3
1 /7482 05s3
1> s2
1=⇒ 1\2 s2
2
s3
1 /7482 05 ¾ ] Ê^8!:TJ.E3*#WA^@%$E%$*(=1b# .E3*#WA^@%$E%$*(=I !9*(=@%2%$%$'( )%$*m!%$E:K$!Ao<C%$;@G/g\JTJlU%$;=&%$*i_'&%$E%23G%$3!1w;;&"$*&=D!P*J%2:E!9T-!!*#/gj:3'&"$*&IS%$%$\#%$SRw'1]h\J;=sK2"$;@%$' J:E3<C%$**&=(!-%$E3<C"$'&%$I;=(:E33\J3*J'
S0i = RT& 03!<+:!G.<C%$**&=(!G"$;Q9*& J*_RW#j:3\J@*MÈ"2 a−c
2b
'&a9" %23!'1T@>?9%$*!*(%$E3P<C*i'(L(w!o%$E3<C%$G"2*#%$;&3LD0E y \#@]L;=D%$!<Q!*J:\JTq'@a7@%$3'&%$9NrG"$0)"23 1%$*U]&<Q!@%$
FK2a'('@%$*#N](r325'&%$ +!*(%$E3P<QEBS=(SR'
r%$ R3:E:3 33 UT,O%$j:3\J]#\#;=M_S%$*&3!n%$ RW:EB:'@*(@%$E3;@%$*#9T
¾ ] ÇÍ b|fbo|pt_@qJÍ-"$;@%233E"$;!j\J<+9'(EHS∞ 6= ∅ ]<Q!&%$J %$;=&%2?G"$0)"2%$':*i<Q!*#/g!;I<+E/i%$G%$<mTCC33E"$;!jC\J<+9'm*#%$<!=@%$*#%$'1]1Q\#%$=(9%$ *#%$<
!=@%$*6%$;=g39%$E35UT#,.!9*&=&OS%$%$&"2%$'('@%$U"$@%H9*& J*S∞ 6= ∅ %$G"$E3\#%$NT
¾ Ë ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌ ¾ ] Ë68!T#,O%$E33E*#/m@%$E%$*(= S:E3@%$E%$*(=!"$C:jj+*`0!jK2*U].**J'%$E3<C"$'@"$+@%232!'Y<C%$;#]h'(!*#%$'
pi:E
jK2*&=( %$;=&%$*GN6:EH%2%$"$*<C%$;@32!''@a9a33w'D'@%$E%23G%$BTU C%$GN9Nm0)"$G/#RW%$GaG"2"$\#%$ \J*G@*M'(! )(K2::
qi'@a"$;
ci(qi)T
ci(0) = 0]&ta32'&%$E%2%$
Q(P )]!*&&%$E39%
P (Q)T£Ä;=
N = 1, 2 si = pi Si = R+
ui(p1, p2)−
0\J
si > sj
siQ(si)− ci(Q(si)
) \Jsi < sj
siQ(si)2 − ci
(Q(si)2
) \Jsi = sj .
¾ ] Ì68!T).E3*JWA^@%$E%$*(=1bG%$;=@%$*iX'@%$E%2%$3!ow#;;@"$*(=I!*#%9:E3!Gci(qi) = cqi p = a− bQ
_'(! y 9%$NXw;;&"$*&=ui(qi) = (a− c)qi − bqiq−i − bq2i>? )Nk/%$E3!:X'0R'DQ39"$GN&"$E3"$'&%$9b
a− c− bq−i − 2bqi = 0.£Ä;=M^G%$;7RW 6:j3bBRi(q−i) = max
(a− c)
2b− q−i
2, 0
ú-3\gAÄ%$;=&%$*#3u= ¾ ] ¾ <;QU1=>@?bBAµh
s∗ C hxqD8E`dbo|npcGF\xqHhxaapnpj s∗ ∈ S∞
I letnpg!6Jz]\xqÄ*#/g!E%$'(-<C(/* !9*&=(L3#*'1Ton%$;=(w'mS%$^])\#;=C:L:*#%$<!;&@T>?'('@E ∃ k
]g\#;=s∗ ∈ Sk
1 × . . .× Skn
]/%s∗ ∈ Sk+1
1 × . . .× Sk+1n !%$E: %$ R3:E:3 1k/&/@* ∃ s′i ∈ Sk
i ⊆ Si]g<C%$=(E%b
ui(s′i, s−i) > ui(s
∗i , s−i) ∀ s−i ∈ Sk
−i£Ä;= ∃ s′i ∈ Si]n<C%$=(E%
ui(s′i, s
∗−i) > ui(s
∗i , s
∗−i)]%$M%$G%$*(3<C*#/**J'1]\#;=
s∗ú-3\gAÄ%$;=&%$*#3u=@T ¾ ] ¾ È68!:T).E3*#WA^@%$E%$*(=-:E22'D<C%$;IC.E3*#WA^&%$E%$*(=(E%HQú-3\gAÄ%$;=@%$*#3u=(9T
ÇmÈ -È Ñ Þ@à^ÊUáÄÞ.)âã6Ñ ÌgßÌÓ!Ñ ß ¾ Ì
¾ ] ¾ KPLML6T(q∗1, q
∗2)C hxqD8Eb|pbg$q®c µu$npg]nxqfhpg hxaapnpjHNDLh
q∗i ∈ BRi(q−i) ∀ iE`j@b
BRiEa^bm]®m@jBhpd\xqfnpc Oxh
BRi(qj) =
a−c2b −
qj2
Dhqj ≤ a−c
b0
acGPpg"Q@bgh C hxqD8Eb|pbg$q®c h q1 = BR1(q2)
_@qq2 = BR2(q1)
R qBtml(dQdbm]zjlohdQloh^]o]`
q∗1 = q∗2 = q∗HNJì|^
q∗ =a− c
2b− q∗
2
q∗ =a− c
3b.
S P8QTQ@qBtTPpjPxq C hxqD8Eb|pbg$q®c k%$'(!*&3w'M_j: !#R3:"$'&9b
¾ ] ¾¾ 68!:T¾ \ ¿ s12 s22s11
S¿g] ¿ SÈg] Ès21
SÈg] È S¿g] ¿RWoj:3\J]#\#;=MtR3:"$'(*J'D'@"$ú-3\gAÄ%$;=&%$*#3u=I0)*(SR3X*
(s11, s12)
"2(s21, s
22) .
¿È ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌ ¾ ] Ã Í bo|fb|pt_@qJ:E/#G<H )*+;=&'(E*XSE/g#P*)'X%$GN\J*Gr%2%$%$'1]%$9%$'@%$:j: J*D*#E3<+:'('^]g37 J:=@'(')'@%$9%$ Rw'1T(q'&a2G%$'@%2/"23 1%$*IG%$\#%$RW Er\JR"2 JEX\JR%$@%$ "$E3&"$*&=&%23L%$*!^]g<Q!*#/6X'@%$3NQ<H5'&a&/g!'1]#/%*#%$<Z%$;=g39%$E3E%Tp'&a9&%$3'@%$9N'( 1%$*<C%$;!G3<C%$E3w*''@"$9]UD'@a2;&9/#:;*( J*Yj0(@%$NQR3:"$'&9]
<C%$=@%$'m_'&"2N !1S%WRW%$9%$%$'( )%$*muREX<C%$;CuREX%$G%$<Q9"2%$!*'D:E3;=(!oG%232*#%$'1b ¾ ] ¾ ¿68!:T).E3*#WA^@%$E%$*(=1%$E3<C%$G"23!U&%$E%$*(=!"$XK2"$;i@%$E%$*(=@%$Dm0!jK2*UTk '(! 1&K2::3tRW%$Ga Rw'
qiA^@%$O"$QM'@a"$;w;;A
&"B*(=(w*'IG%$;=@%$*ci(qi)
TÉ'@"$%$E3<C"$'DG"$*(=@%$;@"$ )%$*6;=&*J@]JHa32'@%$E%23G%$Q(P )
]!*(@%$E39%r01%$/!P;
P (Q)TJ _:EtQ'@"$ %$E3<C"$'M'(! )(K2:::*J'Dw;;@"$*(=@%TJ%$E3<C%$GN'
0E y SR'D<+7g!<+!7:j::EQaE%$'@%2/g*#%$'1TN = 1, 2
si = qi S1 = S2 = R+
u1(q1, q2) = q1p(q1 + q2)− c1(q1)
u2(q1, q2) = q2p(q1 + q2)− c2(q2)7ui(q1, q2) = p(s1 + s2)si − ci(si)
¾ ] ¾9Á 68!:T)f&K'&%$ )%$E3;A^&%$E%$*&=&T&^f&9%$'(&%$*#K$!j:!G%$E3<C%$G"23!U&%$E%$*&=! -%$GNXK2"$;
q1A^O%$E3<C%$^Tg<+:&/g!'IK2"$;Q%$2O<C%$; y ;=@%$!U"2
q2A^O%$E3<C%$^T
s1 = q1S1 = R+ s2 : R+ −→ R S2
0)%2/g!;I_w;;@"$*(=("$E9Tu1(s1, s2) = p
(s1 + s2(s1)
)s1 − c(s1)
u2(s1, s2) = p(s1 + s2(s1)
)s2(s1)− c2
(s2(s1)
).
¾ ] Æ GÍ b|fbo|pt_@qo`S%$*&3!)01"$G/#:'Ir3*#/#E/I<Q!'(EAÄa'@*#<Q!j!o'@a*(=(@%$'( 1%$*I%$G%$<XA9%$3%$'1TZ-a'@"$G%$%2I0!jK$!t&%$E%$*&=(I"$%$G%$2w*'vS%$^]O'('@EM:jj'K2'`:E'&*Q'&%$E%2323w#3#/R3:'_ )%BS=(:*!)t0!jK29T@>?'('&E9]\Jt*!*#K2!*&%$E'&K$!GHK2"B;@%$''&a2a9]#X0E y <+7g!<+!7:j: /a*("23!U0E G"$<+:2:Q:!'1T
Ö6× U× =<>B¥G7:Hñ! G/R'M<C%$;MQG: # !k<+:3E3!P37?/3.R3:"$'@C/<Q!*J:*#-33E"$;!j:'6%$!<Q!PA*J:j::9^
s1 s2 s3t1 (6, 1) (3, 4) (4, 2)
t2 (7, 2) (1, 1) (2, 5)
t3 (4, 3) (2, 3) (3, 4)
ÇmÈ VÈ+ߤÉÊ@ÞËÌXW&@à$6Y:Ï@â<ÌÞÉ3ß ¿ ¾
øB G/R'I<C%$;+Hj: !U<+:3E3!P37n/39"$EAÄa39%$;5QR3:"$'& X/<Q!*J: 33E7A"$;!j:'ME7%$!<Q!*J:j::9^
t1 t2 t3 t4 t5 t6s1 2 2 2 2 2 2
s2 6 4 2 2 3 3
s3 6 5 3 3 4 4
s4 6 9 9 2 3 4
s5 6 5 6 1 4 4
s6 6 5 5 0 4 4
ý! k%$'(!*&3w'6Xj: !JR3:"$'&9Jln:3'D 1%]#\#;=6X;=&%$*;@"$*i/<Q!*J:-33E"$;!j:'%$!<Q!*J:j:CE:*M'&"$'(wGa*& 1a9NC%$E%2/g<C"$*(=(E%R3\J3*'1
BT \ BT l zÍ
(5, 1) (4, 0)4(6, 0) (3, 1)(6, 4) (4, 4)
ç! ln:3'm )%])\#;=iC;=&%$*;@"$*/<Q!*J:r33E"$;!j:'MS'&7t'(!'(w#39a )aG"2%X%$;=*#%$<E%$C\J"$'@*(=M%$;=@%$*#3u=(\#r@%$9%$t_G: # !#R3:"$'( J*Ub
l 4 zÍ
(3, 3) (0, 3) (0, 0)>(3, 0) (2, 2) (0, 2)(0, 0) (2, 0) (1, 1)
¨[Z! n%$'(!*(3w'Ij: !1'(! y 9%$"23!)<+:3E3!P37U/3kR3:"$'&9(4i%$=+33E"$;!j:'C<+7AE/g*J'mX/<Q!*J:t33E"$;!j:'M%$!<Q!*J:j:X:*:MH46!n_3!G32Qú-3\gAÄ%$;=&%B*J3u=!M
l z
(2, 0) (1, 1) (4, 2)4(3, 4) (1, 2) (2, 3),(1, 3) (0, 2) (3, 0)
Ö6×]\× © E$FG7:9IH58=IÙW=_^Ú657xE$ ¾ ] ¾ È6OQP!RLS(TU(VWS(XZY) ¾ TR3:"$'@ !2*&=&%$E%$<C"$*&=&%b
α1 = sups1∈S1
!*gs2∈S2
u(s1, s2).
¾ ] ¾¾ 6OQP!RLS(TU(VWS(XZY) ¾ TR3:"$'@s∗1 ∈ S1
33E"B;!j:R3: 0E3#/%$*#3*#%$'D*#%$@%$22w'1]\Jα1 = inf
s2∈S2
u(s∗1, s2).
¿¿ ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌ ¾ ] ¾ ¿6OQP!RLS(TU(VWS(XZY) ¿gTR3:"$'& !2&%232%2"$;@%b
α2 =!*g
s2∈S2
sups1∈S1
u(s1, s2).
¾ ] ¾9Á 6OQP!RLS(TU(VWS(XZY1s∗2 ∈ S2
33E"$;!jX0E3#/%$*#9]\J
α2 = sups1∈S1
u(s1, s∗2).
¾ ] ¿<;QU1=>@?Y&dQl(gBFbgÂa^_m](qBtbd_c b@qt_ja`qbPxq@qBtb|6cm\^]`_@ajBhα1 ≤ α2
I letnpg!6Jz]\xqo%$;=g39%$E35#%$*i/(/g!'MQ/% y *LGK$!G )^T ¾ ] ¾ Ç6OQP!RLS(TU(VWS(XZYJ\J
G'&"$39%$<C"$=@%22"$E3Ja32%$;5CR3:"$'(E
α1 = α2]J'('@E%$2
G"$E3"$'&"$*#%$'D*#%$@%$22w'1]
\Jα1 < α2
]#'('&E 2<C*#/R'1]GA^*#%$'I*!*#K2 "$E3"$'@%T
¾ ] ¾ Ç68!:ZY1\2 c#%WR £ÄE:cJ%R
(+1,−1) (−1, 1)£ÄE:(−1, 1) (+1,−1)
RW%$G%$*m%2%$3 1%$*α1 = −1
]α2 = 1
]g%$\J:-tR3:"$'(*J'I*!*#K2 "BE"$'@%T ¾ ] ¾9Ã OQP!RLS(TU(VWS(XZYg%B;=
(s∗1, s∗2) ∈ S1×S2
33E"$;!j0J:E3h*(=&%BE%B;01*(3*J'Q*#%$@%$2w*'1]\J
∀ s1 ∈ S1"2 ∀ s2 ∈ S2 : u(s1, s
∗2) ≤ u(s∗1, s
∗2) ≤ u(s∗1, s2).
¾ T ¾ ¾ ] ¾ edf=)P!YNR
Gm\^]`_@ag:hxa ∃ v
_jm]`_@a^b _@q(s∗1, s
∗2)uja`fFbg$qu$npg]nxqfhpgJhxaapnpjHgDLh
(s∗1, s∗2)g!bj@bo|u$npg]:Hs_@q b@aapnpj
v = u(s∗1, s∗2)_Aµh
GE`gWb@ag!l(g:kq_jf]`_@a^bHshxaapnpjg!baEjBb|uLnpg] mhql(g:kq
I letnpg!6Jz]\xqk%$;=(w'sS%$^bv = α1 = α2
"2(s∗1, s
∗2)0E3#/%$*#+33E"$;!j:'1T>?'('@EI
0E3#/%$*#"$;+/% y *LGK$!GR3Q<Q!G9b
sups1∈S1
u(s1, s∗2) = v = inf
s2∈S2
u(s∗1, s2) ¾ T ¿
"2u(s∗1, s
∗2) ≤ v ≤ u(s∗1, s
∗2),
7v = u(s∗1, s
∗2),
ÇmÈ VÈ+ߤÉÊ@ÞËÌXW&@à$6Y:Ï@â<ÌÞÉ3ß ¿ Á
!**#%$*v/% y *LGK$!GR3: )n'&a9&%$3'@%$2!' ¾ T ¾ BT
cJE/gL3gbk%$;=(w'+S%$(s∗1, s
∗2) G
*(=&%BE%B;01*(SR3g]#%$HX'@a7@%$3'&%$9N?RW%$G%$*(3!^b
sups1∈S1
u(s1, s∗2) = u(s∗1, s
∗2) = inf
s2∈S2
u(s∗1, s2) ¾ T Á
α1"2
α2/% y *LGK$!GR3Q<Q!j39b
sups1∈S1
u(s1, s∗2) ≥ α2
"2inf
s2∈S2
u(s∗1, s2) ≤ α1 .
¾ T Á ^A^ 1α2 ≤ u(s∗1, s
∗2) ≤ α1
T146!&%Bα1 ≤ α2
<Q!*#/!P;m!;&@]o%$9"$E3u(s∗1, s
∗2) = v
T>?'('@E
s∗1, s∗2
0#E3#/%$*# 33E"$;!j:'1TBh_T >T T ¾ ] ¾9Ã 68!:ZYJ?:E3 JR"$0J:E3 JRW9r0!G32=(:2)%$;=C/3:9@G:;EJ<C%$;!*#/g1%$;=(<+:?S%BG"'&*#3*#%$ )%2"$;;@%$G%$;=@%$*
t = 0A^ J*Y%$DM:2&G:;#b ¾ ]
t = 1A^ )%$*d0)%2/g!;ÈBTn[j:j3!
0)*(:;@?RW%$Ga RW%ai(t)
"2OG%$;=&%$*DSE/gL3*6E:*(=@ H:9@G:;;&^]g\JQH:9@PA:;CÈg]Hj:G!1G32L*(5#"$;XG%$;=&%$* ¾ TL-Hj<C%$=(!'CS"$1%$j: R3QH<+:3!'9]
+1_*(=@%$E%2"$;@%]\JQNj: R3:'M%$^] −1 "2\JQ%$*'(!U%$<j:^]JÈgTcJE3<+!7:mI01"$G/#:9bS1 = S2 = [0, 1]
]si33E"$;!j6I'@a7@%$3'&%$9NRW%$G%$*(3!^b
iR3:"$'@t = si
!G/N0)*(3 J*GN])\JD+<+:3!'m*#%$<GN3-<C"$;#]1\JI+<+:3!'mS"$kGN39]1"2*#%$<Zj:9]J'('&E
t = 1A^!;C'@%$o:E3*!k_Ga7@"2%$^]#<Q!'&E<+:E !9*(=@t_j:j9T£Ä;=
G = (S1, S2, u)]J\#^b
u(s1, s2) =
2a1(s1)− 1\J
s1 < s2a1(s1)− a2(s2)
\Js1 = s2
1− 2a1(s1)\J
s1 > s2
-I01"$G/#:s1 = s2
]U'('&EP = a1(s1)
(1 − a2(s2)
) G32L*&5#"$;;@%$.K2' ¾ Tj:^]<QL;I¿gTg*#%$<mTF@Uj:3\J]#\#;=MtR3:"$'(*J'I*6"$E3"$'@%r"2*&=&%$E%$;0)*(SR3gb
α1 = 2a1(s∗1)− 1 = 1− 2a2(s∗2)
<Q!@%$ktR3:"$'D32!<Q<C%$3E3!'(#s∗1 = s∗2
α1 = α2 = 2a1(s∗)− 1 = 1− 2a2(s∗)
¿Ç ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌÖ6×&iµ× =<>B¥G7:H¨J¨ f&7:<C R'+'(!
α1<+7g!<Q!*m"2
α2<Q!*!<+7D"$E3"$'&%$'@%$rj: !1'(! y 9%$"23!1<+:WA
E3!P'(Eg
A =
(1 23 4
)B =
(1 34 2
)C =
2 4 6 36 2 4 34 6 2 3
D =
3 2 2 12 3 2 12 2 3 1
¨ Ï n%$'(!*(3w'M_j: !U<+:3E3!P37k/39"$EAÄa39%$;5CR3:"$'&9t1 t2 t3
s1 3 4 7
s2 6 5 2
s3 1 9 4-:E22'6<C%$;α1AÄ%$9]
α2A^-"2C\#27:R'mE39I0E3#/%$*#33E"$;!j:'9JÍ.*
R3:"$'(*J'D"$E3"$'&%M¨gÔ! f&7:<QL3'D'(!nHj: !U<+:3E3!P'('k/39"$EAÄa39%$;5CR3:"$'&'D"$E3"$'@"$9
A =
5 3 6 24 1 5 10 4 2 −14 4 5 3
B =
1 2 −3 30 3 1 20 4 2 −11 2 2 2
¨(Ý 4i%$'('@E*(=@%$E%2"$;@%$k3#/r<+;&:*)';&E*(:*!#.BTR3:"$'@Oj: !R3:"$'( J*U]\JX'@%$&%$E333E"$;!j:t'<+xM
BT \ BT t1 t2 t3s1 (1, 0) (6, 4) (0, 9)
s2 (2, 1) (0, 2) (3, 0)
s3 (3, 7) (2, 3) (4, 0)
¨ì" /#3%$;=69"$EAÄa39%$;5mR3:"$'A<+:3E3!P37^To46!=&%$**&=&%$E%$<C"$*&=(r3#/i;&7A
E*(:*!.<+;&:*J'YDBT#R3:"$'&MI>?"$E3\J%B!%$2_3!G32633E"$;!j:2(MM46!H%$;=&%$*v3BT]g\#;=MBTg%$<Q<Q!'&"$006%-'0RW*m%$**#"$Ua 1%$M
A =
(1 23 4
)
¨gñ! k%$'(!*(3w'Dj: #!h<+:3E3!P37/3 R3:"$'&9UÍ.*3!G326ú-3\gAÄ%$;=@%$*#3u=!MÍ.*D/<Q!*J:*#33E"$;!jM-4i%$'('&Er*&=&%$E%$<C"$*&=(O3#/g*J'+ !23L*!1<+;'(*J'D%$;=@%2R3:"$'@'!M
BT \ BT t1 t2s1 (1, 4) (3, 1)
s2 (4, 0) (2, 2)
ÇmÈ jÈ Òp@áßgß@Þ)â<Ë ¿ Ã
¨#øB rj: !U<+:3E3!P37k/39"$EAÄa39%$;5CR3:"$'( J*M<C%$'('&E_*(=@%$E%$<C"$*(=(3#/;&E*(:*!U<+;&:*J'6rBTR3:"$'@M
¿ Á ¾¾Ê à ¿
¨gý! %$E%23w'I<C%$;IHj: !n9"$EAÄa39%$;5CR3:"$'@'I*(=@%$E%$;01*&SR3!9
t1 t2 t3s1 −4 5 6
s2 −1 2 4
s3 0 7 5
:t1 t2 t3 t4
s1 −5 2 1 19
s2 6 4 4 8
s3 −5 4 0 −6
Kt1 t2 t3
s1 1 −12 6
s2 0 −4 1
s3 3 −7 2
s4 3 −4 2
s5 5 −4 7
¨gç! £ÄESR'+S%$k_j: !#R3:"$'D*#E3<+:kj'R3:9]#"2'@%$E%23w'I<C%$;IX*(=@%$E%$;01*&'9"$R3:"$'@Q%$;=g39%$E3E%D<HSR36S%$ ¾ ]¿6;= Á RR3:9]"2Q3BT?**&=(! y 9%$CBT A^*#%$'1]<C%$**(=(!kRR3'Da39%$;@%TÏ Z! %$E%23w'I<C%$;IX32!;@E3u6ú-3\gAÄ%$;=&%$*#3u=@'tHj: !#R3:"$'&'( J*U
t1 t2 t3s1 (0, 2) (1, 1) (1, 1)
s2 (1, 5) (2, 2) (1, 0)
s3 (1, 8) (1, 1) (0, 0)
Wt1 t2 t3
s1 (6, 2) (5, 6) (3, 4)
s2 (5, 1) (5, 7) (5, 6)
s3 (4, 1) (4, 7) (4, 6)
s4 (−3, 1) (4, 7) (5, 5)
Ï ¨ %$E%23w'I<C%$;I ¨J¨ S%$j/#3 J*i/<+:3E3!P'I*(=@%$E%$;01*&SR3!9ÏJÏ
a<Q!=&%B*6"$E3"$'&%$!E%G%23H_j: !#R3:"$'(*J'D32!;@E3u6ú-3\gAÄ%$;=&%$*#3u=&M
a 6 2
−1 a −7−2 4 a
Ï Ô! /R*'i01"BG/#:_=&*mRW:"B'(EBg]U\#;=@%$*;@"$*s/<Q!*J:_33E"$;!j:''&SR3:'ú-3\gAÄ%$;=&%$*#3u=(9Ï Ý Í.*s%$;= ¾ '(<ö\#32us33E*#/o]n<Q!*YM33E*#/G'%$;=&%$*G%$%2%B*d32j*J'%$ST"$^;=(!G*U])C33E*#/G'6\\#_<C%$**#%$'1])<C%$=(!'M'&a9%$G%$ 6*UT#Í:j32*'\#%$=@%$t_^;=(!P9'(*J'1O46!nXú-3\gAÄ%$;=&%$*#3u=!M :O46!n_:E/#<Q!k03!<H<¤M
¿ Æ ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌÖ6×*ks× l =ID6=IÙ7 ;$7ÙWG7:9_^6«BFH ¾ ] ¾9Æ 68!:T&4i*&3<CE3 G9;C0)"$G/#:R3>?;=60JCQ'@a7@%$3'&%$9NR3:"$'&tR3:* R3C y :*J'1T# X0JQ%$G/g;I%$;=6E*&=(01"$*2
j<C%$=(!'v'&%$9"$ )%]O y u*J''(!'&%$ j:*!^]h\#;=v<C%$=(!'( 1%TZ-s y u`%$j: R3R<+'0UTL-Qr0JQH Jo'@%$9"$ 1%-/g;X"2%$j: R3 ¾ E*&=(9]g<QL;Q;=&*#%$RW M'&"$r%2%$"$*i¿QE*(=@T#4:3E3!Pg JXLPEgb
y u \ 0J J RW J
(1,−1) (0, 0)RW (0, 0) (2,−2)
S%$jj3 J*I%$;=(!'Q33E"$;!jr%$<p/<Q!*J: R3H<+:3!'7]&7t%2//g!;!J<C%$;@G/#:WA<C&/g#*'1]JX/<Q!*J:t33E"$;!j:'M/g!<C%$*2!RWQ*#%$<Z<H5'&a&/g!'1T>? 1%$*sQ%2%$3 )%$*:j32'i+'@a7&%$3'@%$9NI<C%$;@G/#:39Ton%$'(!*(3w'6u;=@])<Q!*(3\J
'g39ER3:37*J:*'DG%_-R3:"$'& "2t:3j;@ *&=&%$E%$<C"$*&=(w*'&%$-'E3*J:*'M<+7g!<+!PA7:*!^]7.<Q!*#/%$*X3E"$;!j:\#.E%$*#/%$ Rw*'rG32L*(5#"$;@%$"22k(!93;&: R'1]9\#;=%$H_G32L*(5#"$;Q<C%$'('&E_G%$;=@%$*U]\#;=MH:3j;@*&=&%$E%$<C"$*&=(w*'D<+7g!<+:!G%BA;=&%B*nTk%B'(!*(3w'M_j: !U<+:3E3!P9b
y u \ 0J b2 j2b1 (a, α) (b, β)
j1 (c, γ) (d, δ)
\J+X0JpG32L*(5#"$;;@%$
b2A^.RW:3$!'M<QL;D y u
b1A^9]1'('@E- y u6:E3\J+*&=&%BA
E%2"$;@%bu1(b1) = pa+ (1− p)b\JX0)%2/g!;
j1A^?R3:32!'1]# y uU]#'('&E _*(=@%$E%2"$;#b
u1(j1) = pc+ (1− p)d
k%$'(!*&3w'IuREC_<+:3E3!P9by u \ 0J b2 j2
p 1− p
b1 w wpa w(1− p)b
j1 (1− w) (1− w)pc (1− w)(1− p)d
y um:E3\JQ*&=&%$E%$<C"$*&=&%b
u1(w, p) = wu1(b1)+(1−w)u1(j1) = w[pa+(1−p)b]+(1−w)[pc+(1−p)d] ¾ T Ç
ÇmÈ mÈ Ë:f&@à@Þ ÞUàßÞ6Y:Ñ ÊË ¿@Ê
y um*&=&%$E%$<C"$*&=&%Hww;;@"$*(=@%]L;=
wA^ 1%$*m'@%$kX39"$GNQ"$E3"$'&%$'&%$E%23*&w*#'1b
∂u1∂w
= pa+ (1− p)b− pc− (1− p)d = 0 ¾ T Ã
pA^E%-<C%$;@G/g
p =d− b
a− b+ d− c.
¾ T Æ -*G@*iH0J:EQ'(!G39:<CC/(/g!'
w =γ − δ
β − α+ γ − δ
r%$E%2/%$3!US%Bj/# JX(!G37\#%$=@%$3%23L3&%bp = w = 2/3k%$'(!*&3w*'D*#"$\J:*&=D'&a2!3<C%$E3 !<+:3E3!PQR3:"$'&9b
¾ T ¾ Ê^68!:Zå18j<_ ge@\#"WR3-R3:"$'[n"$:j%$;=H!G/N )%$*C"$E3'&%B2!'X :0j:G"$'(\#@]&<C%$=H'@%$3NRw'(*#%$'H*#%$<x%$G%$;@%$*#/NT
46!*#/g'&"$t:j 37:<+:EQ'&"$33EB"$;!G+:nE%$*#/%$'@%$9"23E%TJ4i%$;'(LG"$E3!U<C%$;@39%$E%$2*!?%$ RW%2_:0j:G"$'&Â\#"WR3r;= )"$'&"2%$*d<C%$;@322*!;&j<_ :BTI-6<Q!*#/g'@%$3%$*;&j<H i33E"$;!j:R3: '@a7@%$3!'1]J'('@E;=(*CRWo*#%$<Zj'(*J'1]#/%j<C%$**(=(!U:0j:7AG"$'(\#R3*J'1TL-_%$;=(!'+;&j<H m33EB"$;!G:'@a7@%$9]'('&EH\#"WR3_33E"$;!j:'&a9&%$NRW1R3:E9]1a9&"_G%23CQ%$;@"23Q"$G%$G%$<mT)-+<Q!*#/g'@%$3%$* ^\#"WR3:']o'('@E-'&<C=i"$E3wG"23'&$\)3*J'M%$;=(<+:3*J'M_:0j:G"$'&"$E3S=(3'(w9/%$G%$<E:*UTF@%$G%$<C%$22w'6_j: !U'(! y 9%$NQ<+:3E3!P37ktR3:"$'&9b
A \ B;&j<H \#"WR3
;&j<_ (1, 1) (0, 2)\#"WR3(2, 0) (−10,−10)
¾ T Æ ^AÄr'&"$0G%$3 )%Q\#%$=&%$3%23L3@%bp = w = −10−0
1−10−2 =1011
]n7 1011
G32L*(5#"$;;@%$;&j<H q33E"$;!j"2 1
11
G32L*(5#"$;;@%$-\#"WR333E"$;!jmRW%$G%$*&3!-i03!<+:!G33E"$;!j:9T ¾ T ¾ Ë68!:Zå18t=(:2_*&=(u#R3:"$' ;&<C%r?K\!GK'@%$*:ÉR3:"$'%$;=F&<C%2 %9* y <H 1N'&a2!G3<C%$E39bo'@"$ y G0)3X9&%$E%BA
*(=@%$@Tg>?;=I37'/"$'QS%$G"tE\J**J'D"$O<Q!G%$GN3 H37'/"$'( J_%23*#%r]('(!;E7A*J'1T &%232L9]#'(!o'&E: J*m;E3!'D'(!^T E:9 R'XR3:"$'&9$n%$'(!*(3w*'_%$3^]\#;=rG%$\#%$WAÄ%O'(! y 9%$"2%$'@%$?E%$*#/%$*!HS%$j/#3\#@T oøépB÷ ¡þ8òøúòðòø÷ñî ò* ú& B÷Iî ðrqsô¡ïõüxò¡îXsóñ ÷ 8tÿauvw
¿Ë ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌlU%$;=&%B*
K x '(!;E3!'T x 9: \)G/
\ K T
K (−10,−10) (−25, 25)T (25,−25) (−100,−100)
S%$j/#3*J'XRWoj:3\J@*6*!*#K2/<Q!*J:*# <C%$;@G/#:gT
Ö6×&yµ× =<>B¥G7:HÏ ì" n%$;=(w'C%$^](\#;= ¾ T7R3:"$'&.%$;=
4× 3 A.'(! y 9%$"23!)<+:3E! 37U/3R3:"$'( J*&%$3E"2329Tg4i%$=(!'+&%$'(EOE%$0E%$9%$*(: R3C_'@a7@%$3'&%$9N_'&%$@%$E333E"$;!j:9b#Q¿gTg3!G3233E"$;!j:\J%$<x:j32SR3g] a !?01%2/g!;r%$;=@%$*GN-G32L*(5#"$;;@%$G"$0!(M46!=&%$*&"BG%$3G%$*m<C(/39%$E3E%$n3#/g*J:Q%$2 X'@%$&%$E333E"$;!j:&"$;E%$\JR*!(MÏ ñ! 46!k
A<+:3E3!P37/3 9"$EAÄa39%$;5IR3:"$'M%$;=@%$*#3u=(!n'@%$&%$E3 33E"$;!j0J:7A
Eg]9\JtB<+:3E3!P37/3)R3:"$''@%$&%$E3kú-3\gAÄ%$;=@%$*#3u=( ((5
7,2
7
);
(4
7,3
7
)) M
A =
2 2 2 2 2 25 4 2 4 3 36 6 3 5 6 66 9 0 −3 3 46 5 6 1 4 46 5 5 0 4 4
, B =
(3 56 1
).
Ï øB 463*'I0)"$G/#: =(*2× 2 2"$EAÄa39%$;5+R3:"$'(Eg]#\#U<Q!*#/%$*mG(/g!o'&%BA&%BE33E"$;!jQ/<Q!*J: R3
s1AÄ%$9]#"2
s2/<Q!*J:n<Q!*#/%$*mG(/g!o'&%$@%$E3 33E"$;!j:9
Ö6×&zµ× =IÙW;=<5b^6ÛæE$FG7:9IH8H ¾ ] ¾9Æ OQP!RLS(TU(VWS(XZY.>?;=
G =1, 2, S,ºi
32!;@E3uJ*@%$E%$*;@N]?\Ji J:E3<C%$=a ∈ S
]b ∈ S a º1 b
0)*(*i'('@Eb º2 a
T ¾ ] Ê^8Í bo|fb|pt_@qk_S%$*&3!n/% y *LGK$!+301%2K$!G:!G %2%$%H_9"$E3# a39%$;5QR3:"$'1T ¾ ] ¾ Ê^yOQP!RLS(TU(VWS(XZYhlU%$;=@%$*
G =(1, 2), S, u
32!;@E3uJ*@%$E%$*;@NDRW:"B'1]?%$'('@Es∗1 ∈ S1
<+7g<Q!*!<+:G ¾ TR3:"$'@3Eg]\Jgb
mins2∈S2
u1(s∗1, s2) ≥ min
s2∈S2
u1(s1, s2) ∀ s1 ∈ S1
ÇmÈ |È&@àÌY$Ð ÉÊÞËâ<Ë ¿Ì
"2\J*G@*s∗2 ∈ S2
<+7g<Q!*!<+:GI¿gTR3:"$'&3Eg]g\Jgb
mins1∈S1
u2(s1, s∗2) ≥ min
s1∈S1
u2(s1, s2) ∀ s2 ∈ S2.
¾ ] ËÍ bo|fb|pt_@qQ<+7g<Q!*!<+:G+<+7g!P<+!7: R3I9'-X'(! y 9%$"2%$'&%$9])<C%$=@%BA'&%BttR3:"$'@'I;&E*(:*!n3#/g*J'D<+;'(*J'1T£Ä;=I0)"$G/#:
s∗1X<C%$;@G/#:C
maxs1
mins2
u1(s1, s2)A^*#%$'1T
¾ ] Á <;QU1=>@?Y!&b|pbgG(N,S, u) |N | = 2 H<qBtmlw|npj®$hpg~Opbj@qbg|vsm\^]`_@a^¬ aapnpj
maxs2∈S2
mins1∈S1
u2(s1, s2) = − mins2∈S2
maxs1∈S1
u1(s1, s2)
R8ts2 ∈ S2
db|npcGF\xqfhhmaxs2
mins1
u2(s1, s2)u$npg]nxqfhpg1hxaapnpjHDLhdb|npcGF\xqfhh
mins2maxs1
E`gWb@a^I letnpg!6Jz]\xq1%$39NG%$;@%2w;;&"$*&=(E%!;&@b
min(−f) = −max f
argmax f = argmin(−f)S%$*&3!na39%Bw;;@"2%$'( )NU3E3!(!j:!G*i/&/g!'1b ¾ ] Ç<;QU1=>@?Y!&b|pbg
G = (N,S, u)qBtmlw|npj®$hpg~Opbj@qbg|vsm\^]`_@a^ ¬ aapnpj
Aµh(s∗1, s
∗2) G C hxqD8Eb|pbg$q®c hHhxaapnpj s∗1 dhw^dQl(g!lrd\xce~ 1 E`j@bµ_@q s∗2 dhTpdQl:Eg!l(d\pcw~2E`j@b
Aµh(s∗1, s
∗2) G C hxqD8Ebo|pbg$q®c hHhxaapnpjmaxs1
mins2
u1(s1, s2) = mins2maxs1
u1(s1, s2) = u1(s∗1, s
∗2)
_@q ∀ C hxqD8Ebo|pbg$q®c `^|phpg:htf]Ghyal ¨Ctbm]`_@qm]bjBbFpd_g!bBtml Aµh
maxs1
mins2
u1(s1, s2) = mins2maxs1
u1(s1, s2)H
s∗1dhTpdQl(g!l(d\pcw~
1E`j@b_@q
s∗2dhTpdylrg!l(d\pcw~
2E`jBbHZhaapnpj
(s∗1, s∗2) G C hxqD8Ebo|pbgEq®c h
Á È ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌI letnpg!6Jz]\xqo%$GN39aE ¾ T"2 ¿gT#lU%$;=@%$*
(s∗1, s∗2) G
ú-3\gAÄ%$;=&%$*#3u=&gT#>?'('@E9bu2(s
∗1, s
∗2) ≥ u2(s
∗1, s2) ∀s2 ∈ S2,
;=I<Q!&%$u2 = −u1
u1(s∗1, s
∗2) ≤ u1(s
∗1, s2) ∀s2 ∈ S2,
<Q!&%$u1(s
∗1, s
∗2) = mins2
u1(s∗1, s2) ≤ maxs1
mins2(s1, s2),
\J*G@*u1(s
∗1, s
∗2) ≥ u1(s1, s
∗2) ∀s1 ∈ S1
"2u1(s
∗1, s
∗2) ≥ mins2
u1(s1, s2) ∀s1 ∈ S1,L;=
u1(s∗1, s
∗2) ≥ maxs1
mins2
u1(s1, s2),L;=
u1(s∗1, s
∗2) = maxs1
mins2
u1(s1, s2) ∀s1 ∈ S1,"2L;=
s∗1<+7g<Q!*!<+:G1A^E%T
-*G@*m )%$j:3\J]#\#;=s∗2<+7g<Q!*!<+:GI¿7A^E%
Á Tg!;&9j::\#@]#G%$;=&%$*u∗ = max
s1mins2
u1(s1, s2) = mins2maxs1
u1(s1, s2)T
i%$GN9NY:L:I<Q!j39bmaxs2
mins1
u2(s1, s2) = −u∗Th46!&%$
s∗1<+7g<Q!*!<+:G
¾ A^E%bu1(s
∗1, s2) ≥ u∗ ∀ s2 ∈ S2,
"2s∗2<+7g<Q!*!<+:GI¿7A^E%b
u2(s1, s∗2) ≥ −u∗ ∀ s1 ∈ S1,%$ )N/&/g!'
u1(s∗1, s
∗2) = u∗
"$u2 = −u1,
L;=(s∗1, s
∗2) Gú-3\gAÄ%$;=&%$*#3u=&gT
¾ ] Ì¤Í b|fbo|pt_@q-:L:3 1 '@a7&%B'@%$2!'1]?\#;=32!;@E3uJ*&%$E%$*;@NmR3:"$'@'ú-3\gAÄ%$;=&%$*#3u=&!oS%$K2%$E"$\#%$N'1])7r\J
(s11, s12)
"2(s21, s
22)ú-3\gAÄ%$;=@%$*#3u=&'1]#'('@E
(s11, s22)
"2(s21, s
12)!G9'1T
¾ ] ¾ ÈÍ bo|fbo|pt_@qn\Jmins2maxs1
u1(s1, s2) = maxs1
mins2
u1(s1, s2)]J'('@E %B_-R3:"$'
"$E3"$'&%T
Ö6×@ÖNµ× =<>B¥G7:HÏ ý! %$E%23w'I<C%$;IHj: !JR3:"$'I'@%$&%$E3ú-3\gAÄ%$;=@%$*#3u=(:9
(2, 1) (0, 2)
(1, 2) (3, 0)
ÇmÈ ÇÈ Ò^áßgß@Þ)â<Ë Ág¾
Ï ç! %$E%23w'I<C%$;IXú-3\gAÄ%$;=&%$*#3u=@'tHj: !U'&&E/g!*J:K$!GhR3:"$'( J*U, F
,(2, 2) (0, 0)F(0, 0) (1, 1)
ÔZ! %$E%23w'I<C%$;IHj: !JR3:"$' Á ú-3\gAÄ%$;=@%$*#3u#P=(:9t1 t2
s1 (1, 1) (0, 0)
s2 (0, 0) (2, 2)
Ôo¨ G/R'I<C%$;I_j: !#R3:"$'@9t1 t2
s1 (2, 5) (3, 2)
s2 (1, 0) (6, 1)
Ô Ï 463'+<C%$;#]\#;=mHj: !#R3:"$'&'(*J'I*!*#K2'&%$@%$E3ú-3\gAÄ%$;=&%$*#3u=('1
(−1,−1) (−9, 0)(0,−9) (−6,−6)
: l 4 zÍ
(1, 0) (1, 2) (0, 1)(0, 3) (0, 1) (2, 0)
K l z
(0, 4) (4, 0) (5, 3)4(4, 0) (0, 4) (5, 3),(3, 5) (3, 5) (6, 6)
ÔJÔ! -:E22'i<C%$;m+j: !?*#%$<C%$'i\JEK9HR3:"$'i%$;=&%$3G%$*'@%$&%$E3ú-3\gAÄ%$;=@%$*gA3u=(:9
j:*&=
y u<C%2K2K2 <C2!
<C%2K2K2(3, 2) (1, 1)<C2!(1, 1) (2, 3)Ô#Ý n%$'(!*(3w'M_j: !U'@(E/g!*J:K$!G0E G"$<+:9b
, F,
(1, 1) (0, 0)F(0, 0) (1, 1)qR3:"$'m2t0)"$G/#:27g])\#;=M<Q!*#/%$;=@]J<C%$=(!'mG/#G*<C%$;=(w*'6Hu*U])K2'6
HS*(9]g\#;=D<Q!*#/%$*'(!U;=&*#3<C%$*RW%$*UT46!)t03!<+:!G.33E"$;!jg]@\Jr%$;=Q=(*+E37:; Jr'@%$E3ww*'1](\#J*#%$<p3#/R'1]
\#;=I<C%$=(!'MG/#G*m'&%$o<C%$**!(M
Á ¿ ÇmȤÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌÔ)ì" ¿i'&%$E%23'@%2/N:E3 R3i;=(*J2+i%$E3<C"$'&%$9T?>?;=g"$;'&a"$;#bÇ Ã È # ]%$ RW%2+:E9bÆ ÈÈ # ]h/g!G32'@*(I:E9b à ÈÈ # T?Íh* ¾ ÈÈ::E3G]h'(!*#%$< aEN(/g!'i:E3EBS]h"2+%$;=&"BG%$3G%$*#39%$E35#%$*i:j323 13 J*6@%232!U<C%$;I_:E39T#Íh* ¾ ¿ÈX::E3G])'(!UK2'XK2a'('&%$*&%$3t:E*m\JRj*#/Q::E*!^T#46!n_'&%$E%23'@%2/N'D03!<+:!Gt33E"$;!j:R3MÔJñ! n%$'(!*(3w'M_j: !U;=&:9_*(=(uR3:"$'@9
(1, 1) (0, 2)(2, 0) (−3,−3)
.Íh*gAÄ%3!G32Xú-3\gAÄ%$;=&%$*#3u=!M :G-:E22'D<C%$;I_'@%$&%$E3ú-3\gAÄ%$;=&%$*#3u=(9K?46!1G32L*(5#"$;@%-**J'1]@\#;=I'@%$&%$E3.ú-3\gAÄ%$;=@%$*#3u=( J*M&%$E%$*&=(9N'
"$G%$3 )%$*6<+E/g*J'!M/O4i%$=(!'I%$;=&%$*#3u=M/R3C_G%$;*J;=@ # m\J32*#trBTR3:"$'&3*J'!M
Ô1øB j: !g9"$EAÄa39%$;5R3:"$'( J*_'&%$E%23w'r<C%$;rO03!<+:!G'&%$@%$E3?33E"$;!j:9
A =
2 1 02 0 3−1 3 3
ÔJý! £ÄESR'6S%$.D'&Ne@0J0LEWe&G+R3:"$'@H*#E3<+:Oj'( J*U]n"2rj: R'<C%$;Mú-3\gA%$;=@%$*#3u#P=(;@*#/j3!nu*UÔJç! XBTJE37:;MA^ J*"2-,hA^ )%$*:<+/g\J9To46!*#/g'&"BE37:;*J'6K2'6%$;=m\JEK$!%$;=g"$;@%Q*UTn CBTUE37:;6*&=&%$E9]o\J6I:jM:<+/3X RW%$'(3<CH*#%$<ö&"2/g!'1T r RW%$'(3<C'M"$E3"$'@%b
vA > vBT#46!nXú-3\gAÄ%$;=&%$*#3u=!M
°¾½¿Às¿Á´Â¿ÂÃ
, ¸ -/, Ä · _? º · 10¸¹¸
Ø×@Ö6× © E$FG7:9IH=@s7:=<5¥@D ÞfÙ)?AFGE$R3:"$'+^:R3:*J'D%$G%$<C%$!^b
¾ T<?G%$'1T¿gTK2<C01*&'&*i-R3:"$'@'I<C%$;S%$G%$%$"2%TÁ T r"$G%$'M"2 tR3:"$'@'D/a*("2%$!*#%$'D<C%$;S%$G%$%$"2%TÇ#TK2<C01*&'m"$"$G%$'ME3E%$*#/RW%TÃ T r"$G%$'DG32L*&5#"$;+%$G32j:gT
S¿g] ¾ QOQP!RLS(TU(VWS(XZYh ^6;=@a'&%$E%I%$;=d=&*K$<C01*(9]<C%B=&%$X*!*#K2X<C%$;@%$GN9NK2<C0)*(9])7<Q!*#/%$;=(!'MK2<C01*& <C%$;@%$GN2!^T
S¿g] ¿6OQP!RLS(TU(VWS(XZYJÍ-"$;@NQK2<C0)*(9])<C%$=(*#%$'I*!*#K2E:'@a7@%$3'&%$9NRW%TcJ%B%$&"2%$!*' '&"2N m%$*(=(\LSRw'DN'&%$Bb¾ TK2<C01*&'m39:<+X&"$;@%29T¿gT>?;=D;=@a'&"$EG"$%$2!'1TÁ TÉ;=&a'@"$E3Nkr%$;=&%2@"$;@NCK2<C0)*('(!;+K2'D%$;=Iu@%$9%$9T
Á Ç ÎÈp$Ë&ÌÓ!Ñ ÊUáÑ µÉÊÞËâ<ËS¿g] ¾ Í bo|fb|pt_@q? _%$;=&%2.R3:"$'@'m/a*("2%$!'6E:*C<Hu3E+'(wGa*( )a9NI<C"$EWA"$'( )%$*6%$<QG"$'&%$2\#%$3*#%$'M"2L;=M/a*&"2%$!'D!G%$"$EN&%$'DG%$\#%$3*#%$'1TS¿g] Á 8OQP!RLS(TU(VWS(XZYoF&:"$'M%$ RW%2 %$<QG"$'&%$9%$3%$^b)%$;=QR3:"$'@ %$ RW%2t%B<QG"$'&%$9%$35IR3:"$'&A*J'I*#%$&%$2w*'1]\Jgb¾ T%$;=XRW:"B'&\JX%$<S%BG%WR3!n%$o'&E: !n/a*&"2%$!9]¿gT%$;=rR3:"$'&O\JH%$<pS%$G%WR3!o%$o2t!*gSE3<+:K$!G9]g<C%$=@%$O'&E: !o/a*&"2%BA!*#"$U !E3'&9T
S¿g] ¿6Í b|fbo|pt_@qU%$ RW%2%$<QG"$'@%$9%$35639%$'(&%$*#K$!j:P!G?R3:"$'(EtRWX0)"$G/#C_'('1]\#kR3:"$'@'D%$;=(<+::*mG"$0*#%$'D"2'&E: !UG"$0)"2%$!'D!G3<C%$E3%$'D%$GN33w'1TS¿g] ¾ 8!ZYg'&a9&%$3'&%$2N'( 1%$*C<C%$;!G3<C%$E3'&%2/gw*'+ u;=(*#%$@%$9%$3f&7:2j: u_R3:"$'&A'^]#<C%$=&%$t<C"$;MQ'&"2N !G%$'( )%$*i7: %$G%$<Q$w#*'1T)"$.R3:"$'@3*'m*Ub
(A,B)]
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l l l l l l
(1, 1) (0, 3) (95, 98) (97, 97) (96, 99) (98, 98)
(97, 100)
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r r r r , c c c c
l l l l
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A \ B F I
F (2,−2) (−2, 2)I (−2, 2) (2,−2)
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T ′\J<+7:Eg]1\JC%$;=6K2<C0)*(-E3PA
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S¿g] Á µ8!ZYf&9%$'(&%$*#K$!j:!G;&j<_ ge@\#"WR3XR3:"$'1]UC%$GND<+/#:Er<+:E:0j:'&2!'1]L;=I(/#"$E3'@%$9N6<+:&/g!'<+/#:E3*J''@"$_33E"$;!j:R3M*UbU<+E/;=:2&2!'1TRW%$G%$*G"$@N ¾ <+/#:Et%$'('@E;=I;&j<_ m;=I\#"WR3_<+;&E3:3 &%$\#%$OS%$^T
µ´¶³
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S¿g] Ã ¥OQP!RLS(TU(VWS(XZY?Ä*#/g'@K$!G6\J:3E7S%$G"]ΓR3:"$'E"233R3:"$'!P*Y/% y *!j: R'dmE"23$A
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`∗H^_G%$;\#37 i:;&-
`∗ = 1M"$%$.3E3!(!j:!G*d%$ RW%23w^Tnk%$;=(w'S%$^]U\#;=sD:L:_!;& A^!;#T
Q:j:*#:;M<C%$;@39E3L:+*J"B'(w%$%$\#%$SRw'1])\#;= ¾ T(R3:"$'@t\(uQ%$GN39aE"2
ÎÈ ÎÈ Òp@áßgß@Þ)â<Ë Á Ê
G%$;=@%$*#%$'(m1 . . .mt)
+'@a7@%$NDK2<C0)*('1To,:E3<C%$=(!'iK2<C0)*(3 )!*#/gGE"23$^_\#37_ !9*(=@*mEa7(!G/%$ U]<Q!*(
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GR3:"$'i"$E3"$'@%
v = sup1≤r≤t
vrT1>? ¾ TgR3:"$'@37:<+:E+ !29]
¿gTR3:"$'&3 J:E3<C%$=(!'DE"233R3:"$'( J*m !9*&=&*mG%$;S%$ RW%$ v@%232%2"$;+"$E3\#%$3!^T
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σRW%$Ga!?Q/3-K23u#K2 <C%$;@%$GN9N
K23u#K2'9T
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+'(!E:=
>'(!E:=
+'(!E:=1ToV'(! y 9%$"2%$'I'@a7@%$3'&%$9N'1b ¾ T1!;&7r<C*#/o]n¿gTo%$PS;&/R3 → ¾ T1'0 ¾ AÄ%$9|¾ T*J;=@ Jh<C*#/o]&¿gT%$PS;&/R3 → ¾ T'0
(−2) A^9| ¾ T'(!G%$ )%$<C*#/o]&¿gT@%$PS;&/R3→ ¾ T#'0
(−1)AÄ%$9T-+¿gT*#%$<ZS;&/R3C%$^])'('@E-_S%$*(3!n'(! y 9%$"2%$'M/g0j:9(/g*J'1]"2%$G%$*&"$%2 !E:*&=(uJ'DG%232*#%$'1T
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¼ °¾½¿Às¿Á´Â¿ÂÃ
, ¸û , d - º · 10¸¹¸ Äþd · ?ŸÆí¹Ä
F@%$Ga RW%
N = 1, . . . , n -R3:"$'@'D37:<+:S = S1 × . . .× Sn
X33E"$;!jX\J<+2u : S1 × . . .× Sn −→ R
n _'(! y 9%$NXw;;&"$*&=(
¾ ] ¾ 6OQP!RLS(TU(VWS(XZY)lU%$;=@%$*G%$;=
n39%$<C"$=@%2R3:"$'I*#E3<+:kj'( J*Ub
G = N,Si, ui%$'('@Et
iR3:"$'& !2*&=&%$E%2"$;@%b
αi = supsi∈Si
!*gs−i∈S−i
ui(si, s−i)
¾ ] ¿6OQP!RLS(TU(VWS(XZYs∗i ∈ Si
0E3#/%$*# 33E"$;!j:R3iR3:"$'&3*J'1]g01*&*i'('&E9]g\J
infs−i∈S−i
ui(s∗i , s−i) = αi
¾ ] Á µOQP!RLS(TU(VWS(XZYn%$;=GR3:"$'@G"$*(=@%$;%$G%$**#%$'*#%$@%$2w*'1]o\J
(α1, . . . , αn)'(! y A
9%$"2?E%$C"$E3%BG%$<_ )%$*i*#%$<Z/<Q!*J:9])7
6 ∃ s ∈ S : αi ≤ ui(s) ∀ i"2
αi < ui(s)G%$;&j: i%$;=
iA^E%
Ç&¿ ÇmÈÂÌW&ãËânâ<ÏL@àß@ÞB ÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌ ¾ ] ¾ <Í b|fbo|pt_@q#?E J: R*'X%$;=HE3:1R3:"$'&!*'r'&a9a3?%$G32*!^Tn%$;=(w'rS%$^]\#;=
iR3:"$'@r03!<+:!G*mR3:32g]Jw;;@%$3G%$*(w.Ia !G%$'IR3:"$':?G%$;&j:
αi<C"$E%$35I39%$G%$%$h*&=&%$E9] ∑i∈N
αi = 1T@>?'('@E
(α1, . . . , αn)-E3r03!<+:!GS%$G3$B:gT
É\J32*#:;@'I(!G39*(*#%$<S%$GK2%$E"$\#%$N'mtR3:"$'&'I'@a9a39T ¾ ] Ç6OQP!RLS(TU(VWS(XZY)ú-3\gAÄ%$;=@%$*#3u=>?;=r33EB"$;!G!R3:"$'rú-3\gAÄ%$;=&%$*#3u=&:*Q'@a7@%$3'&%$9Nk"$E3SRw'1b
G(N,S, (º)
) %$;=s∗ ∈ S
ú-3\gAÄ%$;=&%$*#3u=@]\J ∀ iA^E%
(s∗−i, s∗i ) ºi (s
∗−i, si) ∀ si ∈ Si
¾ ] Ã µOQP!RLS(TU(VWS(XZYnj<C%$=s−i
33E"$;!jM@%$'(E3EmiR3:"$'&G%$;7RW s:j39gb
BRi(s−i)b
BRi(s−i) =si ∈ Si : (si, s−i) º (s′i, s−i) ∀ s′i ∈ Si
BRA^ %$'('&E
iG%$;7RW m:j3w;;@"$*(=(*#%$'I*#%$@%$22w'1T
¾ ] ¿Í bo|Bmb|pt_@qtiS%$*&3!S;&G<Q<+tú-3\gAÄ%$;=@%$*#3u=Y%$;=s∗33E"$;!j&%$'(E9]
<C%$=(E%bs∗i ∈ BRi(s
∗−i) ∀ i ∈ N ¾ ] ¾ 68!ZY)c#;@=M/g!G%$<Q<+:EQ%$'(!*&3w'm_'@E: !oS;@=D/!PG%$<Q<+:9T
Ö6×@Ö6× l =ID6=IÙ7 ;$7ÙWG7:9_^6«BFH ¾ ] ¿8!ZYkk%$'(!*&3w'D'&a7@%$3'&%$9NM'@Ne@0J0LEWe&G+R3:"$'@9TnÉR3:"$'( J*s'@"$R3:7A"$'&R3:32!'m_S%$*&!U\J:E<33E"$;!j: 1k:j323Q%$;=&%$9T!£Ä;=D'&%$G%$3'@%$9N+33E"$;!j0J:E'(EC_'&a9&%$3'@%$9N'I!;&7'1beIX0J0LE %$EN%$ m<Q!*&tX'@N]g<C%$E3'&"$0)%2 1%2K2<+;@*!eIX0J0LE;=@%$*;@"$ 6<Q!*&tH]<C%$E3tHGC%$:;7R3C_0J0LE3eIX'@NQ%$EN%$ m<Q!*(trG]<C%$E3t_'&NX'(!GK2E3 LSR3QHGÉ'(! y 9%$NQ<+:3E3!PM%$\J:9b
1\2 K P O
K (0, 0) (−1, 1) (1,−1)P (1,−1) (0, 0) (−1, 1)O (−1, 1) (1,−1) (0, 0)
ÇmÈ ÇmÈ Ë:f&@à@Þ ÞUàßÞ6Y:Ñ ÊË Ç Á
Íh%$;=(w'D"$32E%]\#;=M%$r%$;=D32!<Q<C%$3E3!'(#9"$E3#a39%$;5CR3:"$'1TR3:"$'(*J'I*!*#K2 ú-3\gAÄ%$;=@%$*#3u=(Q¥^3!G3$B:*QR3:393¡r33E"$;!j:'I'&a9a39T
¾ ] Æ 6OQP!RLS(TU(VWS(XZY)lU%$;=@%$*G%$;=D33E"$;!j!JR3:"$'1]
G = (N,S, u) iR3:"$'&O'&%$@%$E3
33E"$;!j:R3H%$;=CG32L*(5#"$;X%$G32j:SiA^*UT- |Si| <∞
]('('&EOr'&%$@%$E3O33E"$;!jσi Si −→ R+
]g<C%$=(E% ∑sk∈Si
σi(sk) = 1T
¾ ] Ê^6OQP!RLS(TU(VWS(Xb#F@%$Ga R%Σi
SiSaGa33!UG32L*&5#"$;+%$G32j:'I\)P<+7:9T
F@%$Ga RW%Σ = Σ1 × . . .× Σn
A^9TF@%$Ga RW%
ui(σi, σ−i)
iAÄ/!P'XR3:"$'@'(! y 9%$"2"$9]
\Ji σi
A^9]<QL;I_a !G%$'σ−iA^?R3:37*J'1T
¾ ] Á Í b|fbo|pt_@q- |S| < ∞ ]n'('@EHD'@%$&%$E3 1N9(L"2r'(! y 9%$Nw;;@"$*(=@%IIE"$;!'(! y 9%$Nw;;@"$*(=&%$'I!*#%9:E3!G'&<H !*J:K$!GR3
ui(σi) =∑
si∈Si
σi(si)ui(s−i, si)
¾ ] Ë6OQP!RLS(TU(VWS(XZYσ∗'@%$&%$E333E"$;!j: ú-3\gAÄ%$;=&%$*#3u=(*J'I*#%$&%$22w'1]g\J
ui(σ∗i , σ
∗−i) ≥ ui(σi, σ
∗−i) ∀ i ∈ N
"2 ∀ σi ∈ Σi
¾ ] ¾ <;QU1=>@?Yσ∗uLnpg]nxqfhpg hxaapnpj C hxqD8Eb|pbg$q®c HNDLh
ui(σ∗i , σ
∗−i) ≥ ui(si, σ
∗−i) ∀i ∈ N
"2si ∈ Si
¾ ] ÌCOQP!RLS(TU(VWS(XZY@lU%$;=@%$*G&"$;@%2JR3:"$'1]'('@E
σi'&%$@%$E333E"$;!j:<C;&::*J'
*#%$&%B22w'1]9*d3!G32633E"$;!j:'\J<+7:9]n<C%$=(E%σi(si) > 0
]n"2supp (σi)
A^&%$RW%$Ga Rw'1T 7@b
supp (σi) =si ∈ Si | σi(si) > 0
¾ ] ¿@;QU1=>@?Y8Aµhσ∗a^baOpbjm])qm]zjBhp]`_|ploh C hxqD8Ebo|pbg$q®c ¥_@q s′i, s′′i ∈ supp (σ∗i ) hxaapnpj
ui(s′i, σ
∗−i) = ui(s
′′i , σ
∗−i)
I letnpg!6Jz]\xqoÄ*#/g!E%$'(u*m !9*&=(LSR'1]%$;=(w'+S%$^bui(s
′i, σ
∗−i) > ui(s
′′i , σ
∗i ) s′i, s
′′i ∈ supp (σ∗i )
ÇÇ ÇmÈÂÌW&ãËânâ<ÏL@àß@ÞB ÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ßÌ%$'(!*(3w'M_'&a9&%$3'@%$9NX'&%$@%$E3 33E"$;!j:9b
σi(si) =
σ∗i (s′i) + σ∗i (s
′′i )
\Jsi = s′i
0\J
si = s′′iσ∗i (si)
'(wGa*& 1%$*[email protected]"$;!j:\#r'01K2G&/Q'(! y 9%$"2 "2 _E"$;!U'(! y 9%$"2'(wGa*& #"$;@%b
ui(σi, σ∗−i)− ui(σ
∗i , σ
∗−i) =
=∑
si∈Si
ui(si, σ∗−i)σi(si)−
∑
si∈Si
ui(si, σ∗−i)σ
∗i (si) =
=∑
si∈Si
ui(si, σ∗−i)[σi(si)− σ∗i (si)] =
= ui(s′i, σ
∗−i)σ
∗i (s
′′i )− ui(s
′′i , σ
∗−i)σ
∗i (s
′′i ) > 0
¾ ] Á 68!ZY1\2 s12 s22 s32s11 (1, 1) (0, 2) (0, 5)
s21 (0, 1) (4, 0) (1, 6)
s31 (0, 2) (1, 1) (2, 1)
1\2 s12 s32s11 (1, 1) (0, 5)
s21 (0, 1) (1, 6)
s31 (0, 2) (3, 1)
1\2 s12 s32s11 (1, 1) (0, 5)
s31 (0, 2) (3, 1)
É01"BG/#: )*m*!*#K2%$*#%$'M/<Q!*J:*# 33E"$;!j:'1T 9*( J*m%$;=g39%$E35637:<QL:n/(/g!'1]\#;=I0)"$G/#:
(s12 + s32)/2 > s22T>?9"$E3
s22%$!<Q!*J:\JT-*G@*
(s11 + s31)/2 > s21T
¾ ] Á ;QU1=3>@?Y@&bo|pbgσ∗ G
E C hxqD8Ebo|pbg$q®c hH8]`bo|pparfbc σ∗(si) > 0H6hxaapnpj
siqm]zjh^]`_|plo\^]gWbdAbc l(dQl(g:\pc mh¤hFnpdQl(g:\pcì]<qm]zjh^]`_o|plo\xa bc l(dQl(g:\kmlo~mhHhthtsi ∈ S∞i
¾ ] ¾ ÈIOQP!RLS(TU(VWS(XZY&/<Q!*J:33E"$;!j:'H!%$E3L_%$!<Q!*J:K$!GR3:*+ '@a7@%$3'&%$9N"$E3WARw'1b
S0i = Si
Sk+1i =
si ∈ Sk
i | 6 ∃ σi ∈ Σi(Ski ) : ui(σi, s−i) > ui(si, s−i) ∀ s−i ∈ Sk
−i
S∞i =∞⋂
k=1
Ski
S∞ = S∞1 × . . .× S∞N
ÇmÈ ÎÈ+ߤÉÊ@ÞËâ<ËÌ#â?à#ãÊUá.ßUáßËLÉ3ß Ç Ã
¾ ] Ç6Í b|fbo|pt_@qn %$ R3:E:*#%$<<Q!*#/%$*6%2%$3 )%$*m'@*#33E3J:\JQ<C%$;#T"$G/#:k'&Ne@0J0LEWe&GR3:"$'(EX!%$E3LM%$!<Q!*J:K$!GC*#%$<Z'@*#33E3J:\JT ¾ ] ¾¾ &OQP!RLS(TU(VWS(XZYd'&%$@%$E3n33E"$;!j:'(EtOG%$;7RW H:j3h<C%$;S%$G%$%$"2%$*Q.j: gA !j' "$E3SRw'1]"2
BRiA^@%$RW%$Ga Rw'1b
!(BRi : Σ−i −→ Σi!!(BRi(σ−i) = arg max
σi∈Σiui(σi, σ−i)
¾ ] Ç<;QU1=>@?Yσ∗ C hxqD3EBbo|pbg$q®c uLnpg]nxqfhpg hxaapnpjHNDLh
σ∗i ∈ BRi(σ∗−i) ∀ i ∈ N A^E%
I letnpg!6Jz]\xq_b&%$GN39aEh 1%$j:SR'1]@\#;= ∃ si33E"$;!j_'@%$&%$E3h )N7(L"23 )%$*U]&<C%$=Q*#%$<
G%$;RW +:j3σ∗−i
A^E%T&>?'('&Eui!*#%9E3!:H<Q!j3
i*#a9&%$*!)3#/R3'(! y 9%$"2"$O=
<C&/*n]U\#;=si\#%$=&%$3XMG%$;7RW s:j32 R3:37g]o<Q!&%$
si*#%$< G%$;RW s:j3
σ∗−i
A^E%TcJE/gL3gb%$;=(w'+S%$ ∃ σ′i
]g<C%$=(E%HX'(! y 9%$"2*J;=& m<Q!*(σ∗iA^ 1%$*UT#>?'('@E
ui!*#%9E3!:Q<Q!j3 '@%$n=&*633E"B;!j:*J'MG%$**!G%]#<C%$=@%$'D'(! y 9%$"2%$!n<+;& J'1]L;=
σ∗iA^ J*m*#%$<Z<Q!*#/%$*633E"$;!jXG%$;RW m:j3T
¾ ] Ã &Í bo|fb|pt_@q(h%$;=@%$*#3u=(!@'@%$&%$E3U33E"$;!j: J*H39%$E%$0GN3E"$;!j:'(*J't;=(*gA2t_'(! y 9%$"23'@%$n%BE%$/<C"B*(=&%$2*!G%T
Ö6×øØ× © E$FG7:9IH6H 5¥Ù)?AF6> 6>BH<E$! (¡ `! £¢¯¤3¤J³^«o³S¬!!¥&¦h±6§h³¥8t=('(E*+ú-3\gAÄ%$;=&%$*#3u=m*#%$<'(!G%$G"$;LNT1"$G/#:hC'&&E/g!*J:K$!GR3:"$'@'( J*ie*#%$<C%$'M\JEK9C33 UTge+\JC27_S%$%$&"2%$n"$\#%$3w*'1]#\#;=mH%$;=&%2R3:"$'@'D7Aj<C%$=@%2!*gSE3<+:K$!G7UE%$*#/%$'&%$2*#%$'1]#'('o'(!G%$G"$;LN m<C%$;@G/#:3ER3\J3*'1T ¾ ] ÇI8!ZYlU%$;=&%$*X'@"$)R3:"$'&3*'1]'(!'r'(wGa*( )a9N.RW%$9"2%$'&%$"232G%$*#%$'1]7<C%$=&%$' )%BS=(: R3:'M_<+:3!'D<+;&E3::9])r"232G%$G"23w'D'0)K2G(/XeI7r'&E3E%$j:e/%*#%$<%$ RW%2<C"$E3"$'( 1%$*UTÉ'(wGN-RW%$G%$'I\J:E<:j01&%$\#%$3*#%$'IS%$^b
x, y, zT
.%$GNR3:"$'&k2k3#/R3g]9\#;= x ;= y, z ]9<QL;<+:(/g!'R3:"$'& x, y ];= z T&>?'('@EOR3:"$'@'C-S%$*&3!J!*gSE3<+:K$!G'Q !E3': J*I:j32SR3:'X<C%$;_33E7A"$;!j:!'9T ¾ ] Ã 68!ZY)n%$'(!*(3w'M_'&a7@%$3'&%$9NX0)"$G/#:9b
Ç Æ ÇmÈÂÌW&ãËânâ<ÏL@àß@ÞB ÉÊÞËâ<ËÂÌ#â?à#ãÊUáhßUáßË:@ß̾ \ ¿ s12 s22s11 (9, 9) (6, 10)
s21 (10, 6) (0, 0)
3!G32Qú-3\gAÄ%$;=@%$*#3u=1b(10, 6)
](6, 10)%$@%$E3 ú-3\gAÄ%$;=&%$*#3u=1b (6
7 ,17
) ] Ëg] Ã Ê_:E3\JQ*&=&%$E%$<C"$**&=&%$^T,.L22*'X<C%$;H%$;=_\JE3<+/g!'C39%$<C"$=(hEj:^]/ RW*QS%$#%$;=X"$E3<C"$9]@\Jt%WR
(s21, s12)]g\JXLE:
(s11, s22)]J'('@E X:E3\JQ*&=&%$E%$<C"$*&=mËgT
k%$'(!*&3w'M_j: !n%$G32j:K2j:/9b¾ \ ¿ s12 s22s11 γ 1− γ/2
s21 1− γ/2È
%$'('@E9(2γ1+γ
)+ 6
(1−γ1+γ
)≥ 10
(2γ1+γ
)+ 0
(1−γ1+γ
)
!;&@]\Jγ ∈ [0, 3/4]\J
γ = 3/4]'('&E+*&=&%$E%$<C"$*&=Ëg] Ê Ã ].<Q!*J;=@ U]<Q!*&I'&%$@%$E3+ú-3\gA
%$;=@%$*#3u#P=(*):ST ¾ ] ¾ ¿¥OyP!RS(TU(VWS(XZY>?;=
G(N,S, u)33E"$;!j!kR3:"$''&E3E%$j:X%$;=&%$*#3u=&:*vm'&aA
&%B'@%$9N "$E3SRw'1be(Ω, π)
&"$;@%$IG32L*(5#"$;!-<C%$9N9]\#Ω%2%$<C"$*(=("$EM"2
πG32L*&5#"$;!
<C"$E3"$'ΩA^*
e ∀ i ∈ NR3:"$'&3EX
Ω Pi0JE33LGK$!GR3:S
iR3:"$'@O!*gSE3<+:K$!GQ0JE33LGK$!GR3
e ∀ i ∈ NR3:"$'@3EQ
σi : Ω −→ Si]#\#
σi(ω) = σ(ω′)\J
ω ∈ pi"2
ω′ ∈ pij<C%$=pi ∈ Pi
]u;=&]&\#;= ∀ i ∈ N A^E% ∀ τi : Ω −→ Si
]\#τi(ω) = τi(ω
′)](\J
ω ∈ pi"2
ω′ ∈ pipi ∈ Pi
%$'('@E9b∑
ω∈Ω
π(ω)ui
(σ−i(ω), σi(ω)
)≥∑
ω∈Ω
π(ω)ui
(σ−i(ω), τi(ω)
)
¾ ] Æ Í b|fbo|pt_@qΩ"2?0JE3LGKB!GR3r%/% y *LGK$!GH39%$E3!*(?*#%$<p%B;@"$*U]\J*#%$<pE"239%
m%$;=@%$*#3u=(*J'1T?4:WS%$GN/% y *LGK$!G*#*J's'(!<C*#/#:g]?\#;=s<Q!*#/%$*0)2!3L%$GNSE/gj:3u6%2%$<C"$*(=(E%Hr!R3:"$'&E%9'@K$!GR3Q03!<+:!G9])r!R3:"$'&3#/#:X!G a !#R3:"$'&33E"$;!j:! y ;=@%$G%$<_ )%r@"$&%T
ÇmÈ ÎÈ+ߤÉÊ@ÞËâ<ËÌ#â?à#ãÊUá.ßUáßËLÉ3ß Ç(Ê
¾ ] Ã ;QU1=>@?Y&Í l(gBFbgG(N,S, u)
m\^]`_@aya^baOpbjm] C hxqD8Eb|pbg$q®c DLnt¥cr_m]`bBtmlra¤npc hpgapnpjjBbce\xcw]bo|^bg$q®c ((Ω, π), (Pi), (σi)
) H¥dbc pjBbÁdQl(gBFbgi ∈ N
m\^]`_@apnxqjBhJhtsiqm]zjh^]`_|plo\xaTDLntµaphu$kqfnpcw~TF~ÂbcwnxqBtmcw\xqQbo|pbg!cwv
σiE&Opbce
I letnpg!6Jz]\xqJlU%$;=@%$*Ω = S
"2hG%$;=@%$*π(s) =
N∏
j=1
αj(si) ∀ i ∈ N "2 bi ∈ siG%$;=@%$*
pi(s′i) = s ∈ S : si = s′i
"2-G%$;=&%$* Pi =pi(si)
TolU%$;=@%$*σi(s) = si ∀ s ∈
STk>?'('@E (
(Ω, π), (Pi), (σi)) '@E3E%$j:Q%$;=@%$*#3u=&]n\!G39%$*v6/% y *LGK$!Gi'(L:*J<+!
%$ RW%23w*#%$'1T
¾ ] Æ p;QU1=>@?YCbo|pbgG =
(N, (s)(u)
) qm]zjh^]`_|plohplm\^]`_@aTHGQB\pjdbc 1apnpjj@bcw\pcì]bo|pbmg$q@®cìplIal ¨Ctbm]`_@qgWb@ayapnpgOpbapnpdpQl(g:\kmlo~mhlrqsapnpjjBbcw\pcì]b|pbg$q®c h
GE`gWb@a^
I letnpg!6Jz]\xq(lU%$;=&%$*#%$'u1, . . . , uk
'@E3E%$j:?%$;=&%$*#3u=(!('(! y 9%$"2%$'_"2(λ1, . . . , λk) ∈
R λi ≥ 0 ∀ i"2 k∑
i=1
λk = 1]
k<Q!*#/%$*C"$E3"$'(\#%$G%$;=@%$* (
(Ωk, πk)(pki )(σki )) '&EWA
E%$j:t%$;=&%$*#3u=@]<C%$=(\#%$Xuk'(! y 9%$"2E32!'1TJ _:j:*#:;I<C%$;@39E3L:
*#"$'(woS%$%$\#%$SRw'1]g\#;=Ωk /g!G33R*'(9T
lU%$;=&%$*Ω =
⋃
k
Ωk "2 ∀ ω ∈ ΩA^E π(ω) = λkπk(ω)| ∀ i
A^E% Pi =⋃
k
Pki
"2
σi(ω) = σki (ω)
'('&E
k∑
i=1
λiui '@E3E%$j: %$;=&%$*#3u=(!o'(! y 9%$"2
¾ ] Æ 68!:ZYJlU%$;=@%$* |N | = 3 "2 _'(! y 9%$"2%$'M_j: !j'1bÁ
s13 s23 s331\2 s12 s22 s12 s22 s21 s22s11 (0, 0, 3) (0, 0, 0) (2, 2, 2) (0, 0, 0) (0, 0, 0) (0, 0, 0)
s21 (1, 0, 0) (0, 0, 0) (0, 0, 0) (2, 2, 2) (0, 1, 0) (0, 0, 3)
>?'('&E9bM ú-3\gAÄ%$;=@%$*#3u=&'(\#x'0)K2G(/ '(! y 9%$"2%$'1b(1, 0, 0)
|(0, 1, 0)
|(0, 0, 0)
T'&E3E%$j:I%$;=@%$*#3u=%2%B"B*nb (1
2(s11 + s21);
12(s
12 + s22), s
23
) Á TUR3:"$'@C*#%$<"$E/%$'&%BC J*U]k\#;= ¾ Tk"2Q¿gTn!*gSE3<+:K$!G\#QR#*J'33E"$;!j:!'s'@(E/g!*J:j:"$E/%$'&"B 1%$*UT
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F (m) = 1,6m−0,2m2 T&4i%2//g!;_*#N_'&a2G%$;@%$GNH\J32*JM,O%$'@a7&%B'@%$2!'&AÄ%6i3E;@"2/g!jg]\J Æ R3:"$'&Q*:MÂ-:*&=R3:"$'@3Ei'&a7@%$3'&%$2!'d )%63E;@"2/g!jMÝoøB ta2G%$;@%$GN \J32*Jgb
F (m) = 1,1m−0,1m2 T7d'&a2G%$;@%$GNk*#%$<É\J32*J:G'\J32*JG%$;=@%$*iÈg] ¿gT#46!'&E:!'I!E3EK$!G*J:!G:CtR3:"$'!Mr46!'&E'@a7@%$3'&%$2!'I )%H_3E;@"2/g!jMÝ)ý! k%$'(!*&3w'Cj: !?'@a2G%$;@%$GN'i0E G"$<+:R3:9o>?;=/3%$E3wG%$%$* ¾ È+;&9/#G%$;@%$%$ ¾ e ¾ %B\J%B*J%B7T46!*#/%$*X%$\#"$*C3u=( ¾ ÈÈÈt'(;#Tp-t%$;=r;&9/#t<C"$;r%$;=H%$\#%$*#%$'(wG/MrG%$;@%$GNE%]#'('&Er5D@"$;@%2"$;@%t<Q!j3H%$\#%$*#%$'I3u=( ¾ ÈÈH'(;A^<+UK$a'('&%$*UT-:*(=%$\#"$*E%.'@a7&%B'@%$2!' 1%O3E;@"2/g!jM46!G"$E3)R3*J'H%$(!G/#:!;MG-:*&=r%$\#"$*X%2%$"$*'(!G%$ 6X'@"$%$\#%$*#%2;&9/#:'D\J32*Jg]<Q!*&tH%$E%2/%$3!k:j013 J*:MÝ)ç! k%$'(!*&3w'M Ý)ý! S%$j/#9#46!*#/%$*MG"$0)"23 1%$*6_'&a9a"B;QuRW**J*M'&%$G%$3'@%$9%$3':E:%$E3\#%$ Rw'_E3Eg]'(!g%$; X'(wG/a3<C"$;%$;=r%$\#%$*#%$h'&a2%$;@%$GNE%4i%2//g!;"$E/%$<C%2 /%$9%$E3:*!(M
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S¿g] ¾ df=)P!YGOp_o|b@qqm]zjBh^]`_o|plohpl:m\^]`_@ag:hxa Oxhpg C hxqD8Ebo|pbg$q®c hÂha^baOpbjm]gQBv6OTJz]`_@qaEQBbgW
!9*&=(L:+r'(*! y g0)*(WA^"$%$G"$@%$ aE3"$*!'1]?%$9"$E3+%$GN39aE+/% y *!j:*'*#"$\J:*(=+S;&<+9TS¿g] ¾ µOQP!RLS(TU(VWS(XZYn>?;=
S ⊆ Rn \J<+C'@*(@%Bo]1\JI%$39NG%$;@%2 s1
A^E%Q"2s2A^E%Q"2
∀ λ ∈ [0, 1]A^E% λs1 + (1− λ)s2 ∈ S]7HH\J<+E3<+27X%$39NG%$;@%2O'&"$
0)*(SR3: a39%$'@aNC37'329TS¿g] ¿6OQP!RLS(TU(VWS(XZY)>?;=
S ⊆ Rn \J<+r7:E39]\JX%$39NG%$;@%2 E73Egb
si∞i=1 si −→ s∗ i −→∞A^ 1NU'@a7@%$3'&%$2!' s∗ ∈ S
7rE3<+27+_\J:E301*(SR3!P7TS¿g] Á 6OQP!RLS(TU(VWS(XZY)>?;=I\J<+H'@<Q0J'(9]\JX'@E3j: "27:E39TS¿g] ÇOQP!RLS(TU(VWS(XZYn>?;=
r : X −→ Y<C%$;%$G%$%$"2r;E:7PR3M
(x, y) | y ∈ r(x)
\J<+@TS¿g] Ã 6OQP!RLS(TU(VWS(XZY)>?;=I<C%$;S%$G%$%$"2 ;E:7PR3_7:E39]\JQX;E:77:E3\J<+@TS¿g] ¾ sÍ b|fbo|pt_@q\J+C<C%$;S%$G%$%$"2tw;;@"$*(=@]1'('@EQ7:E3:;M9*#QS=(A*#:;;&^TS¿g] ¿df=)P! r'(*! y g01*&WA^"$%$G%Bb ¬ |p
r : X −→ Xdbo|fbcebcì]`bm]`_qgWb@a ∃ x ∈
X x ∈ r(x)¨u$npg] mhH¡DLh
Ã È ÎÈ Ìgß8®X¯Þ7Þ@á¾ T
X ⊂ Rn apnpgOpbHapnpdu$hxax]_@qµgWbd jBb@q
¿gTr(x)
gWbdÅjBb@q°QB\pjdbc xE`j@b
Á Tr(x)
apnpgOpb ∀ xE`j@b
Ç#Tr|pj\(mhyt\pjm]`
I letnpg!6Jz]\xqWú-3\gA^"$%$G"k>?GN39aE2×2 AÄ%2)R3:"$'@'(E j:SR'r )%Oú-3\gA^"$%$G"$9T 2×
2AÄ%2?R3:"$'&'D%2%$"$*itR3:"$'@'D33E"$;!j:!-Hj: !j'M39%$E3!*(LE3\JSR'1b
sk1 = αs11 + (1− α)s21
sk2 = βs12 + (1− β)s22L;=M(sk1, s
k2)33E"$;!j:
(α, β) ∈ [0, 1]× [0, 1] 0JEB<C"B%BEE%$U\J:E2\JSR'D<C%$;#T46!*#/%$*dR3:"$'@D%$;RW `:j3i33E"$;!j:R3:I:j32SR3!G3<C%$E%$%$!39%$E3!*&B]7H
BR1(β), BR2(α)0J:E3R3:37:'1T£Ä;=+'@*#33E3J:3*#'I%$;=+G%$'@"$01%$9"23r%$;=&A
"$;*#"$;=(9%$3 )NH%$;=g"$;*#"$;=(2%$3 1%bBR(α, β) = BR1(β)×BR2(α)S¿g] ¾ b;QU1=>@?Y8ht
s∗a^baOpbjf]sqm]zjh^]`_|ploh¤u$npg]nxqfhpghxaapnpj C hxqD8Ebo|pbg$q®c HMDLh s∗
hBR
¨muLnpg] mhHZhthts∗ ∈ BR(s∗)
TI letnpg!6Jz]\xq& !9*(=(LP^RÄ#'1]7\#;=
BR%$;=r'&<Q0J'("2k'@*(&%Br\J<+a*<+;&:EtG
G%$'@"$09"2%r"2 Qr'(*!PA^"$%$US%$"$%$G%$!U%$ RW%23w*#%$'1Te
BR : [0, 1]2 −→ [0, 1]2"2 r%$;=g"$;*#"$;=(9%$'&<Q0J'( "2'@*(@%B
eBR(s)
*#%$<þwE%29].%$i'&a9&%$3'@%$2!'` )^].\#;=BR1(S2)
"2BR2(S1)
*#%$<wE%2%$'1T#>?!;&r<C%$E3tH%$GN-R3:"$'&O'(! y 9%$"2%u1β(α) = αβu1(s
11, s
12) + α(1− β)u1(s
11, s
22) + (1− α)βu1(s
21, s
12) +
+(1− α)(1− β)u1(s21, s
22)
u1βw;;&"$*&=
αA^ J*vS=(*#9]"2+
[0, 1]]h<C%$=(!'d7:E3+!*&%$E3<m]
%$9"$E3u1,β
S%$@%232!O39"$GN6"$E3"$'&%$!QI!*&%$E3<C*UT<£Ä;=*sG%$;&j: v%$;=α∗]g<C%$=D<+7g!<+!7: R3C ¾ R3:"$'&'(! y 9%$"2"$9T
e Á T)S%B"$%$.2RW%$G%$*&3!^]o\#;=i\JM ¾ TgR3:"$'@rIG%$;RW :j37:R3:37gbα∗1s
11 + (1 − α∗1)s
21
"2α∗2s
11 + (1 − α∗2)s
21
"2Q¿+R3:"$'&βs12 + (1 − β)s22
]'('&EXM
λα∗1 + (1 − λ)α∗2 0 < λ < 1A^E%I32!*&"$*Y%$;RW d:j3@]7
BR1(β)'&*&&%BoT&4i!P&%$)<Q!*#/
α1]&<Q!*#/
α2G%$;7RW D:j3@](%$9"$E3O;=(*J2
'(! y 9%$"23-/R3:' ¾ T@R3:"$'@3*J'1T#>?X2hRW%$G%$*&3!^]#\#;=M!*#%9:E3!G '@<_ !*J:K$!GR'!G'&*#3*#9T>?9"$E3
BR1(β)'&*&&%BoT
Ãg¾
eÇ#TS%$"$%$1'@a7@%$%$<C"$*(=@%@]&\#;=BR
;E:7PR3r7:E39T(k%$'(!*&3w'Ir'@a7@%$3'&%$9NE7
Sn = (αn, βn)"2
sn = (αn, βn) ∈ BR(sn)Th46!*#/g'@"$IE7
E3*s∗n = (α∗, β∗)
]k"2s∗n(α
∗, β∗)A^\#@T,O%$j:SR'1]n\#;=
s ∈ BR(s)T
¾ TR3:"$'@3Egb
u1(αn, βn) ≥ u1(α
′, βn) J:E3<C%$=
α′ ∈ [0, 1].
uS=(*#<Q!*#/
αA^ J*U]<Q!*#/
βA^ J*UT#>?9"$E39b
u1(α∗, β∗) ≥ u2(α
′, β∗) ∀ α′ ∈ [0, 1].>? 1N1'&a9&%$3'@%$2!'1]&\#;=
α∗ ∈ BR1(β)]("2.L;=
s∗ ∈ BR(s∗)TL£Ä;=+H;E:7n7:E39T
£Ä;=M'<+2\JSR'mXr'(*!PA^"$%$9]J"2 J:E3<C%$=2× 2 R3:"$'(*J'I*mú-3\gA%$;=@%$*#3u#P=(gT
É'&a9&%$3'@%$9N'( 1%$*M'(!%$ESRW%232SRw'M_ !9*(=(L:39bF@%$Ga RW%b
Σ1 = (α1, . . . , αn−1) | 0 ≤ α1 ≤ 1, . . . , 0 ≤ αn−1 ≤ 1"2 n−1∑
i=1
αi ≤ 1]
7_X33E"$;!j:'Σ132!<Q0G%BM%$G%$<C%$!&%$#RW%$G%$<C%$22w'1T
S¿g] ¿<;QU1=>@?YΣ1apnpgOpbÂ_@qapnpduLhxax]ST
I letnpg!6Jz]\xq'&*&&%B`eC!*#%9:E3!G'@<_ !*J:K$!G'D!*#%9:E3!G'&<H !*J:K$!GR3Q!G9]g!*#%9:E3!G'&<_ !*J:K$!GT'&<Q0J'(De+'@E3j:9]<Q!@%$ ∀ αi
A^E%0 ≤ αi ≤ 1
eC7:E3s]7%$'(!*(3w'αj1, . . . , α
jn−1
E79]αj ∈ Σ1
]9<C%$=(E%αj E3 α∗ A^\#@T>?'('&E
α∗ ≥ 0 "2n−1∑
i=1
α∗i ≤ 1]<Q!*&t_\J:E"$E3"$'M<C%$;E3SR3CH%$;=@%$*GN3G%$*#"$;@%$9T
,O%$j:SR'MQr'(*!PA^"$%$US%$"$%$G%$!*'D%$ RW%23wG"2"$9b¾ TΣi*#%$< wE%29]<Q!&%$%$;=(!'iR3:"$'@C33E"$;!j\J<+7%$< wE%29T?46!@%$
Σi'&<Q0J'(Σ = Σ1× . . .×ΣN
!G.'@<Q0J'(9Tg>?9"$E3BR : Σ −→ Σ
*#%$<[wE%$9]'&*&&%BM"2'@<Q0J'(\J<+9*maE"$*#NQ\#27:E%$*#/%$G"2
¿gT ∀ iR3:"$'&3EI%$'(!*(3w'Ij: !'(! y 9%$"239b
ui,s−i(si) si ∈ Σi
TU46!&%$Σi'&<Q0J'("2
ui,s−i
S=(*#9]1%$9"$E3r+'(! y 9%$"2%$'6\J<+7I'&<Q0J'(9])L;=6"BA%$2!'D<+7g!<H<+gT£Ä;=M
BRi(Σi)*#%$<ZwE%29T
à ¿ ÎÈ Ìgß8®X¯Þ7Þ@áÁ TlU%$;=&%$*
s1i"2
s2i
iR3:"$'@+'@"$+G%$;RW `:j37
s−iA^E%]h%$'('&E
s1i"2
s2i'(! y 9%$"2%iA^*#%$'m9*#9]LP;=D!*#%9:E3!G'@<_ !*J:K$!GR'M"$E3"$'@%!G;=&*J'('&EgT
Ç#Tk%$;=(w'6S%$^]o\#;=snE7H"2
sn ∈ BR(sn)To46!*#/g'&"$HE7r'@*(@%$E3;&:
s∗"2
s∗A^\#@T#4i!P*J/%$*QR3:"$'&3Egb
ui(sni , s−i) ≥ ui(s
′i, s
n−i) ∀ s′i ∈ Σi
É'(! y 9%$NXw;;&"$*&=I!*#%9:E3!G "2%$9"$E3S=(*#9T#>?2"$E3 \J:E"$E3"$'@%$ &"$@%b
ui(s∗i , s
∗−i) ≥ ui(s
′i, s
∗−i) ∀ s′i ∈ Σi
>? 1Ns∗i ∈ BRi(s
∗−i)"2%$9"$E3
s∗ ∈ BR(s∗)]#7__;E:77:E39T
`S%$"$%$G%$'+%$ RW%23wG"2% <Q!j3'<+2\JQHr'(*!PA^"$%$^]gG"$%$2!' y g01*&9]7rG"$%$2!'mXú-3\gAÄ%$;=&%$*#3u=@T
± °¾½¿Às¿Á´Â¿ÂÃ
² º · 10¸¹¸ ,´³ , Äþµ - ¶ d · ºi
Á ] ¾ 68!ZYof&K'&%$ )%$E3;A^&%$E%$*&=k%$;=(w'S%$^]O\#;=
A:jjm'(!PS%WRG%23$%Bm%$;=`uR+%2K\*#G;!j:i"2m%$'@%$9/g!_
;=(:E3:3q1<C%$**(=(!G"$;;@%$^T
BK2"$;_!G.\J*GHS%WRG%232"2%$*D/;@2!'1]r;=(:E3:.<C%$;A
'&%B9/"2%i%$GN3I"232G%$!^]h\#;=<+:3!'vK2"$;d<+:EI%$'&%$9/g%i%$E3<C"$';=&:E3::7]L;=:j32SR3_<C%$;
q2;=(:E33<C%$**&=(!G"$;@"$9T
k%$;=(w'+S%$qi ∈ 1, 2, 3
]"2 _0!jK$!U'&%$E%23G%$ Á %$E3<C"$'1]JX;=&:E3:3!U'@a"$;IÈgT>?'('&E tR3:"$'&L;=M: E:9\JSR'1b
µ´¶³·
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q1 = 1
q1 = 2
q1 = 3
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¾ TN/3 "2 |N | <∞
¿gT[K2<C0)*('@"$;@%2T\J<+7g]U<C%$=@%$'%$;=6^I;E:7S\J:E2*J'<C%$;#]
/3+Z&"$;@NiK2<C01*&'d\J<+7¢S\#**J*v<+:EX*#%$<û!*#/gO'(!O"$(B]
∀ t ∈ T\Z K2<C0)*(3\#X/3'Iw;;@"$*(=@%$'M_j: !U<C&/*Ub
Ã Ç -ÈCߤÉÊ@ÞËâ<Ëw)ÞÌ!8 Ò$â?à#ãÊÉ3ß!r
i(t)R3:"$'&'(!UG"$0U]
!!(MXG%$\#%$"$;@%$G"$01"2%$'I\J<+7A(t)
]!!!(M
n(t, a)IE:'&a9&%$3'@%$9NmK2<C0)*('\J<+7g]o<C%$=&%$'DG%$\#%$"$;@%2
G"$01"2%$'( )Nk/(/g*J'1TÁ T-! y 9%$NQw;;@"$*(=@%$'
ui : Z −→ R
Ç#TÄ*gSE3<+:K$!G0JE33LGK$!Gb ∀ x K2<C0)*(3\#@] h(x) K2<C0)*('H9*H\J<+7g]<C%$=&%$'@%$tiR3:"$'@!G3<C%$E9T#S%$*&3!U0JE33LGK$!GQ'(!G%$G"$;L3!
x′ ∈ h(x) −→ i(x′) = i(x), A(x′) = Ax"2
h(x′) = h(x) Á ] ¿68!ZYof&9%$'(@%$*#K$!j:!G 3*# !
µ´¶³
JJJJc
(1,−1) (−1, 1)
µ´¶³
JJJJc
(−1, 1) (1,−1)
µ´¶³
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lU%$;=&%$*#%$'m_*#%$<Z@"$;@NQK2<C01*&'
µ´¶³X2 µ´
¶³X3
µ´¶³X1
ÃÃ
k%B'(!*(3w'MQ/% y *LGK$!G9b¾ T |N | = 2¿gT
s1 = s2 = F, IÁ T_^X/% y *!j: R3C_&"$;@NQK2<C0)*(' "2
x2"2
x3
x1E:'@a7@%$3'&%$9NRW%
Ç#Th(x1) = x1
"$h(x2) = h(x3) = x2x3
T
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GR3:"$'@*i%Bg%$*2LDSE3<+: J*Q'@a7@%$3'&%$9N
"$E3SRw'1bM-R3:"$'@'
(N)
:DE7'D\J<+7(H)
<C%$=&%$'I'(!G%$G"$;L3!'mHj: !n'@a7@%$%$<C"$*(=@%$'&%$9b!rb ∅ ∈ H!!(b!-
(ak)k=1...K ∈ H"2
L < K]#'('@E
(ak)k=1...L ∈ H!!!(b$-X%$;=I&"$;%$%$*6E73E(ak)k=1...
A^E%(ak)k=1...L ∈ H ∀ L
]#'('&E(ak)k=1... ∈ H
T
Á ] ¿Í b|fbo|pt_@q?H%$G%$<C%$!taE3"$*#"2%$'(*#%$'m\LR'1]#%$;=MaE3"$*#"2t%$;=&%201*&SR3!
0)%2/g!;Dr3:j323 33E"$;!j:'1T Á ] Á 6OQP!RLS(TU(VWS(XZY)%$;=
(ak)k=1...K ∈ H&"$;@NXaE3"$*#%$9]#\J_&"$;%$G%$*U];=
6 ∃ (ak)k=1,...K ∈ H]g<C%$=(E%
(ak)k=1...K+1 ∈ HF@%$Ga RW%Z_&"$;@NXaE3"$*#%$%$'D\J<+7:9T
KD>?;=P : H\Z −→ N
w;;&"$*&=&]k7P (h)
%$;=MR3:"$'&9]k'(!OhaE3"$*#%$
:*6/a*(9T/ ∀ i ∈ N
A^E%r%$;=I0E%BS%$E%$*#K$!jQE%$G:K$!GZA^*UT
tR3:"$'&39':ΓA^RW%$Ga*!^T
Á ] ÇOQP!RLS(TU(VWS(XZYo\J |H| < ∞ HR3:"$'@-&"$;@%23*#%$'m*#%$&%$22w'1])\J ∀ h ∈ H&"$;@%2
\#32uU]#'('&E 2<C*#/R'1]g\#;=mtR3:"$'I&"$;@%2\#E3!9*&3uUT Á ] Á Í bo|fb|pt_@q? ∅ ∈ H
C'@%$9/N01*&9]UCaE3"$*#%$-'&%$9/%$%]1'('&EP (∅) QR3:"$'@9]#'(!kr%$GNXG"$01"23<C%$;%232!^T
ÃÆ -ÈCߤÉÊ@ÞËâ<Ëw)ÞÌ!8 Ò$â?à#ãÊÉ3ß%µ×@Ö6× ¹ s7:=<5¥@D E$FG7:9IH6H5¥Ù)?AF6>6>BHsªQ65 Á ] à µOQP!RLS(TU(VWS(XZYkF@%$Ga RW%
Hi%$;=%Bg%$*2LiSE3<+:RuMR3:"$'%2%$"$*d
iR3:"$'&-!*gA
SE3<+:K$!G!*J'D\J<+7:9b
Hi = S ⊂ T | s = h(t)j<C%$=
t ∈ T, i(t) = ilU%$;=&%B* Ai
iR3:"$'@G%$\#%$"$;@%2/a*("2%B!*#%$'D\J<+7X J:E3<C%$=I!*gSE3<+:K$!GQ\JPA
<+7X<C%$G%$39T Á ] Æ OQP!RLS(TU(VWS(XZY
iR3:"$'&r3!G32m33E"$;!j:R3m%$;=%Bg%$*2LMR3:"$'%2%$"$*
si :Hi −→ Ai
w;;@"$*(=@]g<C%$=(E%si(h) ∈ A(h) ∀ h ∈ Hi
T Á ] Ç8Í bo|fb|pt_@qn33E"$;!jQ%$ 1%$*ir"$E3%$G%$<_ )%$*i%$;=D%$ RW%2K2%$G%$'(@"23!o%$E31T Á ] Ê^6OQP!RLS(TU(VWS(XZY
Γ'@%$&%$E333E"$;!j:R3C_3!G32Q33E"$;!j:'ISaGa33!UE*#/<Q!7:K$!GT
Á ] Ë6OQP!RLS(TU(VWS(XZYΓ'@%$&%$E3(!G%$'@%2/"23!U3E"$;!j:R3gb
σi : Hi −→ 4(Ai),<C%$=(E%
supp(σi(h)
)⊂ A(h) ∀ h ∈ HiS7rE"232 )%$*m!*gSE3<+:K$!G \J<+H%2%$"$*m'(wGa*gA^'(wGa*mE*#/<Q!7:*'!BT
Á ] Ã Í bo|fb|pt_@q?x'@%$&%$E3-33E"$;!jI"$C'&%$&%BE (!G%$'&%2/"23!33E"$;!jD:j:*#WA:; J*m*#%$<9*#9]#K2'DX%$ RW%2%$<QG"$'@%$9%$35+R3:"$'@'D%2%$"$ 1%$*UT Á ] Á 68!ZYof&9%$'(@%$*#K$!j:!G 3*# !ZjS=(:s1 = F, I H¿gT2R3:"$'&3*J'XÇH33E"$;!j:R3*U]<Q!&%$#'&"$h/a*&"23 1NJ:j323\Jh'&"$!*gSE3<+:K$!GQ\J<+_%2%$"$*UT!£Ä;=
1\2 FF FI IF II
F (1,−1) (1,−1) (−1, 1) (−1, 1)(−1, 1) (1,−1) (−1, 1) (1,−1)
7-r¿gT2R3:"$'@3EFF
2nRW%$%$*(3!^]@\#;=FA^:j32<Q!*#/g'&"$hK201E33 J*U]
FI2
RW%$G%$*&3!^]\#;=Mc#%WRW%$:j3$t ¾ TR3:"$'@cJ%WR.\&u7:: "232E%$&"$@%"26£ÄE:3:j32 ¾ T£ÄE:\&u7:: "232E%$&"$@%T Á ] Ç68!ZYof&K'&%$ )%$E3;A^&%$E%$*&=jS=(: ¾ T1K2"$;
q1'(! )(K2::3-:j329]oD¿gT1K2"$;
q2(q1)A^9TW-+\J:E<ÀG%B\J%B"B;@%2'(!PA
)(K2::3!n32!*&*U]#'('&E ¾ T A^*#%$' Á ]<QL;I¿gT A^*J'3× 3 = 9 33E"$;!j:R3X*UT
-È ÎÈ6à$pÉÊ@ÞË Ã Ê
lU%$;=&%$*6<C3qi ∈ [0, 1]
"2p = 1 − q1 − q2
"2c = 0
TJ,:E3<C%$=q′1 ∈ [0, 1]
A^E% ∃ú-3\gAÄ%$;=&%$*#3u=@]g<C%$=( 1%$*ir%$GNCK2"$;q1A^%$E3<C%$^T#lU%$;=&%$*Ub
s1 = q1
s2 =
1−q′
1
2
\Jq1 = q′1
1− q′1\J
q1 6= q′1
7@]@\Js1 6= q1
]&'('&E.H¿gT&K2"$;X**&=(!h%$E3<C%$^]&\#;=QG%$*&=&<R3H-:E'OÈ7A^EgT& s1 = q′1
RWJj:3\J@*Mú-3\gAÄ%$;=@%$*#3u=&]@\!G39%$* ¾ T A^*#%$'C*#%$< "$E3!J<C%$;X%$"$E3*!^]&<C%$E3¿gT33E"$;!j:R3Q%$E%2/g<C"$*(=@%$'&"$006%$'('&E*&=&%$E%2"$;@%HÈgT
%µ×øØ× º 9<;E$FG7:9IH Á ] ÌOQP!RLS(TU(VWS(XYJ>?;=
Γ%Bg%$*2LCSE3<+:RuXR3:"$'
Γ′E"233R3:"$':*mH'&a9&%$3'@%$9N"$E3WA
Rw'1b¾ T
GK2<C0)*(SR3:*J'QE"$32\J<+7:
T ′]@<C%B=Q\Jr%$;=
xK2<C0)*(.E3PA
<+@]J'('&E _'&a9&%$N!!G9]g\Jt ∈ T ′
"2t′ ∈ h(t)
]#'('&Et′ ∈ T ′
¿gTO'(wGa*( )%$*m01%2/g!;G′<+;&X!G%Bg%$*2LISE3<+: J*m<C%$;&/3;EB:7WT
Á ] Ã 68!ZY
µ´¶³
JJJJ
µ´¶³
JJJJ
µ´¶³
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@@
@@S11 S21
µ´¶³
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Ã Ë -ÈCߤÉÊ@ÞËâ<Ëw)ÞÌ!8 Ò$â?à#ãÊÉ3ß%$ )%$*r%2%$3 1%$*i¿_E"233R3:"$'I*U]#-R3:"$'I<+;&Q"2
µ´¶³
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>?**#%$'+E"233R3:"$'(*J'm01%2/g!;D*!*#K2-G&/g!E"233R3:"$'g])<Q!@%$\J+G%$**#%]o'('&ED¿gTR3:"$'@!*gSE3<+:K$!GC\J<+7X'(wGa*J:GCG%$**#%]J<Q! /% y *LGK$!G*'D*#%$<%$*;@%2/M<C%$;#T Á ] ¾ ÈZOQP!RLS(TU(VWS(XZY#>?;=
s∗33E"$;!j_
ΓjR3:"$'QE"233R3:"$'Q'(!G%$G"$;LNX%$;=@%$*#3u=(g]&\J
G J:E3<C%$=DE"233R3:"$':*J'Dú-3\gAÄ%$;=&%$*#3u=&gT
Á ] Æ 8!:ZY1f&K'&%$ )%$E3;AGR3:"$'1jS=(:.,O%$j:SR'1]g\#;=Mr%$;=@%$3G%$*mE"233R3:"$''(!G%$G"$;LNC%$;=&%$*#3u=M
q∗1 = 1/2
q∗2(q1) =1− q12
dE"233R3:"$'&A^'(!G%$G"$;LN %$;=&%$*#3u=@]7ú-3\gAÄ%$;=@%$*#3u=!G9]7<Q!'@E ¾ T9K2"$;q1A^k/a*&9T
>?'('&Eq∗2(q1) = argmax q2
[1− (q2 + q2)
] T>? )Nq∗2(q1) =
1−q12
T-*G@*i ¾ TK2"$;IXR3:G%$;7RW m:j3r;@aE3 )"WRW"$*CR3:32!'1TZE"233R3:"$'&A^'(!G%$G"B;LN%$;=@%$*#3u=&]ú-3\gAÄ%$;=@%$*#3u=YM%$;@"23XR3:"$'(E6!G9]kL;=
q∗2G%$;7RW m:j3q∗1A^E%b
u2(q∗1, q2) = q2
(1− q∗2 + q2(q1)
)= q1 ·
1− q12L;=
u2 q∗2 = 1/2A^ )%$*m&%232!1%$U<+7g!<H<+:9T
%µ×&%µ× ' 5b¥áH6ßC«BàÜD«B;;C6Þf=<>B9R3:"$'I<C%$;@G/#::\#H;=&'(E*m\J:3EU(!G377S%$G"tS%WR3&%.R3*'M%$^T Á ] ¾¾ OQP!RLS(TU(VWS(XZY#>?;=I%Bg%$*2L+SE3<+:EU2<C*#/R'1](\#;=+%$ RW%2O!*gSE3<+:K$!G9]\JX<Q!*#/%$*m!*gSE3<+:K$!GQ\J<+_K2'D%$;=DK2<C0)*( E3<+@T Á ] Æ 8Í bo|fb|pt_@qnR3:"$'&'+a'&"$G%$%2%$*m<C%$; y ;=@%$!'6_'@E: !UG"$01"2%$'@%$9T
-È -È Ñ Ì"Õ:ËLÓ!Ñ Ô&Ñ xpßÒp@á Ã Ì
Á ] ¾ ;QU1=>@?T8Í l(gBFbg ]`bc fb@qÁl(g,mnpjd\kmlo~xq»Op_|b@qÁqtb@aOpbg:kmlo\pc lrqQm\^]`_@ag:hxace_m]`bBtmlra¼ jB_@qBtom\^]`_@a,Ealbmce_|6Jz]vÂbo|^bg$q®c hf½zI letnpg!6Jz]\xq#`E"$33R3:"$'(*J'QG"$%$2!'Cú-3\gAÄ%$;=&%$*#3u=&g]&%$2h\#%$=@%$3%23L3w'Q )%t L;=K2a'('@%$*(%$3R3:"$'( JX"27: M<C%$;@G/g+tR3:"$'@9]!;&93'm_:L:39T Á ] Ê^68!T#k%$'(!*&3w'm_'&a9&%$3'@%$9N-R3:"$'&9b
µ´¶³
JJJJS12 S22
(3, 2) (0, 0)
µ´¶³
JJJJS32 S42
(1, 1) (1, 1)
µ´¶³
¡¡¡¡
@@
@@S11 S21
p¿gT(R3:"$'@t!*#/g!ì¦o%$E%$*#s32"2
s43'&a9a39T1>?'('@E-@"$;%$G%$*'6z "233R3:"$'m'(!G%$G"$;LN
>?;=@%$*#3u#P=D*Ubs∗1 = s11 s∗2 =
(s12 + αs32 + (1− α)s42
)0 ≤ α ≤ 1 T
Á ] ¿¡;QU1=>@?Y[DLh¥heOp_o|\pc(cwhuLn^]nxaPpcGP^]o]dQl(gBFbo|pplra<m\^]`_@apnxqGqBtmlw|npj®¤_jm]`bcebdpQ@bg¾Opbm]o]jBbgBFbBt_@q@qbc&j@bgBFbc a^bBtmlraTHGhxaapnpjQhj@_@qBtom\^]`_@aQalbce_o|6Jz]vÂbo|^bg$q®c bo|p_jm]`bc d°c! Á ] ¾ df=)P!YCSt#\*:Í l(gBFbmg.Op_o|b@q
Γm\^]`_@ag:hxaycr_m]`bBtmlrayj@_@qBtom\^]`_@ayalbce_o|6Jz]vÂb|pbgEq®c h
I letnpg!6Jz]\xq)-:3E7S%$G"aE3"$*#NQ!*#/g'&K$!G9^blU%$;=&%$*i
ΓE"233R3:"$'!*J'D\J<+7gT
Γ 6= ∅ wE%29]g\!G39%$* Γ ∈ Γ T % y *!j: R*'m%$;=DE"232 )%$*6E%$*#/%$9"23ΓA^*UTJlU%$;=@%$*
Γ1"2Γ2E"233R3:"$'
Γ1 ≥ Γ2]
\JΓ2E"233R3:"$'
Γ1A^*#%$'1T
¾ TOG"$01"29bg:j32'I'(!n9't_E"233R3:"$'&'9]g<C%$=(E% 6 ∃ Γ∗ ]\#;= Γ∗ > Γ1¿gTOG"$01"29b)G/R'M<C%$;D%$9%$'@%$rR3:"$'@'9]1Cú-3\gA^"$%$k<Q!j39])G"$%$2*#%$'m<C%$;@PA/#:' \!G39%$*m&"$;@%2%$'!
Á TOG"$01"29bo+ú-3\gAÄ%$;=&%$*#3u=i%$;=@%$*#3u=(!'(! y 9%$"2%$'@%$\#%$=@%$3%23L3!'6 1%]1%$9%$*DR3:7A"$'&'I'(! y 9%$"23!U\#%$P=&%$!E%
Ç#TO!G3<C"$%$ Rw'D<C%$; ¾ T e(¿gT e Á T A9]1//g!;I<QL;CG"$%$2!'IE"$33R3:"$'1T
Á ] Ê^¤Í bo|fb|pt_@q-6%$E%2/g<C"$*&=v*#%$< %$;=@"$E3%$<_5U].76a E"233R3:"$'Y'(!G%$G"$;LN%$;=@%$*#3u#P=D!GG%$\#%$"$;@%29]\JQ-R3:"$'(*J'Ia 6ú-3\gAÄ%$;=@%$*#3u=(_*UT
Æ È -ÈCߤÉÊ@ÞËâ<Ëw)ÞÌ!8 Ò$â?à#ãÊÉ3ß%µ× U× =<>B¥G7:HìZ! -%$%$;=d%$<H 1%$E+6\J/%$E%$; )%]h¿+RW%$aQ*UT 1%$G"$0)"2Q'@a"$;@%b
C = 400]
_a*'@"$*(%2"$;@"$E3hR3:EC )*(#3@bB = 300
TJ>?;=(<+: :*6'@"$E/%$2!'6RW%$Ga%$'@%$9]#\J*ma*'@"$*(%29]#'('@EO&"$;@%]g\J_*!*#K29]'('&EE*J'1]"2O%$E3<C"239%$%2%$*m*!*#K2 )A*(#3@T$ E:9 R'MtR3:"$'@ %Bg%$*2LMj'( J*U]#"2/R'I<C%$;I_E"233R3:"$'&A^'(!G%$G"$;LN%$;=@%$*#3u#P=(7ì1¨ /#3.'&"$O3201%BE\JG<m]"2.aE3"$*!'I%$;=C!*#K$!G/%$*#9TBT&'@"$/;@O%$\#%$9b(;=<C%$GN2!OC!*#K$!G/%$*#39]o;='(!%$ESRW%2323!.I'&*LK#!'(3#39TU Q !.%2%$3 )%$*d3BT1;=(!G37G"$0U].;=vS=(SR3Ys'&*LK#!'(3#D'(!P%BE^R%232"2"$9T>? )%$*É%2%$3 1%$*q 1%$G"$0gA*#%$'C%$;=rR3:"$'( Jg]g\#)\Jr*#%$<[G"$0*#%$'C(!G37<Q!*#/g'@%$3%$*U]'('&Eh*&'(G%9:E3!G.\J: 1E3uG%23@T& E:9 R'sCRW:"B'&9]n"2XM&"$;@NQR3:"$''&"$_%$;=@%$*#3u=(:\#+'&%$E%23w*'¿6/g E"233R3:"$'&A^'(!G%$G"$;LNC%$;=&%$*#3u=(9
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G =(N, (Si), (ui)
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µiT
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j∈N−i
Sj .
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si ∈ Si
iR3:"$'@hG%$;7RW I:j37H/3.@"$G%$'&%$/#"$.<C%$%$37]
\JX*!*#K2=(*m33E"$;!jg]<C%$=D'(! y 9%$"2%*J;=@ mG%$**#%r%$9%$*m:E'&7:<C%$G%$39TÇ#] Á 6OQP!RLS(TU(VWS(XZY)>?;=
si ∈ SiEK$!G*J!7:\)I
G(N,S, u)R3:"$'( J*U]\Jgb
¾ T ∃ ((Xtj)j∈N
)∞t=1
\J<+9'1bXt
j ⊆ Sj ∀ j"2 ∀ t
¿gTiR3:"$'@&"$%$'&%2/"2%b
µ1i]g<C%$=D:<C;&:C
x1iE"232\J<+7
Á T ∀ j ∈ N ∀ t ≥ 1 "2 ∀ sj ∈ Xtj µt+1
j (sj)HR3:"$'&t:E'&7:g]1<C%$P=(*J%B'
:<C;&3:;&IXt+1−j
E"23$\)P<+7C"2%$ RW%23wÇ#T
si
iR3:"$'&OG%$;RW m:j37
µ1i:E'@7:<C%$G%$3
Æ ¿ ÃxÈ6àß!Ó!Ñ â<ÌgßUáÑ ^ÊUá,®gß@Þ:ÔÊYà T
X1i = ∅
"2 ∀ j ∈ N \ i X1i
\J<+7i<Q!*#/#9*s′j ∈ Sj
33E"$;!j:'\J<+7g]#<C%$=@%$'(E% ∃ s−i
µ1i:<C;&:: J*U]J<C%$=(E%
sj = s′j
Æ T ∀ j ∈ NR3:"$'@3Es"2 ∀ t ≥ 1
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µt+1j (sj)
:E'@7:<C%$G%BÊ(T ∀ t ≥ 2 "2 ∀ j ∈ N
Xt
j
\J<+Q<Q!*#/#9*s′i ∈ Sj
33E"$;!j:'6\J<+7g]<C%$=(\#%$ ∃ j<C%$= k ∈ N \j "2 sk ∈ Xt−1
k
"2s−k µt
k(sk):<C;&:3 J*U]
<C%$=(E%s′j = sj
T
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j j ∈ N \ i jR3:"$'@_9*d33E"$;!j:*J'\J<+7g]
<C%$=(\#%$-iR3:"$'&?012!3LXG32L*(5#"$;@%$hE%$*#/%$1R3::E'@7:
µ1i<C%$G%$39]
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si33E"$;!j
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A"2
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¾ TR3:"$'@A33E"$;!j:R3MEK$!G*J!7:\Ji"2H ¾ TR3:"$'&
µ11@"$G%$'&%2/"2%Q0)2!3L
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;=(B,B)
R3:"$':\#@To>?'('@EX12 = X1
3 = A,B T)[¿gT&R3:"$'& µ22(A) "2 µ22(B) :E'@7:!!;&9 R3:'i33E"BA;!j+:j32::9]o<C%$=@%$r ¾ T&R3:"$'&.R3:"$':2'0)K2j-:E'&7:Ij0R3:*\#939T-*G@*m!;&7'
µ23(A)"2
µ23(B)A^E%T
>?QC*#"$;=m:E'@7:t*#%$<S%$"$3G%$*&w9*#t01"$G/#: ¾ T&R3:"$'& :E'@7:7A!)'0)K2j3 J*M"2.*#%$<[S%$"$3G%$*&woE%$*#/%$10)2!3L+G32L*&5#"$;@%$
A33E"BA
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µ22(A), µ23(A), µ
22(B), µ
23(B).
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]U%$'('&Esi ∈ Si
33E"$;!jMEK$!G*J!7:\J]]\J ∀ j ∈ N
A^E% ∃ Zj ⊆ Sj\J<+@]#<C%$=(E%
¾ Tsi ∈ Zi
¿gT ∀ sj ∈ ZjG%$;7RW I:j3-
jR3:"$'@
µj(sj):E'&7::Eg]&<C%$P=(*J%B'C:<CA
;&3:;&ZjE"232\J<+7gT
ÆÁ
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"2(Z ′j)j∈N
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j∈N
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<C%$=(E%(Zj)j∈N
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si ∈ SiEK$!G*J!7:\JIH%$GNQ/% y *LGK$!G+39%$E3!*&9]J'('&EG%$;=@%$*Ub
Zi = si ∪∞⋃
t=1
Xti
"2Zj =
∞⋃
t=1
Xti
∀ j ∈ N \ i .
-CC¿gT/% y *#LK$!G+!;&])'('@EG%$;=&%B*µ1i = µi(si)
"2µtj(sj) = µj(sj) ∀ j ∈ N"2 ∀ t ≥ 2 %$'('&E stj ])<C%$=&%$HC%$GNM/% y *LGK$!G J*'&*#33E3J:*' Zj
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Zi ∀i ∈ NA^E%r9*M33E"BA
;!j:'+\J<+7g]g<C%$=@%$O012!3L+G32L*(5#"$;;@%$1:j32ir'&E3E%$j:%$;=&%$*#3u=( J*UT
>?'('@E_ J:E3<C%$=si ∈ Zi
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a !(R3:"$'&O9*D33EB"$;!G:!1;@%$*#%$E:*J'1](<C%$=&%$'@%$**J'QS%$%$&"2"$@%$(R3:37*J'1]\#;=
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Ç °¾½¿Às¿Á´Â¿ÂÃ
È Ä ¶ d · ¶gÉÊÁÁ - 0æÄÌË
[LbcGPpce_@qΩ = (ω1, . . . , ωn)
]BRW%$Ga RW%'(w(!j:;r:j01!9]<C%$=&%B'H !9*(=(j*#:;&:239%$<H 1%23w*#%$'M39%$E%$0GN!*'M/a*("23\#7'('@E9T Ã ] ¾ s8!:ZYo"$39%$E%$0GN*'i*U]JG%$;=&%$*
Ω = ω1 =%23*!kS;I<+g]
ω2 = 1E3
G%23Hr!G/N]ω3 =
RWX!G/NXG%23 <Q!*#/%$;=(!'M%2%$<C"$*&=M%$;=@%$*GNQG32L*(5i%2//g!;!U!G3<C%$E%$%$!*'M39%$E3!*(BT Ã ] ¾ 6OQP!RLS(TU(VWS(XZY
Ω:j0)3"$E%$*
Θi = θi(ω) | ω ∈ Ω%$;=IG%$'&"$0)%$9"29]<C%$=(E%
!(ω ∈ θi(ω)!!(r\J
ω′ ∈ θi(ω)]?'('@E
θi(ω′) = θi(ω)
]%$'('@EθA^_!*gSE3<+:K$!Giw;;@"$*(=(*#%$'
*#%$&%B22w'1T Ã ] ¾ Í bo|fb|pt_@q
θ(ω)E"232\J<+9'i'(!PS%232L3!'
ΩA^9T"Í ;=m;@*#/'@(/g*'1]1\#;=
θi(ω)
iAÄ/!P'MR3:"$'@r3#/J:m
Ω%$E3<C"239%$3!:Pj013E^Tk
i3jRW/*#:;6;&7A
E*(: R3g]J\#;=irR3:"$'@t!G3<C%$E%$t\J<+7:*J'i%$G%$<C%QC"$*&=(G%$;@%2:j019T<!!(0)%BA/g!;'@*2!G32%$*#K$!ji'(E3!"$E3!<m|%$;=(w'S%$^]k\#;=
ω′ ∈ θi(ω)"2 ∀ ω′′ ∈ θi(ω
′)]/%
ω′′ 6∈ θi(ω)T#>?'('&E
ω:j0)3 J*
ω′′!*'&*2!G32%$*#9]#%$\J:
ω′*#%$<G%$\#%$ "$*&=(G%$;@%2
%$E3<C"239%$3!k:j019T Ã ] ¿68! jS=(:Bb# ¾ TR3:"$'@!*gSE3<+::;&XG%$;=&%$*UbΘ1 = ω1, ω2, ω3 ,
7+HR3:"$'&ORW!*gSE3<+:HEE^]1\#;=Q!G/NR3:E:RW]/%*#%$<3#/6_E3!G/N'D'@a9a3'(wGa*( #"$;@%$%$**!^T Ã ] ¿yOQP!RLS(TU(VWS(XZY.
θ!*gSE3<+:K$!Gw;;@"$*(=S%$G32¢0JE33!GK$!G*J:!jB]?\J ∃ Ω "$E0JE33!GK$!GR3g]k\#;=%$39NG%$;@%2
ω ∈ ΩA^E θ(ω)\J<+DE"239%D**J'dM0JE33!GK$!G*J'1]
<C%$=(*#%$'ω%$%$<C%T
à ] ¾ <;QU1=>@?YI à ] ¾ µ_@q¤ à ] ¿rFbz¨<gJokmlo~ b@a,Ol:Oxhpcebg$q$T
ÆÆ VÈÂÑ ÌWÒ$â?à#ãÊÓ!Ñ Ô<Ò6ÎYY&ÌXWI letnpg!6Jz]\xq1\J_S%$G32]#'('&E%$ RW%23wo!(O"2!!(B]\JM!("2r!!(B]k'('@E9]U\J
ω′′ ∈ θi(ω) ∩ θi(ω′)]U%$'('&Er!!(-<Q!j3
θ(ω) = θ(ω′) =
θ(ω′′)]g!(.<Q!j3 ⋃
ω∈Ω
θ(ω) = Ω]gL;=
θS%$G32T
\×@Ö6× Ï áb¥F6;.ÞwÐb^^D69<5s à ] Á 6OQP!RLS(TU(VWS(XZY
E ⊂ Ω %2%$<C"$*(=(*#%$'I*#%$&%$22w'1T à ] ¿µÍ b|fbo|pt_@qh%$;=i/a*("23\#9]U'(!?3#/R3g]1\#;=
θ(ω) ⊆ E]13#/R3g]1\#;=
ωA^E
E%$GNSE/g^Tk>? )%$*I%2%$3 )%$*v2H<C*#/R'1]U\#;=
ω:j0)3 J*m/a*("23\#9
!G3<C%$E3!EA^9T
à ] Ç6OQP!RLS(TU(VWS(XZY1/3θA^E
i/a*("23\#9Q3#/#:w;;@"$*(=@%
Ki(E) = ω ∈ Ω : θi(ω) ⊆ E ,7- J:E3<C%$=
E%2%$<C"$*(=(E%
K(E)9*m:j0)'I\J<+7g]g<C%$=@%$'( 1%$*mH/#a*&"2WA
\#9Q!G3<C%$E3!EA^9T
à ] ¿<;QU1=>@?YÍ l(gBFbgÁ]`fF\xq@^||6Op_g!albce_|6Jz]zlZht¥hpcw\8QTQl]`cwhFnpg$qf\|nxaph^]`¾ T
K(Ω) = Ω¿gT \JE ⊆ F K(E) ⊆ K(F )Á T
K(E) ∩K(F ) = K(E ∩ F )Ç#TK(E′) ⊆ EÃ TK(E) ⊆ K
(K(E)
)Æ T
Ω \K(E) ⊆ K(Ω \K(E)
) TI letnpg!6Jz]\xqo/&/g!'m à ] Á O"2 à ] Ç % y *LGK$!G'( )^T à ] Á Í bo|fb|pt_@qlU%$\#%$"$;@%2mSE/gL3DuD!G9].7*#%$<íi!*gSE3<+:K$!Gdw;;A&"B*#*&=&%$U/% y *!j:*!_3#/#:w;;@"$*(=(9]g\J*#%$<SE/gL3gbG%$;=&%$*
ωA^ J*
i/a*&"23\#9C!P*E<+:KB!GQw;;&"$*&=&%r_'&a9&%$3'@%$9Nb
θi(ω) = ∩E ⊆ Ω : ω ∈ Ki(E)
.
à ] à yOQP!RLS(TU(VWS(XZY8 M%$;=g39%$E35#"$;'&%2/g@"2"$E3HG%$;=@%$* |N | = 2>?;=
E%2%$<C"$*&=(
R3:"$'&'q'@a9a63#/#::*J'*#%$@%$22w'1]ω ∈ Ω
:j013 J*U] 01*(* '('&E9]\J
ω%$G%$<C%C+j: !?&"$;%$G%$*E7<Q!*#/%$;=(!'i\J<+7:*J'1b
K1(E)]K2(E)
]K1
(K2(E)
) ]K2
(K1(E)
) ]#%$KT
VÈ ÇmÈIÞÕBLÊÒ6ÎYY&ÌXW Æ Ê
à ] ÇQÍ bo|fb|pt_@qO%$D2 RW%$G%$*&!S]n\#;=\J6M'&"$ R3:"$'@H3#/R3EA^9]'('@EX2H!G
3#/R3:'1]g\#;=m_<+:3!'I3#/R3g]#%$KTf&9': _S%$*&3!n/% y *LGK$!G<+:SE3<+: J*m!G'(!<C*#/#*!^T
à ] Æ OQP!RLS(TU(VWS(XZY1>?;=F ⊆ Ω ])<C%$=M<+;&:"$E3%$N(/NDQ'&"$R3:"$'&'&a9a39]#\J
∀ ω ∈ FA^E%
θi(ω) ⊆ F ∀ iA^E%T
>?;=E%2%$<C"$*&=X'&a9a?3#/#:.OR3:"$'&'_'@a9a3
ω ∈ Ω %$E3<C"239%$3!):j013 J*U]\J ∃ F<+;&:k"$E3%$N&/N+%2%$<C"$*&=&]<C%$=(E%
ω ∈ F ⊆ ET
à ] Á 68!T#É<+:(/g!'_R3:"$'&!*gSE3<+:K$!GXw;;&"$*&=&%Θ2 = ω1, ω2, ω3
%$ )%$*r%2%$3 1%$* ω1, ω2 '&a9a3#/#:9]g\JC_%$E3<C"239%$3!:j0) ω1 ;= ω2T
à ] ¾ KPLML6ZY"R8t¥hpcw\8QTQlohxa b@a,Ol:Oxhpcebg$qb@a!(
Ki(E) = E!!(Edh|\^]~pcI_jm]`bm]v3FvHGhm\^]`_@apnxqfnxaQa6PtTP^]o]
!!!(E =
⋃
i
ΘiT
I letnpg!6Jz]\xqUk%$;=(w'IS%$^]#\#;=D!(O!;&@])'('&E ∀ ω ∈ EA^E%
θi(ω) ⊆ E]#%$ 1Nn'&a9&%$WA
'&%B2!'D!P!(B|!!!(^A^ 1NU3E3!(!j:!G*m'@a7&%B'@%$2!'I!(BT Ã ] Á <;QU1=>@?YI Ã ] Ã µ_@q¤ Ã ] Æ rFbz¨<gJokmlo~ b@a,Ol:Oxhpcebg$qI letnpg!6Jz]\xqok%$;=(w'+S%$^]g\#;=
E'&a9a3#/#:
ωA^ J*UT!£Ä;=
E ⊇ Ki(E) ⊇ Kj
(Ki(E)
).
46!&%BΩ&"$;@%2."2.%$G%$<C%-&"$;%$G%$*IE%$;E%232!Ghj:*#K$*J'1]&L;= ∃ F
]@<C%$=(E%Ki(F ) =
F]gL;=
F<+;&:n"$E3%$N(/NT
cJE/gL3gbk%$;=(w'+S%$E%2%$<C"$*&=D'@a9a3#/#: Ã ] Æ O/% y *LGK$!G+39%$E3!*&7T
>?'('&EF ⊆ E Ki(F ) = F ⊆ Ki(E)
%$KT]"2F%$G%$<C%-<Q!*#/%$;=(!'
Ki
(Kj(E)
) A*#%$'1]gL;=
ωT
à ] Ç68!T#?!G32'& EK2'|N | = n
R3:"$'@ *i%$;=M%$E%$<_ )%$*U])<Q!*#/%$*'(!kj:3\JSR3IQa !G%$'iEK9:9])/%XR3:SR3:.*#%$<mTglU%$;&j: M%$;=IEKt0!G32'@9T % y *!j: R'I%$GN39aE
Ω%$E3<C"239%$3!U:j01A
'-\J<+7:9Tx-nR3:"$'@U*U]'('@E
2n−1 G%$\#%$"$;@%2:j0)'H37:<+g]<Q!@%$
Æ Ë VÈÂÑ ÌWÒ$â?à#ãÊÓ!Ñ Ô<Ò6ÎYY&ÌXWG%$;&j: s%$;=EKX0!G32'&BT1>?;=&AÄ%$;=:j0)
ω := (p, p, t, . . . , p)]o\#8
p x 0!G3$A'&9]t x 3!G32B]JRW%$Ga RW% ω
M0!G32'@XEK2'37:<+:9]UL;=ω ≥ 1
Tn>?;=&AÄ%$;=mR3:"$'@j:SR3QXa !G%$'mEK9:
ωi]/%rQR3:SR3:*#%$<m]J7@b
θi(ω) =(p, ω−i), (t, ω−i)
46!*#/%$*CR3:"$'&\J<+7Q'@"$ %$G%$<H5U]#'(!&"$@%-\Jω = 1
TÄlU%$;&j: %$;=I0#!32'& EK*L¡Me+'@a9a3#/#:9T#`RW:"B'm'('&EOS%WRW%$9N(/g!'D )%]\JQj'(!U<+;&:En3#/R3g]#\#;=3!G32X;=I0!G32'@ HEK9gbJ7
∣∣θi(ω)∣∣ = 1 .
46!(aE3"$*!'1]7\J %$*'(!(*#%$< %$<C%$!(%$'@%$9"$?O%$GNSE/gG J*:M46!*#/%$*'(!&S%$uRLSR3!*gSE3<+:K$!G!t"2<C%$;&:j0LSR3g]#\#;=Ia m<Q!*(%$;=D0!G32'@ EK$*J'D'@%$UG%$**!G%T-QtR3:"$' S_%$E3<C"239%$3!k:j01
k0!G32'@ EK2E3<+@])'('@E tR3:"$'
kSE/gGG%$3@T à ] à ¤Í bo|fb|pt_@qtZS%$j/#3 1hRWG:\J]\#;=<C%$;@G/#:D'@a9aC3#/#:*<_u!P'1T à ] à G8!ZY#"$%$E%$;%23O%$;=(<+:31:9@j D: 1E2!'1](*#%$< <C%239%-%$G%$*gA"$;N^T&-D'(wGa*gA^'(wGa*s:<+/g*J'1]n%$&%2323!'MK2:9]n%$;=(w3%2r:<+/#:_%2%$"$*;=&N$*J%B'1T#>?;=(!'D%$E%$;E"23H%$<'ESR3Q%$9"$E3'@(K':23*!kHa*J:GQ:<+/#:39T %$;=(!' %$E%$;E"230JE*#K23*#':E3;=(j:3 Jq 1&K2:3'&93Y%BG%$*#"$;;@%$^T
Y'&"$k%$E%$;%23k0JE*#K23*#'X:E3;=(j:'r<C%$;'@%$9/"2%O%$GN3k<C%$;@%$;=&%B9%$39]\#;=r\JI:E;=&j:'%$E%2/g<C"$*&=(%$G%$*(wh7:E3*J'1]U:<+/g*J'1TU>?'('&EHD:E3;=&j:*sE"232&%Bk0JE*#K23*#'r:E3'(wG/_'@a9ak:<+/#:?\J/g!%$E3@"$&%$#<+:3!'H0JE*#K23*#'(*J'1T **J'IG32L*(5#"$;@%]\#;=M_:E3%$PS;R3:'1b
p, p > 0T
,O%$'&a9&%$3'@%$2!'&AÄ%H_'@a9aB:<+/#:MÉ4:(/g!'I%$E%$;Q0JE*#K23*#'_<C%$;'0R3Qw9%$*#%$%$9]#%$E3EN1:E3O'&%$ ¾ T A^*#%$'
'(wG/%$*!G%]('(wGa*( )%$*m-*#%$< G%$\#%$O !9*&=& )*U](\#;=I¿gT(<C%$;'0Xtw#2%$*#%$%$"2+L;=v*#%$< S;s:<+/g*!^TO46!:* ¾ Th<C%$;'0d6w9%$*#%$%$9]:E3I'@%$-¿gT A^*J''(wG/%$*!^]g\#;=+<C%$;'0C-w9%$*#%$%$9]<C%$E3%B*#*#%$'+\!j:*(=&: J*6¿gTg*#%$<G%$\#%$O !2"2O%$9"$E3*#%$< :<+/o]g%$KT46!@%$1%$;=&%$3G%$*D!*gSE3<+:K$!G :j0)3 J*6%$<[:!'+'@a9a3#/#::QX:<+/#:9]L;=MH%$<+E/oT Ã ] Ê^OQP!RLS(TU(VWS(XZY
µi(ω)A^t%$;=mG32L*(5#"$;!k<C"$E3"$'@%$
SiA^*
(Si =∏
j∈N\i
Sj)
iR3:"$'&@"$G%$'&%2/"2%
ω:j013 J*_a !#R3:"$'&33E"$;!jQ:j32::E^T
S%$*(3!n00JE:3#oS;&<+22'MuREXú-3\M"$%$G"$9bΩ_'&aE3*&=&%$9%$9]J<C%$=( )%$*i-R3:"$'@'_R3:37*J'1T
VÈ ÎÈ Ñ (ã6@à$7Þ Xã66Yâ?á,LÊÑËâ<ÌÓ!ÏÓ!Ñ Ô Æ Ì
!(si(ω)
iR3:"$'&33E"$;!j:R3
ω:j013 J*
!!(µi(ω) i
R3:"$'@&"$G%$'@%2/"2%r%$G%$*gS%$G%B!k33E"$;!j:!E!!!(
θi(ω) iR3:"$'@!G3<C%$E%$%
ωA^ J*UT£Ä;=di33E"$;!ji\J<+2Q'(! )N7(LSRw'd6@"$G%$'&%2/"2%$'\J<+7:9"2_'0R'Y
(s, µ) ∈ S ×M 0J:E39T Ã ] Ç<;QU1=>@?Y S bo|ppagfbc H_DLnB|p
ω ∈ Ω \pcrcwhu$n^]&Qhpg ∀ i\^]`_@a^nq
!(lrqdbjlh¤] P8QTQlm\^]`_@apnxqqm]zjh^]`_|plo\plz]`θi(ω) ⊆ ω′ ∈ Ω | s−i(ω
′) = s−i(ω)
!!((Op_ceb@a^bF_@qbapnpgLtmlrqBtf]`bg$qh]`fF\xqf\6Oxhpce<hµi(ω)
]\pdn@|h^]\xqfh⊂s−i(ω
′) ∈ s−i:ω′ ∈ θi(ω)
!!!(jhkmlonpg:\pc lrqsi(ω) i
cebo|Bn8QTQ$Ox\pcwhxqBthµi(ω)
E`jh¬ aapnpjs(ω) G C hxqD8Ebo|pbg$q®c h
\×øØ× ' ;:?A=IÙW=C7 9<;.?A=_^66>¥F6;«¤H865¥ßC=IÚßC«Bà à ] à ;QU1=3>@?Y S bo|ppafbc HNDLn@|p
ω ∈ Ω \pcrcwhu$n^]&Qhpg ∀ i ∈ Nm\^]`_@apnxqjBh ]`bc fb@qc gWb@aht¥hpcw\8QTQlohxa^
¾ Tilrqdbjlh] P8QTQlm\^]`_@apnxqqm]zjh^]`_|plo\plz]`
θi(ω) ⊆ω′ ∈ Ω : s−i(ω
′) = s−i(ω)
¿gTO&"$G%$<C%$!U'@*2!G32%$*#%$'m!G3<C%$E%$%$!&%$^b
µi(ω) ⊂s−i(ω
′) ∈ s−i:ω′ ∈ θi(ω)
Á Ti&jhkmlonpg:\pc lrqHZhtht
si(ω)ht
im\^]`_@apnxqµceb|mn8QTQ$Ox\pcwhxqBth
µi(ω)E`jh
¬ aapnpj (si(ω)
)i∈N
Gm\p]`_a C hqD8Eb|pbg$q®c h
I letnpg!6Jz]\xqCb Á T<Q!j3si(ω) i
G%$;RW m:j37X@"$G%$'&%2/"2%<C%$G%$39]#<C%$= ¾ 32LPA*(5#"$;5m%2%$<C"$*(=¿gT<Q!j3
s−i(ω′) ∈ s−i : ω
′ ∈ θi(ω) "2
¾ T<Q!j3tHS%$*(3!o\J<+si(ω)
T Ã ] Æ r;QU1=>@?Y S b|ppaefbc |N | = 2 ω ∈ Ω dQl(gBFbo|p^leaµm\^]`_@apnxqjhÁ]`bc fb@qc gWb@aÁhthpcw\8QTQlohxa^
ÊÈ VÈÂÑ ÌWÒ$â?à#ãÊÓ!Ñ Ô<Ò6ÎYY&ÌXW¾ Tælrqdbjlh¤] P8QTQlm\^]`_@apnxqeOp_ceb@a^bF_@qblz]`
θi(ω) ⊆ω′ ∈ Ω, µj(ω
′) = µj(ω)
j 6= i
¿gT4hpOp_ceb@a^bF_@qbmlapnpgLtmlrqtf]`bg$qb@a¤htlrqdbjBbm]`bl:Opbceµi(ω) ⊂
si(ω)
Á Tq3#/R3g]\#;=m_a !#R3:"$'@EK$!G*J:!G J:E3<C%$=ω′ ∈ θi(ω)
A^Esj(ω
′)jR3:"$'&G%$;7RW m:j37
µj(ω′)A^E%
(j 6= i)
>?'('&E(α1, α2)
'&%$@%$E3?33E"$;!j(α1, α2) =
(µ2(ω), µ1(ω)
) '&%$@%$E3?33E"$;!jú-3\gAÄ%$;=&%$*#3u=@TI letnpg!6Jz]\xqnlU%$;=@%$*
s∗i iRW:"B'& 3E3"$;!j:R3
αi = µj(ω):<C;&:^T)>?'('&E-¿gT
<Q!j3 ∃ ω′ ∈ θi(ω)]o<C%$=(E%
si(ω′) = s∗i
| Á T A^ 1h'@a7@%$3'&%$2!'1]o\#;=s∗i i
G%$;7RW :j37
µi(ω′)A^E%]<C%$= ¾ T<Q!j3
µi(ω)T
à ] Ê^<;QU1=>@?Y)n%$;=(w'IS%$^]g\#;= |N | = 2 "2 ω :j013 J*6'@a9a3#/#:\#;=1b¾ T-R3:"$'@'I@"$G%$'&%2/"2%$!o'&*2!G32%$*#%$'I!G3<C%$E%$%$!'('&%$¿gT-R3:"$'@'IEK$!*J:!G'>?'('&E
si(ω)%$G%$<C%HX/<Q!*J: 3E"$;!j:'D%$!<Q!*J:K$!GR3:*J'1]#@"$;@%$E%2/g<C"$*(=@"$*#%$'
S∞iA^*#%$'1T
\×&%µ× Ñ 8=<;[Ò«E$FG7:9IH6H i%2//g!;!ORW%$GaG"2%$'('@%$/3'
i ∈ NR3:"$'@'
Si33E"$;!j:'('^]hj<Q!*(
Ω:j0) \J<+@Tk%$;=(w'mS%$^])\#;=6<Q!*#/%$;=(!'+R3:"$'&tE%$*#/%$'&%$2!'6j<Q!=&%$*G32L*&5#"$;m%BPA
32j:ΩSaGa39]J
ω%2%$<C"$*(=@%$'D )%$'&a9&%$3'@%$2"$!G%$N&%$*UT
à ] Ë<OQP!RLS(TU(VWS(XZY&F@%$&^f&!;*J(nw;;@"$*(=1bBRW%$Ga RW%τi|τi(ω)
iR3:"$'&
ω:j0)3 1%$!
/a*("2%$GN33!lrqdbj@bm]`blz]STF@%$Ga RW%
TiCa39%$-G%$\#%$"$;@%2
τi(ω)"$E3"$'6\J<+7:9]U%$'('&E
TiA^r
iR3:"$'@
3L0#!U\J<+7:*J'D*J%B&%$22w'1T#c#%$%232w'1]\#;=
pi(τ−1i (ti)
)> 0 ∀ ti ∈ Ti-
iR3:"$'&
ti ∈ TiRW%$9"23 "232G%$!^]
τ−1i (ti)_013%$E3!GE3!U\!9T
VÈ -È @ß6W&T¯`ÑxÉÊ@ÞËâ<Ë Ê ¾
à ] Ì6OQP!RLS(TU(VWS(XZY)>?;=D,9=@%2WA^!#R3:"$'1T¾ T
i ∈ N |N | <∞¿gT
ω ∈ Ω |Ω| <∞Á T ∀ i ∈ N Si
33E"$;!j:'Ç#T ∀ i Ti
RW%$3!;*J(^A^\J<+X"2τi : Ω −→ Ti
RW%$3!;*J(^ASw;;&"$*(=Ã T
pi%$G32j:
ΩA^*
iR3:"$'& _0E3!GE3!U&"$G%$'@%2/"2%
po(τ−1i (ti)
)> 0 ∀ ti ∈ Ti
Æ T ∀ iR3:"$'&3E ¹i
0E%BS%$E%$*#K$!jQE%$j:K$!G+S × ΩA^*UT
à ] Æ QÍ b|fbo|pt_@q[S%$*&3!O/% y *LGK$!Gi<C%$;@%$*;@%2/g!^]k\#;=dD%$;=@%2tR3:"$'@''(wGa*gA )a9Ni@"$G%$'&%2/"2%$hE%B*J/%$P'&%$29%$*#%$'Ym%$E3<C"239%$3!:j0)3E^Thf(N9]n\Jim%$E3<C"239%$3!:j0)'m'@a9"HE R'mXa !G%$'M!G3<C%$E%$"$9]1_S%$*(3!n/% y *LGK$!GD'<+t=&*+R3:7A"$'&'I%$G%$<Q9"2"$E%!G9]#<Q!'@Etr%$;=@%2?R3:"$'@'M_a !G%$'I!G3<C%$E%$%$!@%$U*!*#K2%$*#%$'3!G32: J*UT
à ] ¾ È6OQP!RLS(TU(VWS(XZY),9=@%2WA^!R3:"$'&'Dú-3\gAÄ%$;=@%$*#3u=(/3
G = (N,Ω, S, T,Θ,º) A^E%XR3:"$'mú-3\gAÄ%$;=&%$*#3u=&:*Q'&a9&%$3'@%$9N-"$E3WARw'1b
¾ T(i, ti) i ∈ N
]ti ∈ Ti
¿gT(i, ti)
0J:E3EC/3Si33E"$;!jX\J<+
Á Ts∗ º(i,ti) s
0)*(*'('&E9]\JLi(s
∗, ti) ºi Li(s, ti)] \#
Li%$G32j:
S × Ω A^*Ub
((s∗(j, τj(ω)
))j∈N
, ω)=
θi(ω)
θi(τ−1
i (ti)),\J
ω ∈ τ−1(ti)
0,'(wGa*( )%$*
à ] Ê^Í bo|Bmb|pt_@qk ,9=@%2WA^!JR3:"$'@'Mú-3\gAÄ%$;=&%$*#3u=&: J*6<Q!*#/%$*IR3:"$'@tC37:7A<+:EXG%$;7RW i33E"$;!j::j32SR3Q_/3 32!;*J:n"2 Xa !#R3:"$'@3Ek'@3/3 32!;*J:U<C%$G%$33!U&"$G%$'@%2/"2<C%$G%$39T
Ê¿ VÈÂÑ ÌWÒ$â?à#ãÊÓ!Ñ Ô<Ò6ÎYY&ÌXW\× U× =<>B¥G7:Hì#ñ! >?G%$<C%B22w'D-j: !(R3:"$'&9](\#)BT9R3:"$'&3L0#r!G3<C%BE9]&BT AÄ" 1
2
e 12
G32LPA*(5#"$;;@%$
t2]g!G%$3&%
t′2
t2 t′2l z l zÍ
(3, 1) (2, 0) (3, 0) (2, 1)(0, 1) (4, 0) (0, 0) (4, 1)
ì)øB k%$'(!*&3w'`j: !-*#%$< %$ RW%2I!*gSE3<+:K$!G_R3:"$'&9 14
G32L*(5#"$;;@%$rA<+:3E3!P37U/3R3:"$'&kR3:32'1] 3
4
AÄ/%$10)%2/g!;CBA^@%$U/39Tg%$E%23w'C<C%$;
_'&%$@%$E3 ú-3\gAÄ%$;=@%$*#3u=(9
A =(5, 1) (1, 0)
(2, 4) (3, 6)B =
(4, 1) (3, 3)
(5, 6) (2, 3)
ì#ý! G/R'<C%$;Oj: !g%Bg%$*2LHj'( J*X/#39]9*#%$<%$ RW%2U!*gSE3<+:K$!G)R3:"$'&9
r r r r r r r rAAAAA
¢¢¢¢¢rBT
,(2, 1) (0, 3)
AAAAA
¢¢¢¢¢rBT
,(−1, 6) (5, 4)
AAAAA
¢¢¢¢¢r3BT
,(4, 0) (1, 3)
AAAAA
¢¢¢¢¢rBT
,(6, 2) (5, 3)
@@@@@
¡¡
¡¡¡rp1 =
1
3p2 =
2
3
@@@@@
¡¡
¡¡¡rp1 =
3
4p2 =
1
4HHHHHHHHH
©©©©©©©©©rBT ,
ì#ç! Íh*I%$;=C/g#01J0!jKP = 1− q1− q2
'&%$E%23G%$3!#w;;@"$**(=&%$^T&46!*#/g'&"B.%$E3<C%$GN12
G32L*(5#"$;;@%$c]@!G%$3@%
c'&a"$;;@%$JE%$*#/%$'@%$2!'1]&/% K2'CR3:SR3:!G3<C%$E3!^T&46!
X,9=@%2WA^!Uú3\gAÄ%$;=@%$*#3u=!M
Ó °¾½¿Às¿Á´Â¿ÂÃ
È 0 , n º · 10¸¹¸
>?;=@%2.33E"$;!j:'O39':h%$*#%$@%$2*!^]g9'Or33E"$;!j:'9]g<C%$=@%$'(*#"$o-a !G%$'G"$0)"2%$!N.w;;@%$3G%$*(wR3:32*'1]US%$"$3G%$*v33E"$;!j:*J's*#%$@%$22w'1T>?9%$'G%$\#%$3*#%$':j*#/@'iS7H<Q!*#/g!;I;=&*J2.R3:32'!O"2E:K$!G'1])\#nj<Q!=&%$*i%$GNE%Ea;2L%$3h39 ):P=X39%$E3!*&?:2SR'X33E"$;!j:!*'80)"$G/#:#'@"$?33E"$;!j-%2%$"$*:j*#/@*iK2%$E"$;@%$3w*'!BT 9'mY33E"$;!j:'9]O<C%$=&%$'qs<+:3!'dR3:"$'@'!+%$GN9NYG"$01"2%$!N%232!'
w;;@N7@"trR3:hE:'&a9&%$3'@%$9NHG"$0)"239]@S%$"$%$G%2O33E"$;!j:'(*J'C39':h*#%$&%$2*!^T&>?9%$''&a2aO39':O%$'(wGa*L%$*!nrE93y3E3!;;@%$E.33E"$;!j:'"2O%$9%$'@%$*D 1%$wUr;E3!<m]XO!SE"2 XE"$0J7e@ )t33E"$;!j:'9TE23m33E"$;!j//g!;'@(0)%$E3L1]h<QL;dia !G%$'Y!G9]:*J%$;=d w*(%BA
"23!U0)%$E3!G(/g#3!'( 1%T#É;E3!<33E"$;!jX w#*&%$"23!k37'37_&"$;%$G%$*U]J7H\JC'&&0)%$E:K$!GM<C%$;@37'/o]U'('@E-@"$;G%$;#]UIO!tSE33E"$;!jD%2%$"$*U]1\JD+0JE3WA*#%$E3w*'6*#%$<À'@(0)%$E:^]U'('&Et<Q!%$<m]1/%_\JD+0JE33*#%$Er%$'@%$9/'@(0)%$E:*!^]n'('@E<Q!U!G9TÉE"$0J7e@ )t33E"$;!jQ%2%$"$*i_ w*(%$"2%$;=D0)%$E3!G(/g#\#32uUT
iµ×@Ö6× 9_^G7:=<>B=<5¥;C=IÙ«B;:?A9C7:=<>¡7E$FG7:9IH8H 9'R3:"$'@'9]g<C%$=&%$'@%$K2'C%$;=g39%$EkR3:37*J'CG%](%$;=g39%$E.G%WR3:393R3:"BA'&'(*J'ä*#%C3\#r;&<C%B]n9'__R3:"$'&'7]U<C%$=@%$'&%$H%$;=(<+::*a #39aER3:32*'QG%]@!G3<C"$3GN(/"2%2nR3:"$'&'(*J'Q*#%$@%$22w'1Tf&9':O L;=C%$GN@:kR3:"$'&.32gA0)%$ER3:"$'(*J'I!G*#%$&%$2*!^T#n%$'(!*(3w'MHS;@=M/g!G%$<Q<+:9TY@"$;%$G%$*tR3:"$'&'r%2%$"$*XO%$E%2/%$3!R3:"$'( )g37:E3<+2SR'_huRU320)%$ESR3:"$'
33E"$;!jX\J<+7:t"2'(! y 9%$NXw;;@"$*(=@%$!9T Æ ] ¾ OyP!RS(TU(VWS(XZYklU%$;=@%$*
si(t)
iR3:"$'@r33E"$;!j:R3M
tAÄ/g!'i0)%$E3!G(/g#3 J*U|oG%BA
;=&%B*s(t) =
(s1(t), . . . , sn(t)
) 33E"$;!jQ&%$'(EttAÄ/g!'D01%$E3!G&/g#3 J*U]JG%$;=&%$*
h(t)X33E"$;!jC32%$'(E'DE7Ct01%$E3!G&/g#3!;#])7
h(t) =(s(1), . . . , s(t)
) T
Ê7Ç jÈÂÑ (ã67Þ@áÄÞ ÉÊÞËâ<Ë Æ ] ¾ 8Í bo|fb|pt_@q&£Ä;=
h(t)E3<+27+
t01%$E3!G&/g#3!;+<C%$;%$3 a39%2G"$0)"239T
Æ ] ¿Í bo|fb|pt_@qt3E"$;!j%$'('@EC%$;=dG%$'&"$09"2+t ∈ 1, 2, . . . "2C6a39%2G%$\#%$"$;@%2aE3"$*#%$3 )N^T
Æ ] ¿OQP!RLS(TU(VWS(XZYnlU%$;=@%$*iR3:"$'&-'(! y 9%$N+w;;&"$*(=@%$!*#%$'iE7I01%$E3!G&/g#*gA
'&"B*(vi =
(vi(1), . . . vi(t), . . .
) T1>'('&EriR3:"$'&t'(! y 9%$NIw;;&"$*&=&%QI320)%$EWA
R3:"$'( J*ui(vi)
T
Æ ] Á 8Í bo|fb|pt_@qui(vi)
A^E%H:j: J*6'@"$<C%$;\J:E7: 39':9b
¾ Tui(vi) =
∞∑
t=1
δt−1vi(t)]
\#δsu;=(*#%$&%$9%$3/g!G32'&*&i"$*(=@%$9Næ'&"2N É01*&*x/% y *!j: R'!BT
/g!G32'&*&Q"$*(=@%$9NMRW%$G%$*(3\#%$C!G/N0E%BS%$E%$*#K$!j:9];=sS=(:3!G32L*&5gA"$;@%$9T
¿gT39': H:3j;@\J32*#:;C<C%$;\J:E7:9b
ui(vi) = limT→∞
1
T
T∑
t=1
vi(t).
iµ×øØ× 9_^G7:=<>B=<5ÅE$FG7:9IHÔ¥«B;&H66587:F6>BF6; 5¥9<>HrÐ¥> Æ ] Á 6OQP!RLS(TU(VWS(XZYJ<C%$;@%$*;@%2/g\#%$NC'(! y 9%$"2%$'D\J<+7
C = w | w H%$;=g39%$E35QR3:"$'I'(! y 9%$"2%$!*'D'&*&&%BD E3'@GR3 .
Æ ] ¾ 68!ZY)n%$'(!*(3w'M_'&a7@%$3'&%$9N_S;@=D/g!G%$<Q<+R3:"$'@9bA \B t&0 út%$< (0t&0
(3, 3) (0, 4)út%$< (0(4, 0) (1, 1)
%$'('@E
jÈ ÎÈ&6YgÞ@áÌÉÊ@ÞËÆÑ p$Ëâ<ÌÞ^ÊUá@Ê¥ÌW@áËBÎ)á Ê Ã
>? 1%$*`6%2%$3 )%$*<Q!*#/%$;=(!'R3:"$'@+NC
33E"$;!j:C:j323iG%$;&j: ¾ %$;=g"$;C*&=&%$E%$<C"$*&=(\#%$ R9T$£Ä;=D%$**#"$U'@%$&%2%$ )%$n%$;=(!'(w'D%$<Z%$G"$;@32!'I<C%$;#T !G3<C"$3GN(/"2%2RW:"B'(EB_!G/% y *!j: R'm
iR3:"$'@<Q!*!<+7M'(! y 9%$"2"$9b
Æ ] Ç6OQP!RLS(TU(VWS(XZYumi := min
(δ1,...,δi−1,δi+1,...,δN )maxδi
ui(δ).
Æ ] ÇQÍ b|fbo|pt_@qiR3:"$'@_%$2Qm'(! y 9%$"23_<Q!P*J/%$*'@"$001%$*Y3#/R3M;&EB*&:*!^]
%$;=g39%$E35#%$*i_G%$;RW m:j32?R3:32gT
Æ ] Ã 6OQP!RLS(TU(VWS(XZY)lU%$;=@%$*UbD := w | w ∈ C
"2wi ≥ um
i ∀ i
D_G%$\#%$"$;@%2 "2!*#/g!(!G/gJ:!G*6EK$!G*J:!G'(! y 9%$"29T
Æ ] ¿68!ZY) r%$GN9NQ0)"$G/#:S=(3gb
Ê Æ jÈÂÑ (ã67Þ@áÄÞ ÉÊÞËâ<Ë Æ ] Æ 6OQP!RLS(TU(VWS(XZY
E = w | G"$%$2!'Mú-3\gAÄ%$;=@%$*#3u=M w:3j;@'(! y 9%$"2%$ .
Æ ] ¾ df=)P! jS'(GEÕpBT ¬ |p¤bo|pxqBtbjacµm\^]`_@apn^]<lrqd_m]`bc @abOp_|^]`bcebg$qBtbj S b|ppaÅfbc HDLnB|pÁhm\^]`_@apnxqfnxaÂht¤\^]zcwh|nxq°DLhxqBtnpg1al ¨Ctbm]v$@^||6Op_g!^]hpc aphpc dht@t\xa^ ¬ aapnpjE =
DT
I letnpg!6Jz]\xqoÉ !9*(=(L: Á G"$0)"23 1Nk:^b¾ TgG"$01"29b
E ⊆ D|E"2
D/% y *LGK$!GR3: 1n3E3!(!j:!G*m'@a7@%$3'&%$2!'
¿gT7G"$01"29bn%$'(!*(3w'H%$;=r%$39NG%$;@%2?=&*Q33E"$;!jtE79]<C%$=w ∈ D
:3j;@'(! y 9%$"23.%$E%2/g<C"$*&=&%$@T(>?'('@E ∃ =(*+3!G32rú-3\gAÄ%$;=@%$*#3u=(rr320)%$ESR3:"$'(*J'1]\#kX33E"$;!j:'M%E7Q%$;=&%$*#3u=(!^T#>2 '@*#33E3'(3LIu*mj:SR'I 1%bk%$'(!*&3w'Ir'@a7@%$3'&%$9N_33E"$;!j:'9TL-HR3:"$'
t− 1 0)%$E3!G(/g#*m:.<C%$;S%$G%$h′(t)
A^*#%$'1]&'('@ES=(3't01%$E3!G&/g#3 J*+!G9T^-t*#%$<V&h9*#
h′(t)A^&%$^]@'('&E
(!93;&: R'<C%$;h(t)
aE3"$*#%$%$_"2r%$'(!*(3w'+%$GNm%$"$E"239TnlU%$;=@%$*d%$i'('&E
j 6= iA^E%HtR3:"$'
sm−i = arg min(δ1,...,δi−1,δi+1,...δN )
maxui(δ).
,O%$j:SR'1]g\#;=D%$Hú-3\gAÄ%$;=@%$*#3u=&T$-X-R3:"$'@'D%B;=&%$*#3u=(!U33E"$;!j:R':7Aj32SR3:'1]k+%$E%2/g<C"$*(=
h′]n"2_D:3j;@_'(! y 9%$"2
iA^E%
wi ≥ umi
<Q!&%$w ∈ D
BT-i%$"$E39]'('@ES%$"$&%]o\#;=sDa !nR3:"$'&r%$;=&%$*#3u=(!h33E"$;!j:tR3:32!'1]
i37:<+:E+G"$%$2!'i'@%2/g&%$9N U])<Q!*&umi
<Q!*#/%$*01%$E3!G&/g#3 )*UTo>?C2ORW%$G%$*&3!^]1\#;=i:3j;@'(! y 9%$"2%*#%$<Z*J;=@ U]#<Q!P*(
umi
T$£Ä;=Mr%$"$E"2*#%$<Z'@%2/g&%$9NiA^*J%B'1T
Á TgG"$01"29b ∀ w ∈ DA^E% ∃ 33E"$;!jQE79]<C%$=(*#%$'m:3j;@'(! y 9%$"2%b w T
Æ ] Ã 8Í bo|fb|pt_@qnÉ"$%$U*&G:;&gb# <Q! 39%$E39N&/"2%$n%$G"$E3\#%$N]J%BG"$E3\#%$N+a*<+7A;&:.'(!G%$EN37'@GX%$;=@%$9"$;;@%$1!9T( t%$;=&%$*#3u=(!)0J:=&g]g<Q!.G%WR3:37*J'1]&:E3;=(j:7A'_uSR3:*Q\J:E$\)r<C%$;#T@n%$E3<C"239%$%29]@\#;=QOR3:"$'@'Q %aÖ+K2%$*#0)%$E%$<Q!;_%$'E3*J'R3*#!S](%$\\#%$-%$;=&%$9"$;@%B'C'&%$PG%$*#%$'1]gt;=&'(E*M:E3;=&j:'+E:*Dj'('(!^]g\#;=D\#9tR3*)'D%$^T
iµ×&%µ× × 6;ØrÒ=_^6=<5¥;Ù¥>Ô¥«B;&H66587:F6>BF6;;6>4i3t2t_%2%B%B (!93;&: R'1]J<Q!'@E--R3:"$'@'mX'(! y 9%$NQw;;@"$*(=D'@*#33E3J:7Aj:XE:*6/g!G32'@*("$*(=&%B9N @%232*#%$' y ;=&%$G%B<_ 1%T 7@b
ui(vi) =∞∑
t=1
δt−1vi(t).
jÈ -È Ìgß8®X¯`6Y8W&Ì$8Ú)áW.Ñ p$Ëâ<ÌÞ^ÊUá@Êx7ßUá ÊÊ
t%$;=g39%$E35#"$;H'&%2/g@"2"$E3h%$;=(w'XS%$^]@\#;=δ<Q!*#/%$*_R3:"$'@.37:<+:E;=(*J
δ ∈(0, 1)
T Æ ] Æ Í b|fbo|pt_@q
δA^r!*&%$E30E%$:\JSR'u;=!G9]1<Q!*(_C!G/NM0E%BS%$E%$*#K$!jM"$*(=@%$9N
ρ"2r
λR3:"$'6S=(::*J'iG32L*(5#"$;@%_e**J'IG32L*&5#"$;@%]o\#;=i\Jm
R3:"$'t0)%$E3!G(/g#3!;CE39]J'('@EOS=((/g!'1]J7H%$"$E3!
(t+ 1)A^9]L;=
δ = ρλ.
k%$;=(w'S%$^]U\#;=YQR3:"$'TA^ )%$*Y )%BS%WRW%$9N(/a3Q"2XM%$G"$E3_'(! y 9%$"2_9"$E3#S%B
K2'I*#E3<+!7:j: '@"$E/"2%BT#>?'('@E9b∞∑
k=1
[P (T = k)
k∑
t=1
ρt−1vi(t)
]=
=∞∑
k=1
[λk−1(1− λ)
k∑
t=1
ρt−1vi(t)
]=
=∞∑
t=1
(λρ)t−1vi(t).
Æ ] Á 68!ZY)É<+:E(!93;&:S;@=D/g!G%$<Q<+: %$G%$<C%$22w'6HS%$*(3!U"$E3%$<C%$9"23!U'&%BAE%$3 1%B*nbF&:"$'&!*'D'@a7@%2"$'D_'&a7@%$3'&%$9NQ33E"$;!j:'9b
¾ T0)%$E3!G(/g#3 J*6'@(0)%$E: R3*J'(C)
t > 101%$E3!G&/g#3 J* NC
\Jh(t)
G%$;&j: 6%$;=NC
E3<+C
%$;=&"$ '@"$*(9TδG%$;=&%$*Qt/g!G32'@*("$*&=&%$9NTYS%$*&3!33E"$;!jtú-3\gAÄ%$;=&%$*#3u=1bx-thR3:"$'@'
'(!E3*J'mX33E"$;!jX<C%$G%$39])_'(! y 9%$"29b
vi(t) = 3 ∀ t.
/g!G32'@*(:'(! y 9%$"29b∞∑
t=1
3δt−1 =3
1− δ.
Ít!93;&: R*'D<C3 %$;=D<+:3!'D33E"$;!j:9TJ H%$GNR3:"$'&:j37C ¾ T0)%$E3!G(/g#WA J*
NC*#%$<'&&01%$E3L!33E"$;!j:9T1>?QrR3:"$'& 3#/R3g]J\#;=iC<+:3!'
CA^
S;mR3:37*!t%$'('@EÁ\!G39%$*!G3<C%$E3!-33E"$;!j:R3:BTZ£Ä;=Y37:<+:E!G+K2'NC33E"$;!jX:j32:X<+E/oT!£Ä;=M_'(! y 9%$NXw;;@"$*(=1b
vi(t) =
4 t = 1
1 t > 1
ÊË jÈÂÑ (ã67Þ@áÄÞ ÉÊÞËâ<Ë%$'('@EtX/g!G32'@*(:'(! y 9%$"2%
4 +∞∑
t=2
δt−1 = 4 +δ
1− δ
<Q!@%$^b3
1− δ≥ 4 + δ
1− δ
\Jδ ≥ 1/3.
7rK2'I<C%$;S%$G%$GN(%$*6*J;=δ'@"$*(=g39%$E3L3!o'(!nX'@(0)%$E:K$!G9T
Æ ] Ê^6Í b|fbo|pt_@q&-δ −→ 0
]#'('@EOS%$**&E33\J+r%$;=&"$*!G%$;CEK$!G*J:!G'@(0)%BAE3LD33EB"$;!GgT-
δ −→ 1]#'('&E(!G39*&t_'(! y 9%$"2%$'I'&E3j:3j*#'1T
j'(L3'M: X'(! y 9%$NXw;;@"$*(=(9b
ui(vi) = (1− δ)∞∑
t=1
δt−1vi(t)
%$'('@Et_'(! y 9%$"23u;=I%$'(!*&SRw'1]<Q!*& 3u=&93 :3j;@9b
ui(vi) =∞∑
t=1
∆tvi(t)\#
∆t = (1− δ)δt−1"2 ∞∑
t=1
∆t = 1.
>?'('@Etr%$G"$E3\#%$N+/g!G32'@*(:'(! y 9%$"2%$'I\J<+7CT-
δ −→ 1]1'('&Et%$Q:3j;@t'(! y 9%$NCw;;&"$*&=m'(E3!"$E3!<Q\#X'&*&&%$E3;&:^T)>? gA
)%$*iH%2%$3 )%$* Æ ] ¿df=)P!ÁjS'(GEBb ∀ w ∈ D wi > um
i ∀ i ∃ δ∗ < 1]!DLnB|p ∀ δ ∈ (δ∗, 1) wal ¨Ctbm]vpOpb@ax]npjHOxhpcwhpdbc C hxqD8Eb|pbg$q®c DLnt
I letnpg!6Jz]\xqo r%$GN9NXS'(GE"B%BU !9*(=(L::\#H\J*GTiµ× U× º 9<;E$FG7:9IH HH%$;=@%$*#3u=&]#<Q! %2//g!;C'&*#33E3J:3*'1]*#%$<ZE"233RW:"B'D'(!G%$G"$;LNT %$G%$*gS"$1'&"$*&=g39%$E3L"2%t*#%$<\!\#%$N](\!G39%$*M\J_%$"$E3w*'Dr03!<+:!33E"BA
;!j:^]<+;*'D!GE32#RW:E*'1T Æ ] ¾ <;QU1=>@?Y!R8tµlrqd_m]`bcì]m\^]`_@a ∀ C hxqD8Eb|pbg$q®c h¤jB_@qBtom\^]`_@aQalbce_|6Jz]vI letnpg!6Jz]\xq1'@*#33E3J:33E"$;!jXE79]#HE"233R3:"$'(E_!Gú-3\gAÄ%$;=@%$*#3u=(!^]g\!PA39%$*6XE"233R3:"$'MtR3:"$'I&"$;%$G%$*#"$;@%<Q!j3 %$'((!G%$*# H%$E%2/%$3!#R3:"$'('^T
jÈ VÈ Ìgß8®X¯e&Ñ xpßÓ7ßÏÊ ÊÌ
iµ×]\× × 6;ØrÒD«B;;C6ßC;ÚQF6; Æ ] Ê^OQP!RLS(TU(VWS(XZY(dR3:"$'@'C-'@(0)%$E:K$!G.u;=X0E J: R3:'X'(!'&"$*&=g39%$E3L%$*!^]@\#;=r%$G%$<Q!JR3:"$'D%$;=@%$*#3u=(:!G3<C"$3!'D<C%$;#Tk%$'(!*&3w*'s%$;=s%$G%$<Q!R3:"$'&9]n<C%$=(*#%$'*dG%$;&j: %$;=sú-3\gAÄ%$;=@%$*#3u=(gT
% y *!j: R*'Dj<C%$=Mú-3\gAÄ%$;=@%$*#3u=(E_es∗eD_'&a9&%$3'@%$9N9b
σNi
(t, h(t)
)= s∗i ∀ t, h(t).
f&9%$E%$3*#"$*'m%$;=D%$;=@%$*#3u=(!nE7¥h∗h'(!'&"$*&=g39%$E3L%$*!^b
?E J: R'm_'&a9&%$3'&%B9N9b\JC
t− 1 01%$E3!G&/g#*6'@%$E%2323w h∗ aE3"$*(S=(3'1]\JX*#%$<m]J'('@E:j32'MHS%$*(3!σA^9T
Æ ] ¿<;QU1=>@?Y!RÄfbg]zlIapnpg$qm]zja`apkmlo~@®qm]zjBh^]`_o|plohj@_@qBtom\^]`_@a¤bo|pbg$q®c plI letnpg!6Jz]\xq46!*#/%$*Éú-3\gAÄ%$;=@%$*#3u=(!r33E"$;!jv%$;=( 1%$*E"23WR3:"$'q%$;=&%$*#3u=(!!G&"B;%BG%$*R3:"$'(EgT
σú-3\gAÄ%$;=&%$*#3u=` J:E3<C%$=
h%$;=@%$*#3u=(!raE3"$*#%$6%2%$"$*UT
46!&%B_ J:E3<C%$=ÉE"233R3:"$'É%$'((!G%$*#Y%$E%2/%$3!&%$S]%$9"$E3'@*#33E3'&K$!G`ú-3\gA%$;=@%$*#3u#P=( %$E%2/g<C"$*&=&%$@T Æ ] Ç68!ZY)4i&/3L3'MH%2//g!;!oS;@=D/g!G%$<Q<+Q0)"$G/#X'(! y 9%$NQ<+:3E3!P#:9b
A \B C NC
C (3, 3) (0, 4)
NC (4, 1) (1, 0)
>R3:"$'Iú-3\gAÄ%$;=@%$*#3u=(X(NC,C)
33E"$;!jH0J:E9T>?AA^*J'DH<+7g!<+:!GO'(! y 9%BA
"23r/R3gT)£Ä;=6G%$\#%$%$3G%$*dHR3:"$'&3- J:E3<Q!=@%$*<+:33E"$;!jI:j32::EI'&"$*&=&A39%$E3L%$*!^T
iµ×&iµ× 9_^6=<; «B;:?A9C7:>BÛr¥9<;ÛÅE$FG7:9IH6H 2h;@*#/ R'1]@\#;=I@"$;@%2h!G3<C"$3GN&/"235_R3:"$'&'X3jRW/*'&"$00)%$*DE"239%$!1@"$;%BAG%$*+!G3<C"$3GN&/"2%2nR3:"$'&'(*J'1T@>? *!*#K2%$*IL;=@Tgf&7:<C.%2%$3 )%$*+'(!\J32*J:3'1]&\#;=_E"233R3:"$'D@"$;%$G%$*UT Æ ] Á gdf=)P!Y!Cbo|pbg
Gbo|p¾Op_o|b@qQlrq@d¤_m]zcwv3F_@qb@qZm\^]`_@a^ S b|ppagfbc HNDLn@|pÂbcebdQl:m\6E]`_@ag:hxaªOxhpg C hxqD8Ebo|pbg$q®c h¢_@q ht*bo|p_jm]`bc d°c! ¬ aapnpjÂj@_@qBtom\^]`_@aÁalbce_|6Jz]v C hxqD8Ebo|pbmg$q@®cìh
GE`gWb@aTHdbc ¤]hpjm]hpc dht@th¤htQbcebdQlm\^]`_@a¤bo|pbg$q®c \^]`
ËÈ jÈÂÑ (ã67Þ@áÄÞ ÉÊÞËâ<ËI letnpg!6Jz]\xqUtR3:"$'D0)%$E3!G&/# 37:<+:EXaE3"$*#NQ!*#/g'&K$!G9^T
T = 1r!G3<C"$%$hR3:"$'mH%$G%$<Q!#R3:"$'1]J<Q!*#%$'M*6%$;=@"$E3%$<_5iú-3\gAÄ%$;=@%$*gA
3u=(gTk%$;=(w'_S%$^]\#;=Ct:L:h!;&
T −1 A^!P;#T&n%$'(!*(3w'C T AÄ39%$E!G3<C"$%$nR3:"$'@9T46!*#/%$*=(*E"233R3:"$'1]<C%$=(!'<+:&/g!'Y01%$E3!G&/g#3 J*'&%$9/N&/g!'1]?E3<+(T −1) AÄ39%$Eh!G3<C"$%$nR3:"$'&9](<C%$=(*#%$'Q*+E"233R3:"$'Q'(!G%$G"$;LNHú-3\gAÄ%$;=&%$*#3u=&gT£Ä;=%$GN01%$E3!G&/g#MK2%$G%$'(@"2%$!t*#%$< 1%BSP=(: R3:''&a9&%$3'@%$9Ns0)%$E3!G(/g#'9T>?9"$E3rX%$GN+0)%$E3!G(/g#%$;=@%$*#3u=(C!G-CG%$;RW i:j39'( 1?:^T)>?X0)%2/g!;M%$2.RW%BAG%$*&!S]#\#;=Mr%$GNQ0)%$E3!G(/g#3 J*6!GH%$G%$<Q!#R3:"$'Dú-3\gAÄ%$;=&%$*#3u=&-Ra*mG"$3E%T Æ ] à 68!ZYJ(!93;&: R'MHj: !U'(! y 9%$NXw;;&"$**&=&%$U<C%B;&/#3R3:"$'&9b
A \B b1 b2 b3a1 (4, 4) (0, 0) (0, 5)
a2 (0, 0) (3, 3) (0, 0)
a3 (5, 0) (0, 0) (1, 1)
R3:"$'D32!<Q<C%$3E3!'(#9TqR3:"$'(*J'M'&"$tú-3\gAÄ%$;=&%$*#3u=&C*
(a2, b2)"2(a3, b3) (a1, b1)
?E%$+G%$;ARW U]#/%-*#%$<ú-3\gAÄ%$;=@%$*#3u=&TF&:32'D%HtR3:"$'&'&"$39%$E/g!G32'@*(:j:*#"$'(wnX'@a7@%$3'&%$9NQ33E"$;!j:9^b ¾ Tg01%$E3!G&/g#3 J*
a1
Q¿gT0)%$E3!G(/g#3 J*
a2\JQ ¾ Tg01%$E3!G&/g#3 *
(a1, b1)
a3'(wGa*& 1%$*
>?.33E"$;!j(!j:;@*Hú-3\gAÄ%$;=&%$*#3u=-"2UE"233R3:"$'-'(!G%$G"$;!N!G9]2\!G39%$*r<Q!*#/(A%$;=(!'IG&/g!UE"233R3:"$'(EQú-3\gAÄ%$;=@%$*#3u=( %$E%2/g<C"$*&=&%$@T Æ ] Çdf=)P! jS'(GE I \pjdbc ÆOp_o|b@qylrqd_m]zcwv3F_@qacym\^]`_@aÂb@qbm]`_gfHÅDLhÂhthpcwhuxm\^]`_@a,Eg:hxaOxhpg C hxqD8Eb|pbg$q®c hH¥_@qht b|p_jm]`bc d°c8HhxaapnpjÂht lrqd_m]`bcì]<m\^]`_@ag:hxaÁlrqOxhpgbo|p_mjm]`bc d°c jB_@qBtom\^]`_@aQalbce_|6Jz]v C hxqD8Eb|pbg$q®c hI letnpg!6Jz]\xqoÄ*#/g'&K$!G9
t = 1!G3<C"$%$R3:"$'M_j0R3:"$'1]L;=MHS%$"$%$U<Q!j3t_:L:!;&@T
k%$;=(w'iS%$^]U\#;=D:L:H!;&t − 1 A^E%]n%$'('&Er<Q!*#/%$* t
AÄ%2/g!'01%$E3!G&/g#3 J*!*#/gGHR3:"$'ME3<+27D
t − 1 AÄ39%$E !G3<C"$%$.R3:"$'&9]#<C%$=(*#%$'M*i%$;=&"$E3%$<H5%$;=@%$*#3u#P=(gT&>?t%$;=&%$*#3u=QG%$;7RW I:j3--j0R3:"$'( J*U](L;=C%B*#*#%$'QE"233R3:"$''(!G%$G"$;LNC%$;=&%$*#3u=(*J'I'&%$oG%$**!G%Hr%$;@"23R3:"$'(EgT Æ ] Ã Ndf=)P!YW^f(%$%$*:AµhbOp_o|b@qslrqd_m]zcwv3F_@qb@qCm\^]`_@aQhpcwhum\^]`_@ap\pg:hxabOxhpgÁbo|p_jm]`bc d°cbo|pbmg$q@®cìhHGhxaapnpjQbBtQhtQbo|pbg$q®c h8m\^]`_@aydb|npcGF\xqfhdQl(gBFbgu$bjlo~TF6`qTQBhpgW
jÈ mÈ+ßUáË(ßUáãß:^Ê Ë ¾
iµ×*ks× © >HZ6>@?A6CF6;Ü Ý@ `! ©±Þ¬J³S¬.ßI¯àN¤8¦¯o¬á^°7«)¬#®N = 2
:jj'`'(! 1&K2::3D37 J:=&27:'1T.À'(! )(K2::I'@a"$;@%6'&*#WA*#9b
cT H:E9b
P (q) = a− bqlU%$;=&%$*qm = a−c
3b
X<C*#0)n%$E3<C%$G"2 "2um = (a−c)2
4b
_<C*J0)n\J39*lU%$;=&%$*
qc = a−c3b
/g#0)!<%$E3<C%$G"2%."2uc = (a−c)2
9b
.%$;=@%2nK2"$;@%$'\J32*JgT Æ ] Ë8Í bo|fb|pt_@qk+.E3*#WAÄ%$;=@%$*#3u=m%$;=@"$E3%$<_5U])_@"$;@%2?R3:"$'&'(E_G%$\#%$%$3G%$*_'&&01%$E:K$!G]#%$9"$E3&"$;%$G%$*CR3:"$'@(!93;&:*'1T
Ü Ý@ (¡ £âs²o±6ã_áäåı±æ²!ç±²"èh«o±lU%$;=&%B*nb
h(t)<Q!*#/%$*6K2"$; Qm
2
A^%$E3<C%$n<Q!*#/%$*m0)%$E3!G(/g#3 J*lU%$;=&%$*
δX/g!G32'@*("$*&=&%$9NQ<Q!P*J/g'@"$ K2"$;+37:<+:Egb
r%$;=&%$*#3u=( J*m<Q!*#/g'@"$ K2"$;C'(! y 9%$"2%b∞∑
τ=1
um
2δt−1 =
1
1− δ
(a− c)2
8b.
k%$;=(w'_S%$^]\#;=+ %$;=(!'XK2"$; Qm
2
'(! 1&K2:::EH-<+:3!'QK2"$;H=(*+33E"$;!j::j329]g\#;=m<+7g!<+!7: R3Q0E y SR3:9T#>?'('&E _<+7g!<_<Z0E y 9b 9
64(a−c)2
b46!*#/%$*`%$2+'&a9&%$N01%$E3!G&/g#3 J*UTOS%$*&3!tK2"$;s'0R37: E!GI33!'(#.E3*#WA^\J32*J:
uc A^9T!£Ä;=DHa39%2\J32*Jgb9
64
(a− c)2
b+
δ
1− δ
(a− c)2
9b.
>?'('&E H%$E%2/%$3!n33E"$;!jQ%$;=&%B*J3u=I\Jgb
9
64
(a− c)2
b+
δ
1− δ
(a− c)2
9b≤ 1
1− δ
(a− c)2
8b
%$ )Nδ ≥ 9
17.
Æ ] ÌÍ b|fbo|pt_@qtZS%$*(3!0)"$G/#E:<HDE3Eg]\#;=i'&&01%$E:K$!Gd'<+δA
<C%$G%$3S%$**(E33\JT
Ë¿ jÈÂÑ (ã67Þ@áÄÞ ÉÊÞËâ<ËÜ Ý@ ס êéè.²J±mÖ¯o¬i±7¬¤@²)¬!å^² Æ ] ËOQP!RLS(TU(VWS(XZYOÀ32!<Q<C%$3E3!'(#+E"$0J7e@ )D%$;=&%$*#3u=&%$;=.E*#WAGR3:"$'(*J''&"B'(! )(K2::3!U32!*(3%$#RW%$G%$<Q9%B9]J\#^b
qL < qH .
ui(qi, qj)RW%$Ga RW%H
iK2"$;I0E y SR3:
qij<Q!*&
qj'(! )(K$:B: <C%$PG%$39T1n%$;=(w'DS%$^]
\#;=MqLA^"2
qHA^u;=I:j3233'1]\#;=
ui(qL, qL) > ui(q
H , qH).
% y *!j: R'mX33E"$;!j:tX'@a7@%$3'&%$9NX<C(/*Ub¾ T
sRb(1, h(1)
)= qL
eI7r'&%B9/Rw*#'qLA^G%$
¿gTO\JsRb(t− 1, h(t− 1)
) A^/% y *!j:3'1]#<Q!*#/%$*mG%$\#%$"$;@%2aE3"$*#%$3E%])'('&E
sRb(t,(h(t− 1), q(t− 1)
))=
qL\J
qi(t− 1) = σ(t− 1, h(t− 1)
)
qH'(wGa*& 1%$*
7Mm'(! 1&K2::X32!*&SRW%DD%$GN9N+R3:"$''(! 1&K2:::.w;;#TC-66K2"B;@%$'C%$GNLE3r<C%$**(=(!G"$;@%$%BE<C%B"$'+%$N9NM01%$E3!G&/g#3 J*U]k'('@EHI'@a7@%$3'&%$9N'(! 1&K2::tjK2*&=&]g<+:3'(wGa*( )%$*6<+;&9T
Æ ] ÌQÍ bo|fbo|pt_@qG-M<Q!*#/g'&"$ R3:"$'&H%$2Xm33E"$;!j:_:j32SR3g]k'('&E_QR3:"$'G%B:QX'@a7@%$3'&%$9Nb t%$;=&%$*#3u=(!JaE3"$*#%$%2%$"$*
(qL, qL)&"$;!;_%$2kR3:37:'1T-j'(!)%$"$EO%$;=
t − 1 0)%$E3!G(/g#3 J*%$3N^].'('&ED'@a7&%$3'@%$9N01%$E3!G&/g#3 J* (qH , qH)'&a9&%$3'@%$2!'1]
<+RW/i(!G37"$E3*#%$'+jK2*&=6'(! 1&K2::3\#@TU>+I33E"$;!jI !9*(=@r"$E3%$G%B<XA )%$*6_ w*&%B"$3 1%$*m'@(0)%$E3L1]\!G32%$*
qHA^?R3:32!'1TL-X_ w*&%$"23!U!G/N37'+:*
%$<"$E(!G37QHjK2*(=I'(! 1&K2::9]J'('@E X w*&%$"2S=((/g!'1T Æ ] Á <;QU1=>@?Y
δ > 1/2ABjBb¥hyapnnu$bjB\kmlo~alra^_g!xqBtbjaJz]:D$bm]v
I letnpg!6Jz]\xqo33E"$;!jX'&*#33E3'@K$!GR3X<Q!j3'@"$ %B"$E"23'&%$o(!93;&:*!^bsRb(t, h(t)
)= qL; sRb
(t, h(t)
)= qH
qLA^E%
ui
(BRi(q
L), qL)− ui(q
L, qL) ≤ δ[ui(q
L, qL)− ui(qH , qH)
]
jÈ ëÈ Òp@áßgß@Þ)â<Ë Ë Á
qHA^E
ui
(BRi(q
H), qHb)− ui(q
H , qH) ≤ δ[ui(q
L, qL)− ui(qH , qH)
]
[E"$0JmM/g#01P!<ä'(! )(K2::9]nD 1_D@%$E%$*(=i'(! 1&K2::9boL;=qL = a−c
4bqH = a−c
2b )%$\#%$=&%$3%23L3@%ba− c
16b≤ δ
(a− c)2
8b(a− c)2
64b≤ δ
(a− c)2
8b/&/!P'1bδ ≥ 1/2 T
Æ ] ¾ È¤Í bo|fb|pt_@q <Q!@%$1/2 < 9/17
%$i33E"$;!j<+:EIjK2*(=( δA^E!G9]
\J"$'&*&=&]g<Q!*(tXú-3\gA^(!G37K20J:9T
iµ×&yµ× =<>B¥G7:HñZ! 46!n_EK$!G*J:!Gt33E"$;!jC_&"$;@%2'g39E!G3<C"$%$PS;@=D/g!G%$<Q<+: J*:Mño¨ k%$'(!*&3w'C'&a9&%$3'@%$9N_R3:"$'@9o¿HR3:"$'@t*U])<Q!*#/g'@%$3N*#%$'i;=(*JCC'&"$33E"$;!j:R3_*UTL-H'&&01%$E:^]#'('&E Á A/mH<+:3!'(*J'1]g\J_/%$9%$E3:^]J'('@E ¾ AÄ%$/+<+;&:*J'I"2r<+:3!P'rR3:"$'&3*J'Q!G9T£ÄESR'XS%$oR3:"$'@.*#E3<+:Uj'( J*Ug/RÄ#'<C%$;MQE"233R3:"$'&A^'(!G%$G"$;LND%$;=@%$*#3u=(E3EIX%2%BE%]#\JC\J:E<C39E-!G3<C"$3!'M<C%$;HS%$*(3!R3:"$'&9ñ Ï Oj: !g<+:3E3!P37/3nS;@=r/g!G%$<Q<+hR3:"$'@)R3:32'-&"$;%$G%$*X'g39E9b
(1, 1) (5, 0)
(0, 5) (4, 4)
<+:3!'mE23X33E"$;!j:.R3:32!'1]17_'&&01%$E:^]1<QL;D<Q!n!G9])/%H\J+%$;=g39%$Et/%BA9%$E3:*'1]&'('@EN3'&%$9/g@%O<Q!*#/g!;H/%$9%$E3:^T@46!=@%$*C/g!G32'@*(?"$*&=&%$9N-<C%$G%$3"$E/%$<C%2*#%$'(w*'I!GE93r33E"$;!j:R3:37*!^]!G%$3@%&"$;!;C/%$9%$E3:*!(MñJÔ! Oj: !g<+:3E3!P37/3nS;@=r/g!G%$<Q<+hR3:"$'@)R3:32'-&"$;%$G%$*X'g39E9b
(3, 3) (1, 4)
(4, 1) (2, 2)
r%$G%$*gS"$nO!OSE 33E"$;!j:R3:32!'1]#7H%$N39aE'&&01%$E:n"2:*JX<Q!*#/g!;2G"$0!^])<Q!-H%$G%$*gS%$G%_H%$GN9NQ'&aE3 )%$*UTJ46!=&%$*6/g!G32'@*(:j:3!U"$*(=@%$9NC<C%$G%$3"$E/%$<C%2E93r33E"$;!j:¥j:/ ñ Ï S%$j/#kR3:37*!^]!G%$3@%&"$;!;C/%$9%$E3:*!(M
ì °¾½¿Às¿Á´Â¿ÂÃ
- _í ,(Á Ë , ÄÚªíÚË
C%$@u#K$!GR3:"$'&'6<Ca;@a3_I !GG;!j!.%$@u#K$!GM'@*#K2%$01K$!GR3I\(u9&/g!'1]n7I%$;=@%2/%$'C !GG;!j!j;_<C%$;\J:E23O<C&/*I'&a9&%$3!'C-%B;=(!'+9;=Q<+:3!P'C33E"BA;!j:"2-3uG"$G"23!G32L*&5#"$;w'i'(wGa*& 1a9NI<C"$E3"$'(5sQ%$;=@%233E"$;!j:'(*J'6<C%$;AS%$G%$GN&%$*UTta2;&9/#:;!o01"$G/#:2U"$&%H;&9/#:;!U"$G%$39%$E%$0GN!o*&*J'M%$;=&%2%2%BA
<C"$*(=@%$'( 1N?"2%$**#%$'6<C%$;S%$G%$GN(%$*'&E3E3!;&: R3:'6<+;&E3:3'9TUx'&E3E%$'@K$!G-<C%BAK\)*!$<_#j i9*& J*U]#<Q!*( -R3:"$'Iw%$<C%TÉ'&a9&%$3'@%$9N'( 1%$*6HS%$*(3!G%$'I*J;=&*6%$;=g39%$E35m%2%$"$(!93;&: R'1blU%$;=&%$*
G32!<Q<C%$3E3!'(#R3:"$'1]#7 |N | = n
]Si = Sj = s1, . . . , sn ∀ i, j
Tk%$;=(w'vS%$ 7: J:g].\#;=`%$;=@%2_R3:"$'@'<C%$;@G/#:!'MK2'`3!G32
33E"$;!j:'D'@aE"$ 1%$*m'@%$E%23w'1TR3:"$'@'MQR3:@"$G%$'&%2/"2%$!P'( 1NU*&*J'1T r%$&u#K$!G?R3:"$'&'I'@"$'@*#K2%$01KB!GEC"$0w*#%$'1b
eC<_:K$!G]e+39%$G%$'&K$!GT
<H:K$!GDQE%$*#/39%$E3 1%$!k7R$]J%$XQ7R'(!"$E3L3!k<Q!*#/g!;+j<C%$=(!'CR3:"$'@3-'&a9&%$%$333E"$;!j:R3: 1^]J7r01%$E33E3 J: R3CX0E G"$<+:9T39%$G%$'@K$!G+H%$;=@%2'&%2/g@%$9N m'(! y 9%$"235633E"$;!j:'IS%$**<+E/#::hR%$G%$*(3!^T
ks×@Ö6× î C=<>B=IH6ßC«Bàðï ADZF6>B6;ñ¥«B5¥8?A«HZFZE$'&a9&%$3'&%$2N'( 1%$*D%$;=(w'CS%$^](\#;=D R3:"$'( J*D*!*#K2O<_:K$!GTF@%$Ga Rw'D9*QR3:"BA'&'DE:*(=&:
xiA^&%$^]#'(!'
si33E"$;!j:R3:37*J' n∑
i=1
xi = 1T
>?'('&Esi33E"$;!jC:3j;@'(! y 9%$"2%
u(si, x)]#\#
x = (x1, . . . , xn)]&<Q!@%$k
R3:"$'+32!<Q<C%$3E3!'(#u1(si, sj) = u2(sj , si) = u(si, sj)
.a39%2O33E"$;!jX:3j;@
Ë Æ mÈ fJâ?á,Ú!Ó!Ñ Ô6Y8W&Ì$8Ú)áW'(! y 9%$"2%b
n∑
i=1
xiu(si, x) = u(x, x).
si33E"$;!j-RW U]<Q!*&t_:3j;#]J'('@E
u(si, x) > u(x, x)T
^Ê(] ¾ OQP!RLS(TU(VWS(XZYr32%$G%$'&K$!G<C%2K\J*!2<HJ9bxiE:*(=X01*&*M'('@E*#a7@%$'g32!'1]
\JsiRW m<Q!*&t_:3j;#]J7@b
3;* (xi(t)
)=3;* [
u(si, x)− u(x, x)].
^Ê(] ¿6OQP!RLS(TU(VWS(XZY1X39%$G%$'&K$!GQ/g!*J<Q!':R3gbxi
xi= u(si, x)− u(x, x)
\Jxi = 0 t = 0
A^ J*U]k'('&ErG%$;=&%$*xi = 0
<Q!*#/%$*d!G/N0)%$E3!G(/g#3 J*U]7+\J%$*'(!U%$<xR3:37
si33E"$;!j:9])'('@E%$rL;=I!G<+E/oT
^Ê(] ¾ <;QU1=>@?Y n∑
i=1
xi = 1dQl(gBFbg lFvu!bjlo~TF6`qTQBhpgW
I letnpg!6Jz]\xq ∑xi =
∑xiu(si, x) −
∑xiu(x, x) = u(x, x) − u(x, x)
]JL;= ∑xi =K2*#39]<Q!@%$ ∑
xi = 1'(!!*#/gj:3'@E %$9"$E3 K2*#3 x ¾ T
^Ê(] Á 6OQP!RLS(TU(VWS(XZY)>?;=σ33E"$;!j: 3 !k:j0)3*J'M*#%$@%$2w*'1]
\Jxi = σ(si)
A^E% dx
dt= 0.
^Ê(] ¿e;QU1=>@?Y_Aµhσqm]zjBh^]`_o|ploh hqBtml(dQdbm]zjlra,`qya^baOpbjm] C hxqD8Ebo|^bg$q®c _qydQl(gBF6Ebo|pplra8m\^]`_@apnxqm\^](q@qthHZhxaapnpjqm]h8Ql(cC\pcrceh@u$n^]`
I letnpg!6Jz]\xq)ú-3\gAÄ%$;=&%$*#3u=( J*I<Q!*#/#%B*QR3:"$'@.!*#/g!ì¦o%$E%$*#33E"$;!j:!o:<C;&7A:: J*UT!£Ä;=1b
u(si, x) = u(x, x)T
^Ê(] ¾ 68!ZY)n%$'(!*(3w'M_'&a7@%$3'&%$9N-R3:"$'@9b1 \ 2 s12 s22s11 (0, 0) (1, 1)
s21 (1, 1) (0, 0)
mÈ ÇmȤp$@áËLÓ!Ñ ÔÆòß^ÊUáßp Ñ Ìgßnã6Ñ ËÊÉ3ß Ë@Ê
xs1
xs1
= xs2 − 2xs1 xs2 = (1− xs1)(1− 2xs1).
a*#*&=&%$*mj:3\J]#\#;=xs1 > 0
\J0 < xs1 < 1/2
"2xs1 < 0
]g\J1/2 < xs1 < 1
]%$ )N/&/g!'
xs1 = 1/23 !!:gT
^Ê(] Ç8OQP!RLS(TU(VWS(XZY1>?;= Ä3 !ìH:j01t3 !^]#\J ∀ ε > 0A^\# ∃ δ > 0
\#;=M\JCS=&<+
δ:9@G:;EQ!*#/g¥Ä3 !ìH:j0)3^]1'('&E<Q!*#/g!;
εE3<+:*&=( J*
<+E/oT^Ê(] Ã QOQP!RLS(TU(VWS(XZY>?;=
σÄ3 !ìM:j01C32!<Q03!'(#*v3 !^]k\J ∃ δ > 0
]\#;=DHS=&<+
σA^\#'@*(&%$E3;&:^](\J_G%$;S%$ RW%$
δ:2&G:;E_!*#/gn%$;=&%$*#3u=(!
:j0)3 1^T^Ê(] Æ COQP!RLS(TU(VWS(XZY>?;=
σÄ3 !ì.:j0);G J:!G*X3 !^]9\J S=&<+?GÄ3 !ì
:j0)3\#H'&*&&%$E3;&:U J:E3<C%$=D'@%$9/N+:j0)3 1^])\#xi > 0 ∀ si
A^E%T^Ê(] ¾ <df=)P!Y"Aµh h ¼ qm]h8Ql(c½\pc(cwhuLn^]qm]h8Ql(c HZhxaapnpj C hxqD8Eb|pbg$q®c I letnpg!6Jz]\xq!*#/g!E%$'(+ !9*&=(LSR'1T-*#%$< ú-3\gAÄ%$;=&%$*#3u=@].'('@E ∃ si
]?<C%$=(E%ui(si, σ)− u(σ, σ) = k > 0
T>?'('&E ∀ ε > 0]g\#;= ∀ α
A^EεA^*J:o'(!G%$ M:9@G:;Egb
∣∣u(si, x)− u(si, σ)∣∣ < k/4
"2 ∣∣u(x, x)− u(σ, σ)∣∣ < k/4.
>? )N^b∣∣u(si, x′)− u(x, x)
∣∣ > k/2.
46!&%$σ3 !^]J%$9"$E3 \J
x σA^\#H%$G"$;+'&a9%$U*U]#'@*(@%$E3;&:^T£Ä;=
x(0) = (1−1/2δ)x+ 1/2δ si
A^@%$n37:<C
dxi
dt= xi
(u(si, x)− u(x, x)
)≥ xi
k
2≥ δ/2k
2
%$ )Nx(t) −→∞.
>?H%$G%$*&3<C*#/#:9T^Ê(] ¿68!ZYJ%$'(!*&3w'm_'@Ne@0J0LEWe&GrR3:"$'&9T
K P O
K (0, 0) (−1, 1) (1,−1)P (1,−1) (0, 0) (−1, 1)O (−1, 1) (1,−1) (0, 0)
ËË mÈ fJâ?á,Ú!Ó!Ñ Ô6Y8W&Ì$8Ú)áWÍt!93;&: R'
xK"2
xPE:*(=@'
xO = (1− xK − xP )
u(x, x) = xK [−xP + xO] + xP [xK − xO] + xO[−xK + xP ] = 0
xK
xK= −xP + xO
xP
xP= xK − xO.
'@%$&%$E3I33E"$;!jsú-3\gAÄ%$;=@%$*#3u=(1/3sK , 1/3sP , 1/3sO)
i%$;=&%2/gw!8Ä3 !ì:j0)9TJút"$<Q!o(!93;&:j:*i/(/g!'1]g\J;=D%$*#%$<3 !^T
ks×øØ× ó áµ7:F6ßC«Bà^Ê(] Ê^OQP!RLS(TU(VWS(XZY
x:j01O%B&u#K$!GEH3 !J\J<+Q>.ff$B]&\J ∀ y 6= x ∃ ε
]<C%$=(E%b
u(x, (1− ε)x+ εy
)> u
(y, (1− ε)x+ εy
)∀ 0 < ε < ε.
^Ê(] ¾ Í bo|fb|pt_@qUH3 !)\J<+^!<Q<_*!G¡ <_:K$!GEg](\!G39%$*mr<H:K$!G_'(! y 9%BA"2%rjK2*(=& UT^Ê(] Á <;QU1=>@?Y
x>.ffuLnpg]nxqfhpg hxaapnpjHNDLh ∀ y 6= x
Oxh|p;=
¾ Tu(x, x) > u(y, x)¿gTu(x, x) = u(y, x)"2u(x, y) > u(y, y).
^Ê(] Ç<;QU1=>@?Y!Aµhx>.ff)]&hxaapnpj C hxqD8Ebo|pbg$q®c
I letnpg!6Jz]\xqo>?;=g39%$E35#%$*/&/g!'mX/% y *LGK$!G 1^T^Ê(] Ã <;QU1=>@?Y!Aµh
xqBtmlw|npj® C hxqD8Ebo|pbg$q®c HhxaapnpjQ>.ff)T
I letnpg!6Jz]\xqo>?;=g39%$E35#%$*6'@a7@%$3'&%$2!'DQ/% y *LGK$!G )^T^Ê(] Á 8!:ZY.n%$'(!*(3w'viS;@=v/g!G%$<Q<+MR3:"$'@9]hiS%$*&3!'&"$+:L:C<Q!j3D/%$9%$E3:j: 33E"$;!jQ0J:EG%23_r>.ff@A^!G9T^Ê(] Ç8!ZYn%$'(!*(3w'.;&j<_ ge@\#"WR37AGR3:"$'@9].'@%$&%$E3U33E"$;!jú-3\gAÄ%$;=&%$*#3u=HS%$*(3!n:L:'D<Q!j3tH>hf#f)T
mÈ -È Òp@áßgß@Þ)â<Ë ËÌ
ks×&%µ× =<>B¥G7:Hñ#Ý >?;=H%BEwG%B%B*C\#"WR3:'H"2;&j<H 1'X"$*#%$'1] \#"WR3:'_EB:*&=(
α] ;&j<_ )'&"O0)%2/g!;
1− αT"$<+/#:Ej:'&7:'@Et_j: !U<+:3E3!P6/R3_<C%$;I_*(=@%$E%$<C"$*(=@%$'&%$9b
8(−25,−25) (50, 0)8(0, 50) (15, 15)-:E22'D<C%$;Ir%$&u#K$!G*63 !n%$;=@%$*#3u=(9
ñ)ì" 46!'MX3 !k:j0)'m_j: !U<+:3E3!P37k/3'&&E/g!*J:K$!GR3:"$'( J*:Ms1 s2
s1 (50, 50) (49, 0)
s2 (0, 49) (60, 60)
>?9%$'D'@a2wU<C%$=@%$'m'@3*J'M>.ff@A^MñJñ! n%$'(!*(3w'M_j: !U<+:3E3!P37k/3S;@=D/g!G%$<Q<+R3:"$'@9
(3, 3) (1, 4)(4, 1) (2, 2)
46!'MX3 !k:j0)'!MrÍ.*m>.ff$Mñ1øB n%$'(!*(3w'M_j: !U<+:3E3!P37k/3R3:"$'@9
(−49,−49) (2, 0)
(0, 2) (1, 1)
46!'MX3 !k:j0)'!MrÍ.*m>.ff$M
ô °¾½¿Às¿Á´Â¿ÂÃ
õ û , d - º · 10¸¹¸ · - _© , ÄÌË , , 0æÄ ÄÌË ,
r%2//g!;+39':?RW%$GaG"2%$'('@%$U"$&%-G%$;=&%$*G(N,S, u)
%$;=g39%$E35CR3:"$'1TSËg] ¾ OQP!RLS(TU(VWS(XZYnrR3:"$'@'m%$;= ∅ 6= C1 ⊂ N
\J<+7:'@@LGK$!G*J'6*#%$@%$22w'1]C1 = i1, . . . , ic1
TSËg] ¾ Í bo|fbo|pt_@q
C'&@LK$!G%$;=&%$3G%B*R3:"$'&3*J's37:<QL+CR3:"$'( J*UT?G%$;A
E37 C%2%$C'&@LGKB!G J*CE"2323&%$@NOR3:"$'@'_37:<+:E@]\J-ta !R3:"$'@
a39%BS;i%$G%$*&w's"2_%$;=C2'@@LGK$!G J6a<CaE3w^]k%$'('@E_%$;='&"$39%$<C"$=@%2 R3:"$'&
'0*'C1 ∪ C2 = N C1 ∩ C2 = ∅
TSËg] ¿OQP!RLS(TU(VWS(XCTYI-m
GR3:"$'DR3:"$'&!E%$*#/gE%
C1, . . . , Cn'&@LGK$!G JMa<CaA
E3w*#%$'Ci ∩ Cj = ∅
⋃
i
Ci = N]J'('@Et
Ck = i1k . . . ink'&@PLGK$!G+33E"$;!jQ\J<+9
Si1k× . . .× Sink
%2K9E3%2E7'(! y 9%$NXw;;&"$*&=&%rk∑
j=1
uij .
É9: !j'( J*m(!93;&: R'M_'&"$'@@LGK$!G%2%$%$9]J7_C"2
N \ C '&@7ALGK$!G'9T>?'('&E _'&"$39%$<C"$=@%2R3:"$'@'(*J:n39':.S;&<+'uREX )%$&%$9%B\#%$SRw'1T
SËg] Á 6OQP!RLS(TU(VWS(XZY)lU%$;=@%$*G(N,S, u)
'&&01%$E3LXR3:"$'1]%$'('@E9bG%$;=&%$*
m1 = maxs1
mins2
u1(s1, s2)
m2 = maxs2
mins1
u2(s1, s2).
Ì¿ ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áSËg] ÇOQP!RLS(TU(VWS(XYJ>?;=+'&"$39%$<C"$=@%2R3:"$'+'E'(%$E3!G323!'(#.w;;@"$*(=@"$*mHG%$\#%$"BA;@%2t\J<+9'(\#QE%$*#/%$-<+7g!<Q!*"$E3"$'@%$"$E3SRw'1T)x'E'(%$E3!G323!'(# w;;@"$*(=(νA^@%$RW%$Ga Rw'D"2
ν(∅) = 0 TSËg] ¿yÍ bo|fb|pt_@q"$39%$<C"$=&%2X9"$E3#a39%$;5R3:"$'s%2%$"$*v6<+7g!<Q!*v"2Qm<Q!PA*!<+7v<C%$;@%$;=&%$2!P' j:/ob?*(=@%$E%$;01*&B]h*#%$< 9"$E3#a39%$;5R3:"$'v%2%$"$*0)%2/g!;Y<+7g!<Q!*i_ !2*6<C%$;@32%$E%$2\#%$N+'(! y 9%$"29TSËg] Á 8Í bo|fb|pt_@q
ν(N) = maxsi∈Si
[n∑
i=1
ui(s1, . . . , sn)
]i = 1, . . . , n.
SËg] ¾ <;QU1=>@?Yν(M ∪M ′) ≥ ν(M) + ν(M ′) ∀ M,M ′ ⊂ N, M ∩M ′ = ∅.
I letnpg!6Jz]\xq13E3!(!j:!G*i/(/g!'MQ/% y *LGK$!G )^TSËg] ¿<;QU1=>@?Y ⋃
k
Mk ⊆ N"2
Mk ∩Mj = ∅ j 6= k]#'('&E9b
ν
(⋃
k
Mk
)≥∑
k
ν(Mk).
I letnpg!6Jz]\xqok%$ RW%2!*#/g'@K$!G7kr%$GN9N+:L:3 1k/(/g!'1TSËg] ¾ 88!:ZY)\J:E<pR3:"$'& 3*# !2!'1]#\J+_%$G/g;@3t01"$*9"$E3<C"$'1b ¾ T#0J:Et37:7A<_uO/g;@3%$^]¿gT&0J:E37:<Hu/g;@3%$^] Á T&0J:E37:<Hu/g;@3%$U'('@EO%$*'(!%$< y 9%$9T¾ T0J:EC37:<HuC/g;@3+%$^]¿gTk0J:EC37:<_uC/g;@3C%$^] Á T0J:E3j*37:<Hu
/g;@3 %$^]J'('@E Á T y 9%$ ¾ T A^*#%$'D"2 ¿gT A^*J' ¾ e ¾ SE3!*&9T¾ Tg0J:E37:<Hu/g;@3 %$^]#¿gT(0J:E3j*637:<_u/g;@3%$^] Á Tg0J:E3j*637:<Hu
/g;@3 %$^]J'('@E ¾ T y 9%$ Á T A^*J'M"$ ¿gT A^*J' ¾ e ¾ SE3!*(9T¾ T0J:EC37:<HuC/g;@3+%$^]¿gTk0J:E3j*37:<_uC/g;@3C%$^] Á T0J:EC37:<Hu
/g;@3 %$^]J'('@E ¿gT y 9%$ ¾ T A^*#%$'D"2 Á T A^*J' ¾ e ¾ SE3!*&9T¾ T0J:E3j*37:<HuC/g;@3+%$^]¿gTk0J:EC37:<_uC/g;@3C%$^] Á T0J:EC37:<Hu
/g;@3 %$^]J'('@E ¾ T y 9%$t¿gT A^*J'M"$ Á T A^*J' ¾ e ¾ SE3!*(9T¾ Tg0J:E3j*637:<Hu/g;@3 %$^]#¿gT(0J:E37:<_u/g;@3%$^] Á Tg0J:E3j*637:<Hu
/g;@3 %$^]J'('@E Á T y 9%$ ¾ T A^*#%$'D"2 ¿gT A^*J' ¾ e ¾ SE3!*&9T
Ì Á
¾ Tg0J:E3j*637:<Hu/g;@3 %$^]#¿gT(0J:E3j*637:<_u/g;@3%$^] Á Tg0J:E37:<Hu/g;@3 %$^]J'('@E ¿gT y 9%$ ¾ T A^*#%$'D"2 Á T A^*J' ¾ e ¾ SE3!*&9T¾ Tg0):E3j*637:<_u/g;@3t%B^]J¿gTg0J:E3j*i37:<Hu /g;@3 %$^] Á Tg0J:E3j*i37:7A
<_u/g;@3t%$^]#'('&E%$*'(!U%$< y 9%$9T£Ä;=M_'(! y 9%$NXw;;@"$*(=
Ás13 s23
1 \ 2 s12 s22 s12 s22s11 (0, 0, 0) (1, 1,−2) (1,−2, 1) (−2, 1, 1)s21 (−2, 1, 1) (1, 1,−2) (1,−2, 1) (0, 0, 0)
%$'('@E
u1(s11, s
12, s
13) = u1(s
21, s
22, s
23) = 0
u1(s11, s
12, s
23) = u1(s
11, s
22, s
13) = u1(s
21s22, s
13) = u1(s
21, s
12, s
23) = 1
u1(s21, s
12, s
13) = u1(s
11, s
22, s
23) = −2-*G@*
u2"2
u3A^EgT-Q¿gT e Á T'&@LGK$!Gt'@ ¾ T%$G%$*U]J'('&E
1 \ 2− 3 s1,1 s1,2 s2,1 s2,2s1 (0, 0) (1,−1) (1,−1) (−2, 2)s2 (−2, 2) (1,−1) (1,−1) (0, 0)
¾ T A^E%s1"2
s232!<Q<C%$3E3!'(#9]g%$9"$E3
ν(1) = 0\J*G@*mra !(R3:"$'&3Eg]g7
%$ )%$*r%2%$3 1%$*m*#%$<Z"$E/%$<C%2'&@LGK$!G'&a3*!^TSËg] Ã 6OQP!RLS(TU(VWS(XZY)>?;=I'E'(%$E3!G323!'(#w;;@"$**(=@%$U<C%$;&/3hR3:"$'>?;=
GR3:"$'&?'E'(%$E3!G323!'(#w;;&"$**&=&%$<C%$;&/3nR3:"$'(*J'_*#%$&%$2w*'1]\J
/3ν(M)
G "BE"$'(5Dw;;&"$*&= ∀ M ⊆ NA^E%T
SËg] Æ OQP!RLS(TU(VWS(XZY>?;='E'(%$E3!G323!'(#rw;;@"$**(=@%$h<C%$;&/3tR3:"$'@r'&*#3*#a39%$;5*#%$'I*J%B&%$2w*'1]\Jgb
ν(M) + ν(N −M) = ν(N) ∀ M ⊆ NA^E%
.
SËg] ÇÍ bo|fb|pt_@qUS%$*(3!)'&*#3*#Oa39%B;5J"B;_S;&G<[w;;@%$3G%$*mr'&E: !j'( J*\J32*J:j3^]17QQ*#E3<+:?j'( J*6<C%$;&/.R3:"$'(*J:n\J32*J:\J1'&%$3N'&a2a%$<Q<Q!=&%$*m'0)K2j*!*#K29
ÌÇ ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áSËg] ¾ .df=)P!Yb&b|pbg
νht
Nj@_@qBtaDLhpc dhthpl(göFbz¨<g!lo\pcì]Oxhpcw~xqJ_jm]`_@a,c÷@^||6Op_g!Hdbc pj@b
¾ Tν(∅) = 0 T¿gT ∀ M,M ′ ⊆ N M ∩M ′ = ∅ b
ν(M) + ν(M ′) ≤ ν(M ∪M ′),hxaapnpj ∃ Gdbc pgWb@a
νhyaphpjBhxax]`bjlrqBtf]zlra,`q@^||6O^_g!b
Á TÂAµh ∀ M ⊂ NE`j@b
ν(M) + ν(N \M) = ν(N)lrqlw|htaHhxaapnpj ∃ G
apnpg$qm]hpg$qPxq@qBtbo|6c¥m\^]`_@aTHd¤bc pgWb@aνaphpjBhxax]`bjlrqBtf]zlra,`q@^||6Op_g!b
I letnpg!6Jz]\xq^f&9"$0ge&cJE3;@Ij0R3:*:Bb#4i%$;'@*#33E3J: R'GA^9T
lU%$;=&%$*Ubν<C%$=(E% ¾ TU"2H¿gT1%$ RW%23w^TUlU%$;=@%$*
G\#
Si
M ⊆ N<Q!*#/#9*
E"232\J<+7! ):^]J\#i ∈M
TqR3:"$'& 33E"$;!j:-u;=M:j329]J\#;=M'(!:j32t%$;=
M\J<+29T)F@%$Ga RW%_%$2
sMT>?'('@Et_'(! y 9%$Nw;;&"$*&=DG%$;=&%B*iX'@a7@%$3'&%$9Nb
ui(sM1, . . . , sMn) =
1|Mi|
ν(Mi)\J
Mj =Mi ∀ j ∈Mi Mi ∈ si
νi(i) %$;=&"$ '@"$*(
(i = 1, . . . , n).
C%$;=g39%$E35#"$;m'&%2/g@"2"$E3r*#%$@%$22w'iR3:"$'@3H
[sM1, . . . , sMn ]
33E"$;!jnAÄ%23E%*#"$2&%r'(!3w*&%$%$33*#%$'1]J\J
Mj = Mi<Q!*#/%$*
j ∈ MiA^E%TJ>?G%$*'&%$9NI%2%$3 )%$*
*#%$<Z'(!3w*&%$%$39TlU%$;=&%$*
νI%BGN d/% y *!j:
GR3:"$''E'(%$E3!G323!'(#w;;&"$*&=&%C"2
M ⊂ N%$39NG%$;@%29TJÄ;&9*!oS;R'1]g\#;=ν(M) = ν(M)
<Q!*#/%$*M ⊂ N
A^E%T# M = ∅%2%$3 )%$*M*&=(!:*
ν(M) = ν(M) = 0T(lU%$;=&%$*
M 6= ∅ "2.%$;=(w'CS%$^]&\#;= Mj =M<Q!*#/%$*j ∈ M
A^E%]7m<Q!P*J/%$*j ∈ M
R3:"$'&CsM
33E"$;!j:C:j323gTk%B'(!*(3w'D%$'('&E
GM =
∏
j∈M
Sj ,∏
k∈N−M
Sk;∑
j∈M
uj ,∑
k∈N−M
uk
'&"$39%B<C"$=&%2t'@@LGK$!GR3:"$'&9To>? 1%$*dXR3:"$'( J*ssMj
(j ∈ M)33E"$;!j:'
MA^ 1%rE39HR3:"$'&'M%$;=
sM∗33E"$;!j:R3:t'@"$01%$2!'1TJlU%$;=&%$*
sL
N −M'&@PLGK$!GAGR3:"$'&'6%$;=M%$39NG%$;@%233E"$;!j:R3gT)>?'('@EC'(! y 2%$Nw;;&"$*(=@%$'6"$E3%$PA<C%$9"2%Hj0R3:*MS%$**J:
∑
j∈M
uj(sM∗ , sL) =∑
j∈M
1
|M |ν(M) = ν(M) .
ú =(!:*(G@*S%$**J:rν(M) ≥ ν(M)
%$;=@%$*GN3G%$*#"$;GM
R3:"$'ν(M)"$E3"$'&"BE%T
Ì Ã
N \M '&@LGK$!GH%$'(!*(3@%C!3H!Gr%$;=(w'S%$^]o\#;=<Q!*#/%$*
k ∈ N \M A^E%Mk = N −M
]U7Q<Q!*#/%$*k ∈ N \M R3:"$'@H
sN−M33E"$;!j:r:j32SR3gT
lU%$;=&%B*sL∗
%$rH'@a9a33E"$;!jgTlU%$;=@%$*sM ′
M'&@LGKB!GQ%$;=+%$39NG%$;@%233E7A
"$;!j:R3gTUlU%$;=&%$*7: J:j ∈ M
%$;=i'(!3w*(%$%$3 R3:"$'@_(sM ′ , sL∗)
33E"$;!jnAÄ%23E%.@*J3'&9@* ;=(*!G
sM ′ |M | 37:<_uQ33E"$;!j: sL∗01%2/g!; |N \M | 37:<Hu%$;=&SE3<+C33E"$;!j:hRW%$G%$*&B]J"2G%$;=&%$*
r ∈ Mj%$39NG%$;@%29T)>?'('@E 32w'g"$;'&"$00)%$*
r ∈ M]U;=(*!GH\J
r ∈ N \M G%$**#%]'('@EXsL∗
33E"$;!j6"$E3%$<C%$9"2%D39%BAE3!*(
Mr = N \ M"2+LP;=
Mj = Mr = N \ MG%$**#%]O<Q!9*( J*
j ∈ Mj
(j ∈ M)<Q!j3HG%$\#%$%$3G%$*UTk>? 1Nh'&a9&%$3'@%$2!'1]o\J;=\J
j ∈ M'(!3w*(%$%$3 R3:"BA
'&Q(sM ′ , sL∗)
33E"$;!jnAÄ%23E%]?'('&E
Mj ⊆ MTÍt!j:;@X7: J:g]\#;=s\)
j1, j2 ∈ M'(!Pw*&%$%$3HR3:"$'@'
(sM ′ , sL∗)A^E*#"$2&%]h'('@E
Mj1
"2Mj2
;=9*#'1];=D/g!G33R*'(\J<+9'1TV<C*#/3'j0R3:*d
M\J<+QS%$ )*(3\Jm/g!G33R*'(-\J<+9'a39%BA
;@"$E%bM =Mj1 ∪Mj2 ∪ . . . ∪Mjα ∪ r1 ∪ . . . ∪ rβ ,\#.
j1, . . . , jαR3:"$'@'i'(!3w*(%$%$3 R3:"$'@'1]n
r1, . . . , rβR3:"$'@'0)%2/g!;
*#%$< '(!3w*(%$%$3%$'(sM ′ , sL∗)
A^E*#"$2@%T M'@@LGK$!G J*E"2323&%$@NMR3:"$'@'
'(! y 9%$Nw;;@"$*(=&%H_'@a7&%B'@%$9N'&"B0#06j'(∑
j∈M
uj(sM ′ , sL∗) = ν(Mj1) + . . .+ ν(Mjα) + νr1+ . . .+ νrβ .
>?CQ"$E3"$'2*( J*s+"$%$¿gTJS%$"$%$G%X'@a7@%$3'&%$2"$ )%$**#%$<*J;=& ν(M)
A*#"$^T.>? )Nt01%2/g!;v9**Jtj:3\JY
ν(M) ≤ ν(M)%$;=&%$*GN3G%$*#"$;#T.46!*(3\#;=
M%$39NG%$;@%2_\J<+I@9]M'@E: !O%$;=&%$*GN3G%B*J"B;;@%$%$;=(w3
ν(M) = ν(M)/&/!P'D<Q!*#/%$*M ⊂ N
A^E%Tzt:"$E3w*'Q "$%$<+:(/g!'X:L::*J'X!P;&9j::EgT@cJ%$%$@"23w*'X39%$E3!*&S%$**J:
ν(M)+ν(N \M) = ν(N)
%$;=&%$*GN"B;_<Q!*#/%$*M ⊂ N
A^E%T&`"$%$)%$GNrE"239"$\#%$/% y *!j:'(! y 9%$Nw;;@"$*(=&%B'+"$%$<+:&/g!'6E"239"$*#%$'6 !9*(=(L::\#Q*#%$<<C%$;AS%$G%$GN&%$'1T@>?\\#%$ uRk'(! y 9%$Nw;;&"$*&=&%$'@%$O/g*'_<C%$;#](<C%$=@%$'( 1%$*QS%$\J32*J: R'C%$GN9N'@%$ !G9b
u′i(sM1, . . . , sMn) = ui(sM1
, . . . , sMn) +1
n
(ν(N)−
n∑
i=1
ui(sM1, . . . , sMn)
)
(i = 1, . . . , n) .
9**Jj:3\J]\#;= n∑
i=1
u′i = ν(N)]7iMuR_'(! y 9%$Nw;;&"$*&=&%$'('@%$'&*#WA
*#a32%$;5â'&@LGK$!GQR3:"$'É:X%$GNTt 26'&%$_K230J:*<C%$;<_3*&*#'1]t\#;=%$\\#%$`YR3:"$'(\#vE39
ν ′'E'(%$E3!G323!'(#w;;@"$*(=<C%$;@%$;=@%$2!'
νA^&%$^T
ν ′(∅) = ν(∅) = 0 %$;=&%$*GN"$;C<C3!G*(=(!P:*(GT
Ì Æ ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@álU%$;=&%$*
M ⊂ N%$39NG%$;@%2"2 %$;=(w'M%$^]J\#;=m<Q!*#/%$*
j ∈ MA^E%
Mj = M]
7MA^ )%$*XG%$&NhR3:"$'&'r'(!3w*&%$%$3%$'1T>?'('@E.L;=_%$GN@:G
M∗ "2? N \M'&@PLGK$!GQ%$39NG%$;@%2sL33E"$;!j:R3:EXS%$**J:^b
∑
j∈M
u′j(sM∗ , sL) = ν(M) +|M |n
(ν(N)−
n∑
i=1
ui(sM∗ , sL)
).
-*G@*U]#<Q!*(t%$GN U]1N\J<+2 !GS%$LE3\JSR'm/g!G33R*'( E"232\J<+9'
a39%$;@%$'@"$*(N =Mk1
∪ . . . ∪Mkρ ∪ r1 ∪ . . . ∪ rσ ,\#
k1, . . . , kρX'(!3w*(%$%$3.R3:"$'@'9]
r1, . . . , rσ01%2/g!;DQ*#%$<'(!3w*&%$%$3.R3:7A
"$'&'RW%$Ga!'M(sM∗ , sL)
33E"$;!jnAÄ%23E%-&*J3'@9@*UT
>?'('&EOS%$**J:kn∑
i=1
ui(sM∗ , sL) = ν(Mk1) + . . .+ ν(Mkρ) + ν
(r1
)+ . . .+ ν
(rσ
)≤ ν(N)
%$;=@%$*GN3G%$*#"$;#TJ>? 1Nk9**J∑
j∈M
u′j(sM∗ , sL) ≥ ν(M) ,
"2@"$;wkν ′(M) ≥ ν(M)
%$;=@%$*GN3G%$*#"$;X/(/g!'X<Q!*#/%$*M ⊆ N
A^E%T@46!*(3\#;=Q'@@LGK$!G)R3:"$'_'&*#3*#a39%$;5U]%$9"$E3
ν ′(N \M) = ν(N)−ν ′(M)]S%$**J:7: J:
ν(N \M) = ν(N)−ν(M)
%$;=@%$*GN"$;#]oL;=ν ′(M) ≤ ν(M)
%$;=@%$*GN3G%$*#"$;@%$H*&=&%$ESRw'6<Q!*#/%$*M ⊆
NA^E%TJ>?2ta39%$&%$3@%H_%$GN !%$;=@%$*GN3G%$*#"$;;@%$
ν ′(M) = ν(M)%$;=@%$*GN"$;@%$
*(=@%$ESRw'I<Q!*#/%$*M ⊆ N
A^E%TSËg] Ê^ OQP!RLS(TU(VWS(XZY>?;=
ν'E'(%$E3!G323!'(#Iw;;@"$**(=@%$-<C%$;&/3XR3:"$'( J*%$;=
x = (x1, . . . , xn)&%$'(E3%$G32:3*J'1!<Q03!G*:O*#%$&%$2w*'1]\J
¾ T ∑
i∈N
xi = ν(N)
¿gTxi ≥ ν
(i)∀ iT
SËg] Ã Í b|fbo|pt_@q?H%$G32:S;&<++2hRW%$G%$*(3!^]#\#;=D<Q!*#/%$*IR3:"$'@ %$;&j: '('&E*(=@%$E%$<C"$*(=(.'0U]&<Q!*(O%$;=&%$/w)'0)3O@*J¤S¿gT BT#.&/#:.*&=&%$E%$<C"$*&=I377A0)ELPB:01%2/g!;C*!*#K2 ¾ T BT
Ì@Ê
SËg] Ë<OQP!RLS(TU(VWS(XZY&>?;=ν'E'(%$E3!G323!'(#w;;@"$**(=@%$<C%$;&/3UR3:"$'&G"$*(=@%$;@%2WA
*#%$'i*#%$@%$2w*'1]o\Jν(N) >
∑
i∈N
ν(i) To>?G%$*'&%$9Nm%2%$3 )%$*d2<C*#/R'1]o\#;=
R3:"$'I*#%$<ZG"$*&=&%$;@%29T
SËg] Á ¡;QU1=>@?Y[Aµh ∀ M,M ′ ⊆ N M ∩M ′ = ∅ E`jh ν(M)+ν(M ′) = ν(M ∪M ′)Hhxaapnpjhm\^]`_@agWbd ce_g!bo|b@q_@q¡mnxjF6Jz]Oxh
ν(N) =∑
i∈N
ν(i) H<hxaapnpj ∀ M
HM ′ ⊆ N
M ∩M ′ E`j@bν(M) + ν(M ′) = ν(M ∪M ′).
I letnpg!6Jz]\xqo/&/g!'mX/% y *LGK$!G )^TSËg] Æ 8Í bo|fb|pt_@qnÉ*#%$<ZG"$*&=&%$;@%2?R3:"$'&'( J*m*#%$<Z"$E/%$<C%2'&&01%$E:*!^TSËg] Ê^yÍ b|fbo|pt_@q>?;=G"$*(=@%$;@%2-R3:"$'(*J'&"$;%$G%$*v's%$G32:6*UT>?;=*#%$<G"$*&=&%$;@%2R3:"$'(*J'I(!G39*& K2'I%$;= (
ν(i)) T
SËg] ÌOQP!RLS(TU(VWS(XZYo>;=ν'E'(%$E3!G323!'(# w;;@"$**(=@%$k<C%$;&/3.R3:"$'
x"2
y%$PA
32:'M ⊂ N
'@@LGK$!GTn>?'('@E_2r<C*#/R'1]o\#;=x/<Q!*J: R36
yA^
MA^E%
*#"$2@%bx ÂM y
]g\J¾ T
xi > yi ∀ i ∈M¿gT ∑
i∈M
xi ≤ ν(M)T
SËg] ËÍ bo|fb|pt_@q ¾ T A^ 1N'@a7@%$3'&%$2!'1]k\#;=M'@@LGK$!G J*<Q!*#/%$*'(!?RW J*R3:E
xAÄ39%$^]g<Q!*(
yA^*J^TJ¿gT A^ )U'&a9&%$3'@%$2!'1]g\#;=mr%$G32:"$*&=(G%$;@%2%$*6%$G"$E3\#%$NT
SËg] ÌÍ b|fbo|pt_@qI/% y *LGK$!G )'@a7@%$3'&%$2!'1]J\#;=M = N
%2%$"$*%$;=(!'i%BG32:%$<Z/<Q!*J:\JSR3+X<+:3!'7T
SËg] ¾ È<OQP!RLS(TU(VWS(XZY@>?;=νR3:"$'( J*
x%$G32:?/<Q!*J: R3
y%$G32:39]\J ∃M ⊂ N
M 6= ∅ ]g<C%$=(E% x ÂM yT
SËg] ¿Ådf=)P!YB&b|pbgxbcwnxqBtf]\xq
νaphpjBhxax]`bjlrqBtf]zlra,`qN@^||6Op_mg! _@q
M ⊆ N (M 6= ∅)apnhpc Jokmlo~∑
i∈M
xi < ν(M)
uLnpg]nxqfhpg hxaapnpjlw|htaH_DLh ∃ yHZhpdbc pjBb
x ÂM y
ÌË ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áI letnpg!6Jz]\xq^f&9"$0ge&cJE3;@Ij0R3:*:>?G"$;@"$;@%2"$;#]\J
y ÂM x]%$'('&E9b
ν(M) ≥∑
i∈M
yi >∑
i∈M
xiT
f&2w'g"$;@%2"$;#bgk%$;=(w'CS%$^](\#;= ∑i∈M
xi < ν(M)]g%$'('&E
∆ = ν(M)−∑
i∈M
xi >
0T(cJ%$**J:19: J:X
ν(N)− ν(M) ≥ ν(N \M) ≥∑
j∈N\M
ν(j) %$;=@%$*GN3G%$*#"$;
"2L;=M∆′ = ν(N)− ν(M)−
∑
j∈N\M
ν(j)≥ 0 %$;=@%$*GN3G%$*#"$;+!G9T
lU%$;=&%$*
yi =
xi +∆|M | ,
\Ji ∈M
ν(i)+ ∆′
|N\M | ,\J
i ∈ N \M,
\# |M | "2 |N \M | M]J!PG%$3&%X
N \M \J<+2 J*G%$&NrR3:"$'@'m37:<+:RW%$Ga!^T 9**JUj:3\J]#\#;= ∑
i∈N
yi = ν(M) + ν(N)− ν(M) = ν(N)"2
yi > xi]
\Ji ∈M
Ty = [y1, . . . , yn]
%$7:PBn%$;=D=(*m%$G32:9]J<C%$=(!'D/<Q!*J: R3X(AÄ%$9TSËg] ¾¾ OQP!RLS(TU(VWS(XZYRW%$a RW%
I(ν)-'E'(%$E3!G323!'(#w;;@"$*(=(\#%$ E39H%$G32:'
\J<+7:9TSËg] ¾ ¿IOQP!RLS(TU(VWS(XZY@lU%$;=&%$*#%$'
ν1"2
ν2'E'(%$E3!G323!'(#kw;;@"$*(=@%$'1]%$'('@E? \#27:7A
R'CE39-R3:"$'&'O!9<CEW*J'I*#%$@%$22w'1]g\J ∃ f : I(ν1) −→ I(ν2)%$;=@"$E3%$<_5
G%$'@"$01%$9"29]\#;=x ÂM y ←→ f(x) ÂM f(y)
S%$**J: ∀ M ⊆ NA^E%T
SËg] ¾9Á OQP!RLS(TU(VWS(XZY1lU%$;=&%$*#%$'ν1"2
ν2'E'(%$E3!G323!'(#w;;@"$*(=@%$'1]J%$'('@E-Q\#$A
7:R'+E9-R3:"$'&'33E"$;!j!j;I%$'((!G%$*#3*#%$'I*#%$@%$22w'1]\J_G"$%$2*#%$'D=(*α > 0
"2β1 . . . βn
'&*#3*#'1]g\#;=
ν1(M) = αν2(M) +∑
i∈M
βi.
SËg] Ç<;QU1=>@?YIhtQb@a,Ol:Oxhpcebg$qmnB|hpcwnpd/b@a,Ol:Oxhpcebg:kfloh¤jBbcw\kmlo~^]l(gBF6`ap\pceSËg] Á _d¥f=)P!Y$a^_m]Caphpjhxax]`bjlrqBtf]zlra,`q¡@^||6Op_g!g!bc:dbo|h3Fn^]o]Lm\^]`_@auLnpg]nxqfhpgÂhxaapnpjµb@a,EOl:Oxhpcebg$qHNDLhletnpdnpjfSËg] ¾ ÇZOQP!RLS(TU(VWS(XZY>?;=
ν'E'(%$E3!G323!'(#w;;&"$**(=@%$)<C%$;&/3R3:"$'@.*#E3<+7A
!7:3*J'D*#%$&%$2w*'1]g\Jgb
ÌÌ
¾ Tν(i) = 0 ∀ i ∈ N¿gTν(N) = 1
TSËg] Ã <;QU1=>@?Y&dQl(gBFbg g:npjdhpc let\pcì]m\^]`_@ayce_g!b|b@qI letnpg!6Jz]\xqCb/3
ν(i) "2
ν(N)!G3<C%$E%$"$ )%$*gb
0 = αν(i)− βi α > 0
1 = αν(N) +∑
i=1
βi
%$;=@%$*G%$3E%$*#/#32%$Et<C%B;@G/g\)TSËg] Æ ¡;QU1=>@?Y[&bo|^bg
ν]`bm](qBtvpcebo|b@q6aphpjhxax]`bjlrqBtf]zlra,`q@^||Op_g!g!bc!dbo|h3Fn^]o]&ce_g!baE|b@q&m\^]`_@a^ ¬ aapnpjhtb@aOl:Oxhpcebg$q&m\^]`_@apnxanxqBtf]\pc \8Qhpgu$npg]nxqfhpgb|pµg:npjdhpc let\pcì]m\^]`_@aOxhpgW
I letnpg!6Jz]\xq1'&a9&%$3'@%$2!'MH%$GN9N+:L:3 )^TSËg] Ê^<;QU1=>@?Y!R4g:npjdhpc let\pcì]m\^]`_@apnxa°DLhpc dhthapnpgOpbI letnpg!6Jz]\xqo/&/g!'mX/% y *LGK$!G )^TSËg] ¾9Ã µOQP!RLS(TU(VWS(XZYn>?;='E'(%$E3!G323!'(#-w;;@"$**(=@%$<C%$;&/3R3:"$'i<+;7R3:*J'*#%$&%B22w'D9*M%$G32:'+\J<+7!9]g<C%$=@%$'&%$.<+:%$G32:.*#%$</<Q!*J:^TF@%$GaPARw'
ν<+;R3:¥K2E%
C(ν)A^@%$^T
SËg] Ç<df=)P!Y ¬ |pνm\^]`_@aydh|mh¤htnpg
xO^b@ax]npjnxaDLhpc dhthHdbc pj@b
¾ T ∑
i∈M
xi ≥ ν(M) ∀ M ⊂ N
¿gT ∑
i∈N
xi = ν(N)T
I letnpg!6Jz]\xq1'&a9&%$3'@%$2!'MSËg] ¿O"$%$ 1N^TSËg] ¾ È<Í b|fbo|pt_@qJ"$%$g<Q!j3?G%B\J%B"B;@%2h<+;/% y *LGK$!GR3:h:WS;&<+2*!^bR3:7A"$'i<+;7R3m9*G%$\#%$"$;@%2r'(! y 9%$"2%$'i\J<+71
(xi)i∈NB]1<C%$=&%$'(E%X*!*#K2r=(*
M'&@LGK$!GC"2
(yi)i∈MG%$\#%$"$;@%2'(! y 9%$"29]\#;=
yi > xi ∀ i ∈MT
SËg] Ë<;QU1=>@?YC(ν)
t\pjm]apnpgOpbpDLhpc dhtI letnpg!6Jz]\xq1'&a9&%$3'@%$2!'MH%$GN9NQ"$%$ )N^T
¾ ÈÈ ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áSËg] Ì;QU1=>¡?YAµh
νbo|pÂgWbdAce_g!bo|b@q
nqBtbd_c b@qm\^]`_@aaphpjBhxax]`bjlrqBtf]zlra,`qB^||6EOp_g!bHGhxaapnpj¥hm\^]`_@aydh|mh b|pbm]zcebgbcwnxqBtmcw\xqTQB~pcC\pc(ce
I letnpg!6Jz]\xqnR3:"$'D*#%$<G"$*(=@%$;@%29]#L;=1b ∑i∈N
ν(i)= ν(N)
T#>?_%$GN9NC"B%Bk<Q!j3
xi = ν(i) r%$;=&%$3G%$*6%$G%$<C%H_<+;*J'1T
SËg] ¾ È<;QU1=>@?Y!Aµhνb|pce_g!bo|b@qµapnpg$qm]hpg$qwPxqBqBtb|6cm\^]`_@aTHGhxaapnpj
C(ν) = ∅ I letnpg!6Jz]\xqolU%$;=&%$*
x ∈ C(ν)T#>?'('@E ∑
j∈N\i
xj ≥ ν(N \ i
)∀ iT
R3:"$'I'&*#3*#a39%$;5U]#7@bν(N \ i
)= ν(N)− ν
(i) ]%$'('&E9b
∑
j∈N\i
xj + ν(i)≥ ν(N) ∀ i ∈ N.
x%$G32: %$9"$E39b
xi ≤ ν(i) T#
νG"$*(=@%$;@%29]L;=1b
∑
i∈N
xi ≤∑
i∈N
ν(i)< ν(N),
%$r%$G%$*(3<C*#/#:9]#\!G39%$*6%$'('@Ex/<Q!*J:9T
SËg] ¿Z8!ZYk%$'(!P*(3w*'+%$;=Q\J:E<C39%$<C"$=@%2O*#E3<+!7:O9"$E3#a39%$;5_R3:"$'&9T>?'('@E;=e ¾ T)"2-¿gT)'& '&@LGK$!G Á TJ%$G%$*"2
(1, 2)*(=@%$E%$<C"$*(=@% ¾ ] Á T#&%$E%2"$;@% −1 ]#L;=r%$G32:
(1/2, 1/2,−1)e ¾ T#"2 Á T#'&'&@LGK$!Gt¿gT%$G%$*U]J%$'('@E H%$G32:(1/2,−1, 1/2)eI¿gT#"2 Á T#'&'&@LGK$!G ¾ T%$G%$*U]J%$'('@E H%$G32:(−1, 1/2, 1/2) T>?'('@ERWoj:3\J]#\#;=D%$;=(!'D%$G32:%$</<Q!*J: R3C_<+:3!'9T
SËg] ¾9Æ ÂOQP!RLS(TU(VWS(XZY >?;=`*#E3<+!7:+R3:"$'q%$;=g39%$E35U]O\J ∀ M ⊆ N'&@LGK$!GE
ν(M) = 0;=
ν(M) = 1%$ R%23w^TC-
ν(M) = 0 MA^H@%232%29]U\J
ν(M) = 1*(=@%$E3%2'&@LGK$!G*J'D*#%$@%$22w'1TSËg] ¾¾ Í b|fbo|pt_@qJsR3:"$'_<+;R3:!*#%9:E3!G%$;=&%$*GN3G%$*#"$;@%$'Q\J:E27:'X<C%$;#]L;=7: !o<C%$;&:j0L:'6_!*#%9:E3!G %$;=&%$*GN3G%$*#"$;@%$'M%$G%$<Q9"2"$ 1N/&/g*J'1TSËg] ¾ Ê^yOQP!RLS(TU(VWS(XZYhF@%$Ga RW% C Ma39%2X'@@LGK$!G'd\J<+7:9T,:E3<C%$= M ∈ C A^E%G%$;=@%$*
RM |M | /g!<C%$*2!G%$'(!G/%232!"$E"2kG%$;=&%$* 1M ∈ R
M 'E'(%$E3!G323!'(#&%B'(E9]#7
(1M )i =
1
\Ji ∈M
0'(wGa*( )%$*UT
¾ È ¾
SËg] ¾ ËGOQP!RLS(TU(VWS(XZY(λs)s∈C
%$;=(w33\J'λs ∈ [0, 1]
'(!G%$;=&%$*#3u=@933u=@'(*J'*#%$&%B22w'1]\J ∀ i
R3:"$'@3Egb∑
s∈Ci∈s
λs1s = 1N .
SËg] ¿Z8!ZYglU%$;=&%$* |N | = 3 (λs)s∈C λs = 1/2]&\J |s| = 2 "2 λs = 0
'(wGa*& 1%$*U]'(!G%$;=@%$*#3u=&93 3u=(\J<+@TSËg] ¾ Ì6OQP!RLS(TU(VWS(XZY)%$;=
G(N, ν)'(!G%$;=&%$*#3u=@939]#\J∑
s∈C
λsν(s) ≤ ν(N)
∀ '(!G%$;=@%$*#3u=&93 3u=(EgTSËg] Ã @df=)P!Y,O*#/%$E7egf&\J0#G%$=!Bb ¬ |p
G\^]Ol1D$bm]vbPxq@qBtbo|6cyapnnu!bjh^]J:O<m\^]`_@asdh|mhuLnpg]nxqfhpg hxaapnpjgWbd jBb@qHNDLhalbo|pbg$q®c ntn^]o]`
I letnpg!6Jz]\xq1lU%$;=&%$*GHS%$*&3!^TlU%$;=@%$*
x%$G32: ∈ C
"2(λM )M∈C
'(!G%$;=@%$*#3u=&A93:;#T#>?'('&E9b
∑
M∈C
λMν(M) ≤∑
M∈C
λMx(M) =∑
i∈N
xi
∑
i∈M
λM =∑
i∈N
xi = ν(N).
cJE/gL3gbn%$;=(w'S%$^]k\#;=G'(!G%$;=@%$*#3u=&939]%$'('&E_*!*#K2%$*#%$'d=&*v3u=@'1]
\#;=∑
M∈C
λMν(M) > ν(N).
L;=M(1N , ν(N) + ε) ∈ R
|N |+1ε > 0
'&*&&%BI\J<+@]J/g!G33R*'( @by ∈ R
|N |+1:y =∑
M∈C
λM
(1M , ν(M)
) \#λM ≥ 0 ∀ M ∈ C
,
L;=(λM )M∈C
'(!G%$;=@%$*#3u#P=&93 3u=&'I"2 ∑M∈C
λMν(M) > ν(N)T
>?'('@E _'&"$'@*(@%BM/g!G33R*'(O\J<+r'&a9a3G"$%$2!'D%$:j32Q\!0)%$E3L'D"2=(**#%$<Z*&jX&%$'(E9b
(αN , α) ∈ R|N | ×R
]g<C%$=(E%
(αN , α)y ≥ 0 > (αN , α)(1N , ν(N) + ε
)
¾ È¿ ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@á∀ yA^E6"2
ε > 0Tk (
1N , ν(N)) M'(u0 J*d*d%$9"$E3
α < 0TklU%$;=@%$*
x = αN−α
T46!&%B (
1M , ν(M))∀ M ∈ C A^E%H_'(u0 J*m*U]
x(M) = x1M ≥ ν(M)"2
ν(N) ≥ 1Nx = x(N) .
xA^\#%$+%$;=<C%$;S%$G%$GNm*#%$<ö*#%$;&3L@%$'(E3_/gD'0R'
ν(N) = x(N)A^9]oL;=
<C%$;'03*'M%$;=D%$G32:39]<C%$=D )%$**#%*itR3:"$'I<+;7R3: J*UTSËg] Ç8!:ZYJn%$;=(w'+S%$^](\#;= Á TR3:"$'&%$;=g"$;*&=(!o*(=@%$E%$<C"$*(=@%$*6322!'1TJ =&A<C&/*n]J\#;=m\JC'&%BNC'&a2ww'm'@@LGK$!Gr'@t'0
α ∈ [0, 1] A^9]J<QL;MC\JE3<+/g!'%$<Q<Q! %$<'0UT>?'('&E9b
N = 1, 2, 3 ν(N) = 1, ν(s) = α\J |s| = 2 "2
ν(i) = 0 i ∈ N
R3:"$'X<+;R3C(ν)
<Q!*#/#9*+S%$G32:'1]@<C%$=(E%x(N) = 1
"2x(s) ≥ α ∀ |s| =
2A^E%T!£Ä;=M
C(ν) 6= ∅ 01*&*'('@E9]g\J α ≤ 2/3 T
yµ×@Ö6× ø «B6ßC«µE$FG7:9IHFG7D«Ø¥=C7:Û/58=IÙW=?A9<5¥56=<>SËg] ¿È6OQP!RLS(TU(VWS(XZY)?!jK$!#R3:"$'M:3(!\#%$NQ*(=&%BE%B<C"$**(=&%$k_'&a9&%$3'@%$9N'&"$00)%$*i/39b!(
N |N | <∞ R3:"$'&'*S:;@%$*#%$'!BT!!(
` ∈ NT r!*0%$E3<C"$'@%$'M37:<+gT!!!(
i ∈ NA^E%
ωi ∈ R`+
T!!
i ∈ N fi : R`+ −→ R+
iR3:"$'&X%$E3<C%$G"23!Ow;;@"$*(=&%
fS=(*#9]
*#a7@%$'(&N]g'@*':9!BT>?;=D!*03@%$'(E
R`+
%$G%$<C%$!^](zi)i∈N
!*03@%$'(EEgb∑
i∈N
zi =∑
i∈N
ωi
%$;=MG':K$!GT>? 1%$*+%2%$3 )%$*+:;@%$*#%$'_"$E/%$'@%$%$'_G%$\#%$3*#%$'X '&&01%$E:K$!G J*U]\J %$E3<C%BA
G"23w'D"2!*0SR3!'I'&<Q0G%$<C%$*&%$E%$'1])'('&E !*0'MK2%$E"WRW"$E%32w'g"$;w'IG%$\#%$9TF@%$Ga RW%HHS%$*(3!R3:"$'&9b
G(N, `, (ωi), (fi)
) T r%$;=&%2G%$\#%$"$;@%2'@@LGK$!G'D*&=&%$E%2"$;@%b
ν(M) = max(xi)i∈M
fi(zi) zi ∈ R
`+
"2 ∑
i∈M
zi =∑
i∈M
ωi
ëÈ ÇmÈ ÏLÑ ßÓ!ÑpÉÊ@ÞËyÊÞ$&Ñ ®W7ÞWÐÌXW&@à$&ã6ÌWÌXW&@á ¾ È Á
%$'('@Et_0!jK<+;7R3:*J'mtR3:"$'I<+;R3:*#%$@%$22w'1TSËg] ¿ ¾ OQP!RLS(TU(VWS(XZY)0!jK-'@<Q01%$3!3LD%$;=&%$*#3u=&Q (
p∗, (z∗i )i∈N) 0J:E9]#\#
p∗ ∈R
`+
!*0t:E3&%$'(E"2
z∗i0)%2/g!;I
maxzi∈R`
+
(fi(zi)− p∗(zi − ωi)
) <C%$;@G/#:.
SËg] ¿¿OQP!RLS(TU(VWS(XZYL-(p∗, (z∗i )i∈N )
'&<Q0)%$3!3LC%$;=@%$*#3u=&]@'('&Efi(z
∗i )−p∗(z∗i −
ωi)
iR3:"$'&%$;=&%$*#3u=(!o'(! y 9%$"2%T
>?G%$<C%$22w'M9: itR3:"$'&9R3:"$'&'%$E3<C%$G"23!gw;;@"$*(=@%$!gG%$;=@%$*#%$'X9*#'H"2K2'H%$;=H!*0G%$;=@%$*UT
lU%$;=&%B*nbω∗ =
∑
i∈N
ωi
|N |t:3j;@?SE3E:9T-
f'&*':21]@'('&E
ω∗!*0h<C%$**&=(!G"$;
<+7g!<+!7: R3m309T?lU%$;=&%$*p∗ f
/%$E3!:SR3ω∗A^ J*UT?lU%$;=&%$*
g=&*U]
\#;=g(ω∗) = f(ω∗)
"2g′(ω∗) = p .
>?'('@E (g(ωi)
)i∈N
%$G%$<C%H_<+;*J'1]\!G39%$*
ν(M) = |M |f((
∑
i∈M
ωi
)/|M |
)≤ |M |g
((∑
i∈M
ωi
)/|M |
)=
=∑
i∈M
g(ωi)
"2
ν(N) = |N |f((
∑
i∈N
ωi
)/|N |
)= |N |f(ω∗) =
= |N |g(ω∗) =∑
i∈N
g(ωi)
(g(ωi)
)i∈N
'(! y 9%$"23O'('&EG%$\#%$O%$G"$E3*!^]&\J<Q!*#/%$*_RW:"B'&p∗AÄ*IK2%$E"$!J )% !*gA
0SR3: 303Eg]I%$;=g"$;*&=(!U!*0p∗#303 JQ'@%$E3w(BT-X%$9%$*_:E*mG"$3E%WRWa*
+K2%$E%])'('&E-iR3:"$'@-
maxz
(f(z) − p∗(z − ωi)
) '(!PS%RW%$9"23zA^E%H<C%$;@G/gg]
ω∗A^ J*M<+7g!<+!9:\JSR3QH'(! y 9%$"239Tg 7 (
p∗, (ω∗)i∈N)∀ i ∈ N
]H0!jKt'@<XA0)%$3!3LM%$;=&%$*#3u=&gT
¾ ÈÇ ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áù `! `! £úâ!U¯k³&û«o±6åt®o¯¦çLègçfå üU®ÅýSËg] ¿ Á 6OQP!RLS(TU(VWS(XZYof& !U\J<+ út%$<+**6"24iE3;@%$*#3%$!*:>?;=
V ⊂ I(ν)\J<+2-
νR3:"$'Dút%$<+**m<C%$;@G/#::*J'M*#%$@%$2w*'1]\J
¾ T J:E3<C%$=x, y ∈ V
A^E%x ¨ y
)%$GNC3 !!:¿gT ∀ x /∈ V
%$G32:3\# ∃ y ∈ V]g<C%$=(E%
y  x'(wGNQ3 !!:BT
SËg] ¾ ¿Í b|fbo|pt_@qh>?;=/3R3:"$'(*J'K2'i%$;=<+;7R3DG"$%$2!'1TU46L;m3 !?\JPA<+7Ia !G-G%$\#%$9]o;=i%$;=%$<mToút%$<+**u;="$E3@%$9]o\#;=i%$;=/3H3 !\J<+2\#E39t'(wGa*& 1a9N-%$G32:'r'(wGa*( )a9Nt<+;&E:3*J'rS%$G%$*#%$'r<C%$;#TSËg] ¾¾ ;QU1=>@?YR¢m\^]`_@adh|mhQdQl(gBFbg qm]h8Ql(cDLhpc dhtsjB_@qBtaDLhpc dhth ¬ |pbm]zcebgÂqm]h6EQl(cfDLhpc dht8qbdñOxhpcw~TFpl&j@_@qBtaDLhpc dhthyb|p¥d\xq@lraqm]h8Ql(cfDLhpc dhtmg:hxa^AµhyhZm\^]`_@aµdh|mhqm]h8Ql(c"DLhpc dhtaHZhxaapnpjybBt¥htybo|pbm]zcebg*qm]h8Ql(c"DLhpc dhtI letnpg!6Jz]\xqJ %$GNr'@"$O:LPB:h'@a7@%$3'&%$2!'Q3 !J\J<+-/% y *LGK$!RW:*)'Q'(wGNH37A !!: '@a7@%$%$<C"$*(=@"$ 1N^TJ\JE3<+/g!'6:L:01%2/g!;DH%$GN !:L:'&a9&%$3'@%$2<C"BA*(=@%T
yµ×øØ× © >Háb¥6CF6;«ÿþ Ñ Ù"^66«B5¥«B5b^ Ø¥6>@?A6Ít!93;&: R'C<C3H'&@LGK$!G_'&"$09N&/"239T>?;=D/3O'&@LGK$!G J*m%$;=HR3:"$'@3*J'D'&A'&E?"232%$E35QE"232&%$**!G%]\Jt'(! y 9%$"2%$!g<C%$;\Jj/R3:'X%$;=_<+:'@@LGK$!G J*+%$G"$E3\#%$N'(! y 9%$"2%$!9T^£Ä;=_%$;=Q/3?'&@LGK$!G J*CE"2323@%$&N.R3:"$'&'H<C%$;S%$*&=&%$;@%$3\#%$3!'Q%$;=&A<+:39]<+RW/m'(!PA^'(!oG%$\#%$N"$;@%$!39%$E3!*&t%$G%$*gS%$*&=&%$;@%$"2%$k"$\#%$9TJ>?2-X;@*#/j3<C%BA*#%$%$SE3<+!7: R'm_'&a9&%$3'@%$9N'( 1%$*UTSËg] ¿Ç6OQP!RLS(TU(VWS(XZY)lU%$;=&%$*
x%$;=D%$G32:
GR3:"$'( J*Ub
e(y,M)
0J:E39]?eY\#y%$;=
MG%$\#%$"$;@%2+'(! y 9%$"2+"2
M%$;=Y'&@LGK$!Ge
iR3:"$'&j%$G%$*!S%$*(=@%$;@%$"2"$*#%$'6*#%$&%$22w'1])\J
i ∈ M"2
j 6∈ M"2
yk > xk
∀ k ∈MT
e(z,K)
0J:E3re\#z%$;=
KG%$\#%$"$;@%2r'(! y 9%$"2H"2
K%$;=i'&@LGK$!GMe
j%$PA
G%$*G"$01"2"$*#%$'m*#%$@%$22w'i (y,M) j
%$G%$*!nS%$*&=&%$;@%$"2"$E%]#\Jj ∈ K
]i /∈ K
zk ≥ xk ∀ k ∈ K \M "2zk ≥ yk ∀ k ∈ K ∩M
T
SËg] ¿ Ã ÂOQP!RLS(TU(VWS(XZY G'@@LGK$!GCR3:"$'q'(#/7:3!r\J<+7:*
x%$G32:'
\J<+7:I"$E3SRw'1]<C%$=@%$'(E%b? J:E3<C%$=iR3:"$'&X J:E3<C%$=
jR3:"$'@C%$G%$*!
(y,M)S%$*&=&%$;@%$"2"$E%jA^*#%$'D*m%$G%$*G"$01"$%T
ëÈ ÎÈ+ßUáË:ÕB)â&^ÊÑ@ßUàXY(ßÑ ÌWÑ ÌY_®gßUáãß: ¾ È Ã
SËg] ¾9Á Í b|fbo|pt_@qÉS%$*(3!U;@*J/j3<C%$*#%$Q'&a9&%$3'@%$9NhRW%$G%$*&3!^b1H%$;=(!'QR3:"$'@u;=m@"$!^]J\#;=iQ/3-%$G32:t<C%$G%$3-3uk'@%$&%2%$ '0U]J<QL;I&"$%$'&%2/"2%_39%$E3!*&%$;=H<+:3!' R3:"$'&3u'9T>?'('@E
i<C%$;'(LG"$E%$#%$;=H<+:3!'H'&@LGK$!Gh'&3*!^]<C%$=
'(!7:ESR3QX<+:3!'_R3:"$'@3"2 \#U<+:E<Q!*#/%$*'(!#RW J*CR3:E9T`'(!7:E3*!)'(L:*&nR3:"$'&h%$'('@Es\J*I%$G%$*G"$0)"2%:j32 R3g]&\#;=+'('@E."$*
*#"$'(wG%2/SE3<+:G''@@LGK$!G9]k\#<Q!*#/%$*'(!hG%B;&j: d=(*mRWoR3:E9]o<Q!*(r'@E: gA J*U]JH:j/m<C%$;*&=&%$E3*!U'(L:*('I0)%2/g!;C<C%$;'0R3:'1]J<Q!'(L*J::U*#%$'(!'1TSËg] Ã 8!: jS=(:Bb:-:E<[R3:"$'&3*'M*U])%$;=g39%$E35ia #"$;!1R3:"$'@.R3:WA37*J'1To46!*(H2r<+:E 1%$<HB33'1]n_R3:"$'i<+;R3IwE%29]U3 !?\J<+7m37:<C*UT H'(#/7:3!U\J<+H%$;=M%$G%$<H5 1/3, 1/3, 1/3 TlU%$;=&%$*
x%$;=C%$G32:9]&%$;=(w'XS%$^]@\#;=
(y,M) iS%$*(=@%$;@%$"2%
jA^@%$139%$<H 1%$*UT
>?'('@E ∃ M = i, h ]#\# h Á TR3:"$'&"2 yh < 1− xi\J
yi > xi =⇒ y(M) =ν(M) = 1
BT \\#@]\#;=jA^*#%$'YG%$;=&%$*%$G%$*G"$01"2%
(y,M)A^E%]32w'g"$;@%29]?\#;=
∀ i, j"2
hR3:"$'&3E
yh < 1− xj\J
yh < 1− xi%$ 1NU(!G39*&'@a7@%$3'&%$2!'1]g\#;=
1− xi ≤ 1− xj;=
xj ≤ xi ∀ i, jL;=
x = (1/3, 1/3, 1/3).
ù (¡ `! £¢¤8¦J³SËg] ¿ Æ 6OQP!RLS(TU(VWS(XZYJlU%$;=&%$*
x%$;=
G'&@LGK$!G?R3:"$'D%$G32:g]gG%$;=&%$*
M%$39NG%BA
;@%2'&@LGK$!G]#%$'('@E M'&@LGK$!GQa G%$"$*i%BK2%2OX'@a7@%$3'&%$9N"$E3SRw'1be(M,x) = ν(M)− x(M).
SËg] ¾ ÇÍ bo|fbo|pt_@q<-e(M,x) > 0
])'('&E-Q2t<C"$E3!^])\#;=m<C%B'('&E+a39%$;EN'&%BG%$*#%
MA^*#%$'dG%$<C*#/#*!^]\#;=
x%$G32:C )%$&%$9%$3\#%$NG%$;=&%$*"2XSE/gL3g]?\J
e(M,x) < 0]'('&E?%$"$E3"$'1]<Q!
MA^*#%$'_<C%$;'@%$g'0*#!Gg]\#;=
x )%$&%$9%$3\#%$N
G%$;=@%$*UT)f&9':r%$GNQ%2%$3 1%$*M:E/#<Q!UE%$*#/M"$E/%$'&"$ )%$*m%$3 :G/7:E^])
<+:&/g!'( J*60)%2/g!;C'&a9&%$%$G"2"$ENU )%239"$*!^T4i&/#3LP3'MSËg] ¿ Æ % y *LGK$!GA^9bSËg] ¿@Ê^OQP!RLS(TU(VWS(XZY&>?;=
M'&@LGK$!G
iS%$*(=@%$;@%$"2%
j%$G%$*#"$ 1%$*
x<C%$G%$39]\J
i ∈M]
j /∈M]xj > ν
(j) T
>?;=K'&@LGK$!G
j%$G%$*G"$01"2%
i MS%$*(=@%$;@%$"2%
j%$G%$*U]\J
j ∈ K]
i /∈ K"2
e(K,x) > e(M,x)T
SËg] ¿ËZOQP!RLS(TU(VWS(XZY ^'@%$E3*#%$ìQ )"$(BTGR3:"$'Q^'@%$E3*#%$ìSAÄ"$*M-a39%2h=(*I%$G32:3
"$E3SRw'1]&<C%$=(E% J:E3<C%$=iRW:"B'&
MS%B*(=&%$;@%$"2% J:E3<C%$=
j%$G%$*
x<C%$G%$39]
jA^*#%$'
*m%$G%$*G"$01"2%MA^E%T
¾ È Æ ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áSËg] ¿ÌOQP!RLS(TU(VWS(XZYJk%$39NG%$;@%2
i, jR3:"$'&3E_"2
x%$G32:3Eg]g<Q!*#/#9*m'@@LGK$!G'
<+7g!<+:!Ga G%$%]#<C%$P=&%$'IE3<+27:'iA^9]/%
jA^*#%$<mb
sij(x) = maxM∈C
e(M,x) : i ∈M
"2j ∈ N \M
.
SËg] Á ÈOQP!RLS(TU(VWS(XZYUGR3:"$'^'@%$E3*#%$ìSAGRW%T)46!*#/#9*%BG32:'
x ∈ X<C%$=@%$'(E%]
tR3:"$'&'I J:E3<C%$=(i, j)
0J:ESR3:Egbsji(x) ≥ sij(x)
;=xj = ν
(j) T
SËg] ¾ ¿<;QU1=>@?YISËg] ¿Ë¥bz¨<gJokmlo~ _@q¤SËg] Á È¥bz¨<gJokmlo~b@a,Ol:Oxhpcebg$qI letnpg!6Jz]\xqonE3!(!j:!G9]#/(/g!'mr%$GN9NC/% y *LGK$!G 1^TSËg] ¾9à yÍ bo|fb|pt_@qCR3:"$'<+;7R36"2QM'(#/7:3!.\J<+@]?D%$;=@%2tR3:"$'@''(! y 9%$"2%$!*#%$'Ca39%$\J*L::E)%$<Q<Q!=@%$*CS%$%$@"23*#%$<V!;@"$*(=@%$9T&`'@%$E3*#%$J/%BAy *LGK$!GR3Q"$U%$29%$k_S%$%$@"2%$^]gL;=M'<+2\J:;&C!G%$E3E%-\J;=(3'&$!'1TSËg] ¾9Æ Í b|fbo|pt_@qH'&%$E3*#%$/% y *LGK$!GR3:Mu;=!*(%$E30E%$:\JSR'1]O<Q!*(m9'm<C%$;@%$;=@%$9"2%$'&%$9]&<C%$=&%$'( )%$*MR3:"$'&'Xu;=QL"$!'Q<C%$;#]&\#;=C%$<p3u)*J;=+:G/A7*#%$<\#2*J'1]#%$<Z3uU*J;=M%$GN*&=(\#aH*#%$<VR3*J'1]G%$;&j: 6!G*#%$<"$E3!U<C%$;r%$;=&%$*#3u=(OS%$ 1E3L*!^TSËg] ¾9Á <;QU1=>@?Y!R a^bjgWbcIht¥hpc a,`fFnt\xql_DLhpc dhtµj@_@qBtaDLhpc dhthI letnpg!6Jz]\xq lU%$;=@%$*
G%$;=v'&@LGK$!GHR3:"$'1]G%$;=@%$*
x%B;=%$G32:M'@%$E3*#%$ 1%$*U]
G%$;=@%$*(y,M)
%$;=IS%B*(=&%$;@%$"29b
i ∈M, j ∈ N \M y(M) = ν(M)"2
yk > xk ∀ k ∈M.-xj = ν
(j) ]'('@E (
z, j)
zj = ν(j) %$G%$*G"$01"2
(y,M)A^E%TC-
xj 6=ν(j) ]J'('@E<Q!&%$
xX'@%$E3*#%$n%$G%$<C%b
sji(x) ≥ sij(x) ≥ ν(M)− x(M)T
lU%$;=&%$*KQC'&@LGKB!G])<C%$=(E%b
j ∈ K]i /∈ K
"2sji(x) = ν(K) − x
]1'('@Eν(K)− x(K) ≥ y(K)− x(K)
]L;=
ν(K) ≥ y(M ∩K)+y(M \K)+x(K \M)−x(K \M) > y(K ∩M)+x(K \M),
<Q!@%$y(M \K) > x(K \M)
T^£Ä;= ∃ z KG%$\#%$"$;@%2'(! y 9%$"2
zk ≥ xk ∀ k ∈ K \M"2zk ≥ yk ∀ k ∈ K ∩M
]%$9"$E3(z,K)
%$G%$*G"$01"2(y,M)
A^E%TSËg] Æ ä8!ZY+S%$*&3!m:L:d<Q!j3 Á T R3:"$'&da #"$;!XR3:"$':*)'V'@%$E3*#%$G%(1/3, 1/3, 1/3)
]h\!G39%$*'(#/7:3!\J<+%$;=@%$G%$<_5"2D<Q!@%$-*#%$<ÚwE%29]L;=DK2'D%$_r%$G%$<ZG%$\#%$"$;@%29T
ëÈ -È+ߤÉÊ@ÞË(ÉÞnãßYLÉ3ßÌWÕ!Ó#áâ?áÕL ¾ È@Ê
yµ×&%µ× © E$FG7:9IH ;=$E7W?A^<E$ þx5sá¥ßC>B=<6>Bá¥;x%WR3<+;I/% y *!j:j::*J:C'@%$E3*#%$/% y *LGK$!GR3: )!*#/g*'m'(!^TJ '&%$E3*#%$k%2%$"$ 1%$*QS%$*(=@%$;@%$"23rX%$;=@%2.R3:"$'&'M%$3"$'1])<QL;MC*(#K$G%2#%2%$"$ )%$*CS%B*(=&%$;@%$"23_'&@LGK$!GQ%232!^TSËg] Ág¾ ¥OQP!RLS(TU(VWS(XZY>?;=
(M,y)0J:E39]\#
M ⊂ N'&@LGK$!G6"2
y%$;=%$G32:
x%$G%$*!oS%$*(=@%$;@%$"23*#%$'D*#%$@%$2w*'1]\J
e(M,x) > e(M,y)( 7
y(M) > x(M)).
>?;=K'&@LGK$!GC%$G%$*G"$0)"2%_
(M,y)S%$*(=@%$;@%$"23E%]\J
e(K, y) > e(K,x)"2
e(K, y) ≥ e(M,x).
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<+7: "$E3SRw'1]#<C%$=(E%<Q!*#/#%B*mS%$*&=&%$;@%$"23\#%$(M,y)
G"$%$2!'M%$;=D%$G%$*G"$0)"29TSËg] ÁÁ 6OQP!RLS(TU(VWS(XZY)k%$39NG%$;@%2
x%$G32:3\#H/% y *!j: R'm
M1, . . . ,M2|N|−1
'&ALGK$!GÉE79]
e(M`, x) ≥ e(M`+1, x) ` = 1, . . . , 2|N |−1"2G%$;=@%$*
E(x)a G%$@%$'(E9]t<C%$=@%$j: !j' 39%$E3!*&/% y *!j:*'1bE`(x) = e(M`, x)
∀ ` = 1, . . . , 2|N |−1]OG%$;=&%$* C \J<+ Ssa39%2D'&@LGK$!G'`\J<+7M=(*0JE33LGK$!GR3
P1(x), . . . , PK(x)]U<C%$=(E%I'@"$_'&@PLGK$!G
M"2
M ′ 0)*(*v'('&EH'&%BAE3w*#%$'v%$;=YK2%$j: Jg]\Je(M,x) = e(M ′, x)
T?k%$39NG%$;@%2M ∈ Pk(x)
A^E%MG%$;=@%$*e(M,x) = ek(x)
]=(<C(/*e1(x) > e2(x) > . . . > ek(x)
T>?'('&En2U<C*#/R'1]2\#;=
E(x)G%Bg!'@;E y '(#*r'(!G%$ r<Q!*&
E(y)]2\J
E`(x)<E`(y)
HG%$;'(!G%$ m=(* A^E%]g<C%$=(E%E`(x) 6= E`(y)
];=D<Q!U%$29%$U%$'((!G%$*#9b∃ k∗
=D<C&/*U]\#;= ∀ k < k∗T ∣∣Pk(x)
∣∣ =∣∣Pk(y)
∣∣ "$ ek(x) = ek(y)O':E
e∗k(x) < e∗k(y);=
e∗k(x) = e∗k(y)"2
(∣∣Pk∗(x)∣∣ <
∣∣Pk∗(y)∣∣) .
SËg] ¾ Ç<;QU1=>@?Y!R ¼ qbz]zdh|f½htnpg1bcwnxqBtf]\xqfnxa°DLhpc dhthHdbc b@ajBbQht E(x) Opb@ax]npjcebplrapnB|pjBh¨Ia,`qfhpg dQl(g!l(d\pc lrqI letnpg!6Jz]\xqOlU%$;=@%$*
x=&*%$G32:9]k<C%$=(E%
E(x)G%Bg!'&;E y '(#*Y<Q!*!<+:!G9T
,O%$j:SR'1]h\#;=xÄ%WR3<+;p6%$G%$<C%b.%$;=(w'YS%$^]\#;=
(M,y)%$;=YS%$*(=@%$;@%$"2
xA
E%]o\#;=e(M,y) < e(M,x)
TUlU%$;=@%$*k∗D<+7g!<+:!G9]o<C%$=(E%b
ek(x) = ek(y)"2
Pk(x) = Pk(y)Tn46!@%$
E(y)G%Bg!'&;E y '(#*s*#%$< '(!G%$ U]o<Q!*(
E(x)]UL;=;=
¾ ÈË ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áek∗(y) > ek∗(x)
;=ek∗(y) = ek∗(x)
"2 ∣∣Pk∗(x)∣∣ ≥
∣∣Pk∗(y)∣∣ T1,:E3<C%$=(!'%2%$WA )%$* ∃ K
'&@LGKB!G]o<C%$=(E%K ∈ Pk∗(y)
])<C%$=(E%ek∗(y) = e(K, y) > e(K,x)
T)£Ä;=e(K, y) < e(M,x)
]gL;=K%$G%$*G"$01"2
(M,y)A^EgT#46!@%$
e(M,y) < e(M,x) M /∈k∗−1⋃
k=1
Pk(x)"2
ek∗(x) ≥ e(M,x),
<Q!@%$e`∗(y) ≥ ek∗(x)
L;=e(K, y) ≥ e(M,x).
k%$;=(w'I<C3S%$^]g\#;=xX%WR3<+; J*m*6"2
E(y)G%Bg!'@;E y '(J*m'(!G%$ U]
<Q!*(E(x)
TklU%$;=@%$*k∗MMG%$;'(!G%$ Y"$E3"$'1]n<C%$=(E%
Pk(x) = Pk(y) ∀ k < k∗"2H;=ek∗(y) < ek∗(x)
;=ek∗(y) = ek∗(x)
"2Pk∗(y) 6= Pk∗(x)
Tn,:E3<C%$=(!'%2%$3 )%$* ∃ M ∈ Pk∗(x)
]g<C%$=(E%e(M,y) < e(M,x)
TlU%$;=&%$*
λ ∈ (0, 1) "2 z(λ) = λx + (1 − λ)y]U%$'('@E
e(J, z(λ)
)= λe(J, x) 6=
(1−λ)e(J, y)]7 J:E3<C%$=
J ∈ C T 2:LSR'1]\#;= (M, z(λ)) %$;=_=(*_S%$*&=&%$;@%$"29]
<C%$=(E%*!*#K2O%$G%$*G"$0)"29T(>?-%$;=XS%$*&=&%$;@%$"29]@<Q!@%$e(M, z(λ)
)< e(M,x)
T& \\#@]\#;=
K%$G%$*G"$0)"2OG%$;=@%$*U]('@%$^b
e(K, z(λ)
)> e(K,x)
"$e(K, z(x)
)≥ e(M,x)
T£Ä;=m\J
e(K, z(λ)
)> e(K,x)
]1'('&Ee(K, y) > e(K,x)
]J<C%$=( 1NK /∈
k∗⋃
k=1
Pk(x)
"2e(M,x) > e(K,x)
T146!&%$T /∈
k∗⋃
k=1
Pk(y)]1%$9"$E3
e(M,x) = ek∗(x) ≥ ek∗(y) ≥
e(K, y).£Ä;=
e(M,x) > e(K, z(λ)
) T# 7*!*#K2 %$%$*G"$01"29TSËg] ¾9Ã <;QU1=>@?YIhyqbz]zdh| hya^bjgWbc&jB_@qBtaDLhpc dhthI letnpg!6Jz]\xq lU%$;=@%$*
x%$;=%$32:9]h<C%$=v*#%$< %$G%$<C%*^'&%$E3*#%$ìSA^*#%$'1T.,O%Bj:SR'1]
\#;=I%$'('@EH%WR3<+;*J'I%$<%$G%$<C%TL-Xr'&%$E3*#%$*#%$'+*#%$<%$G%$<C%]'('&E ∃ i, jR3:7A
"$'&9]<C%$=(E%sij(x) > sji(x)
"2xj > ν(j) T46!&%$ xj > ν
(j)∃ ε > 0
]<C%$=(E%y = x+ ε1i − 1j
%$32:OeI\#1k
_'&AÄ/g!<C%$*2!G %$;=g"$;X&%$'(E.eCG%$;=@%$*ε%$G"$;D'(!GK23!^]J\#;=
sij(y) > sji(y)T)
e(M,x) < e(M,y)01*&*'('&Et!;&@])\J
i ∈M]/%
j /∈M"2
e(M,x) > e(M,y)A^EHSE/gL3gTlU%$;=&%$*
k∗_G%$;'(!G%$ m=(*
k](<C%$=(E%
M ∈ Pk∗(x)"$
e(M,x) 6= e(M,y)Tg46!@%$
sij(x) > sji(x)Pk∗(x)
G%$;Aj: %B;=i=(*'@@LGK$!GE3<+@]1<C%$=(*#%$'
i%$G%$<C%]o/%
j*#%$<mT146!&%$
k < k∗
Pk(y) = Pk(x)"2
ek(y) = ek(x)T-C
Pk∗(x)=&*6'&@LGK$!tE3<+@]J<C%$=(E%
':E.<Q!*#/i, j%$G%$<C%]':E%$;=(!'C%$<m]'('@E
ek∗(y) = ek∗(x)"2
Pk∗(y) ⊂ Pk∗(x)T-t%$GN9N&%$'_*#%$< %$ R%23w*#%$'1]'('&E
sij(y) > sji(y)"2
ek∗(y) < ek∗(x)T46!*#/g'&"$
%2%$3 )%$*E(y)
G%Bg!'@;E y '(#*M'(!G%$ U](<Q!*&E(x)
"2x*#%$<%$G%$<C%r_%WR3<+;*J'1T
SËg] ¾9Æ <;QU1=>@?Y!RGapnhpc Jokmlo~xqm\^]`_@aQqbz]zdh|mhgWbdÅjBb@qrDLhpc dht
ëÈ ÃxÈ Òp@áßgß@Þ)â<Ë ¾ ÈÌ
I letnpg!6Jz]\xqo Ekw;;&"$*&=
kA^ J*MS=(*#9T#>?_X/% y *LGK$!G )k'@a7@%$3'&%$2!'1b
Ek(x) = minM∈Ck−1
maxM∈C\T
e(M,x),
\#^b C0 = ∅ "2 Ck k ≥ 1
k'&@LGK$!G'x\J<+7gTX46!@%$
E1S=(*#
x1 = argminx∈X
E1(x)*#%$< wE%2d"2'&<Q0J'(9T£Ä;= ∀ k ≥ 1
%$;@"232E%xk+1 =
arg minx∈X2
Ek+1(x)TgÄ**#%$*M!*#/g'&K$!G9U<Q!*#/%$*M!=&%$*D\J<+r'&<Q0J'("2O*J%B<ZwE%2
"2L;=X2|N|−1
%WR3<+;#TSËg] ¾ Ê^e;QU1=>@?YNR8t hxc a,`fFnt\xqlDLhpc dht_@q¤hÂa^bjgWbc<Q\pjdbc
Gapnhpc Jokmlo~xqm\^]`_@ajhgWbd j@b@q
I letnpg!6Jz]\xq i:L:+'&a9&%$3'@%$2!'6%$GN9Nd:L:3 1t"2I )^]\#;=%WR3<+;E"232\J<+7+_'&%$E3*#%$*#%$'1]#<Q!UE"232\J<+7+H'(#/7:3!U\J<+2*J'1TSËg] ¾ Ë<;QU1=>@?Y!R
Gm\^]`_@aQqbz]zdh|mh b|pbcebd°c!
I letnpg!6Jz]\xqk%$;=(w'sS%$^]\#;=x"2
y%$G32:'Y%$G%$<C%$!t%WR3<+;*J'1]
E(x) =E(y)
)%$j:SR'1]\#;=M%$'('@Ex = y
Tk%$;=(w'S%$^]\#;= ∃ M∗ '&@LGK$!G]<C%$=(E% E(M∗, x) 6= E(M∗, y)
]?"$Q%$'(!*&WA3w'
1/2x + 1/2y = z%$G32:39Tk46!&%$
Ek(x) = Ek(y) ∀ k ek(x) = ek(y)"2
(Pk(x)
)=(Pk(y)
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e(M∗, x) 6= e(M∗, y) ∃ k∗<Q!P*#!P<+:!G"$EWA
"$'1]\#;=Pk∗(x) 6= Pk∗(y)
T!-Pk∗(x) ∩ Pk∗(y) 6= ∅
]J'('@EPk∗(z) ⊂ Pk∗(x)
]\JPk∗(x) ∩ Pk∗(y) = ∅
]#'('@Eek∗(z) < ek∗(x) = ek∗(y)
T,:E3<C%$=_%2%$3 1%$*
E(z)G%g!'&;E y '(#*Q'(!G%$ U]<Q!*&
E(x)]%$%$G%$*&3<C*#/+**J'1]
\#;=xQ%WR3<+;C%$G%$<C%T
yµ× U× =<>B¥G7:HñJý! lU%$;=@%$*
N = 1, 2, 3, 4 k%$'(!*&3w'mHj: !#R3:"$'&9
v(S) =
1,\J
S = N,
3
4,\J
S = 1, 2, 1, 3, 1, 4, 2, 3, 4,0,
%$;=&"$ '@"$*(9T463'I<C%$;#]\#;=MtR3:"$'D<+;7R3XwE%29ñJç! ¾ T?"2C¿gT)R3:"$'&3*J's* ¾ e ¾ J'@%232=(5RW%]? Á T A^*J'd0)%2/g!;%$;=iRW UT-:E22'D<C%$;ICf&\J0G%$=&AÄ"$E3"$'&%$'@%$9
¾¾ È ëÈ Ëânâ<ÏL@àßÞB ÉÊ@ÞËâ<ËyÊ@Þ$&Ñ ®W7Þ:ÐÌXW&@à$&ã6ÌWÌXW&@áøfZ! >?;=%$j/ä ¾ TJR3:"$'@-0E J:.<C%$;@37 J/g*!%$;=s37:<+:E6a*<+;&: J*v"$EWA"$'(%$G%$*s:E3;=(^]k<C%$=@%$H'@"$r&%$@NS¿gTU"2 Á TR3:"$'@t0E J:h<C%$;@39%$E%$2*!^]k'(!'37:<+:E6:E3;=d"$E3"$'&% ¾ T<-:E22'd<C%$;6'E'(%$E3!G323!'(#Xw;;@"$*(=(9]"2Cf&\J0G%$=&AÄ"$E3"$'&%$'@%$9
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ν(1) = 1ν(2) = ν(3) = 0ν(1, 2) = 5ν(1, 3) = 4ν(2, 3) = 3ν(1, 2, 3) = 15ÉS%$*(3!U'E'(%$E3!G323!'(#w;;&"$*&=( 1Nnj:32!'1]\#;=6-R3:"$'&'mQ^*J;=('@@LPA
K$!G9ìR3:E3*J'6QG%$;7RW J*UT)>?'('&EtQ*(=@%$E%$<C"$*(=DS%$G32:QaE3"$*\#%$-39!G/#E3!G3$A3!'(#*U]
(1/3, 1/3, 1/3)E:*(=( J*U](/%G%$\#%$9]&\#;=C%$2O%$;=@%2nR3:"$'&'X<C"$:*(=(7A
j**J'DE3SR3:'1T46!G"$E3M- ¾ Tn"2 Á TU'@@LGK$!GQ'&9]k'('@E ¾ TU0)%$*#K$!j:!G_*(=@%$E%$<C"$*(=@% Á A^<+^]U<QL; Á T A"m¿7A^@%$*#N39T?>?'('&E9]k<Q!G%$GN3+%$'@%$9/%$*#"$*#%$'v322'&&/g*!^]oRWa*¿gT?"2_S%$jR3:* R3 ¾ T A*#%$'iQ'&@LGK$!G+'&a"239]J\!G39%$*%$'('@Et<Q!*#/C'@%$3%$*IRW )*IR3:E3*J'1T)>?'('@E Á TJuRES%$jR3:* R3XK23j'&7::9]\!G39%$*6%$'('@E _013Gj;@*(=@%$E%$<C"$*(= ¾ ÈX%$;=g"$;+33 UT\J:E<xR3:"$'&.:E3;=(j:3!10)2LGK$!GR3:<C"$ESRw'DH'@@LGK$!G\#rK23j'&7:O'0gA
K2:*MS%$<C%BEwGNQ!;@"$**(=@%$^T#`RW:"B'&'I'@a9a3 Á x Æ '&@LGK$!G+j'(\J '(!nXK23j7A'&9: E3E%$*#/RW"$!G y ;=&%$G%$<H 1%&"$@%b¾ ¿ Á¾rÁ ¿¿ ¾HÁ¿ ÁM¾ÁD¾ ¿Á ¿ ¾
¾¾ ¿ |Ȥ8®gßÏgáwW¯`@à@ÞËk%$;=(w'+S%$^]\#;=M%$9%$'DG"$3E%WRWa3"$*#%$'D<C%$;'(LG"$E3G"2%H%$;=@%$*GNQG32L*(5#"$;5UTJ Æ
%2%$3 )N ¾ T'@"$39%$E!*#/gLSR3QX'@@LGK$!G'&a"23!oS=&<+9]#%$'('&E9b
ν(1)− ν(∅) = 1 X*&=&%$E%$<C"$*&=&%TcJ=(3DI'@@LGK$!G'&a"23!S=&<+7]UQ%$GNM%2%$3 )%$*s¿gT10E J:hK23j'@2*!
_'&@LGK$!G\#@b
ν(1, 2)− ν(2) = 5− 0 = 5 C¿gTR3:"$'&O<+7g!<+:!G\J32*JgT- Á T#K$G'&2!'1]J'('@E9b
ν(1, 3)− ν(3) = 4− 0 = 4 Á TR3:"$'&O<+7g!<+:!G\J32*JgTÉ*J;=('@@LGK$!GQ'(!jj'(j:Q%2%$"$*m%$GN i Á TR3:"$'&K23j'&2!'M 1, 2 '&ALGK$!G\#
ν(1, 2, 3)− ν(3) = 15− 0 1, 2 G%$\#%$"$;@%2<+7g!<+:!G\J32*JgT-*G@*
ν(1, 3, 2)− ν(2) = 15− 0 1, 3 G%$\#%$"$;@%2<+7g!<+:!G\J32*JgTûS%$*&3!_;@*#/j3<C%$*J%$%$a !-R3:"$'&3E!G6&"$;!;`'&a9&%$3@%s'(!G37:<QL3\JSR' a39%2?R3:"$'@:E3\JQ'&a9&%$%$G"2"B7T>?9%$'m_'&a9&%$3'@%$9N'1b ¾ TR3:"$'@9b
1
3
[ν(1)−ν(∅)
]+1
6
[ν(1, 2)−ν(2)
]+1
6
[ν(1, 3)−ν(3)
]+1
3
[ν(1, 2, 3)−ν(2, 3)
]=
=1
3[1− 0] + 1
6[5− 0] + 1
6[4− 0] + 1
3[15− 3] = 35
6.
¿gTRW:"B'&9b1
3
[ν(2)−ν(∅)
]+1
6
[ν(1, 2)−ν(1)
]+1
6
[ν(2, 3)−ν(3)
]+1
3
[ν(1, 2, 3)−ν(1, 3)
]=
=1
3[0− 0] + 1
6[5− 1] + 1
6[3− 0] + 1
3[15− 4] = 29
6.
Á TRW:"B'&9b1
3
[ν(3)−ν(∅)
]+1
6
[ν(1, 3)−ν(1)
]+1
6
[ν(2, 3)−ν(2)
]+1
3
[ν(1, 2, 3)−ν(1, 2)
]=
=1
3[0− 0] + 1
6[4− 1] + 1
6[3− 0] + 1
3[15− 5] = 26
6.
¾¾9Á
>?'('@E (356
,29
6,26
6
) %$G32:3ttR3:"$'mf&\)0G%B=&AÄ"$E3"B'&"$*#%$'M*#%$@%$22w'1T
SÌg] ¾ OQP!RLS(TU(VWS(XZYUlU%$;=&%$* N (N) NA^*"BE%$P<C%$9%$3Ha39%2-G%$\#%$"$;@%2-&"$;@%2OR3:7A
"$'(\#rE%$*#/%$'E'(%$E3!G323!'(#w;;@"$*(=@%$'I\J<+7gTSÌg] ¿OQP!RLS(TU(VWS(XZY N (N) \J<+9*X"$E3%$<C%$9%$3 ϕ(ν) = [ϕ1(ν), . . . ϕn(ν)
] w;;A&"B*(=(f&\J0G%$=+%$G32:3*J'*jw;;@"$*(=(*#%$'!?*#%$&%$22w'1]&\JH%$ RW%23w*#%$'Dj: !)S%$PA"$%$G%$'1b¾ T tml(dQdbm]zjloh ∀ π N −→ N
0)%$E3<_:K$!GEC"2 ∀ ν ∈ N (N) A^E%]\J
ν[π(c)
]= ν(M) ∀ M ∈ C, '('&E
ϕπ(i)(ν) = ϕi(ν) ∀ i ∈ N
¿gTr`dQdQ8m\^]`_@apnxq&-_j<C%$=i ∈ N
R3:"$'&3Egb
ν(M) = ν(M − i
)+ ν(i) ∀ M ∈ C, i ∈M,
'('@Eϕi(ν) = ν(i)
Á TgRgF3Fplz]zl:Olz]\xqf ∀ ν1, ν2 ∈ J(N)A^E%b
ϕ(ν1, ν2) = ϕ(ν1) + ϕ(ν2).
SÌg] ¾ 6Í b|fbo|pt_@qkX/% y *LGK$!GQ'@a7@%$%$<C"$*(=@%$!tHj: !j'M32%$E3!*("$E3%$<C%$2\#%$SRw'1b¾ TO\Jd'@"$XR3:"$'&D*#%$<Ú'(wGa*& 1a2%$3\#%$Nv<C%$;6R3:"$'( J*U]'('@EM"$E3"$'&%$!'!G;=(*J9'1]
¿gTO\JiR3:"$'@.<Q!*#/%$*D'@@LGK$!G_*&=&%$E%2"$;@"$K2'
ν(i) A^&%$)3#/R3r*#a7@%$*!^]'&A
'&E"$E3"$'@%!GG%$;=&%$*m%$@]Á TO\Jt'@"$%$;=(<+:3(w;;@%$3G%$*(wG%WR3:393oR3:"$'&<Q!*(%$;= R3:"$'@?%$'(!*&3w*'1]'('&E r"$E3"$'DG%$;=@%$*i//g!3L1T
SÌg] ¾ gdf=)P!YR N (N) aphpjhxax]`bjlrqBtf]zlra,`q<@^||6Op_g!b@aabch3Fn^]o])m\^]`_@aTDLhpc dhtnpg u$npgE]nxqfhpg1bo|pϕce_m]`bBtmlraTHGhpdbc albce_o|6Jz]zl GE¡h^]G_@qQbBtQhya6P6Opbm](a^b@tvhpcwhxa®!
ϕi(ν) =∑
M⊆N
(|M | − 1
)!(n− |M |
)!
n!
(ν(M)− ν
(M − i
))(i ∈ N).
¾¾ Ç |Ȥ8®gßÏgáwW¯`@à@ÞËI letnpg!6Jz]\xqZ^f&9"$0OehcJE3;@Dj0R3:*: !9*&=(L:3ta iG"$0)"23 1%$*i@"$;@%$22w'1T)46!*gA/%$*#%$'&%BGN3
N<Q!*#/%$*6*J%B<wE%2
LE"232\J<+7:\#_\#27:E%$*#/%$w#*'6%$;=QR3:3'@-
'&a9&%$3'@%$9N'&"$00)%$*Ub
νL(M) =
1,
\JL ⊆M
0,\J
L 6⊆M . L;=q"$E3%$<C%$9%$3νLw;;@"$*(=`'(!G%$G"$;L3!_sÌgT ¿gT % y *LGK$!G ¾ T"2m¿gTOS%$"$%$G"$9T ;=(*!G
νL(∅) = 0]7: J:g]#\J
M ∩ P = ∅ (P ⊆ N)]J'('&E
νL(M) + νL(P ) ≤νL(M ∪ P )
T£Ä;=νL(M)
G J*dR3:"$'@I"$E3%$<C%$@T. 6L;="$E3%$<C%$9%B_R3:"$'&Ea7(!G/%$*
νLR3:"$'(*J'I*#%$@%$22w'1T
<Q!*#/%$*ν ∈ N (N) R3:"$'Q%$GN@:L3\J νL
R3:"$'@'X!*#%9:E3!Gh'&<_ !*J:K$!RW'&"$*(
ν =∑
L⊆N
λLνL ,
\#λL = λL(ν) =
∑
R⊆L
(−1)|L|−|R|ν(R) T>?**#%$'Y )%$j:::\#6G%$;=@%$*
M ⊆ N%$39NG%$;@%29T>?'('@E
λL/3+"$E3"$'@"$*#%$'
S%$\J32*J:j::2∑
∅6=L⊆N
λLνL(M) =∑
L⊆M
λL =∑
L⊆M
∑
R⊆L
(−1)|L|−|R|ν(R)
=∑
R⊆M
∑
R⊆L⊆M
(−1)|L|−|R|ν(R)
=
∑
R⊆M
|M |∑
i=|R|
(−1)i−|R|(|M |−|R|i−|R|
) ν(R)
=∑
R⊂M
(1− 1)|M |−|R| · ν(R) + ν(M) = ν(M) ,
\#U/% y *LGK$!G+39%BE!P*(λ∅ = 0
"$E3"$'('&%$U37:<C3*'1T>?29%$kHO:L:3!;&93'1T :OÄ;&9*!oS;7R'M
ϕ(cνL) (c ≥ 0)w;;&"$*&=M%$;=@"$E3%$<_5#"$;@"$9T
lU%$;=&%$*ϕ%$;= N (N)AÄ%$*q"$E3%$<C%$9%$3M"$E3"$'&w;;&"$*&=&]h9: J: ν ∈ N (N)%$39NG%$;@%29T
H.3jRW/*#:;Ij0R3:*ν =
∑
L⊆N
λLνL .
46!*(3\#;=νL(∈ N (N)
) %B;=iR3:"$'1]?%$9"$E3λLνL
!GC@]\JλL ≥ 0
T?>? 1N'&a9&%$3'@%$2!'1]g\#;=
νS%$LE3\JQ'&"$?R3:"$'I'(wGa*( #"$;@%$'@"$*(
ν =∑
λL>0
λLνL −∑
λL<0
(−λL)νL = ν+ − ν−
¾¾9Ã
"2L;=M Á TgS%$"$%$n39%$E3!*&ϕ(ν+) = ϕ(ν) + ϕ(ν−)
<Q!j3
ϕ(ν) = ϕ(ν+)− ϕ(ν−) .
Á Tg%$"$%$U!G3<C"$%$t'<+7::9k/(/g!'1]g\J;=
ϕ(ν) =∑
λL>0
ϕ(λLνL)−∑
λL<0
ϕ(−λLνL).
lU%$;=&%$*i /∈L %$39NG%$;@%29]/%tEa;2L%$3%$G%$<C% N A^*#%$'1Tg>?'('@E cνL(i)=0 (c ≥
0) νL(M)j0R3:*UT!-
M ⊆ N%$39NG%B;@%2t"2
i ∈M]J'('@E*&=(!:*
L ⊆MA^ 1N
L ⊆M −(i) '@a7@%$3'&%$2!'1]"2L;=
cνL(M) = cνL(M − i
)= cνL
(M − i
)+ cνL
(i)
S%$**J:o<Q!*#/%$*MA^E%]#<C%$=(E%
i ∈MTg>?7:kX¿gT&S%$"$%$U39%$E3!*(<Q!*#/%$*
MA^E%]
<C%$=(!'IE3BP<+27iA^"$E3&"$*&=&%2
ϕi(cνL) = cνL(i)= 0
%$;=@%$*GN"$;#]J"2%$r"$E3&"$*&=&%2<Q!*#/%$*iA^E%]J<C%$=(E%
i /∈ LT
lU%$;=&%$*j, ` ∈ L
'&"$%$39NG%$;@%2R3:"$'&LA^ 1%$*UT
ú =(!:*(G@*vG"$%$2!'NA^*#%$'Y=&*
π0)%$E3<_:K$!GR3g].<C%$=(E%
π(L) = L"2
π(j) = `T&>?'('@Eh J:E3<C%$=
M ⊆ N%2%$"$*
L ⊆ π(M)'('@E."2hK2'+'('@EO:#S%$**U]
\JL ⊆ M
Tk>?7:cνL(π(M)
)= cνL(M)
S%$**J:O<Q!*#/%$*M ⊆ N
A^E%Tk ¾ TS%$"$%$39%$E3!*(-%$'('&E
ϕj(cνL) = ϕ`(cνL)T)>?C2.RW%$G%$*(3!^]J\#;=m J:E3<C%$=
j ∈ LA^E%
ϕj(cνL)"$E3"$'@%H;=&*JQC37:<m]#G%$;=&%$*i%$
αTJ46!*&3\#;= [
ϕ1(cνL), . . . , ϕn(cνL)]
cνL
R3:"$'D%$;=D%$G32:X'@%$UG%$;=@%$*U]#%$9"$E3n∑
i=1
ϕi(cνL) = cνL(N) = c = |L|α ,
7α =
c
|L|T>?29%$kX :O:L:3!G!;&93'1T
K?>? )%$*M01*&3 J*Mϕw;;@"$*(=Q%$;=&"$E3%$<H5#"$;@"$O !9*&=(LSR'1T&q !9*&=(L:
+%$GN9N&%$'sj0R3:*<+:EHK2'%$;=g39%$E35d;@%$ E!O:j'(L:'H!P;@"$*(=&%$STnc#%$\J3$A*J: R'I7: J:C_ !*#<%$;=(w33\J'M'@a7@%$3'&%$9NQa39%$;'@"$0G%$"$9b
n∑
j=t
1
j(−1)j−t
(n− tj − t
)=(n− t)!(t− 1)!
n!, (1 ≤ t ≤ n) .
f&7:<QL3'D'(!ϕi(ν)
"$E3"$'@"$(i ∈ N)
S%$\J32*J:X :^A^9T
¾¾9Æ |Ȥ8®gßÏgáwW¯`@à@ÞËk%$39NG%$;@%2
i ∈ NA^E%-S%$**J:
ϕi(ν) =∑
λL>0
λL
|L| −∑
λL<0
−λL
|L| =∑
i∈L
λL
|L| =∑
i∈L
1
|L|∑
R⊆L
(−1)|L|−|R|ν(R) =
=∑
i∈R⊆N
∑
R⊆L⊆N
1
|L|(−1)|L|−|R|
ν(R)+
∑
i/∈R⊆N
∑
R∪(i)⊆L⊆N
1
|L|(−1)|L|−|R|
ν(R)
=∑
i∈R⊆N
n∑
j=|R|
1
j(−1)j−|R|
(n− |R|j − |R|
) ν(R)
+∑
i/∈R⊆N
n∑
j=|R|+1
1
j(−1)j−|R|
(n− |R| − 1j − |R| − 1
) ν(R)
=∑
i∈R⊆N
(n− |R|
)!(|R| − 1
)!
n!ν(R)−
∑
i/∈R⊆N
(n− |R| − 1)!|R|n!
ν(R)
=∑
i∈R⊆N
[(n− |R|
)!(|R| − 1
)!
n!ν(R)−
(n− |R|
)!(|R| − 1
)!
n!ν(R− i
)]
=∑
i∈R⊆N
(n− |R|
)!(|R| − 1
)!
n!
[ν(R)− ν
(R− i
)].
>?29%$kϕi]!G%$3@%H
ϕw;;@"$*(=D%$;=@"$E3%B<_5#"$;@"$ !;&93'1T
/tÍ-"$;w!;&9 R'1]o\#;=ϕ'(!G%$G"$;L3!O+"$E3"$'&w;;@"$*(=iS%$"$%$G%$!9Tn>?GN39aEr!G
)%'&%$oj:3*&*'1]\#;= [ϕ1(ν), . . . , ϕn(ν)
] νR3:"$'D%$G32:gT
ϕi(ν) =∑
M⊆N
(|M | − 1
)!(n− |M |
)!
n!
[ν(M)− ν
(M − i
)]
≥∑
M⊆N
(|M | − 1
)!(n− |M |
)!
n!ν(i)= ν
(i) n∑
m=1
(m− 1)!(n−m)!
n!
(n−1m−1
)
= ν(i)n1
n= ν
(i).
k9: J:Q%$;=D%$G32:'@<Q01*#%$*#%$!*#%$'Ma39%$;@%'(!o'@%$n/R3Cν(N)
"$E3"B'&"$9Tn∑
i=1
ϕi(ν) =n∑
i=1
∑
i∈M⊆N
λM
|M | =∑
∅6=M⊆N
∑
i∈M
λM
|M | =∑
∅6=M⊆N
λM =
=∑
∅6=M⊆N
λMνM (N) = ν(N) .
|È ÇmÈ®gß@ÞßUá$âã ¾¾ Ê
GQ'@"$G"$01"23 )%$*νL(M)
/#% y *LGKB!GR3C"2νS%$ )*(:X'@%$E3wS%$\J32*J:j:3EgT
zt:"$E3w*'_ ¾ T9S%B"$%$E%TlU%$;=&%$*π
N\J<+O%$;=r0)%$E3<_:K$!GR3g]<C%$=( 1%$*
ν(π(M)
)= ν(M)
S%$**J:U<Q!*#/%$*M ⊆ N
A^E%T>?'('&E
ϕπ(i)(ν) =∑
M⊆N
(|M | − 1
)!(n− |M |
)!
n!
[ν(M)− ν
(M −
π(i)
)]=
=∑
M⊆N
(|M | − 1
)!(n− |M |
)!
n![ν(π(M)− ν
(π(M − i
))] =
∑
M⊆N
(|M | − 1
)!(n− |M |
)!
n!
[ν(M)− ν
(M − i
)]= ϕi(ν) .
¿gTS%$"$%$J%$ RW%23wG"2%ϕπ(i)(ν)
/% y *LGK$!GR3_j0R3:*+j:3\Jr 1%];=(*!Gh%$'('&Er%$;=&%$*GN3G%$*#"$;@%$'M\#%$=@%$3<Q!*#/%$*&w %$;=&%$*GN"$;+"$E3@"$*(=@%29TÍ-"$;w Á TJS%$"$%$?%$ RW%23wG"2%_'&a2@%$3G%$*(w
ϕi(ν)%Bg0!GK$!Hj'R3: )/#&/g!'1T
>?29%$kX"$%$%$ RW%2%$*M!;&93'1T
zµ×@Ö6× G7:6>B8?xf&\J0G%$=&AÄ"$E3"$'D;=&'&E3!n'<+7:3!U%$E3wG%$%$!k_\J<Q!U!*#/%B%$'D37:<QL:g]0)A!3!'!^]!G%$3@%r3799Q%233wG%$%$'( )%$*UbÉ\J<Q!U!*#/%Bm/% y *!j:j:'@EtK2'M_;=@N2%2'@@LGK$!G*J'D3jRW/*L3*'m"$EWA
"$'&%B7Tv;=&N2%2'@@LGK$!G'H'&a2wg01%$/!P;H <Q!*!<+:!G?'&@PLGK$!G'?E3SR'_39%$<x%$GN39T>?GN39aE(!93;&: R'MH%$;=g39%$E35m3797:3!R3:"$'&'746!*#/%$*6%$;=&%$R3:"$'@E%$*#/%$'@%$29%$*9*# 3797:3!U%$EN9&%$^£Ä;=
ν(s) = 0\J
s ≤ mν(s) = 1
\Js ≥ m
>?;=X'@@LGK$!GG"$3E%WRWa3"$*#"$J(!93;&: R'Q2.t\#%$=(9%$%$9](<Q!'@Ej<C%$=-R3:"$'@K23j'@7::9U:!'I*&=&%$E3%2"rH/'@@LGK$!GTSÌg] Á 6OQP!RLS(TU(VWS(XZY)>?;=XR3:"$'&30!@R3:"$'&3*J'I*#%$@%$2w*'1]#\JQX'(E*#G;!'(#* GHK23j'@9H%$;=Q<Q!*!<+:!G.;=@N2%2h'@@LGK$!G\#@T(F@%$Ga RW%
p-0!&nR3:"$'@WA
:;@9TSÌg] Ç6OQP!RLS(TU(VWS(XZY)xf&\J0G%$=&egf&\& !'M!*#/%B%$*i_'&a9&%$3'&%B9NQ'(!PS%WRW%$9"23"$E3SRw'1b
ϕi =pin!
¾¾ Ë |Ȥ8®gßÏgáwW¯`@à@ÞË\#
pi9*iE3E%$*#/%$'!.0)%$E3<_:K$!G'm37:<+g]J\#
i0!@R3:"$'@9T
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<Q!*#/%$*6%$GNSE/gGI%$E3*J3LX\J:E3<+3EQu;=I;=(*'I'&"$0)%2%$'D/a*(%$*!^]\#;=D\J%$;=(!'&%B1RW *J'HBG: R'Q<+:3!'(*J:^]@ <+:3!'\JE<+/g!'(*J:^]@'('@E?32w'g"$;A'&"B0#0)%$*m-%$GNRW *J'Cj: R'IH\JE3<+/g!'(*J:^]#7-%$ R3:E:3*'Á]zjhpgLth^]J:O7]g7:j32*'jBbgBFbBtmg!lr3#/gm%$E3*J3L:'9]'('&E_ !9*(=@*vG"$%$2!P'dG%$;7RW %$G%$<m]J7r:j32*'D%$U3#/R3_<+RW/m/a*&%$*!^]#\#;=I'(!E%r37229*UT46!*(C66S%WRW%$9%$Q'@a7@%$3'&%$9N6S%$G"$ 1N'(!G/%$E3w^]?%$**#"$;=@%$*;@"$ vS%$"$%$!GQ%$G%BA
;@%$*#/NTR8tQb|p_g!lWuj@bfbjBbg:kmlo\xaya^bcebm](a^b@t_@qb46!g*#%$< ;j'&2*'Q%$;=@"$*!0E%BS%$E%$*#K$!j:'X%$E%2/%$"$@%$^]&/3:;*J'_'@%$9%$ Rw'
N'&%B7]*#%$<V(!93;&: R'1]\#;=Q37:<C39%$<Q0)*(\#;=(*Qw#;;ra39%O'(3E:!G9]PAj:3!("2k%$;=&"$ H3E/gLGK$!G'('g33 UTf&7:<H*'(EtO%$;=@"$*Q9*_'@"$01%2"$;@%]9\#;=_.%$GN3%<C%$;*(=(LPGr%$E3*J3L:'j<Q!=&%$*Q"$E3%$G%$<H 1%$*Q'&"$0)%2a39%$\J*L*!^]@/3:;#T"2N M-%$;=@"$*#%$'(ENo2.!GS%$%232w'1]@\#;=C'@"$01%2%B'C<+:.%$;=&"B*I\#%$=(9%$"$ 1% '&"$0gA9%$*!J<+;'9]&"2.2.-\#%$=(9%$%$.R3:SR'('#a39%$\J*L*!^]&!G%$3@%<+:h%$<H 1%$E\#%$=(9%$"$@%$na39%$<C"$E3*!U%$;=I:j32*#/+%$E3*J3LX )%$'&a9&%$3'@%$9"2%%2%$"$*UT&4i%$;ARW%$;=@%$22w'1]\#;=D%$r<+RW/D<Q!*#/%$*6'@*#33E3'(3!(!G3Q%$3!'Qj0gS%$%$@"2%T Ö6×@Ö6× © =_^69<5¥« Ú¥ÙW=<Þf=IÙW=<5¥ßC«BFH bgBFbBt_@qb@a ¾ ] ¾ QOQP!RLS(TU(VWS(XZYh
X\J<+9*v"$E3%$<C%$9%$3Q !*J:E3!GXE%$j:K$!G*v%$;=
R ⊂ X×X\J<+2 "$E3w*'1T ¾ ] ¿6OQP!RLS(TU(VWS(XZY)"$E3%$<C%$22w'
RA^E%HHj: !n3jRW/*#:;@'9b
¾ Tg%$ RW%2"$;#b ∀ x, y ∈ X x 6= y −→ (x, y) ∈ R;=
(y, x) ∈ R¿gTgE%KJ%Bg!(!:9b ∀ x ∈ X : (x, x) ∈ RÁ Tg!E3E%KJ%Bg!(!:9b ∀ x ∈ X : (x, x) 6∈ RÇ#Tg3E*2!3!(!:9b ∀ x, y, z ∈ X : [(x, y) ∈ R ∧ (y, z) ∈ R] −→ (x, z) ∈ RÃ T32!<Q<C%$3E3!jgb ∀ x, y ∈ X : (x, y) ∈ R −→ (y, x) ∈ RÆ T#32!<Q<C%$3E3!jgb ∀ x, y ∈ X : (x, y) ∈ R −→ (y, x) 6∈ RÊ(T#*(3!G32!<Q<C%$3E3!jgb ∀ x, y ∈ X : (xRy ∧ yRx) −→ x = y.
¾ ¿Ë ÇmÈ Ïgà$Òp@à$ÌÓ!Ñ ÊËÏBßUà$7Þ)â"¯`@ár&Ë ¾ ] Á 6OQP!RLS(TU(VWS(XZY
R32!<Q<C%$3E3!'(#E"239%
I(R);=IEa9(!G/%$*
I]#X'@a7@%$3'&%$9N
I(R) =(x, y) ∈ X ×X | (x, y) ∈ R ∧ (y, x) ∈ R
R32!<Q<C%$3E3!'(#E"239%
P (R);=IEa9(!G/%$*
P]#X'@a7@%$3'&%$9N
P (R) =(x, y) ∈ X ×X | (x, y) ∈ R ∧ (y, x) /∈ R
.
ËgTg'(:2!o3E*2!(!:9b ∀ x, y, z ∈ X :[(x, y)∈P (R) ∧ (y, z)∈P (R)
]−→
(x, z) ∈ P (R)ÌgT#K$!'(#:;#b ∀ t ∈ N ∀ x1, . . . , xt ∈ X :[∀ τ ∈ 1, . . . , t− 1 :
: (xτ , xτ+1) ∈ P (R)]−→ (xt, x1) /∈ P (R)¾ ÈgTg\):E3<+ K$!'(#:;#b ∀ x, y, z∈X :
[(x, y)∈P (R) ∧ (y, z)∈P (R)
]−→
(z, x) /∈ P (R)¾¾ Tg'&*2!G32%$*#K$!jgb ∀ t ∈ N ∀ x1, . . . , xt ∈ X
[(x1, x2) ∈ P (R) ∧ ∀ τ ∈ 2, 3, . . . , t− 1 : (xτ , xτ+1) ∈ R
]−→ (xt, x1) /∈ R.
¾ ] ÇsOQP!RLS(TU(VWS(XZYUz %$*#/%$9"23*#%$'6*#%$@%$22w'2rR ⊂ X ×X
A^9])<C%$=(E% ¾ B]<S¿B]ÇS7r%$ RW%29]gE%KJ%BgLD"23E*2!3L!BT ¾ ] Ã OQP!RLS(TU(VWS(XZY1h:2!nE%$*#/%$9"23*#%$'m*#%$&%$22w'62
R ⊂ X ×XA^9]J<C%$=(E%¤S¿
"2ÇsS7rE%KJ%BgLM"23E*2!3L!39':<C"$;Q;=&%$*;@%0E%BS%$E%$*#K$!jXE%$j:K$!G*J'D*#%BA&%B2*!(BT ¾ ] Æ 6OQP!RLS(TU(VWS(XZY
R ⊂ X ×XA^ _'&a9&%$3'@%$9N'&"$00)%$*m*#%$&%$22w'1]\J
1) (2), (4), (7)E"232 1%$*mE%$*#/%$9"2
2) (2), (4), (7), (1)j:*#K
3) (4), (6)%$ENE"232 )%$*mE%$*#/%$9"2
4) (4), (6), (1)%$ENE%$*#/%$9"2
5) (2), (4), (5)%$'((!G%$*#K$!jgT
¾ ] ¾ <;QU1=>@?Y!( ]zjBhpgLtl:Olz]\xq −→ apnpgLtmlrqBtf]`bg:kmlohH<a,Ox\tml]zjhpgLtml:Olz]\xqHZh^kfleaxc lokmlz]\qHDL\pjdhxqhkmlrac `qfnxq@qf\|!!( apnpgLtmlrqBtf]`bg:kmloh −→ hkmlrac `qfnxq@qf\|HNDL\pjdhxq¥hkmlrac `qfnxq@qf\|!!!( a,Ox\tml]zjhpgLtml:Olz]\xq −→ hkmlrac `qfnxq@qf\|HNDL\pjdhxqhkmlrac `qfnxq@qf\|!! hkmlrac `qfnxq@qf\| −→ DL\pjdhxq¥hkmlrac `qfnxq@qf\|
¾ ] ¿<;QU1=>@?YRuLnpg]nxqfhpg hxaapnpj !( qBtml(dQdbm]zjlra,`qH_DLh
R = I(R)
ÇmÈ ÎÈiãßf&Ñ ãÊUáÑ Q@á&ã6Ë ¾ ¿Ì
!!(hxqBtml(dQdbm]zjlra,`qH_DLhR = P (R)
T ¾ ] Ê^6OQP!RLS(TU(VWS(XZY)/3
R1, R2]#'('&E
R1R2_'&"$E%$j:K$!GQ'&<Q0)2LGK$!GR3gb
R1R2 = (x, y) ∈ X ×X] ∃ z ∈ X; (x, z) ∈ R1 ∧ (z, y) ∈ R2.
¾ ] Ë¥OQP!RLS(TU(VWS(XZY/3 (R(τ)
)∞τ=1
R(1) = R Rτ = RR(τ−1) (τ ≥ 2)]k'('&E
T (R) =∞⋃
τ=1
R(τ)A^
R3E*2!3LDG%$7:E::*J'D*#%$&%$22w'1T
¾ ] Á <;QU1=>@?Y!R8tQbcwvtvb@aQBbmgFbz¨<g!lo\pcì]TcebBt\pj\xq¥nu!bj\^]npjHGhtht
!(T (R) ⊃ R!!(R1 ⊃ R2 −→ T (R1) ⊃ T (R2)!!!(T[T (R)
]= T (R)!!
T (∅) = ∅. ¾ ] Ç<;QU1=>@?Y
RQl(g:\xjmleqj@bcw\kmlo~µuLnpg]nxqfhpg hxaapnpj¥]zjhpgLtmlz]J:OwH_DLh
R = T (R)T
¾ ] ¾ <df=)P!YT (R)
hcebo|xalrqbTQTQREø]]hpjm]hpc dht~H]zjBhpgLtmlz]J:O$DLhxcìdht
Ö6×øØ× ó ŵ«@?AF6>B«B;.=<>B=?A=IH ¾ ] Ì6OQP!RLS(TU(VWS(XZY1/3
R ⊂ X ×XG%$;=@%$*
S ⊆ X
µR(S) = x ∈ X : ∀ y ∈ S : (x, y) ∈ RσR(S) =
x ∈ µR(S) : ∀ y ∈ µR(S) : (y, x) ∈ R
.
G(S,R) = S ∩ µR(S) = x ∈ S | ∀ y ∈ S : (x, y) ∈ R IG%$;7RW s%$G%$<C%$'\J<+7gTM(S,R) =
x ∈ S | ∀ y ∈ S : (y, x) /∈ P (R)
t<+7g!<+:!Gh%$G%$<C%$'X\J<+7gT ¾ ] ¾ 8Í bo|fb|pt_@qn39':
G(S,R) = C(S,R)A^*#%$'I!G?RW%$Ga*!^T
¾ ] ¿<df=)P!Y!(
S ∩ σR(S) = G(S,R)!!(M(S,R) ⊃ G(S,R)
"2M(S,R) = G(S,R)
DLhR]`bc fb@qQ_@qj@b b6J:Op
¾ ] Á df=)P!YDLhR ⊂ X × X
hkmlrac lra,`q*_@qS ⊂ X
HS 6= ∅ |S| < ∞ H¥hxaapnpj
M(S,R) 6= ∅ T
¾9Á È ÇmÈ Ïgà$Òp@à$ÌÓ!Ñ ÊËÏBßUà$7Þ)â"¯`@ár&Ë ¾ ] Ã ¡;QU1=>@?Y
G(S,R) 6= ∅ ]8DLh |S| <∞ S 6= ∅ _@q R ]`bc fb@qHjBb b6J:O_@q8hkmlrac `qfnxq ¾ ] Æ <;QU1=>@?Y ∀ R ⊂ X ×X |X| <∞A^E% M(X,R) 6= ∅ T
¾ ] Ê^<;QU1=>@?YDLhRa,Ox\tmlj@bgBFbBt_@q
C(S,R) 6= ∅ HZhxaapnpj C(S,R) =M(S,R)
¾ ] Ë<;QU1=>@?YDLhRa,Ox\tmlj@bgBFbBt_@q
SE`g |S| <∞ HZhxaapnpj
∀ x, y : [x, y ∈M(S,R) −→ xIy]←→[C(S,R) =M(S,R)
].
¾ ] ¾ È6OQP!RLS(TU(VWS(XZYR2A^
R1'(!%$ESRW%232"2"$*#%$'I*#%$&%$22w'1]\J
R1 ⊂ R2 ∧ P (R1) ⊂ P (R2).
¾ ] ¾¾ OQP!RLS(TU(VWS(XZYR1"2
R2'&<Q0J3! !!G%B'C%$;=(<+:^]\J-;=
R2, R1'(!%$ESRW%23$A
"2%]g;=+SE/gL3gT
¾ ] ¾ ¿OQP!RLS(TU(VWSrXZY(S,R)
!*#/g'(3L1]o\J ∀ A ⊂ SA^E%]
A 6= ∅ ]o<C%$=@%$* R%$ RW%2r"2
E%KJ%Bg! ∃ R30E%$*(<
SA^ )%$*MeI7
S ∩ σR(A) 6= ∅T
!#"%$ TÁPLML6ZY S bo|ppa¡fbc HBDLn@|pRa,Ox\tml(jBbgBFbBt_@q_@q
(S,R)l(gBF6`ax]J:OwH&b@aapnpj ∃ R
E`jBbdhTpl(d\pc lrq¥bcebdSEQBbgW
& S'>C@?)(=>@?S+*CS(X¡L6ZY@G%$;=&%$*F*#%$<VwE%2h/g!G33R*'(?\J<+9'X\J<+7gT&>?'('&E
∃ A\J<+@]<C%$=D0)*(*6%$;=D%$G%$<C%$E<+
F<Q!*#/#%B*i%$G%$<C"$ )N^T
, XC $ P!T&P-(?SP!.'ZY1lU%$;=@%$*S%$;=M\J<+@T)>?'('@E ∃ %$;=M !*J:E3!G E%$*#/%$9"2 S AÄ%$*U]\#;=
S<Q!P*J/%$*m*#%$<ZwE%2E"232\J<+7:*J'M*M<+7g!<+:!G %$G%$<C%T
¾ ] Çgdf=)P!YR0/Cnpjg1cebdQdhHGh § l:Ox\pcwhxqBtf]\qlhTplo~pdhÂ_@qyh [~pc jBbgBFbBt_@ql6bc O b@a,Ol:EOxhpcebg$qb@a^ ¾ ] Ã <df=)P!Y ∀ R
a,Ox\tmljBbgBFbBt_@qgWb@a ∃ R∗j@bgBFbBt_@qalz]`bjzfb@qBtf]`_@qb
¾ ] Æ gdf=)P!Y<^f&2<_ER ⊂ X ×X
]RE`gWb@auLnpg]nxqfhpghxaapnpjbOxhpg1jBbgBFbBt_@qal:E]`bjzfb@qBtf]`_@qbHNDLh
RapnpgLtmlrqBtf]`bg$q
ÇmÈ -ÈÊUáßp@ÞpÊÑÒ6ÎYY&ÌXW ¾9Ág¾
Ö6×&%µ× F6>B6;7:F6;«ÂÞwÐb^^D69<5 ¾ ] ¾9Á OQP!RLS(TU(VWS(XZY
C(S,R) XA^*M"$E3%$<C%$9%$3w;;&"$*&=+u;=@]g\#;=D
C(S,R)\JPA
<+r*#%$<ZwE%2 ∀ S ⊂ X, S 6= ∅ A^E%T ¾ ] ¾ <Í bo|fb|pt_@qg<Q!*!<+:!GkS%$"$%$G%
C(S,R)G"$"$*#%$'1]%$ RW%2"$;-"2?E%KJ%Bg!(!:9T
¾ ] Ì<;QU1=>@?Y!AµhR ⊂ X ×X
j@bgBFbBt_@qHZhxaapnpj ∃ C(S,R)
¾ ] ¾ Èp;QU1=>@?YgDLhR ⊂ X × X
H |X| < ∞ jBb 6b6J:Oi_@qÁ]`bc fb@qHQhxaapnpjC(S,R)uLnpg]nxqfhpg hxaapnpj°Fbz¨<g!lo\pcì]
XE`gfH_DLh
Rhkmlrac `qfnxq
XE`gW
hkmlonpg:hpc lz]\xqÍ lhjBhkmlonpg:hpc lz]\q1À:E/#G<H3#/<+:*(=( J*!;@%$*q;=&'(E*`\J32*J:IS;&G<íEK$!G*J!:9]
<C%$=j3_IG%$;'(wGa*( )a9N ^R_RW%$G%$*&"2%$'&%$r39':r"$E3%$*!^TUV7: !j'( J*K2'QK$%$G%$'(&"2%$<C"$G%$3!JEK$!G*J!PB:?S;&<+:h'(L:*R'H\J32*J:*!^T:>?G"$;r!3hK2'47»¨%$ 1%$E9]:t%$E3 )%$E3Hf&!<C*i;=mFwE3;@%$* - )%$E3<+tEK$!G*J!:tS;&<+:Eg])E7AK$!G*J!:r"$E3%$<C%$9"$"$E%Q*!^T tF@E<+:*\J:*(=iK2%$G%$'(@"2%$<C"$G%$9]U**&=(!EK$!GA*J!:S;&G<mTJS%$*&3!n%$<C"$G%$%$'M;=&*#=(*6'( )U:<+32'&&/g*J'6_EK$!G*J!PA:3Eg]J<Q!*(-<Q!^]1Q%$<H 1%$E3!?K2%$G%$'(@"2%$k'0)K2j32w'g"$;@%2t<Q!P*#!P<+:!G-S%$%$@"2': )^Tn>?9%$*Y%$<C"$G%B%B''@"$X30)%$'(3#DG%23CS*&H37:<_*'(EgTnpS;&G<ä"$E3%$PA<C%$9"2"$*#%$'+%$;=(!'+30)%$'(3#j<Q!=&%$*I'@*2!G32%$*#K$!jr'&a7@%$%$<C"$*(=@]&EK$!G*J:!G3*J'<C*#/g*'s%$;=K2%$G%$'(&"239]n%$;=!*&"$2<C"$*(=(!.<_5'@a(/"239]n%$;=37 J:=(9]n\J6a*<+;&:7A*J'Q*#%$<[<C*#/I%$G%$*(9TWút%$<['@%$&%$E%2/g!'CK$!EK$#.(!3!G3#3 Jg]@*#%$<pSE/g1%$GNr 1%$**#%_¿¿7AÄ%2OK20)/#:R3H33 UT BT(qS;&G<[<+:3!'I30)%$'(3#g]g<C%$=(E%t32w'g"$;w*'CG%23@]gj<Q!PS"$G%X<+7g!<+!7:Gm%$1]o7+IK2%$G%$'(&N9]oQ!*&"$2<C"$*(=(9]nI37 J:=(H'('@E<C*#/R'rEK$!G*J:!G3*J'1]\Jt?RW X"2E37 _G%$\#%$N"$;t'&a2wg*#%$<E37 J:j32SR3gTS;&G<S%$*&3!o'&"$301%$'(3#:*M'(L(w1%$E3<C"239%$%2%$*_K2%$G%$'(&"2%$<C"BAG%$%$'m<C"$;D37:<C301%$'(3#:-%$G%$<Q2!'iCEK$!G*J!:3*J'1]1"2Q:*'6(!93;&:30)%$'(3J'?!G?%$E3<C"239%$%2%$*+'('#:E3*&=( J*C(!93;&: R3:'1T>?;=@%$GNE% 9*( J*37:<H#*'(EX%$G"$;+%-S;&G<Z!=@%$*632!*&35m(!93;&:jgT hkmlonpg:hpc lz]\xqHdQl(g]IapnpgLtmlrqBtf]`bg:kmloh7: !j'( J*mH'@*2!G32%$*#K$!j_'@a7@%$%$<C"$*(=(r:j32:3!o%$ R3:E:)'01K2A
j3 J*m:E3;=( R' <Q!*#/%$*67: !o*#%$\#"$9"$;+*#"$'(wnX'@a7&%B'@%$9N'M:WS;&<+2\J7A'MX:G32EB_!GBT"$09%$ Rw'i%$?C'@a7&%$3'@%$9N+\#%$=(9%$%$9]o%$E3*J3L:!*'6'(wGa*( )a9NMK20)E33L:WA
J*'@%$E3w*#%$'%$GN]o"2r%$;=&AÄ%$;=K201E33 )h'(!:j323*'%$;=@%$WAÄ%$;=&%$9]o%$E3<C"239%$%2WA*#%$'_35*!'Ct S%$%$&"29]\#;=Q-:G;&:3!J%$ R3:E:?<Ca;@"%$;=X:G;&:3!#'@*#K2%$01K$!G
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E3:3 37:*#/"$'@2*J'DG%$LE3*!Z^fg<_#%$G* ¾ ÌÇ&Ìg]J E3E9~ ¾ Ì Ã ÌBT>?GN39aEO/% y *!j: R*'I%$;=Q:j32:3!137 J:=+:)G"$3E%$\#93O0#E%B%$E%$*#K$!j:9b2
<C*#/R'1]#\#;=x%$E3*J3L
Rc"$E3%$G%$<H 1%$*i0E%BS%$E:
yA^*Jk39%$<H 1%$*U]#\JXG"$%$2!'
=&*YP%BE*J3LM%$;=(w3%2_I%$E3*J3L:''(wGa*( )a9Nm\J<+7!*J'%$;=<C%$;\J7A:E93I\)P<+7: J*n].<C%$=%2%$"$*
xA^+:j323:'"2
yA^+:j323:'@*Jg]
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∀ x, y ∈ X : (x, y) ∈ T (R∗c) −→ (y, x) ∈ RcT
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∀ S ∈ S :[x ∈ S ∧
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] T ¾ ] Ê^<df=)P!Y
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] T ¾ ] Á 8Í bo|fb|pt_@q
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T
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) T
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Rc =(x, y) ∈ X ×X : x ∈ C
(x, y
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T
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CA_@q
SUA_@q
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T
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AA←→ CAT
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WA | |l | |
WCA | |l ↓ ↓
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i∈M
P (Rai ) −→ x = Ca
(x, y
)∀ a = (Ra
1, . . . , Ran)T
¾ ] Ç&ÈIOQP!RLS(TU(VWS(XZY@>?;=&"$*#%$'_%$;=M ⊂ N
K201E3SR3:<+RW/g*#%$<x/a*(N*#%$'H*#%$@%$22w'x"2
y(!G39*&=(: J*m0)*(*i'('@E9]\J
(x, y) ∈
⋂
i∈M
P (Rai ) ∧ (y, x) ∈
⋂
i∈N\M
P (Rai )
−→ x = Ca
(x, y
)∀ a =
(Ra1, . . . , R
an)T
¾ ] Ç ¾ OQP!RLS(TU(VWS(XY2<C*#/R'1]\#;=M ⊂ N
G'('&G(x, y)
(!G39*(=&: J*U]01*gA*i'('@E9]g\J
(x, y) ∈⋂
i∈M
P (Rai ) −→ x ∈ C
(x, y
)∀(Ra
1, . . . , Ran)T
¾ ] Ç&¿OQP!RLS(TU(VWS(XZYk2-<C*#/R'1]1\#;=M ⊂ N
<+RW/g*#%$<À G'('&G(x, y)
(!G39A*(=&: J*U]0)*(*i'('&E9]g\Jgb(x, y) ∈
⋂
i∈M
P (Ri) ∧ (y, x) ∈⋂
i∈N\M
P (Ri)
−→ x ∈ C
(x, y
)∀ (R1, . . . , Rn).
¾ ] ÇÍ bo|fb|pt_@q.+S%$*&3!S;&<+'H39':-G':!Gr/a*(N]o<+RW/g*#%$< /a*&Nm33 UT*#%$&%B2*!^T ¾ ] Ã QÍ bo|fb|pt_@qM G'('&Gm\J%B=&%$3X39':H%$<Q!G/a*(Ni39_\J32*J:*!^]nj:/s?^T%$= ¾ Ì@ÊËBT ¾ ] Ç Á COQP!RLS(TU(VWS(XZY
M ⊂ N]X∗ ⊂ X
]M/a*(NQ G'('@GnO/3
F'(!:j32:3!
37 J:=(Eg]\JM/a*&N G'('&G ∀ (x, y) ∈ X∗ ×X∗ 0J:E3EgT
¾ ] ÇÇsOQP!RLS(TU(VWSrXZYM ⊂ N
;G J:!G*/a*(NDC37 J:=(EC0)*(*'('&E9]#\J
M/a*(N ∀ (x, y) ∈ X ×X
T
ÇmÈ VÈ ÏBßUà$7Þ)â"¯`@ár&Ë ¾9Á Ì
¾ ] Æ ¤Í b|fbo|pt_@qtj:3\J]h\#;=YS;&<H#*'E%$AÄ%$&%$'`:j:*#3L:!^]hL;=!;&@T ¾ ] ¾ ¿p;QU1=>@?Y6|pbg|b 2 hpj@bm]n6Ebc Oihtnpg:nxqÁht@thpc HeDLnB|p
Nht b|p_gWb@aDLhpc dhth|pcwn8Q\pc lrqfhpg*dhFpgWbdF3Ppg]v
¾ ] ¾9Á <;QU1=>@?Y ¬ jvxq 2 hpjBbm]n6Ebc O¤htfnpg:nxq¥ht@thxc H_DLnB|pN|pcwn8Q\pc lrqfhpgF3Ppg]v
°¾½¿Às¿Á´Â¿ÂÃ
D AÄö10 , ? 0 ,
S¿g] ¾ 6OQP!RLS(TU(VWS(XZY)>?;=D/a*("23!U0E G"$<+:*_'&a9&%$3'@%$9N "$E3SRw'1b¾ TH%$E%2/g<C"$*&=&%$'D@"$;@%2\J<+7! A = a1, . . . , an¿gT"2 A %$;=IE%$*#/%$9"2"$9T
S¿g] ¿6OQP!RLS(TU(VWS(XZY)z %$*#/%$9"2%$;=M=&*mE%$j:K$!G]<C%$=(E%!;&@b!(Î%$ RW%2!!(Î3E*2!3L1TS¿g] Á µOQP!RLS(TU(VWS(XZYC-32*#:;Iw;;&"$*(=
u : A −→ R(A,¹) /a*("23!?0E G"BA<+:9U'&*2!G32%$*#9]g\J ∀ a, b ∈ A b
a ¹ b'('&E"2K2'M'('@E9]g\J
u(a) ≤ u(b).
S¿g] ¾ df=)P!Y3Aµh |A| <∞ hxaapnpjGce_m]`bBtmlra u DLhxqBtmg:nxq@qf\|_@^||6Op_g!H)dbc 8apnpgLtmlrqBtf]`bg$qI letnpg!6Jz]\xq(z %B*J/%$$2w'XO%$E%2/g<C"$*&=&%$'@%$?%$'((!G%$*#K$!j 32:=&'( Jg]7<Q!@%$O%$E%2/(A<C"$*(=@%$'I37:<+H@"$;@%29]g%$9"$E3%$'((!G%$*#O32:=&'@"t!G@Tg>?9%$'&%$32:=&'_%$ RW%2"$;C"2 X3E*2!3!(!:<Q!j3 32!;@E3uJ*m*#a7&%B'&%2/NCE3E%$*#/g )%H:L3\JSR'1TS¿g] ¾ Í bo|fb|pt_@qUr\)3$*J:;Hw;;&"$*&=C*#%$<%$;=@"$E3%$<_5UT>?;=C\J32*#:;_w;;A&"B*(=H%$39NG%$;@%2?<C*#*C3E*#3$SE3<+:K$!GR3tuR3 Q\J32*#:;tw;;&"$*&=(%$E%2/g<C"BA*(=@%$@TS¿g] Ç8OQP!RLS(TU(VWS(XZY)>?;=
(A,¹) /a*("23!U0E G"$<+: J*Q/a*("23\#9 EK$!G*J:!G3*J'*#%$&%B22w'1]U\J6a∗ ∈ A A^r:j32SR3g]n<C%$=<+7g!<+!7: R3iD\J32*#:;&: ;=<Q!n%$29%$n%$'((!G%$*# ∀ a ∈ A a∗ º a
BT
¾ Ç&¿ ÎÈFE<ÌÞ@áã6@á9ÞS¿g] ¿sÍ b|fbo|pt_@q? <C%$**(=(! )%$* |A| @"$;%$G%$*U]o'('@EC\J32*#:;+w;;@"$*(=DS=&A*#:;&:'@%$U<C%$;'(L:**&*'1]#%$\\#%$ AA^*m0)G;!j: '&%$Pn/% y *!j:*!^T
Ø×@Ö6× 2436587:9<;=<>@?A9<>B=C7.ª¥«BC65s7:6>B65¥;F^/=<;=C7:9<5S¿g] à 6OQP!RLS(TU(VWS(XZY) A \J<+H%$;=D%$G32j::*i_'&a9&%$3'&%B9N "$E3SRw'1b
(a1, p1), . . . , (an, pn) . . .u;=@]\#;=
n∑
i=1
pi = 1 0 ≤ pi ≤ 1.
S¿g] Á GÍ bo|fbo|pt_@q |A| <∞ %2%$"$*+!GO:3G"$0w*'I@"$;%$G%$*D/a*&"23!)"$E3 )%]&\!G39%$*D%$G32j:'D37:<+X&"$;%$G%$*UTS¿g] Æ OQP!RLS(TU(VWS(XZYkk%$;=(w'iS%$^]o\#;=
u\J32*#:;mw;;@"$*(= A A^*UTU>?'('&E_ P
%$PA32j:3\#rE%$*#/%$:E3\JQ\J32*#:;#b
u(p) =n∑
i=1
u(ai)pi.
S¿g] Ê^6OQP!RLS(TU(VWS(XZYP'@%$&%$E"$'@%$G32j:*i_'&a9&%$3'&%$2N "$E3SRw'1b
p =(p1, q1), (p2, 1− q1)
7q1G32L*(5#"$;;@%B
p1%$G32j:39])<QL;
1 − q1G32L*&5#"$;;@%$
p2%$G32j:3-:7A
j32SR'1TS¿g] ÇÍ b|fbo|pt_@qDq'@%$&%$E"$'@%$G32j:s"2s%$;=g39%$E35p%$G32j:s%$E%2/g<C"$*(=@%G%$\#%$;=(*J_r%$G32j:9]#/%
p ≺ pT
¾ THGI*CS(X¡L6ZY46!*#/%$*D'@%$&%$E"$'C%$G32j:%$'((!PG%$*#O%$;=+%$;=g39%$E35D%$G32j:Ur@"$;@%2A A^*UT¿gTJGI*CS(X¡L6ZYJ4i*#*!:9b#%$;=(w'+S%$^]\#;=
p1%$G32j:30E%BS%$E: R'
p2A^@%$n32%$<XA
)%$*UT#>?'('&E X'@%$&%$E"$'D%$G3$j:3E(p1, α), (p2, 1− α)
0E%BS%$E:(p1, β), (p2, 1− β)
A^@%$U39%$<_ )%$*U]\)α > β.
ÎÈ ÇmÈFE<ÌÞ@áã6@á7Þ !Ñ &â<ÌXW)ÞßUáßÌ$ÊY7ÞÌ ¾ Ç Á
Á T;GI*CS(X¡L6ZY?n%$39NG%$;@%2a < b < c a, b, c ∈ A ∃ p
%$G32j:p =
(a, α),
(c, 1− α) \#;=MQ/a*&"23\#9C!*#/g!ì¦o%$E%$*#
p"2
b'@a9a39T
Ç#TKGI*CS(X¡L6ZYp-p10E%BS%$E:
p2A^@%$(39%$<_ )%$*U]'('&EU%$39NG%$;@%2
p3A^E "$n%$39NG%$;@%2
'&%B&%$E"$'(w'(E%-!;&@b(p1, α), (p3, 1− α)
≥(p2, α), (p3, 1− α)
.
S¿g] Ã Í bo|fb|pt_@q? :E/#<Q!:j32:'6%$<C"$G%$"$ )%$*%$2r_!E3E%$&%Bj:*#r%$EWA*J3!:'+w;;@%$3G%$*#"$;@"$*#%$'M*#%$@%$22w'1TS¿g] ¿$df=)P!Y6út%$<+**U]4iE3;@%$*#3%$E3*:L&bcì]`_aOpb E?M¢ce_m]`bBtmlraOx\pjDLh^]~.DLhxqBtmg:nxq@qf\|@^||6Op_g!I letnpg!6Jz]\xqlU%$;=@%$*
bDG%$;7RW d"2
wDG%$;E37 s'(!P<C%$*#%$ %$9%$'G"$%$2*#%$'1]U\!PA
39%$* A &"$;@%2BT) Á T1g!G<++<Q!j3 ∃ αa ∀ aA^Eg]1\#;=
p =(b, α), (w, 1 − α)
Tn#/g*'6u;=6\J32*#:;Iw;;@"$*(=(/% y *!j:*!^]1\#;=
u(a) = αT ¾ T1g!G<++<Q!j3
%$mM"$E3"$'s%$;=&"$E3%$<H5UT£Ä;='(!O3#/R's37:<C*! m:E3\)i\J32*#:;@9T4i3<+:E K2'M2 '@%$U )%$j:3*!^]#\#;=mX:E3\JQ\J32*#:;Qw;;&"B*(=D'@*2!G32%$*#9T#lU%BA;=&%B*
p1, p2=(*U]#\#;=
p1 ¹ p2TJ>?'('&E
p1 =(a1, p1), (a2, p2), . . . , (an, pn)
"2p2 =
(a1, q1), (a2, q2), . . . , (an, qn)
Tt*#33E3J: R*'D'@%$&%$E"$'D%$G32:39]g<C%$=mHj: !j'm32%$E3!*( "$E3\#%$NC%$^b
p1 =
(b,
n∑
i=1
piu(ai)
),
(w, 1−
n∑
i=1
piu(ai)
)
"2p2 =
(b,
n∑
i=1
qiu(ai)
),
(w, 1−
n∑
i=1
qiu(ai)
)
¿gTg!G<+X<Q!j39bn∑
i=1
piu(ai) ≤n∑
i=1
qiu(ai)7
Fu(p1) ≤ Fu(p2).
S¿g] Æ Í bo|fb|pt_@qJtút%$<+**ge&4E3;@%$*#3%$E3*C\J32*#:;tw;;&"$*&=&%$'H<+-t'@a2;&$A/#:;*ij0R3:'&"$0)%$2!'1] J:E37:<C0JE/2*mSE3E:gTS¿g] ¾ 6& $ C " * " T?TJ j!GE/9*Ít!93;&: R'M_j: !n/a*&"23!U\#%$=(9%$%$9b
¾ ÇÇ ÎÈFE<ÌÞ@áã6@á9Þa1bg !2*m*(=@%$E3w*' Á ÈÈÈ_SE3!*&
a2b#ËÈQ37:7G"$'IG32L*(5#"$;;@%$U*(=@%$E3w*'DÇ&ÈÈÈHSE3!*(¿ÈQ37:7G"$'IG32L*(5#"$;;@%$kÈHSE3!*(9T j: J*
a10E%BS%$E:
a2A^&%$U39%$<_ )%$*UT#k%$'(!*&3w'm_'&a9&%$3'@%$9N9b
b1b#ÌÈQ37:7G"$'IG32L*(5#"$;;@%$U*(=@%$E3w*' Á ÈÈÈHSE3!*(
b2bJÊ¿Q37:7G"$'IG32L*(5#"$;;@%$U*(=@%$E3w*'DÇ&ÈÈÈHSE3!*(
9;=1bc1bJ¿ Ã 37:7G"$'D32L*(5#"$;;@%$U*(=@%$E3w*' Á ÈÈÈ_SE3!*&
c2bJ¿ÈX37:7G"$'D32L*(5#"$;;@%$U*(=@%$E3w*'DÇ&ÈÈÈ_SE3!*&9T
ta**&=&%$*j:3\J]o\#;=b"2
c%$E3*J3LD0J:E'
aA^ 1?*&=&%$E3w*'i%$;=i!E3E%$G%BA
:*# %$E3*J3LX\#27:'&%$@%$E"2"$&%$^]b]1%2%$"$*s ¾ ÈI37:7G"$'%$<Q<Q!r%$<*(=@%$E3w*'"$HÌÈI37:7G"$'
a])<QL;
c%2%$"$*
Ê Ã 39:7G"$'M%$<Q<Q! %$<*(=@%$E3w*'1TF@(j:3\J]7\#;=H/a*("23\#9'_Ç#T7g!G<+:9g:j: J*_*#%$<'&*2!G3$%B*J%B'1T
v !2"2?G32L*&5#"$;!%2%$<C"$*&=("$E'&a9a33!:SR3:E:k*#%$<V9*#?^\#"$3'&a2*J0!ì"2 ^<+%$<+3!'!ìH"23r37:<+:EgTS¿g] ¿6& $ C " * " T¾ T a*("23!#32!3J:K$!Gb Æ ÈÈ-%$<_ )%$E?*J"B=(*XS%$E3N9"2?%$G"]@<C%$=_\Jj:G?G%$\#%$9T& ¾39"$E3<Ç&ÈÈC%$<H 1%$E3t !2*6<C%$;<C%$*(9]J<QL;D¿ÈÈ+37:<+:EQ*#%$<\):9T)¿39"$E3< ¾ N Á G32L*&5#"$;;@%$%$*'(!*#%$'*#%$<ä\J32*J:O"2X¿N Á G32L*(5#"$;;@%$<Q!*#/%$*'(!*#%$'1T
¿gT a*("23!h32!3J:K$!Gb Æ ÈÈm%$<H 1%$Er*#"$I=&*S%$E3N9"2H%$G"]n<C%$=\)G:GHG%$\#%$9T ¾ 39"$E3<[¿ÈÈ%$<_ )%$E3h<C%$;@aJ"2?Ç&ÈÈ7A^h !9*&=&*C<C%$;<C%$*&9T@ J¿39"$E3<[¿N ÁG32L*(5#"$;;@%$n%$*'(!%$<ZaU<C%$;+"2 ¾ N Á G32L*(5#"$;;@%$U<Q!*#/%$*'(!9T
F@1G:\J](\#;=I'&"$O/a*&"23!)\J%B=(9%$%$'((!G%$*#9]@<C"$;!G\#%$=(9%$O*J;=(<C"$EWA"$'( )%$*m 1%BS=&: R3X0E%BS%BE%B*JKB!j:*'9T
Ø×øØ× =<>B¥G7:Høý! *#/gE:r9*;@*#/'&2!'1]1\#;=6*&=(!G*gAÄ%X%$;=m<C2!9T ¾ ÈÈrÈÈÈ # N'@"WRW%_*UT-t<C%$;*&=(!SR3t<C2!9]&Èg] Áà G32L*(5#"$;;@%$#*&=&%$E Á ÈÈrÈÈÈ7AÄ%$ )%$G%2"$E33&% ¾ ÈÈÈÈÈ7A%$t!GB])"2Èg] Æà G32L*(5#"$;;@%$?%$@%232Lt<Q!*#/%$*(9T1ztR9 R'MS%B?rR3:"$'M^:R3:9)46!%23t *#/gE:9]\J'&&K':73:jG_39%$<C"$=(!G"$;M Í:2WAÄ%tr33E"$;!j:R3:*U]@\JHÈg] ÁÃ\#%$=@%$3tÈg] Á _*(=@%$E"23!n%2"$=!M
ÎÈ ÎÈ Òp@áßgß@Þ)â<Ë ¾ Ç Ã
øç! 4i%$;\!E/%$3w*'M%$;=Iw;=(@"2/g!k:G:37T"$?RW%$G%$*(3'@%$9NQ*Ub>/%r"2lU%2T#>/%rÈg] Ê ÃG32L*&5#"$;;@%$t*&=&%$E3!<C%$;Y0)%$E%$'&%$9ThlU%2E-2+3#/R'1]?\#;=Èg] Ã e(Èg] Ã G32LPA*(5#"$;;@%$U:3j;@WA]g!G%$3@%32:E3w;=(@"2/oT>?;=I32:E3w;=(&"2/MÈg] ÌHG32L*(5#"$;;@%$o*&=&%$E<C%$;Y%$;=01%$E39].%$;=`:3j;@D0)%2/g!;Èg] Ã A^%$^TOÀ<C%$;*&=&%$E3D0)%$E%$'\J32*#:;&d37:7A<_*'(E ¾ ]%$@%232%$3%$'&"ÈgT/g!G32'@*(k"$*(=@%$9NÈg] Ì Ã T4i%$=(!'RW%$%$*(3'&%$9N@%$;=(w'S%$(MýZ! n%$'(!*(3w'Q øç! S%$j/#9@k%$;=(w'HS%$^]\#;=X*C%$;=_G%$\#%$N"$;w*'Q'(!0E J:*!lU%29T#46!%$;=(w*'1]g\JX<C%$;*&=&%$E3!k_0E J01%$E3M
± °¾½¿Às¿Á´Â¿ÂÃ
N POÚ ·ÂÁ º · 10¸¹¸
lU%$;=&%B*6¿H%$E3<C%$GN*'1]H'(! 1&K2::Q(P )
!*&&%$E39%P (Q)
<C*#*mK2a'('&%$*#N]giK2"$;
qiA^%$E3<C%$
ci(qi)'&a"$;@%$*
ci(0) = 0T
tml(d°`cì]\pg al Qnkqf\^]\xqlNOpbjqfbmg!RQ n6`jg:n^] Sqi'(! )(K2::
si = qi Si = R+
ui(s1, s2) = P (s1 + s2)si − ci(si)
tml(d°`cì]\pg \pjaOpbj@qbg!RQ I bjm]zjhpgBFTS[K2"$;@%$'32!<H:*s<C&/*s:j32SR3:''(!
pi:E!'9]nD@%$&N'+GK2 %$EWA
<C"$'@%$:j32SR3:' %$;=&%$*GN"B;I%2%$"$*6E:*(=@*m::E*J'm_%$E3<C%$GN'(N(%$'('&E
si = pi Si = R+
pi:E*#E3<+:kj'( J*
si = pi Si = R+
ui(s1, s2) =
0\J
si > sj
siQ(si)− ci(Q(si)
) \Jsi < sj
siQ(si)2 − c
(Q(si)2
) \Jsi = sj
tb@a,Opbg:kmlo\pc lrqal QBnkqf\^]\xqlNOpbj@qbg!RQ ]hka^bc1QBbj|TS r%$GNQK2"$;
q1'(! 1&K2::3:j329T ¿gT#K2"$;+%B2 "232G%$!n"2
q2A^:j329T
útE3<+:kj'( J*s1 = q1 S1 = R+
s2 : R+ −→ R+w;;@"$*(=
S2w;;&"$*&=("$E9T
u1(s1, s2) = P(s1 + s2(s1)
)s1 − c1(s1)
u2(s1, s2) = P(s1 + s2(s1)
)s2(s1)− c2
(s2(s1)
)
¾ Ç&Ë -ȾY(ß:(ßÊY:ÑpÉÊÞËâ<Ë%µ×@Ö6× U 6áÙW5¥G7E$FG7:9IH þx;C«@?âá¥>¡7:F65 7:=IÙ)?A=<>B9<;« D6=IÙW;=<5sVס `! `! AWgå&¦#«"¤3å±ßI¯àN¤8¦¯o¬.Þ`¯NûJ³Ä³Äçg!J³
ci(qi) = cqi p = a− bQ
ui(qi) = (a− c)qi − bqiq−i − bq2i/%$E3!:(a− c)− bq−i − 2bqi = 0
BRi(q) = max
(a− c)
2b− q
2, 0
Á ] ¾ g;QU1=>@?Y@DLhqi ∈ [q0, q1]
Hhxaapnpj ∀ qi < BRi(q1)_@q ∀ qi > BRi(q
0)FnpdQl:Eg:\pcì]
I letnpg!6Jz]\xqolU%$;=&%$*qi < BRi(q
1)T#lU%$;=@%$*
D(q−i) = ui
(BRi(q
1), q−i
)− u(qi, q−i)
(a− c)(BRi(q
1)− qi)+ b[q−i
(qi − γi(q
1) + q2i −(BRi(q
1))2]
%$'('@ED(q1) > 0
"2 dD(q−i)
dq−i= b(qi −BRi(q
1))< 0
L;=q−i < q1
]D(q−i) > 0
%$9"$E3qiA^
BRi(q1)/<Q!*J: R3g]k
qi > BRi(q0)A^ED!G
\J*G@*m!;&__:L:9T Á ] ¿<;QU1=>@?Y!RÄfbg]zl!m\^]`_@apFnpdQl(g:\pg$qsdbo|nxc7F^\xqfhh
q =a− c
3bI letnpg!6Jz]\xqoÄ%$E: %$!<Q!*J:K$!G9^blU%$;=&%$*
S = [0,∞) &ce_`u!_@q [0, BR(0)
] ]J\#BR(0) =
a− c
2b &ce_`u!_@q [BR2(0), BR(0)
] &ce_`u!_@q [
BR2(0), BR3(0)]
. . .TTT !*(%$E3<C'mGQ\J:E
BR2t(0)](S%$GNX\J:EX01%2/g!;#b
BR2t+1(0)]#\#
BRt(0) = −(a− c
b
) t∑
j=1
(−12
)2
-È ÇmÈÓ(â<Õ)à$Ì#âUÞ¤ÉÊÞËXrp$Ñ ã6Õ)áÄÞ^ÊÌCÞ@à#ã6@áÑ&@àÌXWY ¾ Ç&Ì
S%$*(3!UE7'D'@*(@%$E3;@%$*#%$'M"2 ∃ '&a9a\J:E"$E3"$'(w'1]L;=M%$H_<C%$;@G/#:9T-!G37:<QL3gb ∞∑
j=1
(−12
)j
=1
3
]gL;=
BR∞(0) =a− c
3b.
Il(gWb\pjlrq n6`jg:n^] E`gW_c&] P8QTQ¥htnpg:nxqµ]J u`q®Ák@_||bcöS%$*&3!H:L:3MS;&<+22':9F@%$Ga RW%
q−ida !HK2"$;a32'(! 1&K2:::9b
q−i =∑
j,j 6=i
qjT
>?'('&E r!%$E3LM%B!<Q!*J:K$!Gbf&3E"$;!jQ\J<+
S = [0,∞)>?GNQG"$0)"2[0, BR(0)]4:&/!P'DG"$0)"2[BR(N − 1)BR(0), BR(0)]
(N − 1)BR(0) ≥ a− c
b
L;=BR(N − 1)BR(0) = 0 ,
7- !%$E:K$!G_E:*+*#%$<[aE3"$*!'C%$<Q<Q!^T.'C J*IG%$\#%$3w*'Q !2'1]&\#;=r%$;=&%2K2"$;@%$'D*#%$<%$E3<C%$*#%$'6_<C*#01'(! )(K2::3*J:na 1%$9T
ס `! (¡ ßI¯@àN¤8¦¯o¬X°7«)¬#®¬ÕUÖhÖ çg!J³(N ≥ 2)
lU%$;=&%B*qi i
'(! 1&K2::c(qi)
_%$E3<C%$G"23!U'&a"$;Q<Q!*( %2//g!;
q = (q1, . . . , qN ), Q =N∑
i=1
qi Q−i =N∑
j=1j 6=i
qj
Q%$E3<C%$G"2
P (Q)%$;=@%$*#3u=(!k:E*6"$E3"$'@%23w^b'(! y 9%$"2%$'
ui(q) = P
qi +
∑
j 6=i
qj
qi − ci(qi)
>?'('@ES = R+
Tk%$;=(w'+S%$n_7: !j'( J*U]\J;=
ui(q)'(:2!'@*':9
qiA^ 1%$*
iA^E%<C%$;@G/ggb
maxqi
P (qi +Q−i)qi − ciqi
¾9Ã È -ȾY(ß:(ßÊY:ÑpÉÊÞËâ<Ë%$ )N
P ′(Q)qi + P (Q)− ci = 0
dqidQ−i
=P ′′(Q)qi + P ′(Q)
P ′′(Q)qi + 2P ′(Q)− c′′i (qi)\Jc′′ > 0
"2P ′′ ≤ 0 '('&E ∃ <C%$;@G/#:t"2
−1 <dqi
dQ−i< 0
>?'('@E!G3<C"$ ci(qi) = cqi
]p = a− bQ
!G3<C"$<C%$;'0\JSR'
BRi(Q−i) =a− c
2b− Q−i
2%$ )NX32!<Q<C%$3E3!'(#R3:"$'Dú-3\gAÄ%$;=@%$*#3u=(gb
q =a− c
2b− (N − 1)q
2,<Q! )N
q =a− c
b(N + 1)
"2p = c+
1
N + 1(a− c)
ס `! ס ©±Þ¬J³S¬.ßI¯àN¤8¦¯o¬X°7«)¬#®lU%$;=&%B*
N = 2]%$'('&E
qm =a− c
2b
]um =
(a− c)2
4b
_<C*#01U%BE<C%BG"2t"20E y
qd =a− c
3b, ud =
(a− c)2
9b
X/g#01P!<%$E3<C%$G"2%H"20E y SR3gT
Á ] ¾ sÍ b|fbo|pt_@qQS%$*&3!%$;=&%$*#3u= ∃ "2t%$;=@"$E3%B<_5U]1%$9"$E3-@"$;@%2.R3:"$'( J*%$Q%$;=@%$*#3u#P=&T£Ä;=M_7: !j'( J*m&"$;%$G%$*CR3:"$'@'%$'(!*(3w*'1b%$'(!*(3w'629])\#;=m<Q!*#/g'@"$K2"$; qm
2
A^t%$E3<C%$C9: !j'( J*7h(∞) =
qm
2
T>?;=&%$*#3u=D%2%$"$*m<Q!*#/g'&"B K2"$;#b
∞∑
t=1
um
2δt−1 =
1
1− δ
(a− c)2
8b
'(! y 9%$"23'0UT
k%$;=(w'rS%$^]\#;=_%O33E"$;!j:J%$;=(!' R3:"$'@?%$"$E9]%$'('@E-,Ozv33E"$;!j-%$E%2/(A<C"$*(=@%b
9
64
(a− c)2
b
'(! y 9%$"2,
-È ÎȤÞUàß@Þ6Y:Ñ ÊËÑ (ã67ÞUámÐ_ÉÊ@ÞËâ<Ë7ÞÌ ¾9Ãg¾
L;=MtR3:"$'&a32\J32*Jgb
9
64
(a− c)2
b+
δ
1− δ
(a− c)2
9b>?;=&%$*#3u=I*6'('&E9]g\J
9
64
(a− c)2
b+
δ
1− δ
(a− c)2
9b≤ 1
1− δ
(a− c)2
8b
δ ≥ 9
17 Á ] ¿Í b|fbo|pt_@qV,O%$E33E*#/DR3:"$'(*J:δ >
1
2
]JL;=i3%$;=g39%$E35 +'&&01%$E:K$!G%$G"$E"2%T
%µ×øØ× î 7ÙWG7:9_^6«BFH«B;:?A9C7:>BÛr¥9<;=<; E$FG7:9IH8H=<;=C7:9<5 Á ] ¾ OQP!RLS(TU(VWS(XZYo32!<Q<C%$3E3!'(# E"$0J7e@ 1-33E"$;!jC!G3<C"$%$r.E3*#OR3:"$'(E
qL"2qHbqL < qH
TlU%$;=&%$*
u0(qi, qj) i0E y SR3X<Q!*& %2//g!;#]%$;=(w'IS%$^]g\#;=
ui(qL, qL) > ui(q
H , qH)
lU%$;=&%B*i%$'('@Esrb(t, h(t)
) X'@a7@%$3'&%$9N¾
srb(1, h(1)
)= qL
SjK2*(=D'(! )(K2::o!*#/g*'!t)
srb(t, h(t)
)=
\Jqi(t− 1) = srb
(t− 1, h(t− 1)
)q ∀ i
'('@Esrb(t, h(t− 1), q(t− 1) = qL'(wGa*( )%$*srb(t,(h(t− 1), q(t− 1)
))= qH
7I\JD%$N9Ni01%$E3!G&/g#3 J*Y<Q!*#/g'&"$XK2"$;D%$GNLE3X33E"$;!j:X'&a9&%$3!^]'('@EQtAÄ%2/g!'I01%$E3!G&/g#3 J*6!G9]g\JQ%$3Nn%$;=I;=I'@"$K2"$n%$"$E9]J'('&E<+;&9Tk%$;=(w'+S%$^]g<Q!*#/g'&"$R3:"$'@%$2tX33E"$;!j:R3:37gb\Jj<C%$=(!'rK2"$;
qHA^k:j32SR3g]'('&En<Q!*#/g'&"$K2"$;-'@a7&%$3'@%$9N01%$E3!G&/g#3 J*
qHA^9]J2:*m(!G37"$E3*#%$'
qLA^\#%$H33 UT
4i%$;(!93;&: R'1])\#;=srb<Q!'&E-&%$9%$-E"233R3:"$'6'(!%$G"$;LNMú-3\gAÄ%$;=&%$*#3u=&A
\#@TR3:"$'D39%$E3'&%$9%B%-<Q!j3 '@"$%2%$%$'@%$U(!93;&:*!^b
srb(t, h(t)
)= qL
srb(t, h(t)
)= qH
¾9à ¿ -ȾY(ß:(ßÊY:ÑpÉÊÞËâ<Ë\J
qL'('@E9b
ui
(BRi(q
L), qL)− ui(q
L, qL) ≤ δ[ui(q
L, qL)− ui(qH , qH)
]
\JqH'('@E
ui
(BRi(q
H), qH)− ui(q
H , qH) ≤ δ[ui(q
L, qL)− ui(qH , qH)
]
Á ] Á 8Í bo|fb|pt_@qkX ) 33E"$;!jCÈ_0E y -/M%$;=D0)%$E3!G&/gJ3EgT£Ä;=1b(a− c)2
16b≤ δ
(a− c)2
8b
%$ )N
δ ≥ 1
246!&%B ¾ N¿[ZÌN ¾ Ê(]hL;=v%$33E"$;!j'(!G%B # δA^ !G+<H5'&a&/g!'1]h<Q!*&Mú-3\gA
(!G37K20J:9T
%µ×&%µ× Ñ =IÙ7ÙW65b¹E$FG7:9IH þx;C«@?âá¥>¡7:F65 FÙ D6=IÙW;=<5Vס ס `! A\¯Þ´å&¦«!¦h±ÆÞ !U¯n³&ûh«o±D®¤J±J±xlU%$;=&%B*
pi ∈ N"$
ci(qi) = ciqi9*#!*#%9:E3!G'@a"$;Qw;;@"$*(=@T
>?GNQ!%$E:K$!Gb#%$!<Q!*J: R'mÈ7A^¥Spi = c
/<Q!*J: R3<+:&/g!'D!%$E:K$!Gb ¾ A^¥\JC
pi = c/<Q!*J: R3
nAÄ%2/g!'I!%$E:K$!G
nA^\JC
pi = c/<Q!*J: R3
£Ä;=MH:E*#%$<ZG%$\#%$'(!G%$ U]g<Q!*( H%$;=g"$;E%r%2NX%$E3<C%$G"23!U'@a"$;#T Á ] ÇÍ bo|fbo|pt_@qC-
pi ∈ R+"2
ci(qi) = ciqi ci > 07: EQ*!*#K2 /<Q!*J:*#
<C%$;@G/#:9TlU%$;=&%$*
p1, p2 > 0]
p−i ∈0,min(p1, p2)
T9>?'('&Eui(p
1, p−i) = ui(p2, p−i) =
0
R3:"$'(*J'M9*( J*m*M;=&%$*;@%r/<Q!*J:*#<C%$;@G/#:gb¾ T
p1 < c;=@%$*;@"$*6/<Q!*J:-
p2 = c:^T-
p−i < p1]#'('@E
p1"2
p2!G È_'(! y 9%$"23 %$E%2/g<C"$*(=@%$@T-
p−i ≥ p1]#'('@E
p1*#%$;&3L1]<QL;
p2 = 0'(! y 9%$"23 %$E%2/g<C"$*&=&%$@T
¿gTp1 = c
;=@%$*;@"$*6/<Q!*J:-p2 = c+ ε
:^T
p1'(! y 9%$"2%HÈ ∀ pi
A^E%r"2p2'(! y 9%$"2%$!U01%2/g!;C0)2!3L'm
p−i > p2A^E%T
-È -È !@à@ÞUàßÌ"¤ÉÊ@ÞËXrp$Ñ ã6Õ)áÄÞpÊÌyÊUà&@àÌXWY ¾9ÃÁ
Á Tp1 > pm
S\#pmX<C*#01:E.;=&%$*;@"$*i/<Q!*J:
pm:^T-
p−i <pm_'(! y 9%$"2%$'MÈ7A^'1T!-
p−i ≥ p1pm'(! y 9%$"2%<C%$;\Jj/R3
p1A^9T
46!&%$Q(p)(p− c) p
A^ 1%$*m32!;@E3uJ*m<C*#*m*#a9&%$'@%2/N[c, pm]
A^*U]#9"$E3*!*#K2=&*
p1 ∈ (c, pm] ]k<C%$=;=&%$*;@"$*v/<Q!*J:QG%$**#%T<- p2 > p1]
p1'(! y 9%$"2%
<+;& U]\Jp−i ∈ p1, p2
T-p1 > p2
]h'('@Ep1'(! y 9%$"2%M<+;& U]h\J
p−i > p1T
7_r%$!<Q!*J:K$!GC:*i(c, pm]
!*(%$E3<Z<+E/oT
Á ] Á b;QU1=>@?Y&b|pbgN ≥ 2 ci(qi) = cqi
Qøhtnpg:nxqc l(gWbB\pjlrqa6Ppcì](q_o|@^||6Op_mg!TS,Hb@aapnpj ∃ bo|pbg$q®c HdQlrapnpjµh]`bc fb@qal QBnkqf\^]\xqm] p = c\pjBnpg dbo|6Ox\xqf\pjnpc m\xaTHd\xqfbcwvpc
p 6= cE`j@bµg!l(g:kqµnpc hpg1b|pbg$q®c HdQlrapnpjh¤]`bc fb@qsal QBnkqf\^]\xq$OpbaOxvpjBb¥]hpcw\pce
I letnpg!6Jz]\xqok%$;=(w'+S%$^]g\#;=(p∗1, p
∗2, . . . , p
∗N )ú-3\gAÄ%$;=&%$*#3u=IG%$;=&%B*
p = mini
p∗i
H:j:*#:;+<C%$;@39E3L:X*#"$'(wUG%$;=@%$*p∗1 = p
T +:L:_<Q!*&
p = cTok%$;=(w'6S%$^]1\#;=%$C*#%$<À!;&@TU>?'('&E
p > cp < cG%$\#%$%$3G%$*6\#;=M
c%$'('&E/<Q!*J:*J:
pA^BT
>?'('@E\Jgbp∗2 > p
To>?'('&EHM¿DK2"$;M'(! y 9%$"2%+Èg]1(!G39*&p2 ∈ (c, p)
0)2!3L'(! y 9%$"23H/o]L;=MH%$E%2/%$3!k:j0) *#%$<ZG%$\#%$%$3 %$;=g39%$E35UT
p∗2 = pTs>?'('@Er¿IK2"$;D'(! y 9%$"2% ≤ (p − c)Q(p)/2
TW-p12 ∈ (c, p)
¿+'(! y 9%$"2%(p2 − c)Q(p12)
Tn46!&%$QS=(*#X6¿gTkK2"$;
p12 = p − εA^H:j323QR39(L*!h3#/
\#%$=(9%$"$*UT#k%$\J: %$r%$<ZG%$\#%$%$3 %$;=&%$*#3u=@T
£Ä;=p = c
Tk,O%$j:SR'1]k\#;=p = c
%$;=@%$*#3u=&]n\J ∀ pi = c<Q!*#/%$*Y0E y CÈgTI-
K2a'('@%$*Mj\#kH:E9]#'('@E _'(! y 9%$"2*#%$;&3LDG%23@T$-_*#NXj\#nH:E9]J'('@E**J'M_K2"$;*#%$'Dr%$j/#:!o<C%$;@325**#%$'1T$£Ä;=+*#%$<"$E/%$<C%2%$"$E3*!^]#%$9"$E3
p = c%$;=@%$*#3u#P=&T
Á ] Ç;QU1=>@?Y8&b|pbg]n6Ox\8QTQjBhlrqci(qi) = ciqi
_@qGjBbgBFbBt@tmaµhµk@_|b@a^bf]cia8qBtbmjl(g]`
c1 < c2 < . . . < cN .Aµh
c1 < c2 6 ∃ C hxqD8Eb|pbg$q®c I letnpg!6Jz]\xqolU%$;=&%$*
p<Q!*& %$GN 6"2 H%$GN9NQ !9*&=(L:3t'<+2\JSR'1T
¾9Ã Ç -ȾY(ß:(ßÊY:ÑpÉÊÞËâ<Ëס ס (¡ ©±Þ¬J³S¬] ¤¬¤@²!¦ûv°7«)¬#® Hj0R3:"$'mX39':9b
N ≥ 2 ] u(p) = (p− c)Q(p)]gS%$%232w'
u(p)<C*#*6*#a9&%$'@%2/N
[0, pm]
Á ] Ã 8Í bo|fb|pt_@qnú-3\gA^(!G37K20J:33E"$;!jp = c
È_'(! y 9%$"23t/oT j0R3:"$'É<C%$;@G/#:q%$;=@"$E3%$<_5U]%$9"$E3%$&"$;@%2i!G3<C"$3GN(/"2%2DR3:"$'
%$;=@%$*#3u#P=(_!G9T!£Ä;=DX9: !j'( J*_@"$;%$G%$*m!G3<C"$3GN(/"2%$oS;j'@2*'1Tk%$;=(w'+S%$^]g\#;=mXK2"$;@%$'m
[c, pm]!*&%$E3<_ )n:j32*J'm:E9T
r%$;=&%$*#3u=( J*i_'(! y 9%$"29b∞∑
t=1
u(p)
Nδt−1
-DIK2"$;mQ%$;=&%$*#3u=(!:EtSaG"_<C%$;=@]1<C%$;@325**#%$'%$j/#:!^])L;=6'(! y 9%$"2%CÈgT£Ä;=D%$2*#%$<"$E/%$<C%2<C%$;%$**!G%T-_G%BS%$G""$E%$k_\J:E'&a"$;+j:g]J'('@E*J%B;&LPM%$E%2/g<C"$*(=(%$E3<C%$^T£Ä;=1b∞∑
t=1
u(p)
Nδt−1 ≥ u(p)
%$ 1N
1
1− δ≥ N
δ ≥ N − 1N 7H_S%$*&3!
δ<C%$G%$3G%$\#%$"$;@%2t_'&&01%$E:K$!GT
Á ] Æ 8Í bo|fb|pt_@q\J
N = 2 δ =1
2\JN −→∞ δ −→ 1!
Á ] Ê^yÍ bo|fb|pt_@q>? )%$*CR3:"$'( J*δK2'
Nw;;&"$*&=&%]7M%$<
cA^N^]%$<
pA^No*J%B<w;;#T
%µ× U× î 7:6ß:H8=<>ª=IÙ"^4E$FG7:9IH>?'('@E7b
S1 = R+
S2 : R+ −→ Rw;;@"$*(=("$E
'(! y 9%$NXw;;@"$*(=1bui(q1, q2) = ui
(q1, q2(q1)
)
-È VÈ Òp@áßgß@Þ)â<Ë ¾9ÃÃ
ú-3\gAÄ%$;=&%$*#3u=1bq1 := maxq1
u1(q1, q2(q1)
)
q2 := maxq2u2(q1, q2(q1)
)
%$ )N^bq2(q1) ∈ BR2(q1)
\#BR2
+.E3*#R3:"$'IG%$;7RW m:j37gT
ס 6N `! êéJ±6æ7°7«)¬#®®_åJ³^!^7S¬_ âs²o±6ã¡ám!!¥&¦h±6§h³¥,:E3<C%$=
q1<+:WA^<+:E"233RW:"B'& /% y *!j:^]J7
q1 = maxq1u1(q1, BR2(q1)
)
Á ] ËGÍ b|fbo|pt_@qU ¾ T7R3:"$'&O_.E3*#R3:"$'(\#t'&"B01%23ha 1%$.%BE<C%Bo"2ha *(=@%$E%2"$;\#%$+!GR3\J9]o<Q!*(_¿gT#R3:"$'@9TU 7I6f&K'@%$ 1%$E3;+R3:"$'+%$GNMG"$0)"23%$&N*#%$'D%$GN*&=&a9T
%µ×]\× =<>B¥G7:Hýo¨ lo!*#%9:E3!G '&%$E%23G%$3!ow;;&"$*&=M"2'(w#a*( 1a9NC!*#%9:E3!G '@a"$;Cw;;@"$*(=(5m'&"$:PAjj %2%$"$*m\J:E22'D<C%$;IC.E3*#WAÄ%$;=&%$*#3u=(9ý Ï -:E22'Q<C%$;XH.E3*#WAÄ%$;=&%$*#3u=Q%$E3<C%$G"2"$O"20E y SR3:9]@\J'@"$:jj*U]g*!*#K2'@a"$;+"2
P = 1− (q1 + q2)2
ýJÔ! -*L3'ma39%HI.E3*#WAÄ/g#01G!<À"2tQ<C*#01P!<'(! 1&K2:::9]J\JP = 140−Q
"2C = 20
Tý#Ý "$-:jj-%2%$"$*i\J:E22'6<C%$;mDf&K'&%$ )%$E3;AÄ%$;=&%$*#3u=(9])\JC!*#%9:E3!Gr'&%BE%$3G%$3!ow;;&"$*&=D"2'(wGa*( )a9NC!*#%9:E3!G'@a"$;ASw;;&"$*&=m/39ý)ì" "$_:jj_&%$E%$*&=&%$@]
P(q1, q2) = 1 − q1 − q2]c1 = c2 = 0
TC-*L3'a39%rC.E3*#WAO"2 Cf&K'@%$ 1%$E3;AÄ%$;=@%$*J3u=D'(! 1&K2:::9ýJñ! "$O%$E3<C%$GN*'+*
Q = 100−P'@%$E%23G%$3!)w;;@"$**(=@%$o"2
C = 50'@a"$;;@%$^T
46!nH03!<+:!Gt:E Q,O%$E33E*#/(A^<C&/%2k39%$E3!*&Mý1øB "$O%$E3<C%$GN*'+*
Q = 130−P'@%$E%23G%$3!)w;;@"$**(=@%$o"2
C = 10'@a"$;;@%$^T
p'@%$E%23G%$_D,O%$E33E*J/(A^<C&/%$G.39%$E3!*&H32!'i<C%$;iD'@"$r%$E3<C%$GNM'&a9a39T Á G%$\#%$WA"$;@%2r:E-*UbUÊÈg] à Èg])!G%$3&% ¾ ÈgT)£ÄESR'MS%$HR3:"$'&-*#E3<+:hj'( J*U]o"2-'&%$E%23w'<C%$;I_*(=@%$E%$;01*&9ýJý! "$O%$E3<C%$GN*'+*
Q = 120−P'@%$E%23G%$3!)w;;@"$**(=@%$o"2
C = 40'@a"$;;@%$^T-*L3'Da39%r+.E3*#WA."2 X,O%$E33EB*#/(AÄ%$;=@%$*#3u=6:E:9
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A^E%%$ R%23w)%$EN?0)2!3LX'0)K2j8SPA
n3jRW/*gA:;#]g\J
RN "2 x ∈ H(RN )]g\J
RN1
"2 ∀ i ∈ N"2
y ∈ XA^E%
xPiy −→ xP ii y"2
xIiy −→ xRiiy]#'('&E
x ∈ H(RN1 )T
Ç#] Ã <;QU1=>@?Y ¬ |pH SCF
u$npg]nxqmhpg1hxaapnpjQbjBvqbg*dnpg:n^]npgfH_DLhSPA
Ç#] Ç÷d¥f=)P!YÁ%$G%$; ceb|pbg
Fb|p_jm]`_@a,c
SCFH~DLh
FbjBvxqbgÅdnpg:n^]npgA_@q
∣∣R(F )∣∣ ≥ 3 HZhxaapnpj F Fplrax]h^]~xjmlea`qf
Ç#] ¿ Ã OQP!RLS(TU(VWS(XZYΓ = (Σ1, . . . ,Σn, π)
R3:"$'&SE3<+(GF )
]k\#Σi i = 1, . . . , n*#%$<ZwE%2&"$;@%2\J<+H"2
π%$;=
ΣN −→ Xw;;@"$*(=@T
Ç#] Ç¥Í bo|fb|pt_@qΣi
iAÄ%2/g!'DR3:"$'&r33E"$;!j:R3m"2
πD'(! y 9%$NDw;;&"$*&=&]
ΓA^
%$;=_:j:*#:j32:3!g%$ R3:E:3*J'H%$'(!*&3\#%$SRw'1]\#N:E/#G< ;R3!
XA^ )N
:j32*J'1]#(!j:;@9]#\#;=F SCF
%$'((!G%$*# %$;=(R1, . . . , Rn, F )
R3:"$'&SE3<+:7A^TÇ#] ¿ Æ OQP!RLS(TU(VWS(XZYG%$;=&%$*
Γ(Σi, π)"2(R1, . . . , Rn)
]g%$'('@EΓ"2(R1, . . . , Rn)
AÄ%$'0)K2jR3:"$'&*6_'&a9&%$3'@%$9N "$E3SRw'I*#E3<+:oSE3<+: J*Ub
g(Γ, RN ) = (Σ1, . . . ,Σn, π, R1, . . . , Rn).
Ç#] ¿@Ê^6OQP!RLS(TU(VWS(XZYJG%$;=&%$*RN , QN %rq&!! E3!<0)*(%T 0: g A^E%\J ∀ i ∈ N
F (QN )RiF(QN\i , Ti
)∀ Ti ∈ R.
Ç#] ¿Ë8OyP!RS(TU(VWS(XZY1G%$;=@%$* M ⊂ 2N ] U 6= ∅ ]G%$;=&%$* Γ %$;= GF"2
RN eN ∈ ΣN ]%$;= M %$;=@%$*#3u=(01*(SR3g(Γ, RN )
A^*#%$'1])\J ∀ M ∈ M "2 ∀ yM ∈ ΣM ∃ i ∈ M]
\#;=π(eN )Riπ(e
N\M , yM )TF@%$Ga RW%
g(Γ, RN )M %$;=@%$*#3u=(01*&SR3t\J<+7:h%T 0UT(M,Γ, RN )
TÇ#] ¿ÌGOQP!RLS(TU(VWS(XZY
Γ = (Σ1, . . . ,Σn, π)%B;=
GF"2
RN T- M1 =i | i ∈ N
%$'('@E M1
A^tg(Γ, RN )
ú-3\m%T 0nT AGR3:*J'D*#%$&%$22w'1TÇ#] Ã ¾df=)P!Y^8t! JE/(egfg3%2E33\(~O!%Áceb|pbg
Fbo|p_jm]`_@a,c
SCFAµh
FgWbddhpg!l u`cw\pc DLh^]~¢_@q ∣∣R(F )∣∣ ≥ 3
b@aapnpjFFplrax]h^]~pjlra,`qH¥hDLnpc R(F ) =
x | x ∈F (RN )
RN ∈ R HZhDLnpc R fbcGPpc lht S fbcebf]o]zl<jBbmgBFb@t_qfbaDLhpc dht\^]`
I letnpg!6Jz]\xq1'&"$G"$01"23 )%$*6aE3"$*!'1]#%$GN39aES%$%232w'1]\#;=FX32!;@E3umE%$*#/#%B9"BA
%$'@%$*6"$E3%$<C%$2%$37]#<+RW/M%$**#%$'M%$;L"$;@"$@%$U !9*(=(LSR'MX"$%$9T
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FX32!;@E3u6E%$*#/%$'@%$9"2%$'M%$E"$*6"$E3%$<C%$9%$3-k%$G%$;+"$%$G%
<Q!j3.%$G"$;r !9*(=(L*!^]@\#;=F%$EN%$*C<C*#*U]&%$;=(w'HS%$) %$G%$*'&%$9NRW"$9]
'('&E ∃ RN T F (R) = t]RN ] RN A^ )N1'0R' x
0)2LGK$!GR3:*J'_R32(L::91"2F (RN
1 ) = z z 6= t, xT£Ä;= ∃ y ∈ X i ∈ N
"21 < h ≤ n
]o\#;=Rj = R∗j
]\J
j 6= i"2
x = th(R) = th−1(R∗i ) y = th−1(R) = t(R∗h)
"2Ri|X\x,y =
R∗i |X\x,y:j32'I39"$t_'&a9&%$3'@%$9NQ%2%$%$'&%$9T
!rz 6= y
L;=zRit
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])%$;=(w'MS%$^])\#;=zRit
]1'&A'&E
R*#%$<%$;=&%$*#3u=(!n01*&
g(F,RN )A^E% 46!&%$
F (RN∗)R F (R∗)
B]\J*G@*6\J
tRiz RN∗ *#%$<%$;=@%$*#3u=(!U0)*(g(F,RN∗
)A^E
!!(z = y
'('@E\Jx 6= t
%$'('&EzRit⇐⇒ zR∗i t
<Q!*& %$GN x = t
%$'('&EzRit
"2RN *#%$<Z%$;=&%$*#3u=(!o01*& g(F,R)
A^E%
3BTk%$;=(w'MS%$^]#\#;=F ∗ = F |LH
%$'('@EF ∗*#%$<<+*!0j:\J]o%$'('@E-BT#;@*gA
/j3<C%$*#%$"$&%$U 1%$j:3\J]\#;=F/g!'(E3!'(#G%$;=&%$*
j_/g!'(:E
F ∗A^Eg]
%$'('&Et 1%H'&%$kj:3*(*'1])\#;=j/g!'(:E
FA^E%_!G9T)k%$'(!*&3w'
RN A^"2tG%$;=@%$*z = F (RN )
"2
B =x | x ∈ R(F )
"2xRjy ∀y ∈ R(F )
]) )%_'&%$kj:3*!^]1\#;=z ∈ B
T)Ä*#/g!PAE%$'(-<C(/*%$;=(w'6%$^]1\#;=
z /∈ BG%$;=@%$*
RN∗ ∈ LN ])<C%$=(E% BRj(X \ B)"2
(X \ B)R∗iB ∀ i 6= jS\#
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bRaT 46!@%$
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F ∗A^Eg]
F (RN∗) ∈ B
T@lU%$;=&%$*wk = F (R∗1, R
∗k, R1, . . . , Rk)
k = 0, . . . , nT1Ít!j:;@
w0 = F (RN ) = z /∈ B"2
wn = F (RN∗) ∈ B
To46!&%$ ∃ t1 ≤ t ≤ n
]g\#;=wt−1 /∈ B
"2wt ∈ B
TlU%$;=&%$*
RNk = (R∗1, . . . , R
∗k, R1, . . . , Rk) k = 1, . . . , n
]#\Jt = j
])'('@ERN
t−1*#%$<[%$;=&%$*#3u=(!#01*(g(Γ(F ), RN
t−1
) A^E%]&<Q!@%$xPjy ∀x ∈ B
"2y ∈ R(F )\B T-
t 6= j]1'('@E
RNt
*#%$<%$;=&%$*#3u=(!n01*&g(Γ(F ), RN
t
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M ⊂ N'&@LGK$!G
i /∈ M'('&Et2 <C*#/R'1]#\#;=
MA
*#%$'C%$G%$*G"$0)"2%t*i R′
<+*!0j:K$!GR3:9o39%$<_ )%$*MR0E y Er\J ∃ R′′
0E y ^]\#;=¾ T
k /∈M −→ R′′k = R′"2
¿gTF (Rn)Pi (F | R′′n)
T
Ç#] Ág¾ ZOQP!RLS(TU(VWS(XZY%$;=C%$G%$*G"$0)"2O\!\#%$N]&\Ji R
A^ 1NR′A^EraE3"$*#NH<+*!0j:j:
R′′%$G%$*G"$0)"23E% ∀ j ∈M
A^E%F (Rn)PjF (R
n′),"2
F (Rn′′)PjF (Rn′).
ÃxÈ -È Òp@áßgß@Þ)â<Ë ¾9Æ Ì
Ç#] Á ¿OQP!RLS(TU(VWS(XZYk%$;=F CCR
*#%$< <+*!0j:\Ji%$G%$*G"$01"2%$^]U\J ∀ RN A^E%"2ViA^E%]J\J
F i:<+*!0j:\J]U'('&E ∃ S
'&@LGK$!G]1<C%$=(*#%$'6*i\!\#%$N%$G%$*G"$0)"2%T
Ç#] Æ Ådf=)P!Ycebo|pbgF SCF
H_DLh bcrcebg!ce_`u!_@q@qbcgWbd dhpg!l u`cw\pc DLh^]~ _@q ∣∣R(F )∣∣ ≥3HZhxaapnpj
FFplrax]h^]~pjlra,`q
U×&%µ× =<>B¥G7:HýJç! /3%$;=i/g!G32'(E"$%$G3$j:3u:j32!0)0j:K$!GTn¿g] Ã <Q!!G_RW 1G/#!?*U]Èg] Ã <Q!!G JGG/#!*U]k"2 ¾ <Q!!G@*v**J'YiK2%$*&3E3<_ J*UT?Z'@"$X@%$E%$*(=(9N0J:E3h!GO\J:E<pS%$*&3!J!G/%2G;!jr'&a2wJ:j323\J9T(`:j32'+E3Er3792*J'1]('(!!G/%2G;!j!j;I'&a9%$G%$ :U\#27:R'1T-Q%$;=@%$*GNC:9@G:;EQ**J'1])'('@E&"BG%$WAG%$*#39%$E35#%$*I:j32*J'1T E:9 R'+R3:"$'@h*#E3<+:Uj'( J*U(46!=@%$*+!G/%2G;!j:'&"B0((!G%$ RW%$*#%$'M_0J:E3'!MçZ! k%$'(!*&3w'M ýJç! S%$j/#9# JGG/#!j'M37:<+_G%$;=&%$* ¾ <Q!!G]J RW 1G/#7A!j'@"32!*&"$*U]"2_K2%$*&3E3<_ J*mG%$;=@%$*#%$'M¿H<Q!!G@*UTJ G/R'+<C%$;QuREXrS%$j/#%$9%$'('@%$kH/#'('^ço¨ :j32's!G/%2G;!j!OS%$PS;&::X%$;=s!*&%$E3<Q<+E%$0E%$9%$*&: R'1]<Q!*%$;=@%$*G%$%2%$*m32j*J'+%$^T(Íh*I'&"$.0J:E39]r:j32'IE3E_3722*J'1]g'(!)!G/%2G;!PA!j;C'&a2%$G%$ :o\#27:R'1T#46!=@%$*m!G/%2G;!j:S;*J'I'@"$0((!G%$*!nX0J:E3'!Mç Ï :j32'_!G/%2G;!j!#<C%$;;=@N9N(/"2"$E%$0E%$9%$*&B: R'C
[0, 1]!*(%$E3<Q<+^T
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"2x, y ∈ X
\J(x, y) ∈
⋂
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P (Ri)]
%$'('&E [x ∈ S −→ y 6∈ C(S)
]∀ S ∈ S
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(⋂
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Ri
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]∀ S ∈ S T
à ] ¿6OQP!RLS(TU(VWS(XZY)Ä;&9:;@:;#b#%$;=w : U −→ X
G%$'&"$09"23!;&9:;@:;!n%$(*#%$'*#%$&%B2w*#'1]g\JX'@"$01%-'(:2!oE%$*#/%$9"2
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u ∈ X ×NA^EN
w(u) ∈ XA^E%'@"BA
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¾ Ê Ã
à ] Á 6OQP!RLS(TU(VWS(XZYof&00)%2WA^!U!;&9:;@:;!n%$(wS)
(x, y) ∈ ws(u)←→ ∃ π ∈ ΠN ∀ i ∈ N,u(x, i) ≥ u(y, π(i)
)∀ x, y ∈ X.
à ] ¾ ;QU1=>@?Ywslw|htqf\|nxq@qf\|pl6bc ORQøhtht¥a^_`u$b¥a,Ox\tmlGjBbgBFbBt_@qHjBb b6J:O_@qy]zjhpgEtmlz]J:OsS,HFbµgWbd¹]`bc fb@qlw|htqf\|nxq@qf\|plGbc Op
à ] Ç6OQP!RLS(TU(VWS(XZY),O%$*&3\J<XA^!n!;&9:;@:;!n%$ wB
(x, y) ∈ wB(u)←→n∑
i=1
u(x, i) ≥n∑
i=1
u(y, i), ∀ x, y ∈ X.
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(x, y) ∈ wR(u)←→!( (
u(x, 1), . . . , u(x, n)),Â(u(y, 1), . . . , u(y, n)
);= !!( (
u(x, 1), . . . , u(x, n))=(u(y, 1), . . . , u(y, n)
).
!(^A^*#"$(x, y) ∈ P
(wR(u)
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(wR(u)
) ]o\# Â "2 x I'@a7@%$3'&%$9NRW%$G%$*&3!^b
u1 = (u11 . . . u1n) Â (u21 . . . u2n) = u2 ←→ ∃ r ∈ N
∀ i ∈ 1, . . . , r − 1u1i = u2iu1r > u2r
"2u1 = u2 ←→ ∀ i ∈ N : u1i = u2i .
à ] Æ 6OQP!RLS(TU(VWS(XZY) 9*#:;C%$&%b ∀ x, y ∈ X
[∀ i(x, i)Ri(y, i)←→ ∀ j(x, i)Rj(y, i)
].
à ] ¾ pdf=)P!Yµ^f(%$* ¾ Ì@ÊȾR8t*htnpg:nxq@qf\|Jbc Op_gWb@a¾fbcì]`baOp_@qb*dbcrcebm]o]h 2 hpj@bm]n6Ebc OPxq@qBtbBb|pbBtf]`bm]:D$bm]vÂh `uu$b@qaE`llw|htqf\|nxq@qf\|plGbc OTOpbc_@q∀ x, y ∈ X xPy −→ xwsy.
à ] ¾ 8Í bo|fb|pt_@qk_S%$*&3!k:PL:t_9*#:;+%$P&"$*#%$'I%$%$&"2%-*#"$'(wU*#%$<Z!;&@T à ] Ê^¤OQP!RLS(TU(VWS(XZY?:E33j*#:;.
IMBb ∀ u ∈ U A^E w(u) ws(u)
'(!%$ESRW%232"2%]7
ws(u) ⊂ w(u)"2
P(ws(u)
)⊂ P
(w(u)
) T
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u2=(*#'1]
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u1(x, i) = f(u2(x, i)
)∀ (x, i) ∈ X×N'('&E
w(u1) = w(u2)T
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f!*#%9:E3!GODrS%$*(3!
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w ∀ u ∈ U A^EX"$E3%$<C%$9%$39T Ã ] ¾¾ 6OQP!RLS(TU(VWS(XZY)ckw;;@%$3G%$*#"$;*
IBbg\J
u1, u2=(*#'1]\#;=
∀ x ∈ S ∀i ∈ N u1(x, i) = u2(x, i)]#'('&E
w(u1) = w(u2)T
à ] ¿bdf=)P!:Y6^f&2<_E ¾ Ì@ÊËwB
ht b|pbm]zcebg ]`bc mbq¤lw|htqf\|nxq@qf\|plbc OwHhpdbc albce_o|J(]zlGhtQ\pcì]hpcw\pg:nxqhpc aphpc dhtaDLh^]~xqf\|Hh@^||bm]zcebg$q_o|6HhµuL\pjm]h^]zcwhpg$qf\|_@q¥haphpjFpl:Eg:\pc lrq$Pxq@qBtbDLhxqfnxg!c J(]:DLh^]~xqf\|Mfbcì]`_m]`bceblz]` Ã ] ¾ ¿sOQP!RLS(TU(VWS(XZYkf(%$*ge-<Q<C*#/%$;=&%$*GN"$;#TW-
u"2
x, y ∈ X=(*#'1]J\#;=
∃j "2 k ∈ N]<C%B=(E%b
u(y, j) > u(x, j) > u(x, k) > u(y, k)"2 ∀ i ∈ N \ j, k : u(x, i) = u(y, i)
'('&E(x, y) ∈ w(u)
T Ã ] Á pdf=)P!Yz-<Q<C*J/ ¾ Ì@ÊÌ
wRht1bo|pbm]zcebg ]`bc fb@q lw|htqf\|nxq@qf\|plybc OwH¥dbc albce_o|J(]zlht\pcì]hpcw\pg:nxqhpc aphpc dhtaDLh^]~xqf\|Hh$@^||bm]zcebg$q_o|6HshQu$\pjm]h^]zcwhpg$qf\|*_@qh bgJAµhpdQdnpgBF bo|pbg!cwvxq_|Âbc Opblz]`
à ] Ç<df=)P!YC^f(%$* ¾ Ì@ÊÈ/3x, y ∈ X
A^E%xwsy −→ xwRy
"2ws ⊂ wB
Ó °¾½¿Às¿Á´Â¿ÂÃ
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(0, 0)7___\#%$=(9%$9]J<Q!'@E*!*#K2<C%$;&:j0)(/#:9T
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(x1, x2)]n!ST
(y1, y2)B]
%$;=(w'QS%$^]@\J;=Q<Q!*#/'@"$R323jOE%$AÄ%aÖ+K$!G%2*#9Tg>?'('&E(x1− y1)/x1
\J:*(=(/C'&"$R393j'(wGa*( #"$;@"$*#%$'sE:*&=(gTU +%$GNQR3:"$'@r*&=&%$E%2"$;@"$\#%$+'@"$01%23_)%BA3\#%$*d%$(@%$.D\J:*&=(/'(L:*&BD<C"BE*!.+%$GNQR3:"$'&r%$*;@%2/g<C"$*(=@"$9]+%$GNR3:"$'@ '('&E%23'&*#K2%232!Gt)%$3\#%$*i39%$E3!*&9]\Jgb
(x1 − y1)/x1 ≤ (y2 − x2)/x2
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%$;=(<+:3H3'j:E39E!^]o\JI01%2/g!;6%$IM3'G:E ¾ ]n'('@Er32!<Q<C%$3E3!'(#_%$;&!:E3!#<C%$;@G/#:3En )%239"$w*'1]J<Q!k2RW%$%$*(3!^]\#;=I<Q!*#/%$*'(!k%$;=@%$*GNQE"232'0UTYS%$*(3!(<C%$;@G/#:'&*#K2%$0)K$!G'X%2%B%B*#'@"$*(1R%$G%$*(N%$*X'(wGa*( )a9Nt&"$;@%$E%2/g<C"$*&=(E%
&%B9%$3\#%$3*J%B'1]@"2*#"$<Q!G%$;r%$"$EN-3jRW/*#:;@'(''E'(%$E3!7:\J'1]@%$;=( )%$*+9*gA J*s'@a9a%$'1]U<Q!*#/%$;=(!'(w'sM33#Iq&#A^\#+<C"$E3HE%$j3L\#%$=(9%$3E%CS;&<+29<C%$;6:L::9]1L;=HRW 0)2LGK$!G 1!*#/gGM32w'g"$;'@"$001%$* #39ur"$E3%$G%$<_ )%$*a 1%$'06H322'@(/#:3*J:^T?E/%$<C%2.<C%$;RW%$;=@%$2*(w*'I9*( J*M29]&\J_-S%$G%$'Q'&a9a3."$E/%$'('&*LK#!'(3#.*U]
7s<Q!*#/%$*%$;=&%2mG%$\#%$"$;@%2m<C%$;@G/#:%$E3*J3L:Ev*=&*S"$^] '(!H37:7A<+:EC*6'@%2/g&%$9N 6<C%$;@G/#:yS7_!;&2!?'(#/7:3!n\#%$=(9%$t*:B])'('&E *!*#K2=&*P'(J/7:3!k<C%$;@G/#:9])<C%$=i_%$;=&"$*#%$'637:<+:ECEK$!G*J:!GS7X*#%$<[R3:EWA*J'IE37 ^]<Q!*(t_'(!!*#/gj:3!U\#%$=(9%$3w'( )%$*:"2Ow;;@%$3G%$*iQ33#q&#A^^TYS%$*&!G3<C%$E3%$%$3<C%$;@G/#:'_%$<Q<Q!*#%$<V:L*J'X\#"$3'&a2*J0!g\#%$=(9%$%$'&A
)%$*i'@a7@%$%$3-'(#/7:3!k<+;&E3:3E^]1XS%$*(3!n%$<C"$G%$3!'@%$E%$%$'m"2tQ\#"$3'&a2*J0!<+;&E3: '0)K2j:EkK$'I*J0R3!*'( J*6!*#/g'D<C%$;C(!93;&:j'1T Æ ] ¾ 6OQP!RLS(TU(VWS(XZY) '(#/7:3!o\#%$=(9%$3 1%$*iX'@a7@%$3'&%$9N"$E3SRw'1bJ/3N = 1, . . . , n "2 (d, S) 0J:E¾ T
S'&<Q0J'(9]'@*(@%B ⊂ R
n S r%$G"$E3\#%$NC\J32*#:;@%$'(E'D\J<+7¿gT
d ∈ SdX\J32*#:;Q&%$'(E X33#q&#Q0)*(3 J*:
Á T ∃ u ∈ S]g<C%$=(E%
d < uT
Æ ] ¿OQP!RLS(TU(VWS(XZY(G%$;=@%$*Bn n R3:"$'@.'(#/7:3!J\#%$=(9%$"$*#%$'Q\J<+7g]('('&E <C%$;@G/#:
f%$;=
A ⊂ Bn ] f : A −→ Rn w;;&"B*(=&]7<C%$=(E% f(d, S) ∈ S ∀ (d, S) ∈
AT
¾ ÊÌ
Æ ] ¾ <8!:ZYSRW%$Ga RW%%$;=&%2?%$;=@"$*#%$'_\J32*#:;&:%$;=Q/3h%$E3*J3L-<C%$;A
G3L:X%2%$"$*U]#"2G%$;=&%$*dX33#q&#Q0)*(9T
Æ ] Á OQP!RLS(TU(VWS(XZY/3(d, S)
A^E%+"2u ∈ R
n A^* uA^H!*#/g!(!G/gJ:!G*dEK$!G*J:!G3*J'
<C*#/R'1]#\Ju ≥ d (ui ≥ di i = 1, . . .) u
%$EN%$*i!*#/g!(!G/gJ:!G*iEK$!G*J:!G9]J\Ju > d
2t<C*#/R'1]J\#;=uE%BAÄ03!<+:!G9]1\J
u ∈ S"2 ∀ w ∈ S
]#\Jw ≥ u
]'('&E
w = u]J72<C*#/R'1]\#;=I;=&%$*;@"$*mEK$!G*J:!G9]\J ∀ w ∈ S w > u
T Æ ] Ç6OQP!RLS(TU(VWS(XZY&RW%$Ga RW%
Bn nRW:"B'& '(#/7:3!U\#%$=(9%$"$*#%$'D\J<+7:
Bn1 ⊂ Bn,
<C%$=(E% !( ∀ x ∈ S x ≥ dEK$!G*J!PB:
!!((x ∈ S ∧ d ≤ y ≤ x) −→ y ∈ S
Bn0 ⊂ Bn
1 ,<C%$=(E% !!P!( \J
d < y ∈ S'('&E ∀ i, j ∈ N ∃ x ∈ S
xk = yk ∀ x ∈ N \ i, j "2xi < yi
"2xj > yj .
Æ ] Ã yOQP!RLS(TU(VWS(XZY.2X<C*#/R'1]\#;=v6<C%$;@G/#:Q;=&%$*;@%D?E%$AÄ03!<+:G!G9]?\J∀ (d, S) ∈ B
]f(d, S)
;=&%$*;@"$*6E%$AÄ0!<+:!G9T Æ ] Æ 1OQP!RLS(TU(VWS(XZYQq<C%$;@G/#:s%$EN%$*V;= ;=@%$*;@"$*p!*#/g!(!G/gJ:!G*pEK$!G*J:!G∀ (d, S) ∈ B f(d, S)
!*#/g!(!G/gJ:!G*æ%$EN%$*U]?;=d;=&%$*;@"$*:_EK$!G*J:!GSA^ 1%$*
dA^*#"$^T Æ ] Ê^OQP!RLS(TU(VWS(XZY&G%$;=&%$*
π ∈ Π 0)%$E3<_:K$!G] f <C%$;@G/#:.32!P<Q<C%$3E!P'(JS*#*!<¤B]\J ∀ (d, S) ∈ B π(f(d, S)
)= f
(π(d)
]π(S)
) ]\#
π(S) =(π(u)
)u ∈ S
T Æ ] ËCOQP!RLS(TU(VWS(XZYú-3\gA^<C%$;@G/#:9b
n39%$<C"$=&%2n*#%$<32!<Q<C%$3E3!'(#ú-3\gA^<C%$;@G/#:
i'&a9&%$3'@%$9Niw;;&"$*&=f : Bn −→ R
]<C%$=α ∈ Rn α > 0
A^E ∀ (d, S) ∈ Bn
f(d, S)r%$;=&%$3G%$*6=&*m0)*(9]<C%$=I<+7g!<+!7: R3+ n∏
i=1
(ui − di)A^
SA^ 1%$*UT
Æ ] ¾ _df=)P!YWú-3\ ¾ Ì Ã È ¬ |pdb|npcGF\xq 2 hpj@bm]n6E¡nu:]zl(d\pc lrqH<bjvxqbgl(gBFpl:OlF6`$\pc lrqfhpgjhkmlonpg:\pc lrqy]`bc fb@qcZhtyl(jjBbcebaOx\pg$qhpcì]`bjmg:hp]J:Ox\xae@^||bm]zcebg$q_|b_@q¥q@ap\pcwhÁhswägi]zjhpg$qBtEmnpjd\kmlo~^]~pc3@^||bm]zcebg u$npg]nxqfhpg hxaapnpjHNDLhgWbd qBtl(dQd¤bm]zjlra,`q C hxqD8E`dbo|npcGF\xq Æ ] ÌOQP!RLS(TU(VWS(XZY 3!!:E3!#<C%$;@G/#:9]o%$;=6<C%$;@G/#:33!!:E3!#3*J'i*#%$@%$2w*'1]\J ∃ λ ∈ R
3u=@]g\#;=(d, S)
0J:E3E
f(d, S) = argmax[⟨
λ, (u− d)⟩]= argmax
[〈λ, y〉
]
¾ ËÈ jxÈCßUáË:ÕB)â&^Ê@áã6@á7Þ\# 〈 , 〉 Q3'j:E39E37:?RW%$G%T Æ ] ¾ ÈyOQP!RLS(TU(VWS(XZYh>?;&!:E3!#C<C%$;@G/#:
f%$;&!:E3!#C<C%$;@G/#:9]?\J
fE%$A
03!<+:!G "2\J ∃ λ3u= ∀ (d, S) ∈ B
"2
λi
(fi(d, S)− di
)= λj
(fj(d, S)− dj
)∀ i, j ∈ N-
λ = 1]#'('@E
fA^32!<Q<C%$3E3!'(# %$;&!:E3!# <C%$;@G/#:3*J'D*#%$@%$2w*'1T
Æ ] ¾¾ ZOQP!RLS(TU(VWS(XZYf<C%$;@G/#:.<C2;&:.!*&E3!j:*#9]&\J
a ∈ Rn (d, S) ∈ B f(a+
d, a+ S) = a+ f(d, S)T
Æ ] ¾ ¿6OQP!RLS(TU(VWS(XZYf<C%B;@G/#: \#<C;@"$*U]\J ∀ α > 0 ∀(0, S) ⊂ B0
f(0, αS) = αf(0, S)T
Æ ] ¾9Á OQP!RLS(TU(VWS(XZYk2<C*#/R'1]1\#;=i\JM/3(a, S) ∈ BQ
0
"2(b, T ) ∈ BP
0
"2Q < P
]g\#;=(b, T ) (a, S)
A^ )Nk/&/g!'DuR:;@%$*#%$'D )%$&%$9%$"2"$@%$^]\JÙ!(
ai = bi ∀ i ∈ Q!!(u ∈ S
01*&*i'('&E9]g\J ∃ w ∈ T ui = wi ∀ i ∈ Q.
Æ ] ¾ ÇÁOQP!RLS(TU(VWS(XZYr26<C*#/R'1]\J;=f<C%$;@G/#::;@%$*#%$'( )%$*<C*#*U]t\J
∀ (a, S) ∈ BQ0
"2(b, T ) ∈ BP
0
A^E%]n<C%$=(a, S)
A^ 1N/(/g!'uR_:;@%$*#3*#%$' )%$&%$9%BA"2"$@%$^]
fQi (d, S) ≤ fP
i (b, T ) ∀ i ∈ QT
Æ ] ¿edf=)P!YZO\#<Q0#* ¾ ÌË Á ¬ |p dbo|nxc7F^\xqBE`gÁuLnpg]nxqfhpg¢hxaapnpjbce_o|6Jz]zlalh|pbg|b 2 hpjBbm]n6E¡nu:]zl(dhpc lz]\xqm]:HCht8l(jjBbcebaOx\pg$qhpcì]`bjg:h^]J:Ox\xa@^||bm]zcebg$q_o|_m]:H<\|bg$qb@aQ@bgdnpg:n^]npgfHmnpc ^]npg:nxq¥_@qqBtml(dQdbm]zjlra,`qH¡DLhqBtml(dQdbm]zjlra,`q¥b|hpc lz]\xjmlz\pg`qµdb|npcGF\xq
Æ ] ¾9Ã OQP!RLS(TU(VWS(XZY7RW%$Ga RW%I = (I1, . . . , In)
]&\#Ii = supSi : (S1, . . . , Sn) ∈ S+
\#S+ = x ∈ S : x > d %$'('&E I A^!G/%9:!01*&3*J'D*#%$&%$22w'1T
Æ ] ¾9Æ IOQP!RLS(TU(VWS(XZY@rj!Pegf&<CE(/g!*#3'(=H<C%$;@G/#:9b(S, d)
'(#/7:3!g\#%$=(9%$I!G/%BA
:!G01*&3^])'('@Eµ(S, d) ∈ S
<+7g!<+:!G9]<C%$=(E%µi − diIi − di
=µj − djIj − dj
∀ i, j ∈ N.
Æ ] Á df=)P!Y ¬ |^B0E¡npg*_jm]`bc dbBtbm]o]<dbo|npcGF\xqZuLnpg]nxqfhpgÁhxaapnpj 2 hpj@bm]n6E¡nu:]zl(d\pc lrqHqBtml(dQdbm]zjlra,`qH6l(gOxhpjlo\pg$qht hswägä]zjBhpg$qBtmnpjd\kmlo~xapnpgJ_@q¤\|bg$qb@a^bmg¢dnpg:n^]npgfH<DLhh § hpcwhpl dnpjBnTFpl(g$q@adb|npcGF\xqf
¾ Ë ¾
Æ ] ¾ Ê^6OQP!RLS(TU(VWS(XZYo83\!G%$EWA^<C%$;@G/J:*2tx ∈ S
A^"$E3SRw'1]<C%$=(E%
mini∈N
xi − diIi − di
> minj∈N
yj − djIj − dj
∀ y 6= x y ∈ S+.
Æ ] ¾ 8Í bo|pfb|pt_@qnx83\!G%$EWA^<C%$;@G/#:*#%$<Z<Q!*#/g!;+G"$%$$!'1T Æ ] ¾ .;QU1=>@?Y°DLh
(S, d) ∈ B′′_@q
µ(S, d)bjBvxqbg 2 hpj@bm]n6E¡nu:]zlrd\pc lrqHÂhxaapnpj¢hx h6`L]:DlbjaE`dbo|npcGF\xq ∃ _@q¥htnpg:nxqh § hpcwhpl dnpjBnTFpl(g$q@adbo|npcGF\xq@qfhpce
Æ ] ¾ ËsOQP!RLS(TU(VWS(XZYf(%$*ge-<Q<C*#/%$;=@%$*GN"$;f<C%$;@G/#:
AA^*"2
(S, d) ∈ A I!G/%9:!G0)*(3y ∈ S
]g\J ∃ x ∈ S]<C%B=(E% ∃ i, j ∈ N i 6= j
"2yi − diIi − di
>xi − diIi − di
>xj − djIj − dj
>yj − djIj − dj
"2
y 6= f(S, d)
∀ k ∈ N \ i, j yk − dkIk − dk
=xk − dkIk − dk
.
Æ ] ÇMdf=)P!YZ-G%$<Q!GK\ge& \G%$E3 ¾ ÌËË ∃ Cdb|npcGF\xq Bn0
EQBhpgfHdbc l(gOxhpjlo\pg$qhthswägä]zjhpg$qBtmnpjd\kmlo~xajh |pbg|_g 2 hpjBbm]n6E¡nu:]zl(d\pc lrq _@q]`bc fb@qcj\ bmgJ,AµhpdQdnpgBFbo|pbmg!cwvxqf_|Â_@q¥htnpg:nxqh x h6`L]:Dlbjµdbo|npcGF\xq@qfhpce Æ ] Ã edf=)P!YG-G%$<Q!GK\ge& \G%$E3 ¾ ÌËË$Aµh |S| > 2
Hhtht ht¤hpcì]`bjg:h^]J:Ox\xaqBt\pdhg:h|pn8QTQa^bm]o]vpgW_c Hhxaapnpj 6 ∃ hpc a,`fFnt\xqlCdbo|npcGF\xqHIdbc ¥albce_o|6Jz]zlhthswägÂ]zjhpg$qBtmnpjaEd\kmlo~xajhy] Ppjm]`_g:vQl(gOxhpjlohpg:kmlo\^]:HIh|pbg|bbo|p_g!lCjhkmlonpg:hpc lz]\xqm]:H_@qshÅ@^||bm]zcebg$q_o|bm]htµl(jj@bcebaOx\pg$q¥hpcì]`bjg:h^]J:Ox\xax]~pce Æ ] ¾ ÌOQP!RLS(TU(VWS(XZY?/3
S"2
u%2%$"$*01*&*Y'('&E<C*#/R's29]o\#;="$EWA
/%$'('&*LK#!'(3#*U]g\J∀ y ∈ S i ∈ N,
\#;=yi 6= max
si : (s1, . . . , sn) ∈ u(S)
.
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^]\J ∀ d1, d2 ∈ DA^E%
g(d1) = g(d2)\#
DQ33#q*&'D\J<+7gT
Æ ] Æ edf=)P!YG-G%$<Q!GK\ge& \G%$E3 ¾ ÌËËB]_DLh*h3Fn^]o]shpc a,`fFnt\xql6qBtmlz]`$\kmlo~8Qhpg _jFb@a,Eapnpg)lrax]`q$OxhpgfHGhxaapnpj 6 ∃ db|npcGF\qHZdbc albce_|6Jz]zlGhQ|pbg|byb|p_g!ljBhkmlonpg:hpc lz]\xqQ_@qhyqm]h^]`qVy`$n@^||bm]zcebg$q_|Mfbcì]`_m]`bcrblz]`
¾ Ë¿ jxÈCßUáË:ÕB)â&^Ê@áã6@á7Þiµ×@Ö6× =<>B¥G7:HçJñ! "$R3:"$'&3*J'i'&%$h322*!jm37:QSE3!*&*UTn +%$GNM%23+%$;=R3:*j9]n"2\JHr<+:&/g!'+%$PS;&/R3g]('('&Eh<C%$;'0R3:'+R3:*j3 J*D39%$E%$0GN_a39%$;@%$'&%$9T-*#%$<S;&/R3+%$^]1'('&Et%$;=(!'(w'6%r'0%$<Q<Q!9T146!X%$;=&%$*#3u=(!?R3:*jMX 7AG:; J*6%$2 %$PS;&/g*J:C_<+:(/g!'_R3:"$'&Mç1øB "$HR3:"$'&C!GK$!:\J
(s1, s2)¾~# A^EgT-
s1 + s2 ≤ 1].'('@E+<Q!*#/g'@%$3%$*
<C%$;'0R3:'iC!GK$!SRw'@%$9TW-s1 + s2 > 1
]1'('&Et%$;=(!'(w'6%$<'0%$<Q<Q!9T)46!'i3!G32Q33E"$;!j:'m:PBo<C%$;\J:E23tú-3\gAÄ%$;=&%$*#3u=@'!MçJý!
A"2
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A%23XR3:*j9]
D<+:(/g!'@*d0)%2/g!;B]nM<+:3!';=%$PS;&/R3g]U;=%$3LSR3gT&-D*#!P*JK$H<C%$;A
%$;=@%$9"2XM\JE3<+/g!'*J0)*U]'('&EH%$*'(!.%B< '0d%$<Q<Q!9Tn 0)"$*I<Q!P*J/%$*s*J0G%2"$E3"$'@%$GN(/g!'m-R3:"$'@'D37:<+:EÂ
A→ α]B → β
BT#46!n_EK$!G*J:!G 33EB"$;!GMçJç! >?GN+:E-'@K$!GTJ"$hR3:"$'@!GK$!:%B;=M:E3;=(EgTJBT#37:<+:E+X:E3;=M"$E3"B'&% à ]3BT A^*#%$' Æ ThBT?G%$\#%$"$;@%2+!GK$!SRW%$!^bhÇ#] à ] Æ |hBT AÄ"b à ] Æ ]OÊ(Th 6'0R3:E3;=(9].'(!a 1%$?R3:*^]:jtR3:*G3:E*UT a*(%$3G%$*Q%$%$"$*QE*J'1T£ÄESR'-S%$gR3:"$'*#E3<+:jj'R3:9t;=&*6"$E/%$<C%2!GK$!PB:P*#!rM¨[ZZ! k%$'(!*&3w'X çJç! S%$j/#9]@27J'(wGa*( #"$;;@%$^]\#;=X ;=@N2%2*#%$<VtR3:!GK$!SRW"$ y 9%$3!^]@\J*#%$< -<+:(/g!'XG%$;*J;=@ J9(>?2\LR'X<+:&/g!'+:E.'@K$!G*J'1T G/R'I<C%$;CL;=I!G _S%Bj/#9
ì °¾½¿Às¿Á´Â¿ÂÃ
, ?`OÚs ,(Á ;O · ¹¸
É'@a9"$0!G3'&j: *(=&;@*m3u<_])HS%$j/#'D<C%$;@G/#::\#_32w'g"$;@%23#/#:9b *JL2!G9bh/%$E3!:G:9]h%$;=&A_"2Ca (:9Cw;;@"$*(=@%$'v39"$GN&"$E3"$'&%$!^]<C"$E3*!
Ea39%$;@%TÍ.G32L*&5#"$;@37:<QL:9bJ^'(E3!j:!9]J01%BE<H:K$!G]JS%$"$%$G%2tG32L*&5#"$;#]),9=@%2
"$%$G%]:E3\JC"$E3"$'1T¨ £ÄESR*'rS%$):PBG:*J*+%$;=_'&"$39%$<C"$=@%2oR3:"$'&9]\J |S1| = |S2| = 2 T46!'&E?G%23%$;=IEa;2L%$3t33E"$;!j7A^0J:Et/<Q!*J:*#Mz P- " >@?Y
s12 s22s11
(u111 , u112
) (u121 , u122
)
s21(u211 , u212
) (u221 , u222
)
(s11, s
12
) /<Q!*J:*#9]n\Ju111 > u211
]u121 > u221
]u112 > u122
"$u212 > u222
]nMa !33E"$;!j7A^0J:E3EQ\J*G@*/g\J'D<C%$;+_S%$"$%$G%$'1TÏ 4i%$=(!'_R3:"$'@:G32*):g]\J *mG%$**#%HrBTR3:"$'&M
_b BT \ BT l zÍ
(10, 1) (2,−1)(8, 5) (3, 4)
,-b BT \ BT l zÍ
(7, 3) (4, 4)(2, 5) (8, 8)
z P- " >@?YJÍh%$;=(w'Q"232E%]@\#;=C3BT A^*#%$'X<Q!*#/g'&"$nR3:"$'( J*+*I/<Q!*J:*#.33E"BA;!j:R3g]&A^ J*Il]&,hA^ 1%$*+01%2/g!;_zHT£Ä;=C'@"$.%$;=@%$*#3u=1b&A^ J*Ub:ÍX]3lC$](<Q! ¾ È7AÄ%$/BT A^*#%$'1]&,hA^ 1%$* ]3zsB](<Q!oË7AO/IBT A^*#%$'1T&k%$\J:tYR3:"$'&."$E/%$<C%2.:j32*!^T
¾ ËÇ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<ËÔ! /3'&"$hOÍ AÄK2E3*Jg]&<Q!*#/g'@%$3NH'@"$W^R<H5#E3.3;&:E2\JO%$;=C/3h!G/NA0)*(3 J*U])_*#"$9%$3"$;!kE:*(=@'t_j: !U<+:3E3!PM<HSR3gb
1. \ 2. ck!< f&01E3ck!<
(55%, 45%) (52%, 48%)f&01E3(50%, 50%) (45%, 55%)
>?G%$<C%$22w'6tR3:"$'&946!oaE3"$*!'1]\J(F,F)
"BE"$'@%(66, 34)
A^E%:2!'!Mz P- " >@?Y ¾ Tn/<Q!*J:*#_33E"$;!j:R3gbnck!<m]k¿gT AÄ"bf&0)E3 −→ >;=&%$*#3u=1b
(F, S)T
S%$*&3!U:7:*#%$<Z 1%BS=&: R3QH%$;=@%$*#3u=(9TÝ Ô! S%$j/#3 J*i/3?R3:"$'&'@*(@%$E3: R'M:9"$EAÄa39%$;5CR3:"$'(':g]g"2%$**#%$'%$;L"$;@"$@%$U<+;=(E:22'mHS%$*('013%$E%2/g<C"$*(=(9z P- " >@?Y)`E%$'(j:<C9'Q<Q!*#/I*J;=@ I*#"$9%B"$;!)K2E3*J:\#t<C%$**#%$'1]@L;=_'(! y 9%$"2%$'1b (1
2 ,−12) ] (−12 , 12
) ]g!G%$3&%(0, 0)
]g%$\J:t-R3:"$'I*#E3<+:kj'( J*Ub
1. \ 2. ck!< f&0)E3ck!< (
12 ,−12
) (12 ,−12
)f&01E3
(0, 0)(−12 , 12
) .
ÉS%$*(3!U:7:tK2'D7: i%$EN3L%$3%_C/<Q!*J:*#t33E"$;!j:'9])/%rG"$*&=&%$;!n:PA7:3I*#%$<Ú'@939T.Íh%$;=(w'"232E%].\#;=`9"$EAÄa39%$;5YR3:"$'( J*¿gT A^*#%$'K2';=&%B*#;@"B*6/#<Q!P*):*# 33E"$;!j:R3X*UTì" "$?R3:"$'&!GK$!:\J
(s1, s2)¾$# A^EgT!-
s1 + s2 ≤ 1]#'('&E<Q!*#/g'@%$3%B*6<C%$;A
'0R3:'M_!GK$!SRw'&%$9T$-s1 + s2 > 1
]J'('@E%$;=(!'(w'D%$<'06%$<Q<Q!9TO46!'DQ32!;@E3uJ*6/<Q!*J:t33E"$;!j:'!M :O46!'DX;=@%$*;@"$*6/<Q!*J:t33E"$;!j:'!MKO,.!9*&=(L3'D )%]\#;=I*!*#K2 /<Q!*J:*# 33E"$;!jg
z P- " >@?Yry-3BT ¾ AÄ%$+'&"$E9].'('@EIBT<Q!*#/%$*'@"$00È7A^+'0U]hL;=v*#%$< G%$\#%$32!;@E3uJ*6/<Q!*J:t33E"$;!jgT :rÈg] È ¾ '&"$E"2%C;=@%$*;@"$*Y/<Q!*J: R3imÈM'&"$E"239]o<C%$E3_\J6M<+:3!'Èg] ÌÌ7AÄ%$_'&"$E9]
'('&EORW U]1\J ¾ AÄ%B7]U'('@E0)%2/g!;M;=(*#=(*UTo>?Q<Q!*#/g'&"$R3:"$'@3EI!;&@]o%$9"$E3S%$%$\#%$N](\#;=Dr<+:3!'+%$<[S;QÈ7A^O'@"$E3*!^TL£Ä;=+ ¾ AÄ%$O;=@%$*;@"$*M/<Q!*J: R3Q_Èg] ÌÌgT46!*#/%$*<+:
0,01 ≤ x ≤ 0,99 *#%$<ö;=@%$*;@"$*s/<Q!*J:9]U<C%$E3 xDG%$;7RW :j3
1− xA^E%TKG-_G%$**#%r/<Q!*J:*# 33E"$;!jg]J'('&E<Q!*#/%$*m<+: /<Q!*J:G%$**#%T
¾ Ë Ã
ñ! G/R'M<C%$;MQG: # !k<+:3E3!P37?/3.R3:"$'@C/<Q!*J:*#-33E"$;!j:'6%$!<Q!PA*J:j::9^
s1 s2 s3t1 (6, 1) (3, 4) (4, 2)
t2 (7, 2) (1, 1) (2, 5)
t3 (4, 3) (2, 3) (3, 4)z P- " >@?Yt1 > t3
s3 > s1−→
s2 s3t1 (3, 4) (4, 2)
t2 (1, 1) (2, 5)
t1 > t2−→ s2 s3t1 (3, 4) (4, 2)
s2 > s3−→ s2t1 (3, 4)
øB G/R'I<C%$;+Hj: !U<+:3E3!P37n/39"$EAÄa39%$;5QR3:"$'& X/<Q!*J: 33E7A"$;!j:'ME7%$!<Q!*J:j::9^
t1 t2 t3 t4 t5 t6s1 2 2 2 2 2 2
s2 6 4 2 2 3 3
s3 6 5 3 3 4 4
s4 6 9 9 2 3 4
s5 6 5 6 1 4 4
s6 6 5 5 0 4 4
z P- " >@?Yt5 ≥ t6
t5 ≥ t1
t5 ≥ t2
s3 ≥ s2
s3 > s1−→
t3 t4 t5s3 3 3 4
s4 9 2 3
s5 6 1 4
s6 5 0 4
s5 ≥ s6−→t3 t4 t5
s3 3 3 4
s4 9 2 3
s5 6 1 4
t4 > t5−→
−→t3 t4
s3 3 3
s4 9 2
s5 6 1
s4 > s5−→t3 t4
s3 3 3
s4 9 2
t4 ≥ t3−→t4
s3 3
s4 2
s3 > s4−→ t4s3 3
,
%$\J:t_<C%$;@G/#:9b(s3, t4)
Tý! k%$'(!*&3w'6Xj: !JR3:"$'&9Jln:3'D 1%]#\#;=6X;=&%$*;@"$*i/<Q!*J:-33E"$;!j:'%$!<Q!*J:j:CE:*M'&"$'(wGa*& 1a9NC%$E%2/g<C"$*(=(E%R3\J3*'1
BT \ BT l zÍ
(5, 1) (4, 0)4(6, 0) (3, 1)(6, 4) (4, 4)
¾ Ë Æ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëz P- " >@?Y
D ≥ M −→
l zÍ
(5, 1) (4, 0)(6, 4) (4, 4)
L ≥ R−→
lÍ
(5, 1)(6, 4)
D > V−→l
(6, 4)
D ≥ V −→
l z4
(6, 0) (3, 1)(6, 4) (4, 4)
R ≥ L−→
z4
(3, 1)(4, 4)
D > M−→z
(4, 4)
ç! ln:3'm )%])\#;=iC;=&%$*;@"$*/<Q!*J:r33E"$;!j:'MS'&7t'(!'(w#39a )aG"2%X%$;=*#%$<E%$C\J"$'@*(=M%$;=@%$*#3u=(\#r@%$9%$t_G: # !#R3:"$'( J*Ub
l 4 zÍ
(3, 3) (0, 3) (0, 0)>(3, 0) (2, 2) (0, 2)(0, 0) (2, 0) (1, 1)
z P- " >@?Y
M ≥ L "2 E ≥ V −→4 z
>(2, 2) (0, 2)(2, 0) (1, 1)
D ≥ E−→R ≥ M
z
(1, 1);(3, 3)
BT> (1, 1)
(3, 3)
3BT> (1, 1).
¨[Z! n%$'(!*(3w'Ij: !1'(! y 9%$"23!)<+:3E3!P37U/3kR3:"$'&9(4i%$=+33E"$;!j:'C<+7AE/g*J'mX/<Q!*J:t33E"$;!j:'M%$!<Q!*J:j:X:*:MH46!n_3!G32Qú-3\gAÄ%$;=&%B*J3u=!M
l z
(2, 0) (1, 1) (4, 2)4(3, 4) (1, 2) (2, 3),(1, 3) (0, 2) (3, 0)
z P- " >@?Yl z
(2, 0) (1, 1) (4, 2)4(3, 4) (1, 2) (2, 3),(1, 3) (0, 2) (3, 0)
T > B−→
l z(2, 0) (1, 1) (4, 2)4(3, 4) (1, 2) (2, 3)
R > C−→
l z(2, 0) (4, 2)4(3, 4) (2, 3)
BT.A^G"$0 ⇒ BTzOA^ 3BTl1A^G"$0 ⇒ BT4mAÄ%$BT4mAÄ%$G"$0 ⇒ BTl1A^Ñ3BTJA^G"$0 ⇒ BT.A^;=D4mAÄ%$BT,hA^G"$0 ⇒ BTl1A^Ñ3BTzAÄ%$G"$0 ⇒ BT.A^
⇒
⇒ 3!G32Qú-3\gAÄ%$;=&%$*#3u=@'1b4d]3lCO"2]3zs
¾ Ë@Ê
¨J¨ f&7:<C R'+'(!α1<+7g!<Q!*m"2
α2<Q!*!<+7D"$E3"$'&%$'@%$rj: !1'(! y 9%$"23!1<+:WA
E3!P'(Eg
A =
(1 23 4
)B =
(1 34 2
)C =
2 4 6 36 2 4 34 6 2 3
D =
3 2 2 12 3 2 12 2 3 1
z P- " >@?Y_b
maxs∈S
u(s, t1) = 3, maxs∈S
u(s, t2) = 4 −→
−→ α2 = mint∈T
(maxs∈S
u(s, t))= min 3, 4 = 3,
mint∈T
u(s1, t) = 1, mint∈T
u(s2, t) = 3 −→
−→ α1 = maxs∈S
(mint∈T
u(s, t))= max 1, 3 = 3.
-*G@*Ub#,-bα2 = 3
]α1 = 2
|Jtbα2 = 3
]α1 = 2
| bα2 = 1
]α1 = 1
T¨ Ï n%$'(!*(3w'M_j: !U<+:3E3!P37k/39"$EAÄa39%$;5CR3:"$'&9
t1 t2 t3s1 3 4 7
s2 6 5 2
s3 1 9 4-:E22'6<C%$;α1AÄ%$9]
α2A^-"2C\#27:R'mE39I0E3#/%$*#33E"$;!j:'9JÍ.*
R3:"$'(*J'D"$E3"$'&%Mz P- " >@?Y
maxs∈S
u(s, t1) = 6, maxs∈S
u(s, t2) = 9, maxs∈S
u(s, t3) = 7 −→
−→ α2 = mint∈T
(maxs∈S
u(s, t))= min 6, 9, 7 = 6,
mint∈T
u(s1, t) = 3, mint∈T
u(s2, t) = 2, mint∈T
u(s3, t) = 1 −→
−→ α1 = maxs∈S
(mint∈T
u(s, t))= max 3, 2, 1 = 3.
BT7R3:"$'@α1A^\#%$-E39_0E3#/%$*#33E"$;!j:R3gb
s1](3BT7R3:"$'@
α2A^\#a-E39_0E3gA
/%$*#33E"$;!j:R3gbt1Tα1 6= α2
]L;=MtR3:"$'(*J'I*!*#K2 "$E3"$'@%T
¾ ËË mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<˨gÔ! f&7:<QL3'D'(!nHj: !U<+:3E3!P'('k/39"$EAÄa39%$;5CR3:"$'&'D"$E3"$'@"$9
A =
5 3 6 24 1 5 10 4 2 −14 4 5 3
B =
1 2 −3 30 3 1 20 4 2 −11 2 2 2
z P- " >@?YA : α1 = max
s∈S
(mint∈T
u(s, t))= max 2, 1,−1, 3 = 3;
α2 = mint∈T
(maxs∈S
u(s, t))= min 5, 4, 6, 3 = 3 −→
v = α1 = α2 = 3tR3:"$'D"$E3"$'@%T
B : α1 = maxs∈S
(mint∈T
u(s, t))= max −3, 0,−1, 1 = 1;
α2 = mint∈T
(maxs∈S
u(s, t))= min 1, 4, 2, 3 = 1 −→
v = α1 = α2 = 1tR3:"$'D"$E3"$'@%T
¨(Ý 4i%$'('@E*(=@%$E%2"$;@%$k3#/r<+;&:*)';&E*(:*!#.BTR3:"$'@Oj: !R3:"$'( J*U]\JX'@%$&%$E333E"$;!j:t'<+xM
BT \ BT t1 t2 t3s1 (1, 0) (6, 4) (0, 9)
s2 (2, 1) (0, 2) (3, 0)
s3 (3, 7) (2, 3) (4, 0)
z P- " >@?Y
-
6
´´´´´´´´´´´´
³³³³
³³³³³³
³³
QQQQQQQQQQQQ
´´´´´´³³
È ¾¾¿ÁÇÃÆ
E3
E1
E2
¾ ËÌ
s2A^32!;@E3uJ*6/<Q!*J: R3
s3]%$9"$E3%$2BT%t:j32SR3g]gL;=IBT('&%$@%$E333E"$;!j:R3gb
(1− r, 0, r)TBTg*(=@%$E%2"$;@%rBT(G"$01"2"B*J%B'Iw;;&"$*&=&"$ )%$*Ub
t1 : E1(r) = 1 · (1− r) + 2 · 0 + 3 · r = 1 + 2r,
t2 : E2(r) = 6 · (1− r) + 0 · 0 + 2 · r = 6− 4r,
t3 : E3(r) = 0 · (1− r) + 3 · 0 + 4 · r = 4r.BTn2QM*&=&%$E%$<C"$*&=(_3#/R3M;&E*&:*!^]?<Q!h<Q!*#/g\J:E< %$;=@%$*#%2Xj3_*U]n%$D%$E3wG%$-*6 1%$'@%$E%$%$2&%XQ: E:*UT1>?**#%$'m<+7g!<_<+D
E2 = E10)*(3 J**UT
1 + 2r = 6− 4r −→ 6r = 5 −→ r =5
6−→ _;&E*(:\JQ*(=@%$E%2"$;#b
1 + 2 · 56=
6 + 10
6=8
3
|#H%$2;&E*(:G+33E"$;!jX0)%2/g!;#b (16, 0,
5
6
) T¨ì" /#3%$;=69"$EAÄa39%$;5mR3:"$'
A<+:3E3!P37^To46!=&%$**&=&%$E%$<C"$*&=(r3#/i;&7A
E*(:*!.<+;&:*J'YDBT#R3:"$'&MI>?"$E3\J%B!%$2_3!G32633E"$;!j:2(MM46!H%$;=&%$*v3BT]g\#;=MBTg%$<Q<Q!'&"$006%-'0RW*m%$**#"$Ua 1%$M
A =
(1 23 4
)
z P- " >@?Y
-
6HHHHHHHH
HHHHHHHH
È ¾¾¿ÁÇ
E1
E2
BT33E"$;!j:R3gb(r, 1− r)
]#%$'('@E*(=@%$E%$<C"$*(=@%HQ3BTgG"$0)"2"$Now;;@N(%$*Ubt1 : E1(r) = r + 3(1− r) = 3− 2r; t2 : E2(r) = 2r + 4(1− r) = 4− 2r.q;&E*(:\J_*(=@%$E%$<C"$*(=I )%$'&%$E%$%$9%$3OE"232 )%$*I*U](%$**#%$'C<+7g!P<_<+
r = 0A
J**UT?k%$\J:+BT Á AC3#/Y;&E*(:*! <+;&:*J'1T>?2Is23!G3233E"B;!j:9
3#/R3X;&E*&:*!^T-XBTt1AÄ%$R3:32!'1]#'('&EBTg !$*m*#%$<'0ma )%$ Á A^*J:^T
¨gñ! k%$'(!*(3w'Dj: #!h<+:3E3!P37/3 R3:"$'&9UÍ.*3!G326ú-3\gAÄ%$;=@%$*#3u=!MÍ.*D/<Q!*J:*#33E"$;!jM-4i%$'('&Er*&=&%$E%$<C"$*&=(O3#/g*J'+ !23L*!1<+;'(*J'D%$;=@%2R3:"$'@'!M
BT \ BT t1 t2s1 (1, 4) (3, 1)
s2 (4, 0) (2, 2)
¾ ÌÈ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëz P- " >@?Ynú !*#K2/<Q!*J:*#t33E"$;!jgT#ú !*#K23!G32Qú-3\gAÄ%$;=@%$*#3u=&T
-
6
»»»»»»
»»
ZZZZZZZZ
»»»»Z
ZZZ
È ¾¾¿ÁÇ
E1E2
E1(r) = r + 4 · (1− r) = 4− 3r, E2(r) = 3r + 2 · (1− r) = 2 + r,
max : 4− 3r = 2 + r −→ r =1
2−→ 4− 3 · 1
2= 2,5.
-
6
½½½½½½½½
HHHHHHHH
½½
HH
HH
HHH
È ¾¾¿ÁÇ
F1
F2
F1(q) = 4q + (1− q) = 1 + 3q, F2(q) = 2 · (1− q) = 2− 2q,
max : 1 + 3q = 2− 2q −→ q =1
5−→ 1 + 3 · 1
5= 1,6.
k%B\):.BT&¿g] Ã AÄ%B.3#/C;&E*&:*!)<+;&:*J'+ (12 ;12
) 33E"$;!j:2^](3BT0)%2/g!; ¾ ] Æ AÄ%$(15 ,45
) 33E"$;!j:9^T¨#øB rj: !U<+:3E3!P37k/39"$EAÄa39%$;5CR3:"$'( J*M<C%$'('&E_*(=@%$E%$<C"$*(=(3#/;&E*(:*!U<+;&:*J'6rBTR3:"$'@M
¿ Á ¾¾Ê à ¿
z P- " >@?Ys1"2
s23!G32Q33E"$;!j:2n!GK2'M¿7A^3#/M<+;&:*J'D !23L*!^]J/%
%$**#"$#RW7Ua )%$!G%$G"$E3\#%$9]#\J_'&%$@%$E333EB"$;!G:-'<+@T
¾ Ì ¾
E1(x) = 2 · x+ 7 · (1− x) = 7− 5xE2(x) = 3 · x+ 5 · (1− x) = 5− 2xE3(x) = 11 · x+ 2 · (1− x) = 2 + 9x
47g!<_<E2 = E3
A^ J**Ub5 − 2x = 2 + 9x → x =
3
11
]5 − 2 · 3
11=49
11
Tk%B\): BT ( 3
11,8
11
) '@%$&%$E333E"$;!j:9 4911
AÄ%$3#/M<+;&:*J'D !23L*!^T¨gý! %$E%23w'I<C%$;IHj: !n9"$EAÄa39%$;5CR3:"$'@'I*(=@%$E%$;01*&SR3!9
t1 t2 t3s1 −4 5 6
s2 −1 2 4
s3 0 7 5
:t1 t2 t3 t4
s1 −5 2 1 19
s2 6 4 4 8
s3 −5 4 0 −6
Kt1 t2 t3
s1 1 −12 6
s2 0 −4 1
s3 3 −7 2
s4 3 −4 2
s5 5 −4 7
z P- " >@?Yk¿gT A^*J't1/<Q!*J:*# 33E"$;!j:R3g]#ú-3\gAÄ%$;=&%$*#3u=1b
(s3, t1)T
:8t=@%$*;@%rú-3\gAÄ%$;=@%$*#3u=&'1b(s2, t2)
"2(s2, t3)
TKx¿gT A^*J'
t2/<Q!*J:*#p3E"$;!j:R3gTd8t=&%$*;@%Zú-3\gAÄ%$;=@%$*#3u=&'1b
(s2, t2)]
(s4, t2)"$(s5, t2)
T¨gç! £ÄESR'+S%$k_j: !#R3:"$'D*#E3<+:kj'R3:9]#"2'@%$E%23w'I<C%$;IX*(=@%$E%$;01*&'9"$R3:"$'@Q%$;=g39%$E3E%D<HSR36S%$ ¾ ]¿6;= Á RR3:9]"2Q3BT?**&=(! y 9%$CBT A^*#%$'1]<C%$**(=(!kRR3'Da39%$;@%Tz P- " >@?Y ¾ ¿ Á
¾ ¿ Á Ç¿ Á Ç ÃÁ Ç Ã Æ
BT A^*#%$' Á /<Q!*J:*#r33E"$;!j:R3gTo3BT A^*#%$' ¾ /<Q!*J:*#r33E"$;!j:R3gT)£Ä;=(3, 1)
ú-3\gA%$;=@%$*#3u#P=&T
¾ Ì¿ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<ËÏ Z! %$E%23w'I<C%$;IX32!;@E3u6ú-3\gAÄ%$;=&%$*#3u=@'tHj: !#R3:"$'&'( J*U
t1 t2 t3s1 (0, 2) (1, 1) (1, 1)
s2 (1, 5) (2, 2) (1, 0)
s3 (1, 8) (1, 1) (0, 0)
Wt1 t2 t3
s1 (6, 2) (5, 6) (3, 4)
s2 (5, 1) (5, 7) (5, 6)
s3 (4, 1) (4, 7) (4, 6)
s4 (−3, 1) (4, 7) (5, 5)
z P- " >@?YOtBT A^*#%$'t1/<Q!*J:*#H33E"$;!j:R3g]n32!;@E3usú-3\gAÄ%$;=@%$*#3u=i*#!P*JK$9]
;=&%B*#;@%ú-3\gAÄ%$;=&%$*#3u=@'1b(s2, t1)
"2(s3, t1)
T :IBT A^*#%$'
t2/<Q!*J:*#633E"$;!j:R3g]32!;@E3uÉú-3\gAÄ%$;=@%$*#3u=`*!*#K29];=@%$*;@%
ú-3\gAÄ%$;=&%$*#3u=@'1b(s1, t2)
"2(s2, t2)
TÏ ¨ %$E%23w'I<C%$;I ¨J¨ S%$j/#3 J*i/<+:3E3!P'I*(=@%$E%$;01*&SR3!9z P- " >@?Y__b
(s2, t1)|,-bO*!*#K2M*(=@%$E%$;01*&9|-tb!3m3!*#K29] bO*(=&%$E%$;0)*('1b
(si, t4)]i = 1, 2, 3
TÏJÏ
a<Q!=&%B*6"$E3"$'&%$!E%G%23H_j: !#R3:"$'(*J'D32!;@E3u6ú-3\gAÄ%$;=&%$*#3u=&M
a 6 2
−1 a −7−2 4 az P- " >@?Y
a ≤ 7 A^E%*!*#K2.ú-3\gAÄ%$;=@%$*#:=1| −7 < a ≤ −2A^E% 3!*#K29| −2 < a < 1A
E%M3!*#K29ba = −1 A^E% (s1, t1) ;=&%$*;@%Mú-3\gAÄ%$;=&%B*J3u=1| −1 < a < 2
A^E%MBT A^*#%$'s1/<Q!*J:*#i33E"$;!j:R3g] "2
(s1, t1)32!;@E3uú-3\gAÄ%$;=@%$*#3u=1|
a = 2A^E%BT A^*#%$'
s1/<Q!*J:*#+33E"$;!j:R3g]?/%(s1, t1)
K2's;=@%$*;@%Mú-3\gAÄ%$;=&%$*#3u=1|2 < a
A^Ei*!*#K2ú-3\gAÄ%$;=&%$*#3u=@Tk%$\J: −1 < a < 2
%2%$"$*M*632!;@E3u6ú-3\gAÄ%$;=@%$*#3u=&TÏ Ô! /R*'i01"BG/#:_=&*mRW:"B'(EBg]U\#;=@%$*;@"$*s/<Q!*J:_33E"$;!j:''&SR3:'ú-3\gAÄ%$;=&%$*#3u=(9z P- " >@?Y BT \ 3BT l z
Í(1, 1) (0,−1)(−1, 0) (0, 0)"$32!;@E3uú-3\gAÄ%$;=&%$*#3u=m*UbIÍX]3lC "2Q ]3zsBT %HÍx;=&%$*;@"$*/<Q!*J: R3 A
1 > −1 ] 0 ≥ 0
B]?"2Cl;=&%$*;@"$*/<Q!*J: R3zA^ 1 > −1 ] 0 ≥ 0
$TÍ.;=(!G+ú-3\gAÄ%$;=&%$*#3u=( '@ ]3zs.33E"$;!j7A^0J:E <Q!*#/g'&"$;7R3X;=@%$*;@"$*6/<Q!*J:9TÏ Ý Í.*s%$;= ¾ '(<ö\#32us33E*#/o]n<Q!*YM33E*#/G'%$;=&%$*G%$%2%B*d32j*J'%$ST"$^;=(!G*U])C33E*#/G'6\\#_<C%$**#%$'1])<C%$=(!'M'&a9%$G%$ 6*UT#Í:j32*'\#%$=@%$t_^;=(!P9'(*J'1O46!nXú-3\gAÄ%$;=&%$*#3u=!M :O46!n_:E/#<Q!k03!<H<¤M
¾ Ì Á
z P- " >@?Yh V33E*#/g*J'mS%$G%$%23w'm<C%$;6[0, 1]
!*&%$E3<C9]UBTJ\#%$=@%x1]
3BT1\#%$=@%x2]x1 ≤ x2
]x∗ :=
x1 + x22
]oBT1::E3G!^b[0, x∗]
]o3BT AÄ"b[x∗, 1]
T)-I'&a2%BAG%2/g*#%$'M%$;=(<+:3\#@])'('@Etr!*(%$E3<*#N]L;=mQú-3\gAÄ%$;=@%$*#3u=( J*m;=(*#3**J'1]#H03!<+:!G\#%$=D0)%2/g!;C3E3!(!j:!G*
x1 = x2 =1
2
T :
x1 =1
4
"2x2 =
3
4
|E(<C%$;%$3 u
) =1
8
]g<QL;IXú-3\gAÄ%$;=&%$*#3u=( J* 14
TÏ ì" n%$;=(w'C%$^](\#;= ¾ T7R3:"$'&.%$;=
4× 3 A.'(! y 9%$"23!)<+:3E! 37U/3R3:"$'( J*&%$3E"2329Tg4i%$=(!'+&%$'(EOE%$0E%$9%$*(: R3C_'@a7@%$3'&%$9N_'&%$@%$E333E"$;!j:9b#Q¿gTg3!G3233E"$;!j:\J%$<x:j32SR3g] a !?01%2/g!;r%$;=@%$*GN-G32L*(5#"$;;@%$G"$0!(M46!=&%$*&"BG%$3G%$*m<C(/39%$E3E%$n3#/g*J:Q%$2 X'@%$&%$E333E"$;!j:&"$;E%$\JR*!(Mz P- " >@?Y (1
3, 0,
1
3,1
3
) TU"$G/#:./ d%$;='@(K':9^]U"2r\J ¾ AÄ%$H;=¿7A^H/ U]'('&E- ¾ T)33E"$;!j:OR3:37g]J\J Á A-;=MÇAÄ%$9]o'('&E Á T A9])"2t\J Ã AÄat;=Æ AÄ9]J'('@Et_Ç#T AÄ%$9TÏ ñ! 46!k
A<+:3E3!P37/3 9"$EAÄa39%$;5IR3:"$'M%$;=@%$*#3u=(!n'@%$&%$E3 33E"$;!j0J:7A
Eg]9\JtB<+:3E3!P37/3)R3:"$''@%$&%$E3kú-3\gAÄ%$;=@%$*#3u=( ((5
7,2
7
);
(4
7,3
7
)) M
A =
2 2 2 2 2 25 4 2 4 3 36 6 3 5 6 66 9 0 −3 3 46 5 6 1 4 46 5 5 0 4 4
, B =
(3 56 1
).
z P- " >@?Y1Íh%$;=(w'+"232E%]&\#;=A<+:3E3!Pg 1o%$ R3\J3*'
BA^ 1%-_32!;@E3uJ*M/A
<Q!*J: 33E"$;!j:'ME7 %$!<Q!*J:j::2^b
A −→ s3 > s1, s3 > s2 −→ t4 > t1, t4 > t2, t4 > t5, t4 > t6 −→
−→ s3 > s4, s5 > s6 −→ B.
>? )N/&/g!'1]\#;=mr%$E%2/%$3!JR3:"$'D%$;=@%$*#3u=(gb((0, 0,
5
7, 0,
2
7, 0
);
(0, 0,
4
7,3
7, 0, 0
)).
Ï øB 463*'I0)"$G/#: =(*2× 2 2"$EAÄa39%$;5+R3:"$'(Eg]#\#U<Q!*#/%$*mG(/g!o'&%BA&%BE33E"$;!jQ/<Q!*J: R3
s1AÄ%$9]#"2
s2/<Q!*J:n<Q!*#/%$*mG(/g!o'&%$@%$E3 33E"$;!j:9
¾ ÌÇ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëz P- " >@?Y
t1 t2s1 2 1
s2 5 7BT#:j:*#'&%$@%$E333E"$;!j:R3gb(r, 1− r)
TBTg*(=@%$E%$<C"$*(=@%rBT(G"$01"2"$Now;;@N&%$*Ubt1 : 2 · r + 5 · (1− r) = 5− 3r; t2 : 1 · r + 7 · (1− r) = 7− 6r.
5 − 3r > 2"27 − 6r > 1
<Q!*#/%$*0 < r < 1
A^E%]n"25 > 5 − 3r "2 7 > 7 − 6r<Q!*#/%$*
0 < r < 1A^E%T£Ä;=Y%$ RW%23w*#%$'iS%$j/#3 J*/3+S%$P"B%BG%$'1]O<C%$=@%$'
%$E3<C"239%$%2%$*i%B'((!G%$*#%$'M27^]\J;=s2/<Q!*J:*# 33E"$;!jgT
Ï ý! %$E%23w'I<C%$;IHj: !JR3:"$'I'@%$&%$E3ú-3\gAÄ%$;=@%$*#3u=(:9(2, 1) (0, 2)
(1, 2) (3, 0)
z P- " >@?YUBTR3:"$'&33E"$;!j:R3gb(α, 1− α)
]3BTR3:"$'@33E"$;!j:R3gb(β, 1− β)
Tu1(α, β) = 2 · α · β + (1− α) · β + 3 · (1− α) · (1− β) =
= 2αβ + β − αβ + 3 + 3αβ − 3α− 3β =
= 3 + 4αβ − 2β − 3α ∂−→∂α
4β − 3 = 0 −→ β =3
4
u2(α, β) = α · β + 2 · α · (1− β) + 2 · β · (1− α) =
= −3αβ + 2α+ 2β ∂−→∂β
−3α+ 2 = 0 −→ α =2
3.
k%B\):t_'@%$&%$E3ú-3\gAÄ%$;=&%$*#3u=D33E"$;!j:!^b (23,1
3
) "$ (34,1
4
) TÏ ç! %$E%23w'I<C%$;IXú-3\gAÄ%$;=&%$*#3u=@'tHj: !U'&&E/g!*J:K$!GhR3:"$'( J*U
, F,
(2, 2) (0, 0)F(0, 0) (1, 1)
z P- " >@?Y1vR3:"$'Q%$ RW%2%$*D32!<Q<C%$3E3!'(#9](u;=(\#;=QS%$%$\#%$SRw'1]&\#;=IR3:"$'&A'I;=&*#=(*mG32L*&5#"$;;@%$n:j32SR3:'MH%$;=@%23E"$;!j:'7T
u1(α, α) = 2·α·α+1·(1−α)·(1−α) = 3α2−2α+1 d−→dα
6α−2 = 0 −→ α =1
3.
k%B\):t_\J:E<%$;=&%$*#3u=1b ((13,2
3
);
(1
3,2
3
)) ],-],O"2F#]3FBT
¾ Ì Ã
ÔZ! %$E%23w'I<C%$;IHj: !JR3:"$' Á ú-3\gAÄ%$;=@%$*#3u#P=(:9t1 t2
s1 (1, 1) (0, 0)
s2 (0, 0) (2, 2)
z P- " >@?Yk¿X3!G32Xú-3\gAÄ%$;=&%$*#3u=1b(s1, t1)
"2(s2, t2)
T
u1(α, β) = α·β+2·(1−α)·(1−β) = 3αβ−2α−2β+2 ∂−→∂α
3β−2 = 0 −→ β =2
3.
-*G@*U]α = 2
3
Tk%$\J:t_'&%$@%$E3ú-3\gAÄ%$;=&%$*#3u=1b ((2
3,1
3
);
(2
3,1
3
)) TÔo¨ G/R'I<C%$;I_j: !#R3:"$'@9
t1 t2s1 (2, 5) (3, 2)
s2 (1, 0) (6, 1)
z P- " >@?YnO!G32Qú-3\gAÄ%$;=@%$*#3u#P=&'1b(s1, t1)
"2(s2, t2)
Tu1(α, β) = 2αβ + 3α(1− β) + (1− α)β + 6(1− α)(1− β)
= 4αβ − 3α− 5β + 6 ∂−→∂α
4β − 3 = 0 −→ β =3
4;
u2(α, β) = 5αβ + 2α(1− β) + (1− α)(1− β) = 4αβ + α− β + 1∂−→∂β
−→ 4α− 1 = 0 −→ α =1
4.
k%B\):t_'@%$&%$E3ú-3\gAÄ%$;=&%$*#3u=1b ((14,3
4
);
(3
4,1
4
)) TÔ Ï 463'+<C%$;#]\#;=mHj: !#R3:"$'&'(*J'I*!*#K2'&%$@%$E3ú-3\gAÄ%$;=&%$*#3u=('1
(−1,−1) (−9, 0)(0,−9) (−6,−6)
: l 4 zÍ
(1, 0) (1, 2) (0, 1)(0, 3) (0, 1) (2, 0)
K l z
(0, 4) (4, 0) (5, 3)4(4, 0) (0, 4) (5, 3),(3, 5) (3, 5) (6, 6)
¾ Ì Æ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëz P- " >@?YkOBT33E"$;!j:R3gb
(α, 1− α)]BTg33E"$;!j:R3gb
(β, 1− β)T
u1(α, β) = (−1) · α · β + (−9) · α · (1− β) + (−6) · (1− α) · (1− β) =
= −αβ − 9α+ 9αβ + 6α+ 6β − 6− 6αβ =
= 2αβ + 6α+ 6β − 6 ∂−→∂α
2β + 6 6= 0;3!G32Qú-3\gAÄ%$;=@%$*#3u=1b ] BT :4Ú32!;@E3uJ*Y/<Q!*J: R3izA^9]n%$9"$E3X2X%$< :j32SR36mBTR3:"$'&9T£Ä;=
BT33E"$;!j:R3gb(α, 1− α)
]#3BT33E"$;!j:R3gb(β, 1− β, 0)
T
u1(α, β) = α · β + α · (1− β) = αβ + α− αβ = α∂−→∂α
1 6= 0;3!G32Qú-3\gAÄ%$;=@%$*#3u=1bÍX]34 BTK,-]3zs?;G J:!GO<+7g!<_< <Q!*#/g'&"$R3:"$'@3*J'1](L;=C3E3!(!j:!G*D*!*#K2O'@%$&%$E3
33E"$;!j:'m'@3Qú-3\gAÄ%$;=&%$*#3u=@TÔJÔ! -:E22'i<C%$;m+j: !?*#%$<C%$'i\JEK9HR3:"$'i%$;=&%$3G%$*'@%$&%$E3ú-3\gAÄ%$;=@%$*gA3u=(:9
j:*&=
y u<C%2K2K2 <C2!
<C%2K2K2(3, 2) (1, 1)<C2!(1, 1) (2, 3)z P- " >@?Y
u1(α, β) = 3 · α · β + α · (1− β) + β · (1− α) + 2 · (1− α) · (1− β) =
= 3αβ + α− αβ + β − αβ + 2− 2α− 2β + 2αβ =
= 3αβ − α− β + 2∂−→∂α
3β − 1 = 0 −→ β =1
3;
u2(α, β) = 2 · α · β + β · (1− α) + α · (1− β) + 3 · (1− α) · (1− β) =
= 2αβ + β − αβ + α− αβ + 3− 3α− 3β + 3αβ =
= 3αβ − 2α− 2β + 3 ∂−→∂β
3α− 2 = 0 −→ α =2
3.
k%B\):t_'@%$&%$E3ú-3\gAÄ%$;=&%$*#3u=D33E"$;!j:!^b y uUb (23,1
3
) ]gj:*(=1b (13,2
3
) T
¾ Ì@Ê
Ô#Ý n%$'(!*(3w'M_j: !U'@(E/g!*J:K$!G0E G"$<+:9b, F
,(1, 1) (0, 0)F(0, 0) (1, 1)
R3:"$'i2-01"$G/#:27g]1\#;=m<Q!*#/%$;=i<C%$=(!'6G/#G*<C%$;=(w*'_u*U]oK$'HS*(9]g\#;=D<Q!*#/%$*'(!U;=&*#3<C%$*RW%$*UT46!)t03!<+:!G.33E"$;!jg]@\Jr%$;=Q=(*+E37:; Jr'@%$E3ww*'1](\#J*#%$<p3#/R'1]
\#;=I<C%$=(!'MG/#G*m'&%$o<C%$**!(Mz P- " >@?Y#ú !*#K2/<Q!*J:*#33E"$;!jg](!G39*('&"$3!G32-ú-3\gAÄ%$;=&%$*#3u=H!G?*U]L;=M_'&%$&%BE33E"$;!j:'(\#r'&%$1S=&<C(/g*&#*'1b
u1(α, β) = α ·β+(1−α) ·(1−β) = 2αβ−α−β+1∂−→∂α
2β−1 = 0 −→ β =1
2,
\J*G@*`/(/g!'α = 1
2
T?n%$\J:I6iG%$;RW U]?\J6S%$G/ *'Y%$;=d01"$*29]"2+**J'D<C%$;S%$G%$N+G/#G*m<C%$;=(w*'1TÔ)ì" ¿i'&%$E%23'@%2/N:E3 R3i;=(*J2+i%$E3<C"$'&%$9T?>?;=g"$;'&a"$;#bÇ Ã È # ]%$ RW%2+:E9bÆ ÈÈ # ]h/g!G32'@*(I:E9b à ÈÈ # T?Íh* ¾ ÈÈ::E3G]h'(!*#%$< aEN(/g!'i:E3EBS]h"2+%$;=&"BG%$3G%$*#39%$E35#%$*i:j323 13 J*6@%232!U<C%$;I_:E39T#Íh* ¾ ¿ÈX::E3G])'(!UK2'XK2a'('&%$*&%$3t:E*m\JRj*#/Q::E*!^T#46!n_'&%$E%23'@%2/N'D03!<+:!Gt33E"$;!j:R3Mz P- " >@?Y
(C
,C) : u1 = u2 =
100
2· (600− 450) = 7500;
(C
,
) : u1 =100
2· (600− 450) = 7500,
u2 =100
2· (500− 450) + 120 · (500− 450) = 8500;
(
,&) : u1 = 8500, u2 = 7500;
(
,
) : u1 = u2 =100 + 120
2· (500− 450) = 5500 −→
_'(! y 9%$"23!U<+:3E3!P6_'&a9&%$3'@%$9Nb1 \ 2 C &
(7500, 7500) (7500, 8500) (8500, 7500) (5500, 5500)
¾ ÌË mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ë <Q!*J:*# 33E"$;!jX*!*#K29T#"$3!G32Xú-3\gAÄ%$;=&%$*#3u=1b
(C
,
)"2(
,C)T
u1(α, β) = 7500αβ + 7500α(1− β) + 8500(1− α)β + 5500(1− α)(1− β) =
= −3000αβ + 2000α+ 3000β + 5500 ∂−→∂α
−3000β + 2000 = 0 −→ β =2
3,
t32!<Q<C%$3E3!jt<Q!j3α = 2
3
]%$\J:h03!<+:!G33E"$;!j7A^0J:E9b ((23,1
3
);
(2
3,1
3
)) TÔJñ! n%$'(!*(3w'M_j: !U;=&:9_*(=(uR3:"$'@9
(1, 1) (0, 2)(2, 0) (−3,−3)
.Íh*gAÄ%3!G32Xú-3\gAÄ%$;=&%$*#3u=!M :G-:E22'D<C%$;I_'@%$&%$E3ú-3\gAÄ%$;=&%$*#3u=(9K?46!1G32L*(5#"$;@%-**J'1]@\#;=I'@%$&%$E3.ú-3\gAÄ%$;=@%$*#3u=( J*M&%$E%$*&=(9N'
"$G%$3 )%$*6<+E/g*J'!M/O4i%$=(!'I%$;=&%$*#3u=M/R3C_G%$;*J;=@ # m\J32*#trBTR3:"$'&3*J'!M
z P- " >@?YkO"$3!G32Qú-3\gAÄ%$;=&%$*#3u=1bz_]3O"2Q]BT :
u1(α, β) = α · β − 3 · (1− α) · (1− β) + 2 · (1− α) · β =
= −3 + 3α+ 5β − 4αβ ∂−→∂α
3− 4β = 0 −→ β∗ =3
4,
X32!<Q<C%$3E3!jQ<Q!j39bα∗ =
3
4
]g%$\J:t_'&%$@%$E3%$;=&%$*#3u=1b ((34,1
4
);
(3
4,1
4
)) TK
P("$G%$3 1%$*m<+E/g*J'
) = 1−P((H,H)
)= 1− (1−α∗)(1−β∗) = 1− 1
4· 14=15
16.
/u1(α
∗, β∗) =3
4· 34· 1 + 1
4· 34· 2− 3 · 1
4· 14=9 + 6− 3
16=12
16< 2,
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+ 3(1− α− β)(1− µ− ν) = 2αµ+ αν + 2βµ+ 3β − 3βµ− 3βν−
− µ+ αµ+ βµ+ 3ν − 3αν − 3βν + 3− 3α− 3β − 3µ− 3ν + 3αµ+
+ 3αν + 3βµ+ 3βν = 6αµ+ αν + 3βµ− 3βν − 4µ− 3α+ 3.
∂u1∂α
= 6µ+ ν − 3 = 0, ∂u1∂β
= 3µ− 3ν = 0 −→ µ = ν =3
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C = 400]
_a*'@"$*(%2"$;@"$E3hR3:EC )*(#3@bB = 300
TJ>?;=(<+: :*6'@"$E/%$2!'6RW%$Ga%$'@%$9]#\J*ma*'@"$*(%29]#'('@EO&"$;@%]g\J_*!*#K29]'('&EE*J'1]"2O%$E3<C"239%$%2%$*m*!*#K2 )A*(#3@T$ E:9 R'MtR3:"$'@ %Bg%$*2LMj'( J*U]#"2/R'I<C%$;I_E"233R3:"$'&A^'(!G%$G"$;LN%$;=@%$*#3u#P=(7z P- " >@?Y
r
r
r
r
r
r
r@@@@@@
@@@@@@
@@@@@@
BT BT'(!:E '(!:E P1 =1
2
(−100, 0) (0,−100) (0,−400)
(−400, 0)
a*'&"$*&%2 a*'&"$*&%2P2 =
1
2
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r
r
r
r
-@@@@@@
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BT BT'(!P%BE^R%232 '(!%$ESRW%232
(0, 0) (1,−1)
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BT \ BT *('(G%9:E3!G\J: )E3u (!G37G"$01"2*&'(G%9:E3!G\J: )E3u (−L,−L) (−L,−L)(!G37G"$01"2
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2
e 12
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t′2
t2 t′2l z l zÍ
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z P- " >@?YB32E%$&"$%$G%$'1b&BT@*&=&%$E%$<C"$*&=&%w;;@%$3G%$*D3BT3L0#:^|&<Q!*#/g'@"$h3L0#3u3BT A^*#%$'r*Q/<Q!*J:*#?33E"$;!j:R3Q
t2 → L]t′2 → R
BT>?9"$E3%$'(!*(3\#%$SRw'Hu;=@]\#;=3BT 1
2
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A =(5, 1) (1, 0)
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u1(α, β) =1
4· uA1 +
3
4· uB1 =
1
4
(5αβ + α(1− β) + 2β(1− α)+
+ 3(1− α)(1− β))+3
4
(4αβ + 3α(1− β) + 5β(1− α) + 2(1− α)(1− β)
)=
= −14αβ +
1
4α+ 2β +
9
4
∂−→∂α
−14β +
1
4= 0 −→ β = 1,
u2(α, β) =1
4· uA2 +
3
4· uB2 =
1
4
(αβ + 4β(1− α) + 6(1− α)(1− β)
)+
+3
4
(αβ + 3α(1− β) + 6β(1− α) + 3(1− α)(1− β)
)=
= −94αβ +
7
4β − 3
2α+
15
4
∂−→∂β
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7
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2
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4p2 =
1
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z P- " >@?Y(A, p1)
:;@*63BT,A^G"$0U]<C%$E33 > 1
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|(B, p1)
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|(B, p2)
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3 · 3 + 23 · (−1) = 1
3
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4 · 5 = 2T#k%$\J:tBT#,hA^
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4
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P = 1− q1− q2'&%$E%23G%$3!#w;;@"$**(=&%$^T&46!*#/g'&"B.%$E3<C%$GN
12
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¿ÈÌ
z P- " >@?Yπ1(q1, q2) = (1− c1 − q1 − q2)q1, Eq2 =
1
2q2 +
1
2q2 −→
−→ Eπ1 = (1− c1 − Eq2)q1 − q21∂−→
∂q11− c1 − Eq2 − 2q1 = 0 −→
−→ q1 =1− c1 − Eq2
2−→ Eq1 =
1− Ec− Eq22
,
E%$*#/%$2&%2Eq1 + Eq2 = 1− Ec
]#Q32!<Q<C%$3E3!jX<Q!j39b
Eq1 + 2Eq2 = 1− Ec −→ Eq1 = Eq2 =1− Ec
3−→
−→ q1(c1) =1− c1 −
1− Ec
32
=2− 3c1 + Ec
6,
q2(c2)\J*G@*i/&/g!'1T
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(1, 1) (5, 0)
(0, 5) (4, 4) <+:3!'mE23X33E"$;!j:.R3:32!'1]17_'&&01%$E:^]1<QL;D<Q!n!G9])/%H\J+%$;=g39%$Et/%BA9%$E3:*'1]&'('@EN3'&%$9/g@%O<Q!*#/g!;H/%$9%$E3:^T@46!=@%$*C/g!G32'@*(?"$*&=&%$9N-<C%$G%$3"$E/%$<C%2*#%$'(w*'I!GE93r33E"$;!j:R3:37*!^]!G%$3@%&"$;!;C/%$9%$E3:*!(Mz P- " >@?Y?V/g!G32'&*&-"$*(=@%$9NORW%$Ga Rw'
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3r33E"$;!j:R3:32*'1b
4 + 4 · α+ 4 · α2 + . . . = 4 · (1 + α+ α2 + . . .) = 4 · 1
1− α,
<C%$E3α < 1
Tú =&%$E%$<C"$*&=(w*'1]\J_&"$;!;+/%$2%$E3B:P#*'1b5 + 1 · α+ 1 · α2 + 1 · α3 + . . . = 5 + (α+ α2 + α3 + . . .) = 5 +
α
1− α;
4
1− α≥ 5 + α
1− α−→ 4 ≥ 5− 5α+ α −→ 4α ≥ 1 −→ α ≥ 1
4;
%$\J:h\Jα ≥ 1
4
]&'('@EE93 33E"$;!j:."$E/#%B<C%2kR3:37*!^]\Jα < 1
4
]&'('@Eh0)%2/g!;"$E/%$<C%2&"$;!P;+/%$9%$E3:*!^TñJÔ! Oj: !g<+:3E3!P37/3nS;@=r/g!G%$<Q<+hR3:"$'@)R3:32'-&"$;%$G%$*X'g39E9b
(3, 3) (1, 4)
(4, 1) (2, 2) r%$G%$*gS"$nO!OSE 33E"$;!j:R3:32!'1]#7H%$N39aE'&&01%$E:n"2:*JX<Q!*#/g!;2G"$0!^])<Q!-H%$G%$*gS%$G%_H%$GN9NQ'&aE3 )%$*UTJ46!=&%$*6/g!G32'@*(:j:3!U"$*(=@%$9NC<C%$G%$3"$E/%$<C%2E93r33E"$;!j:¥j:/ ñ Ï S%$j/#kR3:37*!^]!G%$3@%&"$;!;C/%$9%$E3:*!(Mz P- " >@?YJú =&%$E%$<C"$*&=(w*'XE23 33E"$;!j%2%$"$*Ub
3+3 ·α+3 ·α2+ . . . = 31−α .ú =&%$E%$<C"B*(=(w*'1]\J_&"$;!;C/%$9%$E3:*'1b
4 + 2 · α+ 2 · α2 + . . . = 4 + 2α1−α
T3
1− α≥ 4 + 2α
1− α−→ 3 ≥ 4− 4α+ 2α −→ 2α ≥ 1 −→ α ≥ 1
2.
k%B\):α ≥ 1
2
%2%$"$*v6E23D33E"$;!jCRW U]n<QL;\Jα < 1
2
]'('@EX"$E/%$<C%2&"B;!;+/%$9%$E3:*!^Tñ#Ý >?;=H%BEwG%B%B*C\#"WR3:'H"2;&j<H 1'X"$*#%$'1] \#"WR3:'_EB:*&=(
α] ;&j<_ )'&"O0)%2/g!;
1− αT"$<+/#:Ej:'&7:'@Et_j: !U<+:3E3!P6/R3_<C%$;I_*(=@%$E%$<C"$*(=@%$'&%$9b
8(−25,−25) (50, 0)8(0, 50) (15, 15)-:E22'D<C%$;Ir%$&u#K$!G*63 !n%$;=@%$*#3u=(9
¿ ¾¾
z P- " >@?Yo E:*(=I'('&EO3 !^]&\JXr\#"WR3:'+"2r;&j<H 1'+:E3\J_*&=&%$E%$<C"BA*(=@%+<C%$;@%$;=@%$2!'1]k<C%$E3_'(wGa*( )%$*M*J;=& d\J32*#:;uYK201E3X%$'&%$9/d370)AE&/*!^T
uH(α) = −25α+ 50(1− α) = 50− 75α, uG(α) = 0 · α+ 15(1− α),
uH = uG −→ 50− 75α = 15− 15α −→ 35 = 60α −→ α∗ =7
12.
ñ)ì" 46!'MX3 !k:j0)'m_j: !U<+:3E3!P37k/3'&&E/g!*J:K$!GR3:"$'( J*:Ms1 s2
s1 (50, 50) (49, 0)
s2 (0, 49) (60, 60)
>?9%$'D'@a2wU<C%$=@%$'m'@3*J'M>.ff@A^Mz P- " >@?Y
u1 = 50x + 49 · (1 − x) = 49 + x]u2 = (1 − x) · 60 ] u ~? . =
x · u1 + (1− x) · u2T ¾ T33E"$;!j:hR3:39'Ia G%$3*&=&%$E%$<C"$*(=@%b
F (x) = x · (u1 − u ~? . ) = x ·(u1 − x · u1 − (1− x) · u2
)=
= x(1− x)(u1 − u2) = x(1− x)(61x− 11);
F (x) = 0]#\J
x = 0]#;= ¾ ]#;= 11
61
TJn%$\J:C3 !:j0)'1bx∗ = 0
]x∗ = 1
]x∗ = 11
61
T>?;=M3 !k:Pj01 >.ff)]#\JF ′(x∗) < 0
T
F (x) = −61x3 + 72x2 − 11x −→ F ′(x) = −183x2 + 144x− 11.
F ′(0) = −11 < 0]F ′(1) = −50 < 0
]F ′(1161
)≈ 9 > 0
T)k%$\J:x∗ = 0
"2x∗ = 1>.ff)]#/%
x∗ = 1161
K2'D3 !n:j019TñJñ! n%$'(!*(3w'M_j: !U<+:3E3!P37k/3S;@=D/g!G%$<Q<+R3:"$'@9
(3, 3) (1, 4)(4, 1) (2, 2)
46!'MX3 !k:j0)'!MrÍ.*m>.ff$Mz P- " >@?Y
uK = 3 · x+ 1 · (1− x) = 1 + 2x]uD = 4 · x+ 2 · (1− x) = 2 + 2x
]uK − uD = −1 T F (x) = x · (1 − x) · (−1) ] F (x) = 0
]n\Jx = 1
]U;=x = 0
Tk%B\):
x∗ = 0"2
x∗ = 1I3 !?:j01'1T
F ′(x) = −1 + 2x ] F ′(0) = −1 < 0]
F ′(1) = 1 > 0])%$\J:
x∗ = 0>.ff)To <Q!?*#%$<À<C%$;G%$0)N]o\!G39%$* ] t/<Q!*J:*#
33E"$;!j7A^0J:E9T
¿ ¾ ¿ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëñ1øB n%$'(!*(3w'M_j: !U<+:3E3!P37k/3R3:"$'@9
(−49,−49) (2, 0)
(0, 2) (1, 1)
46!'MX3 !k:j0)'!MrÍ.*m>.ff$Mz P- " >@?Y
u1 = −49 ·x+(1−x) · 2 = 2− 51x ] u2 = 1−x]u1−u2 = 1− 50x
TF (x) = x · (1 − x) · (1 − 50x) | F (x) = 0
]o\Jx = 1
;=x = 0
;=x = 1
50
Tk%B\):
x∗ = 0]x∗ = 1
"2x∗ = 1
50
C3 !?:j01'1TF ′(x) = 1 − 102x + 150x2
−→ F ′(0) = 1 > 0]F ′(1) = 49 > 0
]F ′(150
)= −0,98 < 0
T#n%$\J:x∗ = 1
50
>.ff)TñJý! lU%$;=@%$*
N = 1, 2, 3, 4 k%$'(!*&3w'mHj: !#R3:"$'&9
v(S) =
1,\J
S = N,
3
4,\J
S = 1, 2, 1, 3, 1, 4, 2, 3, 4,0,
%$;=&"$ '@"$*(9T463'I<C%$;#]\#;=MtR3:"$'D<+;7R3XwE%29z P- " >@?Yn,.!9*(=(L3*'I!*#/g!E%$'(9]
(x1, x2, x3, x4) ∈<+;JT>?'('@E
x2 + x3 + x4 ≥3
4−→ x1 ≤
1
4−→ x2 ≥
3
4− 14=1
2
"2x3 ≥
1
2
"2x4 ≥
1
2,
%$r%$G%$*(3<C*#/#:9TñJç! ¾ T?"2C¿gT)R3:"$'&3*J's* ¾ e ¾ J'@%232=(5RW%]? Á T A^*J'd0)%2/g!;%$;=iRW UT-:E22'D<C%$;ICf&\J0G%$=&AÄ"$E3"$'&%$'@%$9z P- " >@?Y
v(1, 2, 3
)= v
(1, 3
)= v
(2, 3
)= 1
]<Q!*#/%$*<+:S ⊂
1, 2, 3 A^E v(S) = 0 −→ 123 132 213 231 312 321 −→ ϕ3(v) =46 =
23
]ϕ1(v) = ϕ2(v) =
16
TøfZ! >?;=%$j/ä ¾ TJR3:"$'@-0E J:.<C%$;@37 J/g*!%$;=s37:<+:E6a*<+;&: J*v"$EWA"$'(%$G%$*s:E3;=(^]k<C%$=@%$H'@"$r&%$@NS¿gTU"2 Á TR3:"$'@t0E J:h<C%$;@39%$E%$2*!^]k'(!'37:<+:E6:E3;=d"$E3"$'&% ¾ T<-:E22'd<C%$;6'E'(%$E3!G323!'(#Xw;;@"$*(=(9]"2Cf&\J0G%$=&AÄ"$E3"$'&%$'@%$9z P- " >@?Y(rE'(%$E3!G323!'(#nw;;@"$*(=1b
v(1, 2, 3
)= v(1, 2
)= v(1, 3
)=
1]H%$;=@"$ '&"$*&YÈgT
123 132 213 231 312 321 −→ ϕ1(v) =4
6=
2
3
]
ϕ2(v) = ϕ3(v) =1
6
Tø)¨ f&7:<C R'D'(!nCf&\J0G%$=&AÄ"$E3"$'&%$'@%$ '@"$?R3:"$'&%2%$"$*U
¿ ¾9Á
z P- " >@?Yϕ1(v) =
(1− 1)!(2− 1)!2!
v(1)+(2− 1)!(2− 2)!
2!
(v(1, 2
)− v(2))=
=v(1)+ v(1, 2
)− v(2)
2,
\J*G@*ϕ2(v) =
v(2)+ v(1, 2
)− v(1)
2.
ø Ï f&7:<C R''(!.if&\J0G%$=&AÄ"$E3"$'@%$'&%$ Á R3:"$'&H%2%$"$*U]U\Jv(1, 2, 3
)= 1
"2v(i)= 0
<Q!*#/%$*iA^E%T
z P- " >@?Yϕ1(v) =
(1− 1)!(3− 1)!3!
v(1)+(2− 1)!(3− 2)!
3!
(v(1, 2
)− v(2))+
+(2− 1)!(3− 2)!
3!
(v(1, 3
)− v(3))+
+(3− 1)!(3− 3)!
3!
(v(1, 2, 3
)− v(2, 3
))=
=1
6v(1, 2
)+1
6v(1, 3
)+1
3
(1− v
(2, 3
))=
=v(1, 2
)+ v(1, 3
)− 2v
(2, 3
)+ 2
6,
\J*G@*
ϕ2(v) =v(1, 2
)+ v(2, 3
)− 2v
(1, 3
)+ 2
6,
ϕ3(v) =v(1, 3
)+ v(2, 3
)− 2v
(1, 2
)+ 2
6.
øÔ!AS ¾ '(<XA^E%]
B^67: !¿Q'(<XA^E%rS%$'g32!'mQ(L$SE3E:3^T#ÍtL2&%$9%$"$'@%$
"$0L%$*#%$'1]\#;=(*637:'+<C%$;I_'&a"$;@%$Mz P- " >@?Y
v(A) = 1]v(B) = v(A,B) = 3
]g'&"$?R3:"$'@%2%$"$*
ϕi(v) =v(i)+ v(i, j
)− v(j)
2−→
−→ ϕA(v) =1 + 3− 3
2=1
2, ϕB(v) =
3 + 3− 12
=5
2,
¿ ¾ Ç mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ë;=(!G
1 : 5E:*(=( J*m'&%$U%$G32*!k_'&a"$;@%$9T
øgÝ "$Uo2!'-7g!^T2 y 9%$%$*#/N a39%$; à ÈÈÈck9T2 ¾ T$nuoÇ&ÈrA:*J:37:o'(!^T!t;=&*m37:'I<C%$;I_'&a"$;@%$Mµ£ÄESR'+S%$n_'E'(%$E3!G323!'(#w#;;@"$*(=(9z P- " >@?YkrE'(%$E3!G323!'(#w;;@"$*(=1b
v(1)= 2000
"2v(1, 2
)= v
(2)
= 5000T
ϕ1(v) =v(1)+ v(1, 2
)− v(2)
2=2000 + 5000− 5000
2= 1000;
ϕ2(v) =v(2)+ v(1, 2
)− v(1)
2=5000 + 5000− 2000
2= 4000;
%$\J:t ¾ Tg ¾ ÈÈÈQckWAÄ y 9%2%$*U]JQ¿gT0)%2/g!;CÇ&ÈÈÈQckWAÄ9Tø#ì" /R*'-01"$G/#:k%$;= Á 0J:E3n0JE3j<C%$*&3E%]\#gtf&\J0G%$=&AÄ"$E3"$'r<Q!*#/%$*_0J:E33E13
]/%6i<+*#/#:3<XAEB:*&=&'iG%$\#%$NG%$;7RW J*`%$"$E3*#%$'13727:C%$;=g39%$E35a #"$;;@%$U7R!'1]#"2 Ã X )%WR:3!U'(w#39a UTz P- " >@?Y
A = B = 47,5%]C = 5%
]#<C%$E3ABC ACB BAC BCA CAB
CBA]\#J-?!@WAÄ%$G%$<C%$'X**J'Qj:\&u2g]L;=
ϕA(v) = ϕB(v) = ϕC(v) =26 =
13
Tøñ! /R*'-01"$G/#:%$;= Á 0J:E3n0JE3j<C%$*(3E%]\#
C0J:E3?Ètf&\J0G%$=&AÄ"$E3"$'r<C%$G%$3
_G%$\#%$NCG%B;*);=@ 6<+*#/#:3<XAE:**(=(nE%$*#/%$'@%$2!'1]\J 23
AÄ Q3727:9z P- " >@?Y
A = B = 33,5%]C = 33%
](<C%$E3ABC ACB BAC BCA CAB
CBA] −→ ϕC(v) =
06 = 0
]ϕA(v) = ϕB(v) =
36 =
12
TøJøB G/R'H<C%$;H j: !gS%$j/#?\#37?37:<Cj:?*#"$'(w^/3%$;=_a0J:E30JE3j<C%$*(X%$;=g39%$E35da #"$;!O3797:^Tn <+*#/#:3< E:*(=@'dM'@a7&%$3'@%$9N'1b= 26%
]#,= 22%
]J= 19%
] = 18%
]>= 15%
T46!'MCf&\J0G%$=&AÄ"$E3"$'@%$'!Mz P- " >@?Y,:E3<C%$=(!' Á 0J:E3ka #"$;@%$h'@9]/%.%$<C%$=(!'r'&"$k0J:E3?%$< "$E3!g%$#Ã ÈrAÄ9Tg>?9"BE_f(\J0G%$=&AÄ"BE"$'@%$'(*#%$'+<C%$;_'@%$1%B;=&%$2*!w'1T( 7
ϕA(v) = ϕB(v) =ϕC(v) = ϕD(v) = ϕE(v) =
15
Tøý! *#/gE:r9*;@*#/'&2!'1]1\#;=6*&=(!G*gAÄ%X%$;=m<C2!9T ¾ ÈÈrÈÈÈ # N'@"WRW%_*UT-t<C%$;*&=(!SR3t<C2!9]&Èg] ÁÃ G32L*(5#"$;;@%$#*&=&%$E Á ÈÈrÈÈÈ7AÄ%$ )%$G%2"$E33&% ¾ ÈÈÈÈÈ7A%$t!GB])"2Èg] ÆÃ G32L*(5#"$;;@%$?%$@%232Lt<Q!*#/%$*(9T1ztR9 R'MS%B?rR3:"$'M^:R3:9)46!%23t *#/gE:9]\J'&&K':73:jG_39%$<C"$=(!G"$;M Í:2WAÄ%tr33E"$;!j:R3:*U]@\JHÈg] ÁÃ\#%$=@%$3tÈg] Á _*(=@%$E"23!n%2"$=!M
¿ ¾9Ã
z P- " >@?Y
r¡¡¡
@@@
r
r
»»»»»»
»»»»»»
XXXXXXXXXXXX
r
r*(=(!
*#%$<*(=(!
P(*(=@%$E
) = 0,35
P(&%$32L
) = 0,65
¾ ÈÈrÈÈÈ #
Á ÈÈrÈÈÈ #
È #
E = 0,35 · 300 000 + 0,65 · 0 = 105 000 > 100 000]%$\J:<C%$;*&=(!SR3gT
E′ = 0,3 · 300 000 + 0,7 · 0 = 90 000 < 100 000](%$\J: %$ )%$*ir%2%$3 )%$*m*#%$<
*(=(!SR3_<C%$;#Tøç! 4i%$;\!E/%$3w*'M%$;=Iw;=(@"2/g!k:G:37T"$?RW%$G%$*(3'@%$9NQ*Ub>/%r"2lU%2T#>/%rÈg] Ê ÃG32L*&5#"$;;@%$t*&=&%$E3!<C%$;Y0)%$E%$'&%$9ThlU%2E-2+3#/R'1]?\#;=Èg] Ã e(Èg] Ã G32LPA*(5#"$;;@%$U:3j;@WA]g!G%$3@%32:E3w;=(@"2/oT>?;=I32:E3w;=(&"2/MÈg] ÌHG32L*(5#"$;;@%$o*&=&%$E<C%$;Y%$;=01%$E39].%$;=`:3j;@D0)%2/g!;Èg] Ã A^%$^TOÀ<C%$;*&=&%$E3D0)%$E%$'\J32*#:;&d37:7A<_*'(E ¾ ]%$@%232%$3%$'&"ÈgT/g!G32'@*(k"$*(=@%$9NÈg] Ì Ã T4i%$=(!'RW%$%$*(3'&%$9N@%$;=(w'S%$(Mz P- " >@?Yn>/%\J32*#:;&XG%$;=@%$*6>]#lU%2&"r0)%2/g!;+lT
E(L| 32:E ) =∞∑
t=0
0,95t(0,9 · 1 + 0,1 · 0) = 0,9
1− 0,95 = 18,
E(L| :3j;@ ) =∞∑
t=0
0,95t(0,5 · 1 + 0,5 · 0) = 0,5
1− 0,95 = 10,
E(L) = 0,5 · 18 + 0,5 · 10 = 14;
E(E) =∞∑
t=0
0,95t(0,75 · 1 + 0,25 · 0) = 0,75
1− 0,95 = 15.
15 > 14]g%$\J: >/"$&%$;=(w'CS%$^T
ýZ! n%$'(!*(3w'Q øç! S%$j/#9@k%$;=(w'HS%$^]\#;=X*C%$;=_G%$\#%$N"$;w*'Q'(!0E J:*!lU%29T#46!%$;=(w*'1]g\JX<C%$;*&=&%$E3!k_0E J01%$E3M
¿ ¾9Æ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëz P- " >@?YP (32:E | *&=&%$E ) = P (
32:E) · P ( *(=@%$E | 32:E )P (*&=&%$E
)=
0,5 · 0,90,5 · 0,9 + 0,5 · 0,5 =
0,45
0,7=9
14
E(L) =9
14· 18 +
(1− 9
14
)· 10 = 162 + 50
14=212
14>210
14= 15.
k%B\):9]\J_<C%$;*(=@%$E3!kX0E J0)%$E39])'('&ElU%2 "$E/%$<C%2S%$@%$**#!STýo¨ lo!*#%9:E3!G '&%$E%23G%$3!ow;;&"$*&=M"2'(w#a*( 1a9NC!*#%9:E3!G '@a"$;Cw;;@"$*(=(5m'&"$:PAjj %2%$"$*m\J:E22'D<C%$;IC.E3*#WAÄ%$;=&%$*#3u=(9z P- " >@?Y
p = 1− q1 − q2, c1(q1) = c1 · q1, c2(q2) = c2 · q2;
π1(q1, q2) = p · q1 − c1 · q1 = q1(p− c1) = q1(1− q1 − q2 − c1) −→
−→ ∂π1∂q1
= −2q1 − q2 − c1 + 1 = 0 −→ 2q1 + q2 + c1 = 1,
\J*G@*
2q2 + q1 + c2 = 1 −→ q2 = 1− c1 − 2q1 −→ 2(1− c1 − 2q1) + q1 + c2 = 1 −→
−→ q∗1 =1− 2c1 + c2
3−→ q∗2 =
1− 2c2 + c13
.
r%$E%2/g<C"$*(=D'@a2;&9/#:;!j;I"$E3%$<C%29]\J1 > 2c2 − c1
"21 > 2c1 − c2
Tý Ï -:E22'Q<C%$;XH.E3*#WAÄ%$;=&%$*#3u=Q%$E3<C%$G"2"$O"20E y SR3:9]@\J'@"$:jj*U]g*!*#K2'@a"$;+"2
P = 1− (q1 + q2)2
z P- " >@?Yπ1 = q1 · P, π2 = q2 · P, Q := q1 + q2;
∂π1∂q1
=∂
∂q1(q1 − q1Q
2) = 1− (Q2 + 2q1Q) = 0,
\J*G@*∂π2∂q2
= 1− (Q2 + 2q2Q) = 0;
¿ ¾ Ê
X32!<Q<C%$3E3!jQ<Q!j3q1 = q2 =: q → Q = 2q
]
1− (Q2 + 2q1Q) = 1−((2q)2 + 2 · q · 2 · q
)= 1− 8q2 = 0 →
→ q =1√8→ q∗1 = q∗2 =
1√8
π1 =(1− (q∗1 + q∗2)
2)q∗1 =(1−
(1√8+
1√8
)2) 1√8=
=
(1− 4
8
)1√8=
1
2√8→ π∗1 = π∗2 =
1
2√8.
ýJÔ! -*L3'ma39%HI.E3*#WAÄ/g#01G!<À"2tQ<C*#01P!<'(! 1&K2:::9]J\JP = 140−Q
"2C = 20
Tz P- " >@?Y.E3*#9b
π1 =(140− (q1 + q2)
)q1 − 20q1 = 120q1 − q21 − q1q2
∂−→∂q1
120− 2q1 − q2 = 0,
X32!<Q<C%$3E3!jQ<Q!j39b
120− 2q2 − q1 = 0 −→ q1 = q2 = 40 −→ QC = 40 + 40 = 80.
4i*#01!P#<mb
(140−Q) ·Q− 20Q = 120Q−Q2d−→dQ
120− 2Q = 0 −→ QM = 60.
k%B\):QC > QM
Tý#Ý "$-:jj-%2%$"$*i\J:E22'6<C%$;mDf&K'&%$ )%$E3;AÄ%$;=&%$*#3u=(9])\JC!*#%9:E3!Gr'&%BE%$3G%$3!ow;;&"$*&=D"2'(wGa*( )a9NC!*#%9:E3!G'@a"$;ASw;;&"$*&=m/39z P- " >@?Y ¾ Tr`@%$9%$N]r¿gTr`'@a7@%$N]
P = 1 − q1 − q2]c1(q1) = c1 · q1
]c2(q2) = c2 · q2
|
¿ ¾ Ë mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ë
π2 = (1− q1 − q2 − c2)q2 = (1− q1 − c2)q2 − q22∂−→
∂q2
−→ 1− q1 − c2 − 2q2 = 0 −→ q2 =1− q1 − c2
2,
π1(q1, q2(q1)
)=
(1− c1 − q1 −
1− q1 − c22
)q1 −→
−→ (1− q1 + c2 − 2c1)q12
=1− 2c1 + c2
2q1 −
q212
d−→dq1
−→ 1− 2c1 + c22
− q1 = 0 −→ qs1 =1− 2c1 + c2
2,
qs2 =1− 1− 2c1 + c2
2− c2
2=1 + 2c1 − 3c2
4.
ý)ì" "$_:jj_&%$E%$*&=&%$@]P(q1, q2) = 1 − q1 − q2
]c1 = c2 = 0
TC-*L3'a39%rC.E3*#WAO"2 Cf&K'@%$ 1%$E3;AÄ%$;=@%$*J3u=D'(! 1&K2:::9z P- " >@?Y.E3*#9b
π1 = (1− q1 − q2)q1∂−→
∂q11− 2q1 − q2 = 0 −→ q2 = 1− 2q1,
π2 = (1− q1 − q2)q2∂−→
∂q21− 2q2 − q1 = 0 −→
−→ 1− 2(1− 2q1)− q1 = 0 −→ q1 =1
3−→ q2 =
1
3.
f&K'@%$ 1%BE;Á ¾ TJ_&%$9%$NBbπ2 = (1− q1 − q2)q2
∂−→∂q2
1− q1 − 2q2 = 0 −→ q2 =1− q12
;
π1 = (1− q1 − q2)q1 =
(1− q1 −
1− q12
)q1 =
q1 − q212
d−→dq1
−→ 1
2− q1 = 0 −→ q1 =
1
2−→ q2 =
1
4.
¿ ¾ Ì
QS =1
2+1
4=3
4, QC =
1
3+1
3=2
3−→ QS > QC.
ýJñ! "$O%$E3<C%$GN*'+*Q = 100−P
'@%$E%23G%$3!)w;;@"$**(=@%$o"2C = 50
'@a"$;;@%$^T46!nH03!<+:!Gt:E Q,O%$E33E*#/(A^<C&/%2k39%$E3!*&Mz P- " >@?Yn,O%$E33E*#/ob
P1 = P2 ⇒ D1 = D2 =100− P
2
T
πi = (P − C) ·D = (P − 50)100− P
2= −1
2P 2 + 75P − 2500 d−→
dP
−→ −P + 75 = 0 −→ P = 75.
ý1øB "$O%$E3<C%$GN*'+*Q = 130−P
'@%$E%23G%$3!)w;;@"$**(=@%$o"2C = 10
'@a"$;;@%$^Tp'@%$E%23G%$_D,O%$E33E*J/(A^<C&/%$G.39%$E3!*&H32!'i<C%$;iD'@"$r%$E3<C%$GNM'&a9a39T Á G%$\#%$WA"$;@%2r:E-*UbUÊÈg] à Èg])!G%$3&% ¾ ÈgT)£ÄESR'MS%$HR3:"$'&-*#E3<+:hj'( J*U]o"2-'&%$E%23w'<C%$;I_*(=@%$E%$;01*&9z P- " >@?Yn46!*#/g'&%$3NIÊÈgb
D =130− 70
2= 30 −→ π = 30 · (70− 10) = 1800.
46!*#/g'&%$3N Ã Ègb
D =130− 50
2= 40 −→ π = 40 · (50− 10) = 1600.
>?;=(!' Ã Èg]<+:3!'mÊÈgb
D = 130− 50 = 80 −→ π = 80 · (50− 10) = 3200.¾ Ègb#46!*#/%$*6%$%$3 1%$*
π = 0]g<C%$E3
π = D(P − C) = D(10− 10) = 0 TBT \ 3BT P2 = 70 P2 = 50 P2 = 10
P1 = 70 (1800, 1800) (0, 3200) (0, 0)
P1 = 50 (3200, 0) (1600, 1600) (0, 0)
P1 = 10 (0, 0) (0, 0) (0, 0)
(50, 50)32!;@E3umú-3\gAÄ%$;=&%B*J3u=@]
(10, 10);=&%$*;@%ú-3\gAÄ%$;=@%$*#3u=&]
(70, 70)r0gA
3!<+:!Gt_%$E3<C%$GN'M37:<+:EgTýJý! "$O%$E3<C%$GN*'+*
Q = 120−P'@%$E%23G%$3!)w;;@"$**(=@%$o"2
C = 40'@a"$;;@%$^T-*L3'Da39%r+.E3*#WA."2 X,O%$E33EB*#/(AÄ%$;=@%$*#3u=6:E:9
¿¿È mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëz P- " >@?Yn,O%$E33E*#/obπi = (P − 40)
120− P
2= −1
2p2 + 80p− 2400 d−→
dP−P + 80 = 0 −→ PB = 80.
.E3*#9b
P = 120−Q = 120− (q1 + q2) −→ π1 =(120− (q1 + q2)− 40
)q1 =
= 80q1 − q21 − q1q2∂−→
∂q180− 2q1 − q2 = 0 −→ q2 = 80− 2q1,
X32!<Q<C%$3E3!jQ<Q!j3
80− 2q2 − q1 = 0 −→ 80− 160 + 4q1 − q1 = 0 −→ q1 =80
3−→
−→ q2 =80
3−→ pC = 120− 2 · 80
3=200
3.
k%B\):PC < PB
TýJç! /3%$;=i/g!G32'(E"$%$G3$j:3u:j32!0)0j:K$!GTn¿g] Ã <Q!!G_RW 1G/#!?*U]Èg] Ã <Q!!G JGG/#!*U]k"2 ¾ <Q!!G@*v**J'YiK2%$*&3E3<_ J*UT?Z'@"$X@%$E%$*(=(9N0J:E3h!GO\J:E<pS%$*&3!J!G/%2G;!jr'&a2wJ:j323\J9T(`:j32'+E3Er3792*J'1]('(!!G/%2G;!j!j;I'&a9%$G%$ :U\#27:R'1T-Q%$;=@%$*GNC:9@G:;EQ**J'1])'('@E&"BG%$WAG%$*#39%$E35#%$*I:j32*J'1T E:9 R'+R3:"$'@h*#E3<+:Uj'( J*U(46!=@%$*+!G/%2G;!j:'&"B0((!G%$ RW%$*#%$'M_0J:E3'!Mz P- " >@?Y
, F,
(2; 2) (0,5; 3,5) (1; 3)(3,5; 0,5) (2; 2) (1,5; 2,5)F(3; 1) (2,5; 1,5) (2; 2)
BT,A^G"$0 −→ BTgtA^BT A^G"$0 −→ BTgFA^BTFA^G"$0 −→ BTgFA^3BT,A^G"$0 −→ BTtA^3BT A^G"$0 −→ BTFA^3BTFA^G"$0 −→ BTFA^
=⇒
F#]3Ft3!G32Dú-3\gAÄ%$;=&%$*#3u=@T1Í.;=(!GrI0J:E3'(*J'IRW 1G/#!!G/%2G;!j:H"$E/%$<C%2'&"B0((!G%$*!^TçZ! k%$'(!*&3w'M ýJç! S%$j/#9# JGG/#!j'M37:<+_G%$;=&%$* ¾ <Q!!G]J RW 1G/#7A!j'@"32!*&"$*U]"2_K2%$*&3E3<_ J*mG%$;=@%$*#%$'M¿H<Q!!G@*UTJ G/R'+<C%$;QuREXrS%$j/#%$9%$'('@%$kH/#'('^
¿¿ ¾
z P- " >@?Y , F,
(2, 2) (1, 3) (2, 2)(3, 1) (2, 2) (3, 1)F(2, 2) (1, 3) (2, 2)Q]3_3!G32ú-3\gAÄ%$;=&%$*#3u=@T?k%$\J:Ii0J:E3'vm!G/%2G;!j! K2%$*(3E3<H JS;*J'
\#%$=@%$2'&%2/g*!^Tço¨ :j32's!G/%2G;!j!OS%$PS;&::X%$;=s!*&%$E3<Q<+E%$0E%$9%$*&: R'1]<Q!*%$;=@%$*G%$%2%$*m32j*J'+%$^T(Íh*I'&"$.0J:E39]r:j32'IE3E_3722*J'1]g'(!)!G/%2G;!PA!j;C'&a2%$G%$ :o\#27:R'1T#46!=@%$*m!G/%2G;!j:S;*J'I'@"$0((!G%$*!nX0J:E3'!Mz P- " >@?Yh46!*#/g'@"$0J:E3HQ!*(%$E3<ÀS%$G%$9N0)*(SR3:*J'<C%$;S%$G%BGNM!G/%2G;!j:S;7R3_'&"$0((!G%B*!^T#Ä*#/'(j:3j:/6 Ï Ý S%$j/#<C%$;@G/#::*J:^Tç Ï :j32'_!G/%2G;!j!#<C%$;;=@N9N(/"2"$E%$0E%$9%$*&B: R'C
[0, 1]!*(%$E3<Q<+^T
"$O0J:E33*J'IEa;2L%$3!G/%2G;!j!n:j:30)*(SR3_*Ub(0,2; 0,7)
TLt7r\#%$=&%$$'&%2/RW%$*m\JE3<+/g!'+0J:E3<+E/"$'MÊ-3!9%2/%2.01*&.'&a2w^]&\#;=C<+7g!<+:!G.G%$;=@%$*Mr:j3$A!*J'M37:<+MrÉ:j32'D%$G32j::*J'M35E35#"$;w;;&"$*&=&%b
-
6
´´´´´´ Q
QQQQQQÈ Èg] Ã ¾
¾¿
z P- " >@?Y):j32'Q37:<+H-%$E3wG%$3*#%$'XS%$G%$#<C%$;#T&út"$22w'X<C%$;XrÈg] Ã %2%$%$9É:j32'M_ 1%$'@%$E%$%$9%$3 %$E3wG%$%$*6\#%$=@%$2'&%2/g*#%$'M%$^T
-
6
´´´´´´ Q
QQQQQQ
È Èg] ÃÈg] ÁÃ Èg] Æ ¾
¾¿
T0,5 = 1−0,35 · 1,4
2− 0,4 · 1,6
2=
= 0,435.
a !"$E3"$'(E%I\)*G@*v37:<C\JTT0,1 < T0,9 < T0,8 = 0,1
|T0,3 = 0,375
|T0,4 = 0,415
|T0,6 = 0,435
]?%$\J:T0,5 = T0,6
G%$;*J;=@ UTZ£Ä;=\JE3<+/g!'0J:E33*J'mQÈg] à ;=MQÈg] Æ 0)*(3 JQ"$E/%$<C%2\#%$=&%$2'@%2/g*!G%TçJÔ! >?;=d%233wG%$3*#%$'d%$;=s 5*(w;=( )%$*v'&%$O/a*(%$*!G%T?Z%233wG%$ Áà rA39%$E3!*&+:/gG3-*#%$< 5*#a9]o¿ÈZ 5*#a3*#%$'6E3SR3+"2-37:<_59%$"2339%$E%$3*#%_ w*(%$"23*#%$'1]Ç Ã x01%2/g!;Qr'(!&"$;9"23hBE3SR3_G%$;RW I w*(%$"23*#%$'1TLt;=&*I&%$9"$*&=&%$ RW%--%$*#a'X3797:39]g\JC2 'ESR3g]\#;=MX:/gG3 37:<H529"$'!M
¿¿¿ mÈ Òp@áßgßÞnã66Yâ?á,LÊâ<Ëz P- " >@?Y)>?GN39aEh<C%$;@3792SR3g]@\#;=Q 5*#aWAÄ%](\# ÆÃ e ÁÃ E:*&=( J*+'(!\#$A7:'Q 5*#a3*#%$'1T&4R/+3792O w*(%$"2SE3<+:R3:E^T( '(!)39%$E3!*&.*#%$<p 5*#a9]g*(=(!P:*I*#%$<pS;X'(!&"$;2"23E%t3722*!^]&L;= ÃÃ e@Ç Ã E:*(=( J*Mr37:<_59%$"23hS;7R3:'<C%$;@3722*!^T?>?29%$ 6<C&/39%$E3E%$ M%$*#a'd%$G"$E3\#%$3!^]\J;=d6G%$;*#"$0#39%$E353G%$*#%2 &"BG%$<C"$*(=D;=@N29a*UTç#Ý >?;=IE37:; Á :G32'&aE39%$3E%32!'1TBT('@aE39%$3 1%$*Mj'(!'DrE37:;XÇ&ÈrAg]J!3_G'&' ÃÃ rAim,V0J:E33E6372@]nÇ Ã 01%2/g!;ix0J:E33EgTk3BTU'@aE39%$3 1%$*dj'(!'I¿ÈI]1%$**#%$' Æ ÈrAI,0J:E33!^]JÇ&ÈrA+01%2/g!;m0J:E33!^T1V3BT#'@aE39%$3 1%$*i!GtÇ&Èj7A'(!'1]1'(!'(*#%$'i¿ÈrAI,0J:E33!^]1"2ËÈrA^'0J:E33!^T1/R*'m'@"$t:j32:3!?%B R3:E:39]<C%$=@%$'D'(wGa*( )a9NC%$E%2/g<C"$*(=(E%r@%$9%$3*#%$'1z P- " >@?Y&-H:j32'@aE2%$%$*'&"$*( 3722*J'1]#'('@E X,0J:E3*(=@%$E9]g<C%$E3'&"$'&aE39%B )%$*6NQ*&=&%$E9]gE:/#:3n%$__'&"$'@aE39%$trE39:; Æ ÈrA:-/R3gT-3!<+6!G3:C3727:Q*m%$;@"23mE37:; J*U]h'('&E+p*&=&%$E9]<C%$E3
B =0,4 · 0,55+0,2 · 0,6+0,4 · 0,2 = 0,42 "2 C = 0,4 · 0,45+0,2 · 0,4+0,4 · 0,8 = 0,58 Tç)ì" >?;=H:j32:*QÇ.RW%$Ga?!*#/g^b_],-]@t] Tv:j32'HÇ&ÈrA:*J'_0E%BS%$E%$*#K$!GE%$*#/%$9"2%b
A > C > B > D]1<+:3!' Á ÈrA^*J'1b
B > D > C > A]U"2HD<+E/"$'
Á ÈrA^*J'1bD > C > B > A
T1/R*' Á :j32:3!?%$ R3:E:39]o<C%$=&%$' Á '(wGa*( )a9N;=&N$%$3 %$E%2/g<C"$*(=@%$2*#%$'1z P- " >@?Yo3729j01*MK2'Ir0E%BS%$E%$*#K$!jHE%$*#/%$9"2%$GNX%$G%B<C"$'@%$JS%$3w*gA%$3*!^T#>?'('&E _;=&N2%29T %$GN-'@"$%$G%$<C%B'&%$PgS%$3w*&%$3*!^]@"2¿g]!G%$3@% ¾ 0)*(?'0*J'1T
A = 2 ·0,4 =0,8|B = 2 ·0,3 = 0,6 | C = 1 ·0,4+1 ·0,3 = 0,7 | D = 1 ·0,3+2 ·0,3 = 0,9 TÍ.;=(!G%$'('@E _;=&N2%29T _%$GNC\J:E<%$G%$<C%Bt'&%$US%$3w*&%$3*!^]1"2 Á ])¿g]#!G%$3&% ¾ 01*&t'0*J'1T
A =3 · 0,4 = 1,2 | B = 1 · 0,4+3 · 0,3+1 · 0,3 = 1,6 | C = 2 · 0,4+1 · 0,3+2 · 0,3 = 1,7 |D = 2 · 0,3 + 3 · 0,3 = 1,5 T#n%$\J: %$'('&EtqX;=@N2%29TçJñ! "$R3:"$'&3*J'i'&%$h322*!jm37:QSE3!*&*UTn +%$GNM%23+%$;=R3:*j9]n"2\JHr<+:&/g!'+%$PS;&/R3g]('('&Eh<C%$;'0R3:'+R3:*j3 J*D39%$E%$0GN_a39%$;@%$'&%$9T-*#%$<S;&/R3+%$^]1'('&Et%$;=(!'(w'6%r'0%$<Q<Q!9T146!X%$;=&%$*#3u=(!?R3:*jMX 7AG:; J*6%$2 %$PS;&/g*J:C_<+:(/g!'_R3:"$'&Mz P- " >@?Y1ú-3\gAÄ%$;=&%$*#3u=1b
(99, 1)]&<C%$E3O%$2O%$;=QEBKB!G*J:!GO<+:&/g!'(*J'+%$)'&%$P
S;&/g*!jg])\!G39%$*1 > 0
T) G:; J*¿È7e Á ÈCSE3!*&j3-*#%$<39'('iR3:*j%$PS;&/g*!^]<C%$E3<C"$:*(=(j**J'ME3SR3:'1Tç1øB "$HR3:"$'&C!GK$!:\J
(s1, s2)¾~# A^EgT-
s1 + s2 ≤ 1].'('@E+<Q!*#/g'@%$3%$*
<C%$;'0R3:'iC!GK$!SRw'@%$9TW-s1 + s2 > 1
]1'('&Et%$;=(!'(w'6%$<'0%$<Q<Q!9T)46!'i3!G32Q33E"$;!j:'m:PBo<C%$;\J:E23tú-3\gAÄ%$;=&%$*#3u=@'!M
¿¿ Á
z P- " >@?Y(s1, s2)
!GK$!30J:Ehú-3\gAÄ%$;=&%$*#3u= ⇔ s1+s2 = 1]<C%$E3\J
s1+s2 = 1]
'('&E-\JDj'(!%$;=@G/#uJ*d%$"$EH+%$;=@%$*#3u=(^]n'('@E;=i'@%$&%2%$ )%$r'0U];=+%$;=(!'(w'I%$< '0M%$<Q<Q!9Tg46!*#/%$*M<+:33E"$;!j7A^0J:E3E3!(!j:!G*M*#%$<ú-3\gA%$;=@%$*#3u#P=1j:/mH%$GN !U;@*#/j3<C%$*#%$t<C%$;SE/gL::BTçJý!
A"2
B322'@(/g!' ¾°# AÄ*UTJ _%$GN+"2tQ\JE3<+/g!'m*J0)*
A%23XR3:*j9]
D<+:(/g!'@*d0)%2/g!;B]nM<+:3!';=%$PS;&/R3g]U;=%$3LSR3gT&-D*#!P*JK$H<C%$;A
%$;=@%$9"2XM\JE3<+/g!'*J0)*U]'('&EH%$*'(!.%B< '0d%$<Q<Q!9Tn 0)"$*I<Q!P*J/%$*s*J0G%2"$E3"$'@%$GN(/g!'m-R3:"$'@'D37:<+:EÂ
A→ α]B → β
BT#46!n_EK$!G*J:!G 33EB"$;!GMz P- " >@?YJ8-*#/'&&/R*'_(!G377S%$G"T Á T*J01*
B<+:E J:E3<Q!%$ S;&/o]@È7A^!G9T
%r Á Tg*J01* ¾$# ;=&*J**(=(! "$EAA^*J'1]g<Q!*( Q¿gT*J0)*
α# ]LP;=
B03!<+:!G
R3:*jQQ¿gT(*J01*Ub(α, 1− α)
T %rX¿gTg*J0)*(1− α)
# ;=(*J**(=(!"$EBA^*#%$'1]
<Q!*(_ ¾ To*J01*β(1 − α)
T£Ä;=IDEK$!G*J:!G9]U\JA ¾ To*J01* (
1 − β(1 −α), β(1− α)
) A^ R3:*^]#"2B%$PS;&/R3gT
çJç! >?GN+:E-'@K$!GTJ"$hR3:"$'@!GK$!:%B;=M:E3;=(EgTJBT#37:<+:E+X:E3;=M"$E3"B'&% à ]3BT A^*#%$' Æ ThBT?G%$\#%$"$;@%2+!GK$!SRW%$!^bhÇ#] à ] Æ |hBT AÄ"b à ] Æ ]OÊ(Th 6'0R3:E3;=(9].'(!a 1%$?R3:*^]:jtR3:*G3:E*UT a*(%$3G%$*Q%$%$"$*QE*J'1T£ÄESR'-S%$gR3:"$'*#E3<+:jj'R3:9t;=&*6"$E/%$<C%2!GK$!PB:P*#!rMz P- " >@?Y
5 6 7
4 (0; 1) (0; 0) (0;−1)5 (0; 0,5) (0; 0) (0;−1)6 (−1; 0) (−0,5; 0) (0;−1)BTbgÇ+"2 Ã %$'((!G%B*J9]"2;=@%$*;@"$*i/<Q!*J: R3:' Æ AÄ9TJ3BTb
5 ≥ 6 > 7TÍh;=(!G X:E3;=
"$E3"$'&"B*J"Bk"$E/%$<C%2'@%$&%2%$ )%$L;@"$E3*!^T¨[ZZ! k%$'(!*&3w'X çJç! S%$j/#9]@27J'(wGa*( #"$;;@%$^]\#;=X ;=@N2%2*#%$<VtR3:!GK$!SRW"$ y 9%$3!^]@\J*#%$< -<+:(/g!'XG%$;*J;=@ J9(>?2\LR'X<+:&/g!'+:E.'@K$!G*J'1T G/R'I<C%$;CL;=I!G _S%Bj/#9z P- " >@?Y
5 6 7
4 (0; 2) (0; 2) (0; 2)
5 (0; 0,5) (0; 1) (0; 1)
6 (0; 0) (−0,5; 0) (0; 0)BTb)ÇM"2 Ã %$'((!G%$*#%$'1]1"2t;=&%$*;@"$*/<Q!*J: R3:' Æ AÄ9T1BTb Æ "2HÊ+%$'((!G%$*#%$'1]1"2;=&%B*#;@"B*6/#<Q!P*): R3:' Ã AÄa9Tk%$\J:!3 "$E/%$<C%2t_:E3;=M"$E3"$'@"$tR3:*j*!^T
È suOÚ º ,(Á Ë .0¸
) ¾a+ GL6CTTb# "% P $ =<Y&bBkf]`jBb@q¥npg x hxd¤b S D$bnpjf¨s%233(!G%$~?E%29] ¾ ÌËÌgT) ¿ + GIL6CTTb "% P $ = , åZCTWP $ IS?=µ $ =<Y8%2/g!EB]_AµhpgBF8QBnnxa*n x hpdbS D$bBnpjV<lz]:D ¬ k@npg:npdQlokRZuuc lok@h^]zlonpg$q H H H# <C3%$E/#<m](lU*#/*U]gút%$~
E3'M*#/Mk'(=@b#>?G%$(!G%$E9] ¾ ÌÌ¿g] ¾ ÌÌ Ã ] ¾ ÌÌÌgT) Á,+ GI*CP! $J" C+ "% P $ =<Y S D$bbaOxnpc `L]zlonpgÁnk@nnu$bmjh^]zlonpgW(út%$~ E3'1b&,3!GK ,O&'g9]
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