# $ %&' %(() ! " 1 * + $ , " - " 0 0 0 / " 0 " " " " / " 0 0 " 0 " " / 0 1 0 " " " ." . " / 0 " 0 " " " " / 2 " " 2 ,3 0 4 " " " " " " " 5 / 0" 0 4 " " " " ' /0/' / / 6 4" 0 3 * 789:'…' 8 " <2 3 8 / " <2738 <2 38( <2>3 8 " - ;8 ' " 7 " " ' " " 0 " 0 ." " " = 27'<3/ 4 * 5 <2>3 –" - / 4 . " 0 <2 3 ?<2 3 / <2>3 . <2 3@<2 3A<2 3/ . 0 ." 0 " 0 " / 5 * 5 , –" " / 4 " 6826:'B'6 3 27'<3 0 6 !' C 8:6 8<273' C 6 ?<2 3 7/ 0 " " " " . / 6 –" 0 5 4 <2>3 <2 " " 2 7 9D;3 . <2 3 A<2 <2>3 " " " / " " / " 4 ' " 0 0 . / 3 9D;3 . <2 3 27'<3 . 7 # ' 0 " 0 + " 0 " " " " " ' " " / / " " " 0 / 8 , 0 G6 G 5 0 , 0 " " 2F- 3 4 , # ! 6 " 0H $ I 2 ,3 " " / 0 , / 2F 3 " "$ " 0 E $ 0 0 " 0 F- / 9 G" " F- " " ' 0 1 " " " ’ 0 ' " , " " " / " 1 " " , " / " " . , / 0 " " / 10 , ,– " " , / " "" G 0 / 0 0 ,/ " " 0. " 0" 0 " ' " " ' 0 / " 0 ." 1 0 $/ 11 , " ! * " ! * 2:JJ:3 2:JJF3 5 2:JJK3' 5 2%(((3' 5 2%(((3 ' " $ 2%((%3 " ' '* " . H "" " 2%(()3 L $ 0 2#-3 2%(()3 , L 02 2%((&3' #-3 12 ! 2 !-3 " " M ' 0 # 0 / $ 6 $ / . / " 6 6 " 0 / 13 !/ 7 G" 7 89:'…' ;/ / " " "$ . 5 "" 0" 0 8 I(. 4 E % " 1 I " 14 !-2" ' ' 7' " /3 " I(@ 0 GN I "$ / !" " O+ /P 15 0 . 0 . " 7 " " $ " ' 6" . 03 " / !- " / " " L 2L . / 16 ! " - " – "$ "$ + " 8: . 9 4: ( 8 0( 8( !-- 2734 . . 2- 3 ' 6 2:JQR3 4 $ = ;8% 8% 23 2(3 S 8('B 8('B' 9I E @0 ; 8% 2 3 8('B' 2 3A 2(3 8:'B' 23 0 8('B' 17 !-- 273 I:E 0: I%E 0% B I E 0 T273 0 6 2I(@I:@B@I 3 E 20:@B@0 3 7/ 9:'B' T273; : " IE 0 " 18 - - " . U GN 0 0 4 I(@I:@B@I T273 0: @ 0%@B@0 T273/ V "" ' "" 4 " 0" O 0P GN W T273 I 0/ . . / 19 G6 ! ( : % F & R 4 I 0 IE 0 :(( :% R% &( &( :Q( ( :/)R R : (/& (/J .. K/QK82%/K%3% :(/&82F/%%3% &(82K/F%3% :((8:(% %((82:&/:&3% I(@I:@///@I 0(@0:@///@0 .. K& %&/F82&/JF3% %K/F 20 G6 2" /3 – - - " 9:'%;/ ! : % ('&'Q…/ ! F & ('Q':K'…/ ! R (':K'F%…/ " " G :K , " 4 I( 0 $ / :K :K & $ 2 :8 %8&3 Q $ 2 F8 &8Q3 $ 2 R8:K3 $/ " & $ 2 (8&3/ 21 - - 0 0 2- QR " KX / 6 ! :JQR/3 " " 0 " " " /# 0 / ' %X 2 U / ! . QR/3 " " 22 , " H 1 " . 2%((&34 " 5 " !!2%(()3 . " "" 0 –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– " 0 . " / 0 " . " / 0 " " " U / !" " 00 ' " ’ 0 31 , " ! M/M ' / 2#-3 – " 1 " ' M/ * " 0 " $2%((K3 0/ 32 , " - 0 " , ' " " " / G" " 2 " 3/ " " # " 0 " “ 0” ' " 6 . " " / $ 0 Y 33 5 " 789:'…' ; 7 " " 0 - G" 6 " " = / 8 E [:/ :8 (W : % … 6 " W @:8( 34 , “ 0” 2 38 ' " " 2 38 / 2 38 2 3E 2 3/ " " ' " 0 " 0- " <2 38 2 3E 2:. 2 338 2 3E 2 2 3. 2 33/ <2 38(/ 27'<3 " 0 / 35 " + <2 3 <2 3 <2 3[<2 3 <2 3 ' <2 3 <2 <2 3[ <2 7 " 3 <2 3/ 3[<2 3/ <2 3 ' / /' ' . <2 38<2 3' <2 3[<2 3 8 " <2 3W<2 3/ 36 " <2 3 . <2 3 ' / /' <2 0 " 3 <2 3@<2 3' " " 0 . 0 / 37 Is the cost function <2•3 concave ? "$ 2 !-3 " <2 38 2 3E 2 2 3. 2 33 " " 0 " ' 0' " ' . / "$ 0 Y 7 \ 2 6 3 " " 38 How do we prove that (N,V) is balanced? , " # " " " ' 0 27'<3 0 4 # " 2:3 #2738<273 2%3 #2 3 <2 3 7 2F3 #2•3 " " 2 . ' 0 27'#3 " " 27'#3 " " " / / <2•3 #2•3 04 3 " / " 27'<3/ 39 The game (N,W) #2 38 #2 3 " 9<2 3 4 7; " #2 3 8 (' #273 8 <273' #2 3 <2 3 7/ ' #2•3 #2 9 ;3 #2 3 2 ' < 2 •3 3/ " . " 1 / 0' / /' 40 Is W(•) concave? ! " ' #2 38 #2•3 . #2•3 " " , " 9<2 3 4 Y Y 7; . / 41 The Good News 4 27'#3 27'<3 0 0 " " 0 #2•3 % = " " . " 0 / " 4 " 0 " 27'#3/ !"# " " !$# & 42 How to find an allocation in the core? 7 6 " 82 :' %'…' 5 0 ' ( " ) *! !' " 2 " 3 " +!…! ' 27'#3 " 27'#3 4 / " 2: 3 ,# % " ) *! +!…! # ! % *, !"# ! !$# & " 6 2:J):3 \ / " 0 " 2:JQ:33 43 , " + :' : 0 " 7' 0 % %' T 9<2 3 4 <2 : %38<2 :3/ 0 7; " ' T / / 44 , " 0 T 4 ' 7/ 4 #2 3 T/ (4 T8 = ’87] / " 8 6 ’= :4 # <2 T3 <29" ;3 T T 9" ;' ’ ’ ] 9" ;/ ’ " 8 6 ’/ ' " 8( 0 %/ G / %4 #2 38<2 T3/ ! #2 3 T/ 45 G6 < 9:; 9%; 9F; 9:'%; 9:'F; 9%'F; 9:'%'F; J R : :& :( K :R :( :( :( %( %( %( F( (/J (/R (/: (/) (/R (/F (/R T J 9:'%'F; : 9%'F; :E J 9F; )E F 9:'%'F; : 9:'%'F; FE ) 9%'F; : 9:'%'F; # : FE ) :E J : : FE ) : 46 G6 - – 6 " " 6: 6% 6F :'%'F : ( ( :'F'% : ( ( %':'F &E ) FE ) ( %'F': &E ) FE ) ( F':'% QE J ( :E J F'%': &E ) %(E KF :E J 47 + " " # $ " U / 48