ࡱ> (X G : c6"Ordinal solutions6"Ordinal solutionsN:Ordinal path-valued solutions8$Ordinal guidelinesH-4Shapley s ordinal solution8.$Shapley s solutionH04Shapley s ordinal solution43 Ordinal solution/ 00DDavidew RomanTTh{ܖ 0ܖDArialew RomanTTh{ܖ 0ܖ DTimes New RomanTTh{ܖ 0ܖ0DSymbolew RomanTTh{ܖ 0ܖE(.2  @n?" dd@  @@`` ( x {.Kn|/ $33-HBZ^wHYf'x*o/G- "V4L8 '?4'(5I$CIR%7" 2 ^n0+3 ,!#"2r.7k;I=$T)O}3&&%2??-(($%/4iJOP  +0/6%&>/&Q*74, :')A( *"0>3%+&(0.\0/>0F1ZI8<I= ?,@9A6BDERVcHBI=J ?L{ 0e0e    A Ao L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab f3@<^ ʚ;X8ʚ;g4JdJd 00ppp7<4!d!d k 0TD{<4dddd k 0TD{ <4BdBd l 0TJ^0___PPT10 ___PPT9-.03? %O =puZConstructing F$  The solution Ya:  /Pp   0` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>>L0  (    60݉ P  T Click to edit Master title style! !  0߉   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0 ``  X*  0 `   Z*4  0L ` ` H@___PPT9"@ l*$"T  <޽h ? ̙33 Default Design 0 0P(  P P 0| 0   ^*  P 0 C X0  `* d P c $ ?<   P 0 H .*  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S P 6    ^* D P 6h C X` H@___PPT9"@ v*.&  H P 0re"f ? ̙3380___PPT10.W!9 @d(  d d 06  0  6  ^*  d 0\;  C X0 6  `*  d 6;    ;  ^* D d 66  C X` ; H@___PPT9"@ v*.&  H d 0re"f ? ̙3380___PPT10.W!a:D 0 L0 SK04(  4Q 4 3 rfԔ L!?19    ]A family of ordinal solutions to bargaining problems with many players Z. Safra, D. Samet^EAAACEE E* @  0 ] 0]^V 4  `hfԔL!?"+ 0S#,$D   0 www.tau.ac.il/~samet$(2g, 0 4  f fԔL!?"6@`NNN?N d  0H 4 0޽h ? ̙33y___PPT10Y+D=' ; = @B +2" 0L0 z (    S &L   ~4Shapley s ordinal solution    fd13Ԕ s|9)?"6@ NNN?N$ ,$D  0 yShapley proposed a solution to three-player bargaining problems which is Ordinal Efficient Symmetric Individually rationalJ 26% KEEE EE EEE. 0R   f=3Ԕ s|9)?"6@ NNN?N@ |i,$D  0 J^In Safra, Samet (2004) we extended Shapley s solution to any number of players 3. Here, we construct Shapley s solution in a way that lends itself to the construction of a continuum of solutions Ya for a [0,1], (where Y1 is Shapley s solution), with these properties, for any number of players 3. 0 2NAEEEftEEEEAEEEEEE1EEEfEE` < s C3 0 06   `(ZGHԔL!?"6@ NNN?N  ,$D  0 4A solution is ordinal if it is covariant with respect to monotone transformations of each player s utility.Bl 2EEWE  3 rx`GHuoL!?"0@NNN?N] D]F,$D  0 ABy the way, you can click underlined words and see the reference. B(2BEAH  0޽h ?/  ̙33@8___PPT10.Q P+Cg]zD ' h= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D+' =%(D' =A@BBBB0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D+' =%(D' =A@BBBB0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*++0+0 ++0+0 ++0+0 ++0+0 +H 0L0  +(   $ Hs ?"   ,$ 0 A.$(2g0  c $x0   ~4Shapley s ordinal solution l  >   >  ,$D 0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S" B  TDo?" B  TDo?"   H| ?"C>8 A3$(2g  H@w ?" K  E  A2$(2g  H ?"   A1$(2g  < ?"    ?a" 2gB % TDoL!?"0@NNN?N FF ,$D   0B (  `DoL!?"0@NNN?Nh = ,$D  0  HH?" j,$ 0 8VConsider a three player bargaining problem (a,S ) with a disagreement point a . . . . W 2,eEEEEee e,(   l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cRqWONy2L#1 % jH v$obQ5( @            `S" ,$D  0 !  ` fԔL!?"6@ NNN?NPr,$ 0 0... and a bargaining set with Pareto surface S. P1 2,EEAEI " # l fԔL!?"6@ NNN?N,$  0 ;S 2E # N ԔL!?"6@ NNN?NY q +,$ 0 We consider S as the graph of a function 3(x) defined on the plane where 3 s utility is fixed at a3 .h 2 EEEeoeaeaeema & H ?"  ,$  0 A.$(2g03 ' ZToL!?"0@NNN?Np F ,$  0 =x (2E ) H ?" ,$  0 A.$(2g0| + # loL!?"0@NNN?N? ,$ 0 t 3(x)N(2eoeH  0޽h ? ̙33&++___PPT10*+$۔D(' = @B D(' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-g6B fade*<3<*!D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*"D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*"D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-g6B fade*<3<*#D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-g6B fade*<3<*'D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bwipe(down)*<3<*%D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-u6Bwipe(right)*<3<*(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-g6B fade*<3<*)D' =%( D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-g6B fade*<3<*+++0+$0 ++0+0 ++0+!0 ++0+"0 ++0+#0 ++0+&0 ++0+'0 ++0+)0 ++0++0 +W 0L0 ?(7([,@'(  ,B :, TDjJ?"V,$D  0B ;,@ TDjJ?"L,$@  0B J, TDjJ?"3 ,$D   0B K,@ TDjJ?",$@  0z   N,  ,$D   0B O, NDjJ?"6  B P,B NDjJ?"< l    W,  ,$D  0B S, NDjJ?"~ W B T,B NDjJ?" |   ,  0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    , c $x0   ~4Shapley s ordinal solution B  , TDo?" B  , TDo?"    ,  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cRqWONy2L#1 % jH v$obQ5( @            `S"  , H ?">C8 A3$(2g , H ?"K  E  A2$(2g , HH ?"  A1$(2g , <D ?"   ?a" 2g 9, H ?",$ 0 A.$(2g0n <,  0e0e    B8CDE(FA Ao 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 5kxoXA#8 @   S"(,$D  0 L, # 0e0e    BCxDE@F  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||x(JQG{eE0 @     S"0 ,$D   0 M, H%  ?"NU ,$ 0 A.$(2g0 Q, HH%  ?" ,$  0 A.$(2g0z R, 3 0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"( ,$D   0z U, 3 0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0`x Qaq7 @   S" @ ,$D  0 V, H %  ?"  ,$ 0 A.$(2g0 X, H % ?"'=G,$ 0 HConsider 3 equi-valued lines on S & % 2 eeoeee  Z, # lH%  fԔL!?"6@ NNN?N ;S 2EH , 0޽h ? ̙33..___PPT10.+ ܺDy-' % = @B D4-' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X,%(D"*' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*9,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*9,D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;,%(D' =-s6Bwipe(left)*<3<*;,D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:,%(D' =-s6Bwipe(left)*<3<*:,D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<,%(D' =-o6Bwipe(up)*<3<*<,D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*M,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*M,D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*K,%(D' =-s6Bwipe(left)*<3<*K,D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J,%(D' =-s6Bwipe(left)*<3<*J,D' =%( D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L,%(D' =-o6Bwipe(up)*<3<*L,D#' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*Q,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Q,D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*N,%(D' =-s6Bwipe(left)*<3<*N,D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*R,%(D' =-o6Bwipe(up)*<3<*R,D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*V,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*V,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*V,D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*W,%(D' =-s6Bwipe(left)*<3<*W,D' =%(|D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*U,%(D' =-o6Bwipe(up)*<3<*U,++0+9,0 ++0+M,0 ++0+Q,0 ++0+V,0 ++0+X,0 +}R 0L0 22=H0 2(  HB H@ TDjJ?"] ,$@  0B H TDjJ?"  ,$D   0B H TDjJ?" ,$@  0B H@ TDjJ?"+ u}q ,$D  0 1  H# 9 ,$D  0B  HB NDjJ?"1 B  HB NDjJ?"a B  H TDjJ?" 5=),$@   0B  H@ TDjJ?"  ,$D   0V H  0e0e    BC@DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| @ y8g @   S" H8,$D  0V H  0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"h 0 ,$D  0 H  0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"   H c $d:% 0  %  ~4Shapley s ordinal solution B H TDo?" B H TDo?"   H  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cRqWONy2L#1 % jH v$obQ5( @            `S"  H H>%  ?">C8 A3$(2g H HA%  ?"K  E  A2$(2g H HE%  ?"  A1$(2g H <I%  ?"   ?a" 2gF H 3 0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"( ` H # 0e0e    BCxDE@F  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||x(JQG{eE0 @     S"0 p H  0e0e    BCxDE@F  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||x(JQG{eE0 @     S" P ,$D 0@ H # 0e0e    B8CDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 5kxoXA#8 @   S"(P H  0e0e    B8CDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 5kxoXA#8 @   S"X  ( ,$D 0F H 3 0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0`x Qaq7 @   S" @   H HN% ?"(,$ 0 :X& and their projections on the plane x3 = a3- 2e"eeemmeem +B#style.visibility<* H%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bwipe(up)*<3<*HD' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bwipe(up)*<3<*HD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bwipe(up)*<3<*HD' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bwipe(up)*<3<*HD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bwipe(up)*<3<*HD' =%( D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD' =%( D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-o6Bwipe(up)*<3<* HD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-o6Bwipe(up)*<3<* HD' =%( D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*H+8+0+ H0 +  0L0 VV6EXDV(  X" X  0e0e    BC@DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| @ y8g @   S" H8" X  0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"h 0  X  0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"   X c $4u% 0  %  ~4Shapley s ordinal solution B X TDo?" B X TDo?"   X  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cRqWONy2L#1 % jH v$obQ5( @            `S"  X H\y%  ?">C8 A3$(2g X H}%  ?"K  E  A2$(2g X H%  ?"  A1$(2g X <D%  ?"   ?a" 2gF X 3 0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"( ` X # 0e0e    BCxDE@F  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||x(JQG{eE0 @     S"0 < X  0e0e    BCxDE@F  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||x(JQG{eE0 @     S" P @ X # 0e0e    B8CDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 5kxoXA#8 @   S"( X  0e0e    B8CDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 5kxoXA#8 @   S"X  ( F X 3 0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0`x Qaq7 @   S" @ F %X  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.~`@  S" ,$D  0f &X  0e0e    BCPDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  8H`pD@HbP @    S"@ ,$D  0V 'X  0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %7sHM4x @   S" ,$D  0V (X  0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0u (|1 @   S" ,$D  0f )X  0e0e    B CDE4F1 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4!@ {8P%W  @    S" ,$D  0l  x :X x,$D 0B !XB HDo?"  B "X HDo?"  #X H@%  ?"3x A1$(2g $X H%  ?"   A2$(2g +X <(%  ?" E ? ?a" 2gl   X ,$D 0B /X HD?"  B 0X HD?" l   ?X ,$D 0B 1X HD?" B 2X HD?" 3X H(%  ?"r ,$ 0 A.$(2g, 4X H%  ?"X ez8 ,$ 0 A.$(2g, 5X Hd%  ?" Z ,$  0 A.$(2g, 6X H%  ?"h pH,$  0 A.$(2g, 7X  >0e0e    BHCDE|F4 jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||'8Ohp+ SMx@l0(k(xPHO=PH@          S" ,$D  0 8X HL% ?": JZ F  2e  9X H% ?"Pz,$ 0 dThese projections form a family of Pareto surfaces for 1 and 2 parameterized by the value of 3 , that is, player s 3 utility.L 2eaeeeeeeeeeeoee e& ;X Bؿ% ?"{x,$  0 r6For each surface in the family choose the ideal point. 7 27e AX N|% GH?" 8 N,$D 0 nThe path crosses the surface S at exactly one point L3x8(2eeeD @X T% fԔX)?",$D 0 ~VThese ideal points form a monotonic path p3 parameterized by the utility of player 3 Wncg c g ggemg&ggg BX H% ?"6 ,$  0 Tp36 2em CX # l%  fԔL!?"6@ NNN?N ;S 2Eh EX # l%  fԔL!?"6@ NNN?N  ,$  0 ZL3: 2H X 0޽h ?AX ̙33>>___PPT10>+pn(D[<' % = @B D<' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9X%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*:X%(D' =-o6Bdissolve*<3<*:XD/ ' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%X%(D' =-o6Bwipe(up)*<3<*%XD' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&X%(D' =-o6Bwipe(up)*<3<*&XD' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'X%(D' =-o6Bwipe(up)*<3<*'XD' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(X%(D' =-o6Bwipe(up)*<3<*(XD' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)X%(D' =-o6Bwipe(up)*<3<*)XDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;X%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*+B#style.visibility<*6X%(D' =-o6Bdissolve*<3<*6XD' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*=X%(D' =-o6Bdissolve*<3<*=XD' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5X%(D' =-o6Bdissolve*<3<*5XD' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*>X%(D' =-o6Bdissolve*<3<*>XD' =%( D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4X%(D' =-o6Bdissolve*<3<*4XD' =%( D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*?X%(D' =-o6Bdissolve*<3<*?XD' =%( D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3X%(D' =-o6Bdissolve*<3<*3XD' =%(D' =%(DL' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@X%(D' =-{6Bslide(fromTop)*<3<*@XD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7X%(D' =-s6Bwipe(down)*<3<*7XD#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*BX%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*BXD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*BXD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*AX%(D' =-o6Bdissolve*<3<*AXD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*EX%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*EXD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*EX+0+0+3X0 ++0+4X0 ++0+5X0 ++0+6X0 ++0+9X0 ++0+;X0 ++0+AX0 ++0+@X0 ++0+BX0 ++0+EX0 +0 0L0 >(6(Y\'(  \" \  0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"h 0  \  0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"   \ c $( 0  (  ~4Shapley s ordinal solution B \ TDo?" B \ TDo?"   \  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cRqWONy2L#1 % jH v$obQ5( @            `S"   \ H8(  ?">C8 A3$(2g  \ H(  ?"K  E  A2$(2g  \ Ht(  ?"  A1$(2g  \ < (  ?"   ?a" 2gF  \ 3 0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"( " \  0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %7sHM4x @   S" 2 \  0e0e    B CDE4F1 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4!@ {8P%W  @    S" F  x \  xB \B HDo?"  B \ HDo?"  \ H#(  ?"3x A1$(2g \ H4((  ?"   A2$(2g \ <+(  ?" E ? ?a" 2gjF   $\  B %\ HD?"  B &\ HD?"  +\ Hh0(  ?"X ez8  A.$(2g, /\ H4( ?": JZ F  2e  N\ # ll8(  fԔL!?"6@ NNN?N ;S 2E4 O\ # l=(  fԔL!?"6@ NNN?N   ZL3: 2 W\ HB(  ?"W -7 ,$ 0 A.$(2g,^ X\  `J( 3Ԕ s9p?"6@ NNN?Nc,$D 0 \The solution (a,S) L3 (a,S) is ordinal, and symmetric with respect to 2 and 3. Using L3 we define now an ordinal symmetric solution F.  2EEEEEEEEEE EEEEE E    EE EE,oH \ 0޽h ? ̙33___PPT10+LSKD"' ; = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W\%(D' =-o6Bdissolve*<3<*W\D' =%(D' =%(DL' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X\%(D' =-{6Bslide(fromTop)*<3<*X\+p+0+W\0 ++0+X\0 +~ 0L0 @@78r@(  z     ,$D 0B  NDjJ?"6  B B NDjJ?"< z       ,$D  0B  HD?"  B B HD?" f z f   f  ,$D 0B   HD?"*f B   HD?" "    0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"h 0     0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"     c $@j( 0  (  ~4Shapley s ordinal solution B  TDo?" B  TDo?"     l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cRqWONy2L#1 % jH v$obQ5( @            `S"   Hhn(  ?">C8 A3$(2g  H s(  ?"K  E  A2$(2g  Hv(  ?"  A1$(2g  <hz(  ?"   ?a" 2gF  3 0e0e    BC0DE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0,W]Wc! @   S"( "   0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %7sHM4x @   S" 2   0e0e    B CDE4F1 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4!@ {8P%W  @    S" F  x   xB B HDo?"  B  HDo?"   H0~(  ?"3x A1$(2g  Ht(  ?"   A2$(2g  < (  ?" E ? ?a" 2gjF     B  HD?"  B   HD?"  ! H\(  ?"X ez8  A.$(2g, " H( ?": JZ F  2e B # HDo?"o ,$D   0z   $  ,$D 0B % HDo?"` B & HDo?"* i  ' HD(  ?"X m ,$  0 A.$(2g, ( H(  ?" O d ~ ,$ 0 A.$(2g,Cz   )  ,$D 0 * H(  ?"  A.$(2g0 + H( ?"   Tx36 2em* , HH( ?"ok u,$ 0 pvThe point L3 is attained for utility level x3 of player 3. < 2 eeemm eee6 - H( ?"w },$  0 |dIncreasing 3 s utility to x3 results in a point F.3 2 eeeee eeemEE . T( ?". 5 ,$  0 >F  2 / H( ?"7 ( x W ,$ 0 ZL3: 2 0 # l,(  fԔL!?"6@ NNN?N ;S 2E4 1 # l(  fԔL!?"6@ NNN?N   ZL3: 2h 3 # l$(  fԔL!?"6@ NNN?N,$  0 ZL1: 2h 4 # l(  fԔL!?"6@ NNN?N H4 ,$  0 ZL2: 2 5 H(  ?"l ,$ 0 A.$(2g, 6 H(  ?"W -7 ,$ 0 A.$(2g,z 7  `@( GH1L!?"6@ NNN?N+ 5,$D   0 xIt is easy to see that the projections of F on the plains x1= a1 and x2= a2 are L1 and L2. ] 2*eEMEEMEMEEMEee    eAV 8  `) 3Ԕ s9p?"6@ NNN?N,$D 0 TThe solution (a,S) F(a,S) is ordinal and symmetric, but alas, it does not lie on S. This is fixed now...0k 2EEEEEEEEEE EEEE,QH  0޽h ?7 ̙33<<___PPT10<+r*D9' ) = @B DN9' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bdissolve*<3<*(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-o6Bdissolve*<3<*)DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(down)*<3<*#D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =%(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-o6Bdissolve*<3<*.D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*7D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*7D?' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-u6Bwipe(right)*<3<*$DN ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*3D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*3D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*4D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*4D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bdissolve*<3<*5D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bdissolve*<3<*6D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*7D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*7D' =-g6B fade*<3<*7D' =1:Bhidden*o3>+B#style.visibility<*7%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8++0+'0 ++0+(0 ++0+,0 ++0+-0 ++0+.0 ++0+/0 ++0+30 ++0+40 ++0+50 ++0+60 ++0+70 ++0+70 ++0+80 +Y 0L0 (  sz *~  :,$D 0  T+) fԔX)?"*~ =  2a   H@0) ?"h  |a0 = aZ 2emae  H5) ?"(,$ 0 0:Starting with the problem (a, S) we generate the sequence;e3eaeeeeaeae3  HpA) ?"  ,$  0 0a1 = F(a0, S)(2emamaemeeea  HP) ?"  ,$  0 La2 = F(a1, S)(2gococgogggc   H[) ?"   ,$  0 Ja3 = F(a2, S)(2gococgogggc   Hg) ?"f  : ,$  0 Ha4 = F(a3, S)(2gococgogggc   Hs) ?"2   ,$  0 Ha5 = F(a4, S)(2gococgogggc   H) ?"   ,$   0 Ha6 = F(a5, S)(2g o c o  c g o g g g c    H) ?"  * ,$   0 Ha7 = F(a6, S)(2g o c o  c g o g g g c   N) ?"   ,$   0 Ha8 = F(a7, S)(2g o c o  c g o g g g c   Hd) ?"x ,$   0 =.  2e  H) ?" ,$   0 =.  2e  H\) ?"  ,$  0 =.  2e  T) fԔX)?"J e,$D 0 @ The sequence (ak) converges to a point x on S. The solution defined by Y1(a,S) = x is an ordinal, efficient, symmetric and individually rational solution. tJ  (J geemeeeeeeeeeeeeeeee ee eee eg  6H)  0 ~4Shapley s ordinal solution H  0޽h ? ̙33.;&;___PPT10;+dD8' ) = @B D7' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-o6Bdissolve*<3<*;D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D).' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D#' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D#' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D#' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +l 0L0 ,,!H|,(  | $| S 4) -  )  4Other constructions of L3 b: (| Zx,  3ԔL!?"6@ NNN?N ,$D 0 >To construct Shapley s solution we generated a family of Pareto surfaces for all players but 3 & & construct the ideal points of each surface & & which form a path p3. The point L3 is the intersection of p3 with S. We now generalize the idea of this construction.   2^EEEE AEAEEMA EEEEEMEEEEE0AE$ F +|  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.~`@  S"M b ,$@  0f ,|  0e0e    BCPDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  8H`pD@HbP @    S" R,$@  0V -|  0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %7sHM4x @   S"E ,$@  0V .|  0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0u (|1 @   S" j,$@  0f /|  0e0e    B CDE4F1 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4!@ {8P%W  @    S"E %,$D   0z  x 0| @ 2=,$D 0B 1|B HDo?"  B 2| HDo?"  3| H4,  ?"3x A1$(2g 4| H,  ?"   A2$(2g 5| <,  ?" E ? ?a" 2gz   6| M l ,$D  0B 7| HD?"  B 8| HD?" z  F  9|  X,$D  0B :| HD?" F H B ;| HD?"J z   <| 8 ,$D 0B =| HD?"  B >| HD?" z   ?|  i,$D 0B @| HD?" B A| HD?" B| H',  ?"5, ,$ 0 A.$(2g, C| H+,  ?"4 ,$ 0 A.$(2g, D| H0e0e    BHCDE|F4 jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||'8Ohp+ SMx@l0(k(xPHO=PH@          S" ,$D  0 G| H$7, ?",$  0 Tp36 2emh H| # l<,  fԔL!?"6@ NNN?N! <tA ,$  0 ZL3: 2H | 0޽h ? ̙33K?C?___PPT10#?.9/pc+WDDo=' T, = @B D*=' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(|b%(Db' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0|%(D' =-o6Bdissolve*<3<*0|Dw ' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+|%(D' =-o6Bwipe(up)*<3<*+|D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,|%(D' =-o6Bwipe(up)*<3<*,|D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-|%(D' =-o6Bwipe(up)*<3<*-|D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.|%(D' =-o6Bwipe(up)*<3<*.|D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/|%(D' =-o6Bwipe(up)*<3<*/|DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(|b%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*6|%(D' =-o6Bdissolve*<3<*6|D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E|%(D' =-o6Bdissolve*<3<*E|D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*9|%(D' =-o6Bdissolve*<3<*9|D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D|%(D' =-o6Bdissolve*<3<*D|D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<|%(D' =-o6Bdissolve*<3<*<|D' =%( D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C|%(D' =-o6Bdissolve*<3<*C|D' =%( D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*?|%(D' =-o6Bdissolve*<3<*?|D' =%( D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B|%(D' =-o6Bdissolve*<3<*B|D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(|%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F|%(D' =-s6Bwipe(down)*<3<*F|D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G|%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*G|D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*G|D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(|%(D[' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*H|%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*H|D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*H|DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(| %(++0+(|0 ++0+B|0 ++0+C|0 ++0+D|0 ++0+E|0 ++0+G|0 ++0+H|0 + 0L0 HH#5 H(    c $d, -  ,  4Other constructions of L3 b  Z4l,  3ԔL!?"6@ NNN?N- ,$D 0 Starting with the same family of surfaces, we introduce two paths p3,1 and p3,2 which we call guidelines. We construct the  ideal points of each surface with respect to the guidelines& & this form a path p3 The point L3 is the intersection of p3 with S. The guidelines play the role of the axes in the construction of Shapley s solutionT 2BEE3M3EE3M3E E3EE AE%E E3EEEM EEEMEEUE$A     0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.~`@  S"M b 2   0e0e    BCPDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  8H`pD@HbP @    S" R"   0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %7sHM4x @   S"E "   0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0u (|1 @   S" j2   0e0e    B CDE4F1 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4!@ {8P%W  @    S"E %F  x   @ 2=B  B HDo?"  B   HDo?"    H,  ?"3x A1$(2g   H,  ?"   A2$(2g  <,  ?" E ? ?a" 2g  H,  ?"!  ,$ 0 A.$(2g,  HX,  ?"u U,$  0 A.$(2g,h ! # l,  fԔL!?"6@ NNN?N= v] ,$  0 ZL3: 2x # # l ,  fԔL!?"6@ NNN?No,$  0 jp32 2EM  $  0e0e    BDC} DEpF0 A@ 3 o L!     ?A)BCD|E||}  +M@oPYb ):?XvI(6FD766@         # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab ,$D  0, %  0e0e    BCDEF8 A@ 3 o L!     ?A)BCD|E||""o4\G:2^-*':5317-m !T8lrPW<@           # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab) ,$D  0z ' # l,  fԔL!?"6@ NNN?Ns 4 ,$  0 lp3,12 2E3M3z ( # l,  fԔL!?"6@ NNN?NF af,$  0 lp3,22 2E3M3B ) ZDoL!?"0@NNN?N c ,$D   0B *  fDoL!?"0@NNN?N Z c ,$@   0B + ZDoL!?"0@NNN?N Y p ,$D   0B ,  fDoL!?"0@NNN?Ns ]] ,$@   0B - ZDoL!?"0@NNN?N\ Y lc ,$D  0B .  fDoL!?"0@NNN?N_ nqI,$@  0B / ZDoL!?"0@NNN?N! ( ,$D  0B 0  fDoL!?"0@NNN?N$  7,$@  02 1  0e0e    BCDEdF, jJ L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||J%_/s;IyyS5"}M4` 0@        "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab } :,$D  0 2 H,  ?" . ,$ 0 A.$(2g, 3 H|,  ?"U ,$ 0 A.$(2g, 4 H(,  ?"O p/ ,$ 0 A.$(2g, 5  `8, ԔL!?"0@NNN?N ,$D 0 <The question is how to construct the guidelines ordinally.  =(2=E$1 H  0޽h ? ̙33OO___PPT10N.9/pc+}DcL' , = @B DL' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*k%(D ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bwipe(down)*<3<*%D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*'D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*'D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-s6Bwipe(down)*<3<*$D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*k%(D' =%(D.' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-s6Bwipe(down)*<3<**Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-s6Bwipe(left)*<3<*)D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-s6Bwipe(down)*<3<*,D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-s6Bwipe(left)*<3<*+D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bwipe(down)*<3<*.D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-s6Bwipe(left)*<3<*-D' =%( D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-g6B fade*<3<*2D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(down)*<3<*0D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-s6Bwipe(left)*<3<*/D' =%( D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-g6B fade*<3<*3DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-s6Bwipe(down)*<3<*1D+' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*#DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*!D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*!DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-6B%slide(fromBottom)*<3<*5+h+0+0 ++0+0 ++0+0 ++0+!0 ++0+#0 ++0+'0 ++0+(0 ++0+20 ++0+30 ++0+40 ++0+50 +` 0L0 k*c* )(    S 8,   ,  \Ordinal guidelinesz  x  Z ,$D 0B B HDo?"  B   HDo?"    H<,  ?"3x A1$(2g   H,  ?"   A2$(2g   </  ?" E ? ?a" 2g.    0e0e    BdCDEXF(A o L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||3=gUImoBY. g;dRBd@       "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab  ,$@   0B   `DfjJL!?"0@NNN?N < ,$D   0  H0/  ?"C > ,$ 0 A.$(2g6B   fDjJL!?"0@NNN?N  ,$D  0N   "0e0e    BCDEpF0A o L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||w7NRs.ec7 (u (l.4'Jmh@         "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab |,$D  0  Z / ԔL!?"0@NNN?NW],$ 0 QWe describe how to construct ordinal guidelines for a family of Pareto surfaces. R 2REQF  N/  ԔL!?"6@ NNN?NV*v$ 0TL___PPT9.&@ 6Suppose the guideline has been defined up to point x.x7 2)EE EEEE()     N/  ԔL!?"6@ NNN?N<,$ 0 5It is enough to show the direction of the path at x. B6 22EEE5E  N/  ԔL!?"6@ NNN?Ns  ,$  0 Ux 2E   N$&/  ԔL!?"6@ NNN?N ,$  0 tThe rate of utility exchange between 1 and 2 at x on this surface is the negative of the slope of the tangent at x. pu 2$EEEEEEEEEEEEEEEAEEEEt  N6/  ԔL!?"6@ NNN?N@ F ,$ 0 :The slope of the direction of the path at x is this rate. B; 2*EEE:P  N?/  fjJL!?"6@ NNN?N e ,$D 0 `The ordinality of this construction is shown in O Neill et al. (2000)TG 2/E EEE, # 01F  C xD/ oL!?"0@NNN?N W,,$  0 3 = x3b(2emem  C xK/ oL!?"0@NNN?N,$  0 "Consider the Pareto surface at x. D#(2EEE"H  0޽h ? ̙33*5"5___PPT105. ,+PiD2' Y/ = @B Da2' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(DF' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DE' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dv' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*+0+0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 + 0L0 HO@OP;QN(    c $q/   /  \Ordinal guidelinesF  x  Z B B HDo?"  B  HDo?"   Hu/  ?"3x A1$(2g  Hz/  ?"   A2$(2g  <y/  ?" E ? ?a" 2g    0e0e    BdCDEXF( o L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||3=gUImoBY. g;dRBd@       "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab  B    `DfjJL!?"0@NNN?N <    HL/  ?"C >  A.$(2g6B    fDjJL!?"0@NNN?N       (0e0e    BCDEpF0 o L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||w7NRs.ec7 (u (l.4'Jmh@         "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab |  Nd/  ԔL!?"6@ NNN?Ns   Ux 2E>  Z / oL!?"0@NNN?N ,$ 0 HMore formally, 2ES  N/  oL!?"6@ NNN?NnM,$ 0 cdx2D 2EEMS  NȌ/  oL!?"6@ NNN?NV,$ 0 cdx1D 2EEMB  TDoL!?"0@NNN?Ng,$D 0B  TDoL!?"0@NNN?N  ,$@ 0B   `DoL!?"0@NNN?N  ,$@ 0A  N/  oL!?"6@ NNN?N  ,$  0 Qdx22 2EMA  N /  oL!?"6@ NNN?N R ,$  0 Qdx12 2EM  Nԣ/  3oL!?"6@ NNN?N-,$D 0 The ratio between the marginal changes in 2 and 1 s utility at x, as a result of changing x3^ 2*GGGGGGCGgo+  Nl/  oL!?"6@ NNN?N-;<,$D  0 ;= 2E   N/  3oL!?"6@ NNN?N\24,$@  0 The rate of exchange of 2 and 1 s utility at x along the surface where 3 is fixed at x3Z 2GGGGGGGgo G g  o Gl  W'  'I:~,$@  00 ! N 8Pm,$@ 00 ? N0  oL!?"6@ NNN?N O  l3J 2emB @ TDoL!?"0@NNN?N " 4" 0 A N0  oL!?"6@ NNN?N  W'  lx1J 2emW B N0  oL!?"6@ NNN?N?),$ 0 gdx3H 2Eem+ C NH0  oL!?"6@ NNN?N"b,$ 0 ;[ 2G6+ D N`0  oL!?"6@ NNN?N `,$ 0 ;] 2G6. E N00  oL!?"6@ NNN?N  ,$ 0 >-1  2G. F N#0  oL!?"6@ NNN?NI0,$ 0 >-1  2G- G ND'0  oL!?"6@ NNN?Ne,$! 0 =1/2 2E- H N*0  oL!?"6@ NNN?Nz ] ,$   0 =1/2 2E I  `.0 GGHjJL!?"6@ NNN?N ,$D 0 GThese factors guarantee, that at x3 the path reaches the right surface.\H(2!Eem%El Q C x(50 oL!?"0@NNN?N W, 3 = x3b(2ememH  0޽h ?I ̙33.Z&Z___PPT10Z. ,+5DT' = @B DUT' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D[' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(Di ' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-g6B fade*<3<*+D ' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-g6B fade*<3<*,D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-g6B fade*<3<*-D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-g6B fade*<3<*.D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-g6B fade*<3<*2D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-g6B fade*<3<*3D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-g6B fade*<3<*:D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-g6B fade*<3<*<D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-g6B fade*<3<*>D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =-g6B fade*<3<*BD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-g6B fade*<3<*CD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-g6B fade*<3<*DD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =-g6B fade*<3<*ED8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =-g6B fade*<3<*FD ' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*ID' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*ID' =-g6B fade*<3<*IDw' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*HD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*HD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =-g6B fade*<3<*GD' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*ID' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*ID' =-g6B fade*<3<*ID' =1:Bhidden*o3>+B#style.visibility<*I%(+@+0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0++0 ++0+,0 ++0+-0 ++0+20 ++0+30 ++0+40 ++0+:0 ++0+<0 ++0+B0 ++0+C0 ++0+D0 ++0+E0 ++0+F0 ++0+G0 ++0+H0 ++0+I0 ++0+I0 +$ 0L0 aa@za(    c $ 0   0  \Ordinal guidelines & N0  oL!?"6@ NNN?N,$ 0 eWe can generate infinitely many ordinal guidelines by changing the relative weight of the equations. 2f 2eE^ ' TP0  oL!?"6@`NNN?N-( < ,$ 0 hWe fix a [0,1] and choose the guideline p3,2 to be the solution ofF AEE3M3~ < N0  oL!?"6@ NNN?N;,$  0  (1- a)f 2EEE ? N$0  oL!?"6@ NNN?Nw P  cdx2D 2EEM @ TLإ0  oL!?"6@ NNN?N   ;= 2EF  W'  A  S0 B NЩ0  oL!?"6@ NNN?N O  l3J 2emB C TDoL!?"0@NNN?N " 4" 0 D N0  oL!?"6@ NNN?N  W'  lx2J 2em# E N0  oL!?"6@ NNN?Nd 9#  gdx3H 2Eem F N0  oL!?"6@ NNN?NV  cdx1D 2EEM G TL0  oL!?"6@ NNN?N  ;= 2E H NL0  oL!?"6@ NNN?N  ;[ 2G6 I N0  oL!?"6@ NNN?N  ;] 2G6F  W'  J 8Pm0 K N\0  oL!?"6@ NNN?N O  l3J 2emB L TDoL!?"0@NNN?N " 4" 0 M N0  oL!?"6@ NNN?N  W'  lx1J 2em# N N0  oL!?"6@ NNN?N?) gdx3H 2Eem O N00  oL!?"6@ NNN?N"b ;[ 2G6 P N0  oL!?"6@ NNN?N ` ;] 2G6 Q N0  oL!?"6@ NNN?N   >-1  2G R N0  oL!?"6@ NNN?NI0 >-1  2G S N0  oL!?"6@ NNN?Ne =1/2 2E T N0  oL!?"6@ NNN?Nz ]  =1/2 2EP<z | m U | m,$@ 0 V  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.~`@  S"} \M: W  0e0e    BCPDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  8H`pD@HbP @    S" LM* X  0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %7sHM4x @   S"u M* Y  0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0u (|1 @   S"dE: Z  0e0e    B CDE4F1 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4!@ {8P%W  @    S"uUT  x [# |p, mB \B HDo?"  B ] HDo?"  ^ H0  ?"3x A1$(2g _ Ht0  ?"   A2$(2g ` <|0  ?" E ? ?a" 2g a H3  ?"Q 1  A.$(2g, b H 3  ?"  A.$(2g,< c # lp3  fԔL!?"6@ NNN?Npm   ZL3: 2L d # l 3  fԔL!?"6@ NNN?Ni&  jp32 2EM e  0e0e    BDC} DEpF0 @ 3 o L!     ?A)BCD|E||}  +M@oPYb ):?XvI(6FD766@         # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abM  f  0e0e    BCDEF8 @ 3 o L!     ?A)BCD|E||""o4\G:2^-*':5317-m !T8lrPW<@           # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abY C( g # l3  fԔL!?"6@ NNN?NP F  2M3 B h ZDoL!?"0@NNN?N ] B i  fDoL!?"0@NNN?NT ]B j ZDoL!?"0@NNN?NS j B k  fDoL!?"0@NNN?NW W9B l ZDoL!?"0@NNN?NS f B m  fDoL!?"0@NNN?Nh kyB n ZDoL!?"0@NNN?NQ X B o  fDoL!?"0@NNN?NT g p  0e0e    BCDEdF, jJ L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||J%_/s;IyyS5"}M4` 0@        "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abw: 4M q H3  ?"D ($  A.$(2g, r H 3  ?"O   A.$(2g, s H3  ?"j _  A.$(2g,( t # l#3  fԔL!?"6@ NNN?NU  F  2M3 . = N'3  oL!?"6@ NNN?Nl  ,$  0 >a  2 v C xx+3 oL!?"0@NNN?NZz,$ 0 np3,24(2E3M3 x C x03 oL!?"0@NNN?N0 # P ,$ 0 np3,14(2E3M3 y  f53 oL!?"0@NNN?NB  k ,$  0 /Similarly the guideline p3,1 is the solution ofd0  EE3M3/D z # l,=3 G9H,oL!?"0@NNN?N9,$D 0 <Note that when a=1, the guidelines coincide with the axes and we get Shapley s solution. BZ 2EJ$E  H  0޽h ?z ̙33;;___PPT10:. ,+g8D7' P3 = @B D7' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&D[' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =-g6B fade*<3<*UD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*v%(D' =-g6B fade*<3<*vD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =-g6B fade*<3<*xD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'0%(D' =-g6B fade*<3<*'0D' =%(D' =%(Dm' =A@BBBB0B%()))?D' =0l9 CCBB*<3<*vD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'0F%(D' =-g6B fade*<3<*'0FD' =%(D' =%(D' =A@BBBB0B%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*TD' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*TD' =1:Bhidden*o3>+B#style.visibility<*T%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*SD' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*SD' =1:Bhidden*o3>+B#style.visibility<*S%(Dl ' =%(D' =A@BB5BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*<D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*<D' =-g6B fade*<3<*<D' =A@BB5BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*=D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*=D' =-g6B fade*<3<*=D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*y%(D' =-g6B fade*<3<*yD' =%(D' =%(DM' =A@BBBB0B%()?D' =0l9 CCBB*<3<*xD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*y0%(D' =-g6B fade*<3<*y0D' =%(D' =%(D8' =A@BB*BB0B%()?)?D' =.G7 BBBBBYM 8.33333E-7 -4.44444E-6 L 0.00174 0.14445 *3>*B ppt_xB ppt_y=@0BBAApBB9c:B?=<*=D' =A@BBBB0B%()?)?D9' =.7 BBBBBM -2.77778E-7 -4.07407E-6 C -0.01667 -0.0206 -0.03333 -0.0412 -0.03264 -0.06412 C -0.03194 -0.08726 -0.00208 -0.12569 0.00417 -0.13773 *3>*B ppt_xB ppt_y=B0BB aaApBBnB*F<*<D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*z%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*zD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*zD' =-g6B fade*<3<*z++0+&0 ++0+'0 ++0+<0 ++0+<0 ++0+T0 ++0+=0 ++0+=0 ++0+v0 ++0+v0 ++0+x0 ++0+x0 ++0+y0 ++0+z0 +pO 0L0 VGNG&AF(    c $:   \Ordinal guidelines<F | m  | m   0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.~`@  S"} \M:   0e0e    BCPDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  8H`pD@HbP @    S" LM*   0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %7sHM4x @   S"u M*   0e0e    BCDE(F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 0u (|1 @   S"dE:    0e0e    B CDE4F1 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4!@ {8P%W  @    S"uUT  x !# |p, mB "B HDo?"  B # HDo?"  $ HI ?"3x A1$(2g % HO ?"   A2$(2g & <P ?" E ? ?a" 2g ' HT ?"Q 1  A.$(2g, ( HPS ?"  A.$(2g,< ) # l[ fԔL!?"6@ NNN?Npm   ZL3: 2L * # l` fԔL!?"6@ NNN?Ni&  jp32 2EM +  0e0e    BDC} DEpF0 @ 3 o L!     ?A)BCD|E||}  +M@oPYb ):?XvI(6FD766@         # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abM  ,  0e0e    BCDEF8 @ 3 o L!     ?A)BCD|E||""o4\G:2^-*':5317-m !T8lrPW<@           # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abY C( - # l,f fԔL!?"6@ NNN?NP F  2M3 B . ZDoL!?"0@NNN?N ] B /  fDoL!?"0@NNN?NT ]B 0 ZDoL!?"0@NNN?NS j B 1  fDoL!?"0@NNN?NW W9B 2 ZDoL!?"0@NNN?NS f B 3  fDoL!?"0@NNN?Nh kyB 4 ZDoL!?"0@NNN?NQ X B 5  fDoL!?"0@NNN?NT g 6  0e0e    BCDEdF, jJ L!     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||J%_/s;IyyS5"}M4` 0@        "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abw: 4M 7 Hk ?"D ($  A.$(2g, 8 Hp ?"O   A.$(2g, 9 Hn ?"j _  A.$(2g,( : # lv fԔL!?"6@ NNN?NU  F  2M3  > Nd} oL!?"6@ NNN?N ,$ 0 The pair {p3,1, p3,2} is symmetric with the respect to 1 and 2, and therefore L3 is also symmetric with respect to 1 and 2. t~ 2 EE3M3E3M3##}r ? N oL!?"6@ NNN?N j ,$ 0 In a similar manner we can construct also L1 and L2 such that for each i, Li is symmetric with respect to the other two players.J 2)EEEE    EEEEE4EN @ C xLoL!?"0@NNN?NZz np3,24(2E3M3N A C xxoL!?"0@NNN?N0 # P  np3,14(2E3M3H  0޽h ? ̙33___PPT10. ,+܅@D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-g6B fade*<3<*>D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-g6B fade*<3<*?+p+0+>0 ++0+?0 +  0L0  --$<,(  B /  fDoL!?"0@NNN?Nt CC ,$D!  0B 1  fDoL!?"0@NNN?N2 bH B,$D# 0B 2@  fDoL!?"0@NNN?Na :n ,$@  0B 5  fDoL!?"0@NNN?N p 0 ,$@" 0r  S Lg3 6  3    N0h3  oL!?"6@ NNN?N]'$ 0___PPT9jb@ `For Shapley s solution, the points L1 , L2, and L3 are the projections of a single point which we denoted by F.$p 2"EE A  E E:EE:  L (   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"  ,$@ 0B  ZDo?" ,$@ 0B  ZDo?"  ,$@ 0 ; Hq3  ?" 6 ,$' 0 A.$(2g6   l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cRqWONy2L#1 % jH v$obQ5( @            `S" ,$@ 0  NDv3  ?"4!f.,$ 0 A3$(2g  NЃ3  ?"A ; ,$ 0 A2$(2g  NL3  ?" ,$  0 A1$(2g  B3  ?" ,$  0 ?a" 2gB  NDo?"e ,$@  0   #   ,$@  0B  HDo?"` B  HDo?"* i    N3  ?" 6 K ,$  0 A.$(2g, ! N3  ?" - B t ,$ 0 A.$(2g,  % Z3 ?"  ,$ 0 >F  2 & N03 ?"-  V M ,$ 0 ZL3: 2O ' 3 r@3  fԔL!?"6@ NNN?N,$ 0 ;S 2En ( 3 rܤ3  fԔL!?"6@ NNN?N ,$ 0 ZL1: 2n ) 3 r3  fԔL!?"6@ NNN?N   ,$ 0 ZL2: 2 * N3  ?"lb ,$ 0 A.$(2g, + N3  ?"M  - ,$ 0 A.$(2g, , Z3 oL!?"0@NNN?N ,$ 0 LThis does not hold for the points we have constructed using the guidelines. M 2MELB .@  fDoL!?"0@NNN?NM oz,$@  0B 4  fDoL!?"0@NNN?No pM ,$D 0{ 6 Z3 oL!?"0@NNN?Ng e ,$D$  0 /We choose the minimal coordinate on each axis.  0(20E/ 9 HH3  ?" 1 ,$% 0 A.$(2g6 : H3  ?" ,$& 0 A.$(2g6 < Z3 oL!?"0@NNN?N ,$D(  0 hWe define F to be the point with these coordinates. F5(2 E*E4H  0޽h ? ̙332*___PPT10 .J`y+ -Df' 3 = @B D!' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DTS' =%(DR' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*!D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*!D' =-g6B fade*<3<*!DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*%D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*%D' =-g6B fade*<3<*%DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*&D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*&D' =-g6B fade*<3<*&DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*'D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*'D' =-g6B fade*<3<*'DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =-g6B fade*<3<*(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*)D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*)D' =-g6B fade*<3<*)DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<**D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<**D' =-g6B fade*<3<**DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*+D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*+D' =-g6B fade*<3<*+D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-g6B fade*<3<*,D ' =%(D. ' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*%D' =1:Bhidden*o3>+B#style.visibility<*%%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D~' =%(D' =A@BBBB0B%()?)?D>' =.7 BBBBB?M 0.0 0.0 L -0.02986 -0.02408 *3>*B ppt_xB ppt_y=0BBAA<**D' =A@BBBB0B%()?)?D:' =.7 BBBBB;M 0.0 0.0 L 0.04514 0.04167 *3>*B ppt_xB ppt_y=0BBAA<*+D,' =A@BBBB0B%()?)?Ds' =.;7 BBBBBMM 0.00139 0.00092 L 0.02153 -0.03982 *3>*B ppt_xB ppt_y=@0BBAApBBP$<Bߦ<*!D$' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-o6Bwipe(up)*<3<*.D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-u6Bwipe(right)*<3<*4D$' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-u6Bwipe(right)*<3<*2D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-o6Bwipe(up)*<3<*/D*' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-u6Bwipe(right)*<3<*5D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-u6Bwipe(right)*<3<*1D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*6D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*6D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-g6B fade*<3<*9D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-g6B fade*<3<*:D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-g6B fade*<3<*;D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*<D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*<+x+0+0 ++0+;0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+!0 ++0+!0 ++0+%0 ++0+%0 ++0+&0 ++0+'0 ++0+(0 ++0+)0 ++0+*0 ++0+*0 ++0++0 ++0++0 ++0+,0 ++0+60 ++0+90 ++0+:0 ++0+<0 +  0L0 ^(  r  S 6 <   6     ft6 oL!?"0@NNN?N"oD(h`X___PPT9:2@ The constructing of Ya is done in the same iterative way as in Shapley s solution. That is, Ya (a, S) is the limit of ak = F(ak-1, S).  2EEAEEEEA e  m  a  m    a  e  m  e  e  e  a L ( /    fL"6 oL!?"0@NNN?Nt Some of the points ak may lie above the surface S. (Indeed, for Shapley s solution they oscilate). e 2EemEEEE1EH, (   f.6 oL!?"0@NNN?Nz\TL___PPT9.&@ For points above S the construction carried out before is slightly different. Instead of ideal points with respect to the guidelines, which are defined by the maximal payoff to the players, we take horror points which are similarly defined by the minimal payoffs. The coordinates of the point F are the maximal on each axis rather then minimal.O 2% 2 EEEE>E EfE A:EfA EfAEfA$EfA EfEEfEEfAEfAEf(* 4 H  0޽h ? ̙33___PPT10i.z~K+D=' 6 = @B + 0L0 l~(  l l c $DO6 (  6  @Ordinal solution l # lY6 Ԕ s?"x  $Z. Safra, D. Samet (2004) An Ordinal Solution to Bargaining Problems with Many Players, Games and Economic Behavior, 46, 129-142 Presents an ordinal solution for any number of players greater than or equal three. This solution extends Shpaley s solution for three players.  2aef>eea ea1eef0eef eD '  0 0D l Tv6 3>?"  rback"(2e3- 0H l 0޽h ? ̙33y___PPT10Y+D=' 6 = @B + 0L0 @?(    c $}6 (  6  MOrdinal path-valued solutions:  # l6 Ԕ s?"  hO Neill B, Samet, Z. Wiener and E. Winter, (2000) Bargaining with an Agenda, forthcoming GEB. Axiomatizes a path-valued solution for gradual bargaining. This solution is ordinal.  2)AEfAEEEEVee*2 L 02K  T6 3>?" / rback"(2e3 0H  0޽h ? ̙33y___PPT10Y+D=' 6 = @B +$ 0L0 3+/|(  | *| S x6 Z*  6  $Shapley s solution,e   ,| # l6 ԔA|X)?">QS   Shubik, M., (1982), Game Theory in the Social Sciences: Concepts and Solutions, MIT, Press, Cambridge. Thomson, W., (1994), Cooperative models of bargaining, In Handbook of Game Theory 2, Aumann R. J., and Hart S.Eds, North Holland, p.1237-1248. 2e:e30e eee3>e, -| T6 3>?"   D  rback"(2e30 0H | 0޽h ? ̙33y___PPT10Y+D=' 6 = @B +Z 0  |(  |X | C P<    | S PH .*    H | 0re"f ? ̙33r [&G,u2pD : KM^M8ZeXP@+ަZDwOh+'0 `h  דילמת האסיר Samet Dovs331Microsoft PowerPoint@K0 @t~n!@|KI"G g   (  y---$x x ----$ xx ----$x!x!----$!!x)x)!----$))x.x.)----$..x3x3.----$33x6x63----$66x:x:6----$::x?x?:----$??xBxB?----$BBxExEB----$EExIxIE----$IIxLxLI----$LLxOxOL----$OOxSxSO----$SSxVxVS----$VVxYxYV----$YYx]x]Y----$]]x`x`]----$``xcxc`----$ccxgxgc----$ggxjxjg----$jjxnxnj----$nnxqxqn----$qqxvxvq----$vvx{x{v----${{xx{----$xx----$xx----$xx----$xx---'@Times New Roman-. 2 # A family of o."System9-@Times New Roman-. 2 #Irdinal solutions.-@Times New Roman-. (2 /to bargaining problems.-@Times New Roman-. !2 ;&with many playersb.-@Times New Roman-.  2 K%Z. .-@Times New Roman-. 2 K0Safram.-@Times New Roman-.  2 KH,.-@Times New Roman-. 2 KPD. Samet.---$VV---- $VV----$TT--f-- $TT--'@Times New Roman-. 2 " A family of o.. 2 ! A family of o.-@Times New Roman-. 2 "Lrdinal solutions.. 2 !Krdinal solutions.-@Times New Roman-. (2 .!to bargaining problems.. (2 - to bargaining problems.-@Times New Roman-. !2 9(with many playersb.-@Times New Roman-. 33 2 I'Z. .-@Times New Roman-. 332 I2Safram.-@Times New Roman-.  2 IJ,.-@Times New Roman-. 332 IRD. Samet.-@Times New Roman-. %2 k!www.tau.ac.il/~samet.---$`nn``---- $`nn`----$]kk]]--f-- $]kk]--'@Times New Roman-. 33%2 i#www.tau.ac.il/~samet.-՜.+,D՜.+,t0    On-screen ShowSamet0 DavidArialTimes New RomanSymbolDefault Design^A family of ordinal solutions to bargaining problems with many players Z. Safra, D. SametShapley’s ordinal solutionShapley’s ordinal solutionShapley’s ordinal solutionShapley’s ordinal solutionShapley’s ordinal solutionShapley’s ordinal solutionShapley’s ordinal solutionSlide 9Other constructions of 3 Other constructions of 3 Ordinal guidelinesOrdinal guidelinesOrdinal guidelinesOrdinal guidelinesConstructing The solution Ordinal solutionOrdinal path-valued solutionsShapley’s solution  Fonts UsedDesign Template Slide Titles 8@ _PID_HLINKSAx0458,47,Ordinal solutions458,47,Ordinal solutions%465,28,Ordinal path-valued solutions464,25,Ordinal guidelines!460,2,Shapley s ordinal solution404,21,Shapley s solution!460,2,Shapley s ordinal solution458,19,Ordinal solution_0ss  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgijklmnopqrstuwxyz{|}Root EntrydO)Current User~SummaryInformation(hDPowerPoint Document(DocumentSummaryInformation8vRoot EntrydO)`*@Current User/SummaryInformation(hDPowerPoint Document(      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgijklmnopqrstuwxyz{|}_sametsamet