ࡱ>  :<+,-./0123456;C (X G F Zback&AP Lemma backback"Shapleyback$SprumontPOrdinal Solutions to Bargaining Problemsdhttp://www.tau.ac.il/~samet/ordinal-solutions6.pdf2Bargaining with an agendabhttp://www.tau.ac.il/~samet/bargaining-agenda.pdfbackt"Samet s home page6http://www.tau.ac.il/~sametJ#6A short PowerPoint tutorial6O"He has also shown@{>paper in pdf format in JME sitehttp://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6VBY-405KF40-2-18&_cdi=5939&_orig=browse&_coverDate=08%2F31%2F2000&_sk=999659998&wchp=dGLStS-lSzBV&_acct=C000005078&_version=1&_userid=48161&md5=f7e546c4967713eb191a082f9dfad6ff&ie=f.pdf\The paper in pdf format in Math. Program. sitehttp://link.springer.de/link/service/journals/10107/papers/9084001/90840025.pdf/ 0DDavidewRomanPS-TT@zܖ 0ܖDArialewRomanPS-TT@zܖ 0ܖ" DTimes New RomanTT@zܖ 0ܖ0DWingdingsRomanTT@zܖ 0ܖ@DSymbolgsRomanTT@zܖ 0ܖPDArial Unicode MST@zܖ 0ܖ"`DMT Extracode MST@zܖ 0ܖpDTimesNewRomanPSMTT@zܖ 0ܖDMTSYNewRomanPSMTT@zܖ 0ܖDTimesNewRomanPS-BoldMT 0ܖe(.2  @n?" dd@  @@`` ^ V X, 71n|/ $03-HBZ^wHYf'x*o/G- "V4L8 '?1'(5I$CIR%7" 2 ^n0+3 ,!#"/r. 7k;I=$T )O}3&&%2??-(($%/4iJOP" #% *+0/6%&>!/&Q*, 74, :')-&A'(). * "0>3$%+&( 0e0e    A A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S" ff3@P? ʚ;%ab;ʚ;g4KdKdR 0dppp7<4!d!d k 0Tz<4dddd k 0Tz <4BdBd l 0Tx >0___PPT10 pp___PPT9#O{? %O =uZktmou}v/Pp   0` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>>L0  (    6u P  T Click to edit Master title style! !  0w   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0~ ``  X*  0Ѓ `   Z*4  0$ ` ` H@___PPT9"@ l*$"T  <޽h ? ̙33 Default Design 0 P(  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 60,  C X`  H@___PPT9"@ v*.&  H P 0re"f ? ̙3380___PPT10.C d(  d d 0q  0    ^*  d 0v  C X0   `*  d 6{      ^* D d 64 C X`  H@___PPT9"@ v*.&  H d 0re"f ? ̙3380___PPT10.C> 0L0 _W04(  4 4 3 r fԔ L!?NVZ    +SAn ordinal solution to bargaining problems with many players Z. Safra, D. SametTAEAAACEE EC  4  `dfԔL!?"g Nq ,$D   0 Click left mouse-button to proceed, or one of the links to see Samet s home page (web) a short PowerPoint tutorial (in this document)@ 2 q20 qRegegJ@  0@Q 0QR 0XY# 0Yt 0tuH 4 0޽h ? ̙33Q 0L0 l"d"@q`!(  p l # l3ԔD[Ȝ8c?" ,$D   0 A solution is a function Y which assigns to each problem (a,S) a point Y(a,S) in RN.W 2eeeeaeaeeeee e  eeeeeem(e ^  `3ԔD[Ȝ8c?"/ 2 ,$D  0  S 40<$D  0   GBargaining problemsl  >  \>  ,$D 0   0e0e    BCMDEF Ao 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S" T B  ZDo?"[^ B  ZDo?"^ `   Nt0 ?"4>&8 A3$(2g 2 B,5 ?"    ?a" 2g 7 N3 ?" L F  A2$(2g 8 N; ?"G 9  A1$(2gMl K  ]K ,$D  0  Np? ?" ?S"(2e 6  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"2K ,$D 0 Y HD?"zH ,$D 0 DA bargaining problem for a set of players N is a pair (a,S) where E eeee eeee e Z HXN?"H ,$D  0 a - a status quo point in RN,  ee e eem(ez 8  q ( *h,$D 02 e NfԔ?" 8U a TXGSHMf?"(  H"(2e z b H]f?"gJ`P  VThat is, if x,y S, and x y then x = y. z, 2 eeeeeeeeeeeeeeee4l 8 P k8 P,$D  02 j HfԔ?"08 8 * i NxmG{Hyf?"H P ~This means that for each i the projection of S on RN\i is all of RN\i . (The picture shows only part of the infinite surface.) 2eeeeeem( eem(8ae [ Hy?" H  ,$D  0 S - a Pareto surface in RN.   2eeeeem(e m H\?"h ,$D  0 >   2e n H?"P,$D  0 >   2eH  0޽h ?/ ai ̙33/.___PPT10..+PDa-' = @B D-' = @BA?%,( < +O%,( < +D' =%(D' =%(DA' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =-}6B!slide(fromLeft)*<3<*^D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D_' =%(D' =%(D' =A@BBBB0B%(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<*ZDn' =%(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' =%(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<*[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' =%(DW' =4@BB BB%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*qD' =1:Bvisible*o3>+B#style.visibility<*q%(D' =-o6Bdissolve*<3<*qD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*n%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*nD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*nD' =%(D' =%(DW' =4@BB BB%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*kD' =1:Bvisible*o3>+B#style.visibility<*k%(D' =-o6Bdissolve*<3<*kD{' =%(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<*mD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*mD' =%(D+' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*l+P+0+l0 ++0+Y0 ++0+Z0 ++0+[0 ++0+m0 ++0+n0 + 0L0 a Y P(    # lج3ԔD[Ȝ8c?"k ,$D 0 @If Ui is a utility function of player i, and  i is an order-preserving transformation of R, then  i(Ui) represents the same preferences of i.d 2c% 29 2efae!ae aeg  aeaeaefaegae a  e # a  e  a $$,k(  c $P0<$D  0   KOrdinal transformations  # l$3ԔD[Ȝ8c?" k ,$D 0 8An ordinal transformation of the utility space RN is obtained by applying to each player s utility a continuous, order-preserving transformation. Formally: 2aeaem(PaeaefaH  0޽h ? ̙33  ___PPT10 .+>D ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+0 ++0+0 +c 0L0 [ S ` (    c $0<$D  0   KOrdinal transformations  # l3ԔD[Ȝ8c?"7 ,$D 0 *4A vector ( i) i N of continuous, order-preserving functions from R onto R, defines an ordinal transformation  : RN ! RN, by (x) = ( i(xi)) i N .p 2( ( aegag  maeaeefe aeeaem(c eem(eaeaeaeaeg a  e $$(a((g(,,000m00a0   # l3ԔD[Ȝ8c?"l 7,$D 0 The ordinal transformation  transforms a bargaining problem (a,S) to the problem ((a), (S)), where (S) = { (x) | x S }. f (ee eeaeeeeeaeeeeeaeeee eeeeeeeeaeeaeH  0޽h ? ̙33___PPT10p.+h$D'  = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*+8+0+0 +J 0L0 ).!.p .@-(    c $40   CAn ordinal solution   > d # c ,$D 0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S" B  TDo?" B  TDo?"   H8 ?" AS$(2g  H= ?"D>  A3$(2g   < < ?"  b ?a" 2g    l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    HC ?" K    A2$(2g   HH ?"  d A1$(2g9 l    %  ,$D  0  N|K ?". \  A1$(2g    0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"  r B  ZDo?" G x B  ZDo?" x    NlO ?" o0V (S)^(2gcgc  NU ?" 7  A3$(2g  BZ ?" S t :  r(a)N 2ggg   l0e0e    BpCHDEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%"?fz'%BS>8VNKH}p&n>C H:w`NtIuo#" @            `S" bV   N^ ?"  A2$(2g  L-  ## {L,$D 0  Nb?"L>-  . Y(a, S) (2ggcggggc  NDn?"   V". Y((a), (S)) (2gcgcggggcgcg$  Hhz?"u,$D 0 jjA solution Y is ordinal if it depends only on the preferences of the players and not on their representations by utility functions. Thus, the two problems, (a, S) and ((a), (S)) &  ( eeemeefeaeeeeaefaeaeeeeaeae " H?" ,$D 0 & which represent the same problem in terms of preferences, must have solutions &  Q (Qef  & H?" 8,$D 0 f*that represent the same preferences. I.e., + (+ef( ' # l,3ԔD[Ȝ8c?"u,$D 0 J4(Y(a, S)) = Y((a), (S))( gcaeegeacege  cgaeaeggaeacl 6FL .6FL,$D  0 +  0e0e    BC{DEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 5k$|({@  S"6FL ,  fo?"D   @" 2gH  0޽h ? ̙33\T___PPT104++D8' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D} ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(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' =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?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(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' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-o6Bdissolve*<3<*'D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bwipe(left)*<3<*.++0+0 ++0+"0 ++0+&0 ++0+'0 +s 0L0   )S (   $ # l3ԔD[Ȝ8c?"e ,$D  0 4Shapley constructed a solution Y, for problems with three players, which is, ordinal, efficient and feasible (that is, for each problem (a, S), Y(a, S) S), symmetric (that is, covariant with permutation of players). M 2 2P 2B 2eee efeeeeeefeaeeeeaaeeeeageae ea1efaee 0 % # ld3ԔD[Ȝ8c?" e,$D  0 %aHe has also shown that for problems with two players there is no solution with these properties. Hb 2)e ef.eO 0 0l %A  )%A ,$D  0 '  f>Xp?"%A ,$fD  0 We construct an ordinal, efficient, feasible, and symmetric solution to problems with any number of players greater than two, which extends Shapley s solution. The basic building blocks for the construction are &  2eee eeee ee"aeef&e5a BB ( ND3)?"  H  0޽h ? ̙33@ 8 ___PPT10 +hD '  = @B DG ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-}6B!slide(fromLeft)*<3<*$D' =%(D' =%(DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-}6B!slide(fromLeft)*<3<*%DR' =%(D' =%(D' =4@BBBB%(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<*)+p+0+$0 ++0+%0 +)i 0L0 (& &( %(    c $0   @Pareto functions   `t3ԔD[Ȝ8c?"N ,$D   0 rThe function i : RN ! Ri is i s Pareto function. F:(2eooeeoeeoeeeeae>.  <|& ?"x Z ,$D  0 it is:^ 2aoemZ  <, ?"  ,$D 0  independent of xi~ 2e e3eeoB   ZDԔ?"Tss ,$D  0  H3 ?"P +J ,$D 0 Ax$(2g  Hd8 ?"& *- ,$D 0 A.$(2g0 l  >  #>  ,$D 0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"  t  >  " > ,$D 0B   TDo?" B  TDo?"   H< ?" ?S"(2e  HTA ?"C>8 A3$(2g  <lE ?"    ?a" 2g  HI ?" K  E  A2$(2g  HL ?"   A1$(2g   l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   HXP ?",$D  0 (x1, x2, 3(x))(2gogogog  HLW ?"/,$D 0 A.$(2g0R  H[?"Z,$D 0 4Starting from any point x& H 2eee  H`?"OQo,$D 0 Lchange i s payoff until S is reached. ' 2efeefe efe  H8i?"^,$D  0 <\i s payoff at this point is denoted by i(x). / 2e&eeoeae$$ $ NuGeHHI?" 8 ,$D 0 jS is reached at one point at least because it is a surface, and at one point at most because it is Pareto.Jk(2ee eeeeeee e eeeeea % H?"p,$D  0 A   2e & HP ?"h +b ,$D 0 Dx $(2g ' H ?"v *} ,$D 0 A.$(2g0 ( H ?"5Y ,$D 0 $= (x1, x2, 3(x ))(2gogogogH  0޽h ?$ ̙33BB___PPT10iB+eD?' = @B D>' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-o6Bdissolve*<3<*#D' =%(D' =%(D@' =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<*%(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' =A@BBBB0B%(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' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(Dd' =A@BB BB0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*$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?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =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<*%(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<*%(D' =%(D' =%(D@' =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<*'%(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' =A@BBBB0B%(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' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bdissolve*<3<*(+H+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 +3 0L0 &&!%(    c $0   @Pareto functions   `3ԔD[Ȝ8c?"N ,$D   0 rThe function i : RN ! Ri is i s Pareto function. F:(2eooeeoeeoeeeeae>.  < ?"x Z ,$D  0 it is:^ 2aoemZ  < ?"  ,$D 0  independent of xi~ 2e e3eeo{  <` ?" ,$D 0  decreasing in xj~ 2e e3eeo(  <t ?". N,$D 0 z continuous R 2e e3eB  <DԔ?"  ,$D  0B   TDԔ?"Tss ,$D  0   H ?"P +J ,$D 0 Ax$(2g   H0 ?"& *- ,$D 0 A.$(2g09 z  >    >  ,$D 0    0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    >    > ,$D 0B  TDo?" B  TDo?"   H ?" ?S"(2e  H  ?"C>8 A3$(2g  <p ?"    ?a" 2g  H  ?" K  E  A2$(2g  H ?"   A1$(2g   l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   H ?",$D  0 (x1, x2, 3(x))(2gogogog  H ?"/,$D 0 A.$(2g0   X0e0e    BC]DEF @  o 8c8c     ?A)BCD|E||e] v.L]@  S"> ,$D  0R  H,?"Z,$D 0 4Starting from any point x& H 2eee  H$?"OQo,$D 0 Lchange i s payoff until S is reached. ' 2efeefe efe  H-?"^,$D  0 <\i s payoff at this point is denoted by i(x). / 2e&eeoeae$$  H47?"p,$D  0 A   2eH  0޽h ? ̙330 ( ___PPT10 + D| '  = @B D7 ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D~' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+p+0+0 ++0+0 +z 0L0 22*62(  B  HDԔ?" oQz ,$D  0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $G0<$D  0   < Ideal points   TNfԔX)?" B=,$D 0 hThe ideal point of x is (x) = (1(x), 2(x), 3(x))f5(ae eeeeegemememegcB   <DԔ?"  ,$D  0B   TDԔ?"Tss ,$D   0B   TDo?" B  TDo?"     l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   H<] ?" ?S"(2e  H|a ?"& *- ,$D 0 A.$(2g0 0  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S" ,$D 0  He ?"/,$D  0 A.$(2g0  H$j ?">C8 A3$(2g  H$h ?"K  E  A2$(2g  H8q ?"  A1$(2g  Ht ?"6 = ,$D  0 A.$(2g0  Hx ?"   ,$D 0 A.$(2g0l  p  - p ,$D 0|@ H p  ,H p B  ND1?"T .B  ND1?" B  ND1?"@k B  ZD1?" K B   ZD1?"V VX B ! ZD1?"` B " BD1?"m p B # BD1?"u+  B $ BD1?"H- % H~ ?"   A.$(2g0 & Hp ?" D ,$D 0 (1(x), x2, x3)(2gogogog ' H4 ?"m A ,$D 0 H(x)$(2g ) < ?"   ?a" 2g * Hl ?"P +J ,$D 0 Ax$(2g + H ?" s ,$D  0 (x1, x2, 3(x))(2gogogog . H ?"x @ r ,$D  0 (x1, 2(x), x3)(2gogogog 1 H?"L,$D 0 ?   2e 3 BG4HYr?" ,$D 0 hvFood for thought: Which point is closer to S, x or (x)? <(2efeeeeeeeee 4 H\?"fO ,$D 0 >   2e* 5 H?"A|  ,$D 0 pThe relation between x and (x) is one of three, and it defines direction with respect to S._ 2eeee;eee 6 TfԔX)?" O,$D 0 8 x < (x), x is below S x = (x), x is on S x > (x), x is above S |Q% 2eeeeeeeeeeeeeeeeeeeeeeeeeeeeeecH  0޽h ?3 ̙33GG___PPT10`G.+V xDC' = @B DsC' = @BA?%,( < +O%,( < +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?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(DL' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bslide(fromTop)*<3<*D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(Dt' =%(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<*%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-o6Bdissolve*<3<*&Dx' =%(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' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-o6Bdissolve*<3<*.D1 ' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(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@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-o6Bdissolve*<3<*'D' =%(D' =%(DW' =4@BB BB%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*0D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*1D' =%(D' =%(Dd' =A@BB BB0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*3D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-o6Bdissolve*<3<*3D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bdissolve*<3<*5D' =%(D' =%(DP' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-6B#slide(fromRight)*<3<*6++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+&0 ++0+'0 ++0+*0 ++0++0 ++0+.0 ++0+10 ++0+30 ++0+40 ++0+50 ++0+60 +P 0L0 ++&) ,+(    c $0<$D  0   < Ideal points   H  ?"  ,$D 0 H(x)$(2g   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"  el (Wo  (W(o ,$D 0@ Wo  'Wo B   `Do?" o B   `Do?"] B  TDo?">  B   TDo?" b ,$D 0B   TDo?"I  ,$D 0B   TDo?"  ,$D 0B   HDo?" f ,$D 0B   HDo?"W9 ,$D 0B  HDo?"  ,$D 0  H| ?"(I P ,$D 0 A.$(2g0   l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   Ht#?"L,$D  0 ?   2eB  TDo?" B  TDo?"    H' ?" ?S"(2e )  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S" ,$D  0  H+ ?"! q ( ,$D 0 A.$(2g0  H0 ?">C8 A3$(2g  H4 ?"K  E  A2$(2g  H@8 ?"  A1$(2g  H; ?"  ,$D 0 A.$(2g0  <? ?"   ?a" 2g  HDC ?"( " ,$D 0 Ax$(2gl z  &z ,$D 0  HG ?"6,$D  0 A.$(2g0  H,K ?"s\z ,$D 0 A.$(2g0B   HDԔ?"i ,$D  0B ! <DԔ?"  ,$D  0B " TDԔ?"  ,$D  06 # HhO?"c 1 ,$D  0 |Here is a point above S.H 2eee $ TXfԔX)?" B=,$D  0 hThe ideal point of x is (x) = (1(x), 2(x), 3(x))f5(ae eeeeegemememegc % TifԔX)?" O,$D 0 8 x < (x), x is below S x = (x), x is on S x > (x), x is above S |Q% 2eeeeeeeeeeeeeeeeeeeeeeeeeeeeeecH  0޽h ? ̙33##___PPT10#.+~DG"' i= @B D"' = @BA?%,( < +O%,( < +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<*#D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*"D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*"D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*"D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*"D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(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' =%(DW' =4@BB BB%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*)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?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+P+0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+#0 +Ph 0L0 J1B1$60 0(    c $0   dOrdinality of ideal points B   ZDԔ?" 8 ,$D  0   N ?"  ,$D 0 A.$(2g,?   >   # MM,$D 0    0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    >    > ,$D 0B  TDo?" B  TDo?"   H ?" ?S"(2e  H ?"@> A3$(2g  < ?"    ?a" 2g  H< ?" J   A2$(2g  H ?"   A1$(2g   l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   N ?" < ,$D  0 &y = (x1, x2, 3(x))(2gogogog  Nx ?" { ,$D 0 A.$(2g,B  ZDԔ?" R,$D   0  N ?" ,$D   0 A.$(2g, l   1 ,$D 0 !  0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"R B # ZDo?"R  R B $ ZDo?"[ 8B % N ?"|  F(S)"(2e & N< ?"    A3$(2g ' BD ?" / ) r(a)N 2ggg ( NP ?"G A  A2$(2g ) N ?"; A1$(2g *  l0e0e    BCbDEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%@$D*O`m5-'imn':Nb%3e9i]R3"_"k @            `S"[ Y , N ?"   ,$D 0 A.$(2g,R  Nt ?"b\,$D  0 >(x) = (1(x1), 2(x2), 3(x3))8 (2 gogogogogogog   N ?"` mZ,$D 0 x = (x1, x2, x3)(2gogogog 0 H?"~%,$D 0 ^Consider the problems (a, S) and ((a), (S))& L0 2eaeeeeaeaeeeeaeae 2 HD?",$D 0 vStart with x and change i s payoff until S is reached at y.$< 2 efee efeeefe3ee efeee! 4 H?"$3*,$D  0 Similarly, start with (x) and change i s payoff until (S) is reached at z.$M 2eee eeeeeee eeee&$ 6 H$ ?" $ ,$D 0 b<z = (1(x1), 2(x2), 3((x))) (2gogogogogogH  0޽h ? ̙3366___PPT10n6+l D3' := @B D3' = @BA?%,( < +O%,( < +De' =%(D ' =%(D' =A@BBBB0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*0D' =1:Bvisible*o3>+B#style.visibility<*0%(Dv' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-o6Bdissolve*<3<*1D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-o6Bdissolve*<3<*2D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D1 ' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D1 ' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bdissolve*<3<*6+h+0+ 0 ++0+0 ++0+0 ++0+0 ++0+,0 ++0+0 ++0+ 0 ++0+00 ++0+20 ++0+40 ++0+60 +T 0L0 ,:$:*39(    c $P0   dOrdinality of ideal points B  TDԔ?" 8 ,$D  0  HxT ?"  ,$D 0 A.$(2g,?   >  # MM,$D 0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    >    > ,$D 0B  TDo?" B   TDo?"    HY ?" ?S"(2e   HL^ ?"@> A3$(2g   <db ?"    ?a" 2g   Hf ?" J   A2$(2g  Hi ?"   A1$(2g   l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   HPm ?" { ,$D 0 A.$(2g,B  TDԔ?" R,$D   0  Hp ?" ,$D   0 A.$(2g, z     ,$D 0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"R B  TDo?"R  R B  TDo?"[ 8B  HTu ?"|  F(S)"(2e  Hz ?"    A3$(2g  <x ?" / ) r(a)N 2ggg  H ?"G A  A2$(2g  H ?"; A1$(2g   l0e0e    BCbDEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%@$D*O`m5-'imn':Nb%3e9i]R3"_"k @            `S"[ Y  H ?"   ,$D 0 A.$(2g,L   H@ ?"b\,$D  0 >(x) = (1(x1), 2(x2), 3(x3))8 (2 gogogogogogogl    2  ,$D  0 &  0e0e    B C/DEF 5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||c<7x!  /@  S"    ' H ?" ,$D 0 H(y)$(2g ( H ?" < ,$D  0 &y = (x1, x2, 3(x))(2gogogog ) H ?" $ ,$D 0 b<z = (1(x1), 2(x2), 3((x))) (2gogogogogog6 * Hܱ?"C ,$D 0 |The payoffs of all players other than i are the same in (y) and in z.G 2%efeeeefeeefee , H?"S ,$D 0 |By definition both (y) and z are on the Pareto surface (S). D? 2eeeeeeeeeeeaeeae - No?" N,$D  0 . H?" (W,$D  0 1 H?"Z`,$D 0 |(y) and z coincide also for i, that is, i(i(x)) = i((x))).? 2eaeae aememeaeme,) 3 H ?"` mZ,$D 0 x = (x1, x2, x3)(2gogogogV + N$GGH?". N ,$D 0 (y) = (1(x1), 2(x2), 3(3(x))) z = (1(x1), 2(x2), 3((x)))F(2 emememememememememememeH  0޽h ?+ ̙33___PPT10+fD' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-s6Bwipe(left)*<3<*2D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-o6Bdissolve*<3<*+DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(DL ' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*-D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*-DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D#' =+4 8?nCB!#ppt_x-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*.D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*.D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*.D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*.D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-o6Bdissolve*<3<*1++0+*0 ++0+,0 ++0+10 ++0++0 +  0L0 0(+(    c $ 0   dOrdinality of ideal points  +  `H3ԔD[Ȝ8c?"[00 ,$D   0 For each player i, i(i(x)) = i((x)), and hence, ((x)) = ((x)). (   ( c g o c g o c g c g c g o c g c g c g  c g c g c g c g c g c g c g c ,$H  0޽h ? ̙33___PPT10+ Dm' = @B D(' = @BA?%,( < +O%,( < +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<*++8+0++0 +u 0L0 U1M1):0(  pl   5 ,$D 0`t   4 ,$D 0B - NDo?"]w ] B / NDo?"z\ B .B NDo?"] B ,B NDo?"M | B *B NDo?"n& B + NDo?"3 F H 6 HA?"}0! ,$D 0 (x) = a means that for each i, i(x) = ai . That is, for each i, (x-i , ai ) S.|Y(2eaeewwwefaeaemeaeewwwmwwweaefaeaemaewwwmwwwmegee>#"   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $T[0<$D 0   ZA source for an ideal point "B  TDo?" B  TDo?"     l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H$` ?">C8 A3$(2g   Hd ?"K  E  A2$(2g   H`h ?"  A1$(2g   < l ?"   ?a" 2g  Ho ?"l 9 s ,$D 0 A.$(2g0  Hds ?"cMj ,$D 0 A.$(2g0B  HDo?"O ,$D  0  Hw ?" ' ,$D  0 A.$(2g0xB  BD?" B @ HDo?"I  ,$D   0  H{ ?"? ,$D 0 A.$(2g0  <?" 5 ,$D 0 Ax$(2g  H̃ ?"  ,$D   0 = (x1, x2, a3 )(2gogoggwwwowwwog  H ?"wX ,$D  0 = ( a1 , x2, x3)(2ggwwwowwwogogog  H ?"S M ,$D  0 = (x1, a2, x3) 2goggwwwowwwgog  <t?"  F  2e D ! 0  ?"6]p,$D 0 Is the point a the ideal point of some x? That is, is there x for which (x) = a?R  aewwwaeae aeaeewwwa " 0 ?"   ,$D  0 (x1, x2, 3(x)) 2gogogog # 0  ?",$D 0 (1(x), x2, x3)(2gogogog $ 0 ?" M ,$D  0 (x1, 2(x), x3)(2gogogogB ) HDo?" I ,$D  0| 7 H?" /,$D 0 "Such a point x is illustrated now.# 2 e3eeee3ecl  a 9 a,$D 0  2 H(?" a,$D  0 J (x)$(2g 8 NZ?" l ?="(2e$ :  f=d>X!?" ,$fD  0 LClaim (Shapley): For three players, there exists a unique point x such that (x) = a. R 2G(2g3cgcggf&gg ggcggce 0H  0޽h ? ̙333WCOC___PPT10/C.+tD?' = @B DB?' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bdissolve*<3<*!DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D ' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(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' =%(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' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(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' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bdissolve*<3<*5D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-o6Bdissolve*<3<*9D_' =%(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<*:++0+60 ++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+$0 ++0+70 ++0+:0 +$\ 0L0 q+i+N*(  B  NDo?"\ z ,$D  0B @ NDo?"] ,$D   0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $"0   v A source for an ideal point: n=3:!B   TDo?" B   TDo?"      l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H( ?">C8 A3$(2g  H@- ?"K  E  A2$(2g  H0 ?"  A1$(2g  <4 ?"   ?a" 2g-l c s  Jc s ,$D  0  H8 ?"9 l s  A.$(2g0  H+ ?"cMj ,$D 0 A.$(2g0  H|? ?" '  A.$(2g0l fq  Ffq ,$D 0  NC ?" w q ,$D 0  (x1, x2, a3) 2gogoggwwwowwwg  N(L ?"h ,$D 0  (a1, x2, x3) 2ggwwwowwwgogog  N`U ?"f w{ ,$D 0  (x1, a2, x3) 2goggwwwowwwgog ' Hp^ ?" G ,$D  0 A.$(2g0 ( Hc ?". s ( ,$D  0  (a1, x2, a3) 2ggoggoggog} ; Hq?"|,$D 0 kConsider the three points: (x1, x2, a3), (x1, a2, x3), (a1, x2, x3) projected on S by the required point x.<l(eememeewwwmwwwemeewwwmwwwemeewwwmwwwememe eeee = H<?". 4,$D 0 <Starting from (x1, x2, a3) we reach (a1, x2, x3) as follows.= 2eememeewwwmwwwe aeewwwmwwwememe a I Hr?" ,$D 0 >   2e4 H  fxfԔA|X)?"M,$D  0 \There are several ways to prove the claim. The  trick is to find a proof that can be extended to more than three players. H} 20eeHe L H?"  ,$D 0 VFirst, reduce 1 s payoff to status quo a1 & , 2 eeeeewwwmwwwee M HD?" h ,$D  0 dThen, increase 3 s payoff to 3(a1, x2, a3) = x3.v3 2eeaemeewwwmwwwemeewwwmwwweaeme N  fhԔD[Ȝ8c?"X \,$D  0 xWe call 3(a1, x2, a3), 3 s gains over 1 at (x1, x2, a3). =(aemeewwwmwwwemeewwwmwwwee eeeememeewwwmwwweaeH  0޽h ? ̙33C0;0___PPT100+D.' = @B D-' = @BA?%,( < +O%,( < +D' =%(DA' =%(D' =A@BBBB0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*HD' =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<*HD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*HD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*HD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ID' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*IDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(DR' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*JD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*JD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =-o6Bdissolve*<3<*FD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-o6Bdissolve*<3<*=D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bdissolve*<3<*LDx' =%(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' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bdissolve*<3<*(D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =-o6Bdissolve*<3<*MD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =-o6Bdissolve*<3<*N++0+'0 ++0+(0 ++0+;0 ++0+=0 ++0+I0 ++0+H0 ++0+L0 ++0+M0 ++0+N0 +E 0L0 ,, *8,(  B  NDo?"w ]] ,$D  0B  NDo?"\ z ,$D  0B @ NDo?"] ,$D   0B @ NDo?" M | ,$D  0    0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"     c $0   v A source for an ideal point: n=3:!B   TDo?" B   TDo?"      l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   HD ?">C8 A3$(2g  H ?"K  E  A2$(2g  H ?"  A1$(2g  <  ?"   ?a" 2g;z c s   c s ,$D  0  H ?"9 l s  A.$(2g0  H ?"cMj ,$D 0 A.$(2g0  H ?" '  A.$(2g0z fq   fq ,$D 0  H ?" w q ,$D 0  (x1, x2, a3) 2gogoggwwwowwwg  H|  ?"h ,$D 0  (a1, x2, x3) 2ggwwwowwwgogog  H) ?"f w{ ,$D 0  (x1, a2, x3) 2goggwwwowwwgog  H2 ?"t Tn,$D 0  (x1, a2, a3) (2goggoggog  H< ?" ,$D 0 A.$(2g0  H4A ?" G ,$D 0 A.$(2g0  HE ?". s ( ,$D  0  (a1, x2, a3) 2ggoggoggog}  HR?"|,$D 0 kConsider the three points: (x1, x2, a3), (x1, a2, x3), (a1, x2, x3) projected on S by the required point x.<l(eememeewwwmwwwemeewwwmwwwemeewwwmwwwememe eeee ' H j?". 4,$D 0 >Similarly, from (x1, x2, a3) we reach (x1, a2, x3) as follows.? 2eememeewwwmwwwe aemeewwwmwwweme a ( HT?"  ,$D 0 VFirst, reduce 2 s payoff to status quo a2 & , 2 eeeeewwwmwwwee ) H?" h ,$D 0 dThen, increase 3 s payoff to 3(x1, a2, a3) = x3.v3 2eeaememeewwwmwwweewwwmwwweaeme *  fdԔD[Ȝ8c?"X \,$D  0 Similarly, 3(x1, a2, a3) is 3 s gains over 2 at (x1, x2, a3). @A( aememeewwwmwwweewwwmwwweaee eeeememeewwwmwwweaeH  0޽h ? ̙33___PPT10++1DQ'  = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-o6Bdissolve*<3<*'D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bdissolve*<3<*(Dt' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-o6Bdissolve*<3<*)D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-o6Bdissolve*<3<**+P+0+0 ++0+0 ++0+'0 ++0+(0 ++0+)0 ++0+*0 +FE 0L0 110#'$}1(  $z   $  ,$D  0B $ NDo?"6 w ,$D 0B $B NDo?"< ,$D 0B $ NDo?"w ]] ,$D  0B $ NDo?"\ z ,$D  0B $@ NDo?"] ,$D   0B $@ NDo?" M | ,$D  0  $  0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    $ c $0   v A source for an ideal point: n=3:!B  $ TDo?" B  $ TDo?"    $  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"  $ H ?">C8 A3$(2g $ HT ?"K  E  A2$(2g $ H ?"  A1$(2g $ < ?"   ?a" 2g;z c s  $ c s ,$D  0 $ H ?"9 l s  A.$(2g0 $ H ?"cMj ,$D 0 A.$(2g0 $ H ?" '  A.$(2g0z fq  $ fq ,$D 0 $ H ?" w q ,$D 0  (x1, x2, a3) 2gogoggwwwowwwg $ H< ?"h ,$D 0  (a1, x2, x3) 2ggwwwowwwgogog $ H ?"f w{ ,$D 0  (x1, a2, x3) 2goggwwwowwwgog $ Hx ?"t Tn,$D 0  (x1, a2, a3) (2goggoggog $ H  ?" ,$D 0 A.$(2g0 $ HL ?" G ,$D 0 A.$(2g0 $ H ?". s ( ,$D  0  (a1, x2, a3) 2ggoggoggog} $ H#?"|,$D 0 kConsider the three points: (x1, x2, a3), (x1, a2, x3), (a1, x2, x3) projected on S by the required point x.<l(eememeewwwmwwwemeewwwmwwwemeewwwmwwwememe eeee $ H8 ?" ,$D  0 A.$(2g0  $ TjX>?"e;5 ,$D  0 H3(x1, a2, a3) = 3(a1, x2, a3) = x3\%(2gogoggwwwowwwggwwwowwwgggoggwwwowwwgoggwwwowwwgggo* !$  0e0e    BCDEF Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||=I{@ S"J  ,$D   02 &$  fHVԔD[Ȝ8c?"  ,$D  0 ZNote that 3 s gains over 1 and 2 are the same. That is 3(x1, a2, a3) = 3(a1, x2, a3)W( aeaeaeaememeewwwmwwweewwwmwwweegemeewwwmwwwemeewwwmwwwem$ '$ HU?" ,$D 0 jThis gives rise to the following two conditions which are equivalent to (x) = a .T 2EefeefeeaeeH $ 0޽h ? ̙33___PPT10+F%sD' }= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&$%(D' =-o6Bdissolve*<3<*&$Dv ' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bdissolve*<3<* $D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!$%(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<*'$%(++0+$0 ++0+ $0 ++0+&$0 ++0+'$0 +b 0L0 >>@+> <>(  B =@ NDo?"] ,$D   0z   8  ,$D  0B 9 NDo?"6 w ,$D 0B :B NDo?"< ,$D 0B  NDo?"w ]] B  NDo?"\ z B @ NDo?" M |    0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $0   v A source for an ideal point: n=3:!B  TDo?" B   TDo?"      l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H  ?">C8 A3$(2g   H ?"K  E  A2$(2g   HH ?"  A1$(2g  < ?"   ?a" 2g  H ?"l 9 s  A.$(2g0  HL ?"cMj  A.$(2g0  H ?" '  A.$(2g0  H ?"  A.$(2g0  HP ?" G  A.$(2g0  N@ ?"  ,$D 0 d(x1, x2, a3) S (x1, a2, x3) S (a1, x2, x3) S3 ememeewwwmwwwegemeewwwmwwwemegeewwwmwwwememege  N ?",$D  0 p(x1, x2, a3) S 3(x1, a2, a3) = x3 3(a1, x2, a3) = x3N9 ememeewwwmwwwegememeewwwmwwweewwwmwwweegememeewwwmwwwemeewwwmwwweegemr  ZH'o?"',$D  0 ( H  ?"t Tn,$D  0  (x1, a2, a3) (2goggoggog ) H ?". s ( ,$D 0  (a1, x2, a3) 2ggoggoggogl "6 <6",$D  0R   N?""6,$D  0R * H?""/,$D  0R + H?""2,$D  0R . H?";,$D  0> / Z ?",$D 0 r(x) = aF (2eae 0  f8ԔD[Ȝ8c?" 00,$D  0 Thus, to prove the claim we need to show the existence of a unique (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) E (- Eeememeewwwmwwwege eeememeewwwmwwweewwwmwwwemeewwwmwwwemeewwwmwwwe 1 H?"m Y ,$D 0 bThe last equivalence follows immediately from the definition of the Pareto functions i .rZ 2TeeemeU 2 Tj9?"e;5 ,$D 0 H3(x1, a2, a3) = 3(a1, x2, a3) = x3\%(2gogoggwwwowwwggwwwowwwgggoggwwwowwwgoggwwwowwwgggo* 3  0e0e    BCDEF Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||=I{@ S"J  ,$D  0z fq  4 fq ,$D 0 5 HN ?" w q ,$D 0  (x1, x2, a3) 2gogoggwwwowwwg 6 H$W ?"h ,$D 0  (a1, x2, x3) 2ggwwwowwwgogog 7 H\` ?"f w{ ,$D 0  (x1, a2, x3) 2goggwwwowwwgog ; Hli ?" ,$D  0 A.$(2g0H > NpGf;H?" ,$D  0 This condition says that there exists a point (x1, x2, a3) in S at which 3 s gains over 1 and 2 are the same. s(20eememeewwwmwwweae eeeeee aeH  0޽h ?> ̙33##___PPT10s#+{D"' = @B D!' = @BA?%,( < +O%,( < +D ' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-o6Bdissolve*<3<*/D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*.D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*.D' =%(D' =4@BBBB%(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@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*<D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*<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<*1%(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@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0+P+0+0 ++0+0 ++0+/0 ++0+00 ++0+10 ++0+>0 + 0L0 b9Z9$58(  B . HDjJ?" ,$D  0B / HDjJ?"  ,$D   0B  @ <DfjJ?" ` ,$D  0  N3o?" ,$D  0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $0<$D 0   4(x) = a - existence: n=3eeeewwwg B  TDo?" B  TDo?"     l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   HĦ ?">C8 A3$(2g  HT ?"K  E  A2$(2g   H ?"  A1$(2g   < ?"   ?a" 2g   No?"9 ,$D  0 Reminder: Looking for (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) 2; 2, g"ggogoggwwwowwwggg ggogoggwwwowwwggwwwowwwgoggwwwowwwgoggwwwowwwgd   H?" (,$D 0 PConsider the set D = {(x1, x2, a3) S}>) 2eeeememeewwwmwwwegea$   H?" ,$D 0 jand the functions on D, f1(x1, x2, a3) = 3(a1, x2, a3) f2(x1, x2, a3) = 3(x1, a2, a3) Z 2eeeemememeememeememeemeefmfememeemememeemeemee`   0e0e    B?CSDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %$?]k.P5lylTS @    S"} ,$D  0  H?"F  @,$D 0 ?D" 2g ! HD ?"o ? i ,$D  0  (x1, x2, a3) 2gogoggwwwowwwgB # HDfjJ?" ss ,$D  0B $ HDfjJ?" o ,$D  0 % H ?" !@ ,$D 0 A.$(2g$3 & H, ?" 4d ,$D 0 A.$(2g$ ' H ?" 0x,$D 0 A.$(2g$ ( H<# ?" L o ,$D  0 A.$(2g$ + Z?" $ ,$D   0  (a1, x2, a3) (2ggogoggwwwowwwg , H/ ?" XK ,$D  0 A.$(2g$ - H3 ?"eV,$D 0 A.$(2g$B 0@ BD?"},$D  0 2 H7 ?"d,$D 0 A.$(2g$ 3 H< ?"K,$D 0  (x1, a2, a3) (2goggoggog 4 HG?"p Pxv ,$D 0 `@Computing f1 at (x1, x2, a3) & ! 2 eemeememeeme 5 HP?" X,$D 0 BComputing f2 at the same point & " 2 eefmfeae 1 HZ ?"{g ,$D 0 2Bf1(x1, x2, a3) = 3(a1, x2, a3) " 2gogogoggogoggogoggwwwowwwg " Z(m?"* ,$D 0 .>f2(x1, x2, a3) = 3(x1, a2, a3) 2gfofgogoggogogoggwwwowwwggwwwowwwgH  0޽h ? ̙33UU___PPT10U.+H*oDEQ' = @B DQ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =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<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* DR' =%(D' =%(D' =4@BBBB%(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<* %(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-o6Bdissolve*<3<*4D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bdissolve*<3<*!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' =%(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<*+DH' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*.D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*.D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*.D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*.D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =%( D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bwipe(up)*<3<*0D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-o6Bdissolve*<3<*1D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bdissolve*<3<*5D' =%(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<*'%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-o6Bdissolve*<3<*3DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*#D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*#D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*#D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*#D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-s6Bwipe(down)*<3<*$D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =%( D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-o6Bdissolve*<3<*"+(+0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+!0 ++0+%0 ++0+&0 ++0+'0 ++0+(0 ++0++0 ++0+,0 ++0+-0 ++0+20 ++0+30 ++0+40 ++0+50 ++0+10 ++0+"0 +s 0L0 vInIP-/H(  B  HDjJ?" ,$D  0B  HDjJ?"  ,$D   0B @ <DfjJ?" ` ,$D  0  N3o?" ,$D  0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $P0<$D  0   4(x) = a - existence: n=3eeeewwwg B  TDo?" B   TDo?"      l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H ?">C8 A3$(2g   H ?"K  E  A2$(2g   H ?"  A1$(2g  <l ?"   ?a" 2g  No?"9 ,$D  0 Reminder: Looking for (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) 2; 2, g"ggogoggwwwowwwggg ggogoggwwwowwwggwwwowwwgoggwwwowwwgoggwwwowwwgd  H?" (,$D 0 PConsider the set D = {(x1, x2, a3) S}>) 2eeeememeewwwmwwwegea$  Hp?" ,$D 0 jand the functions on D, f1(x1, x2, a3) = 3(a1, x2, a3) f2(x1, x2, a3) = 3(x1, a2, a3) Z 2eeeemememeememeememeemeefmfememeemememeemeemee`   0e0e    B?CSDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %$?]k.P5lylTS @    S"} ,$D  0  H ?"F  @,$D 0 ?D" 2g  Hp6 ?"o ? i ,$D  0  (x1, x2, a3) 2gogoggwwwowwwgB  HDfjJ?" ss ,$D  0B  HDfjJ?" o ,$D  0  H\@ ?" !@ ,$D 0 A.$(2g$3  HD ?" 4d ,$D 0 A.$(2g$  HH ?" 0x,$D 0 A.$(2g$  HDL ?" L o ,$D  0 A.$(2g$  ZO?" $ ,$D   0  (a1, x2, a3) (2ggogoggwwwowwwg  HZ ?" XK ,$D  0 A.$(2g$  H^ ?"eV,$D 0 A.$(2g$B @ BD?"},$D  0  Hb ?"d,$D 0 A.$(2g$   H(g ?"K,$D 0  (x1, a2, a3) (2goggoggog # Hs ?"{g ,$D 0 2Bf1(x1, x2, a3) = 3(a1, x2, a3) " 2gogogoggogoggogoggwwwowwwg $ Z(?"* ,$D 0 .>f2(x1, x2, a3) = 3(x1, a2, a3) 2gfofgogoggogogoggwwwowwwggwwwowwwg  & H$?" @,$D  0 RFirst edge&  2 e3 ' H?" ` ,$D  0 &Compare f1 and f2 at the edges of D. ' 2eefmfeemeee ( H@ ?"L p ,$D  0 A.$(2g$ ) HЧ?"V  P ,$D  0  (a1, x2, a3 ) 2ggogoggogg * H3?"@0 f ,$D  0 f1 < f2 2emmfeefmf1 % TԔX)?"& V,$D   0 k 6 em , N?"  ,$D  0 xf2(a1, x2, a3) = 3(a1, a2, a3) = 3(a)N= efmfeememeememeewwwmwwweemeemeeemeee - Z?"@ ` ,$D   0 >f1(a1, x2, a3) = 3(a1, x2, a3) 2emeememeememeememeeme6 . Z?"o a ,$D 0 j= a3J eem( + NGIH?"4 ,$D  0 hBecause, (a1, x2, a3) is on S .(2 gggogoggogcggp / Z?",$D  0  3(a )| 2gogggH  0޽h ?+ ̙33))___PPT10).+cDM'' = @B D'' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =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<*(%(D' =%(D@' =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<*%%(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<*%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<*-D#' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-o6Bdissolve*<3<*.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' =+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' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*/D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*/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<**+0+0+&0 ++0+(0 ++0+)0 ++0+*0 ++0+%0 ++0+,0 ++0+-0 ++0+.0 ++0++0 ++0+/0 +s 0L0 LL`/= \L(    7  Hh"?" @,$D 0 TSecond edge&  2 e3B   HDjJ?" ,$D  0B   HDjJ?"  ,$D   0B  @ <DfjJ?" ` ,$D  0   N3o?" ,$D  0    0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"     c $(0<$D  0   4(x) = a - existence: n=3eeeewwwg B   TDo?" B   TDo?"      l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H4 ?">C8 A3$(2g   H8 ?"K  E  A2$(2g   HX< ?"  A1$(2g   <@ ?"   ?a" 2g   NXHo?"9 ,$D  0 Reminder: Looking for (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) 2; 2, g"ggogoggwwwowwwggg ggogoggwwwowwwggwwwowwwgoggwwwowwwgoggwwwowwwgd   H f?" (,$D 0 PConsider the set D = {(x1, x2, a3) S}>) 2eeeememeewwwmwwwegea$   Ht?" ,$D 0 jand the functions on D, f1(x1, x2, a3) = 3(a1, x2, a3) f2(x1, x2, a3) = 3(x1, a2, a3) Z 2eeeemememeememeememeemeefmfememeemememeemeemee`    0e0e    B?CSDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %$?]k.P5lylTS @    S"} ,$D  0   H8r?"F  @,$D 0 ?D" 2g   H ?"o ? i ,$D  0  (x1, x2, a3) 2gogoggwwwowwwgB   HDfjJ?" ss ,$D  0B   HDfjJ?" o ,$D  0   H ?" !@ ,$D 0 A.$(2g$3   H ?" 4d ,$D 0 A.$(2g$   HD ?" 0x,$D 0 A.$(2g$   H ?" L o ,$D  0 A.$(2g$   Z?" $ ,$D   0  (a1, x2, a3) (2ggogoggwwwowwwg   HPC ?" XK ,$D  0 A.$(2g$   H\ ?"eV,$D 0 A.$(2g$B  @ BD?"},$D  0   H ?"d,$D 0 A.$(2g$   HD ?"K,$D 0  (x1, a2, a3) (2goggoggog !  H ?"{g ,$D 0 2Bf1(x1, x2, a3) = 3(a1, x2, a3) " 2gogogoggogoggogoggwwwowwwg "  Z?"* ,$D 0 .>f2(x1, x2, a3) = 3(x1, a2, a3) 2gfofgogoggogogoggwwwowwwggwwwowwwg $  H?" ` ,$D  0 &Compare f1 and f2 at the edges of D. ' 2eefmfeemeee %  H$ ?"L p ,$D  0 A.$(2g$ &  H?"V  P ,$D  0  (a1, x2, a3 ) 2ggogoggogg '  H3?"@0 f ,$D  0 f1 < f2 2emmfeefmfp -  Z?",$D  0  3(a )| 2goggg  6  TԔX)?"& h,$D  0 D " m 8  H$ ?" E 0,$D 0 A.$(2g$~ 9  N*?"  ,$D  0 xf2(x1, a2, a3 ) = 3(x1, a2, a3) = a3*= efmfemeemeemoememeemeemeeeem :  H>?" ,$D 0  (x1, a2, a3 ) 2goggoggogw ;  HLG3?"E ,$D   0 f1 > f2 2emmfaeefmf <  NS?"g  ,$D  0 |f1(x1, a2, a3 ) = 3(a1, a2, a3) = 3(a) >? 2ememeemeemoemeemeemeememeee( =  NiGkHoj?" ,$D  0 hBecause, (x1, a2, a3) is on S .(2 ggoggoggogcggH   0޽h ?=  ̙33&&___PPT10g&.+[%D$' }= @B D:$' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7 %(D' =-o6Bdissolve*<3<*7 D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =%(D@' =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<*6 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*6 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*6 D' =%(DM' =%(D' =K@BBBBPB0B%(/%(D' =1:Bvisible*o3>+B#style.visibility<*6 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*6 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*6 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<*< D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9 %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*9 D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*9 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' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*; D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*; ++0+7 0 ++0+6 0 ++0+8 0 ++0+9 0 ++0+: 0 ++0+; 0 ++0+< 0 ++0+= 0 +\ 0L0 IIp-2=I(   1  fԔD[Ȝ8c?"& h,$D  0 D " mB  HDjJ?" ,$D  0B  HDjJ?"  ,$D   0B @ <DfjJ?" ` ,$D  0  N3o?" ,$D  0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $P0<$D  0   4(x) = a - existence: n=3eeeewwwg B   TDo?" B   TDo?"      l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H ?">C8 A3$(2g   H ?"K  E  A2$(2g  H ?"  A1$(2g  <l ?"   ?a" 2g  Nܴo?"9 ,$D  0 Reminder: Looking for (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) 2; 2, g"ggogoggwwwowwwggg ggogoggwwwowwwggwwwowwwgoggwwwowwwgoggwwwowwwgd  H?" (,$D 0 PConsider the set D = {(x1, x2, a3) S}>) 2eeeememeewwwmwwwegea$  H?" ,$D 0 jand the functions on D, f1(x1, x2, a3) = 3(a1, x2, a3) f2(x1, x2, a3) = 3(x1, a2, a3) Z 2eeeemememeememeememeemeefmfememeemememeemeemee`   0e0e    B?CSDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %$?]k.P5lylTS @    S"} ,$D  0  H?"F  @,$D 0 ?D" 2g  Hp  ?"o ? i ,$D  0  (x1, x2, a3) 2gogoggwwwowwwgB  HDfjJ?" ss ,$D  0B  HDfjJ?" o ,$D  0  H ?" !@ ,$D 0 A.$(2g$3  H ?" 4d ,$D 0 A.$(2g$  H< ?" 0x,$D 0 A.$(2g$  H ?" L o ,$D  0 A.$(2g$  Z ?" $ ,$D   0  (a1, x2, a3) (2ggogoggwwwowwwg  HT+ ?" XK ,$D  0 A.$(2g$  HL/ ?"eV,$D 0 A.$(2g$B @ BD?"},$D  0   H3 ?"d,$D 0 A.$(2g$ ! H7 ?"K,$D 0  (x1, a2, a3) (2goggoggog " HD ?"{g ,$D 0 2Bf1(x1, x2, a3) = 3(a1, x2, a3) " 2gogogoggogoggogoggwwwowwwg # ZU?"* ,$D 0 .>f2(x1, x2, a3) = 3(x1, a2, a3) 2gfofgogoggogogoggwwwowwwggwwwowwwg $ HS?" ` ,$D  0 &Compare f1 and f2 at the edges of D. ' 2eefmfeemeee % H$o ?"L p ,$D  0 A.$(2g$ & Hs?"V  P ,$D  0  (a1, x2, a3 ) 2ggogoggogg ' H|3?"@0 f ,$D  0 f1 < f2 2emmfeefmfp ( Z4?",$D  0  3(a )| 2goggg * H, ?" E 0,$D 0 A.$(2g$ , H?" ,$D 0  (x1, a2, a3 ) 2goggoggogw - H3?"E ,$D   0 f1 > f2 2emmfaeefmf 0 NX?"h   ,$D 0 sThe order of f1 and f2 is different at the two edges of the line segment D. Therefore they coincide at some point. Pt 2 eemmfaeefmfme e&eeeee 2  fԔD[Ȝ8c?"C ,$D  0 This proves the existence of x that solves (x) = a, for three players. Later, we generalize this proof for more than three players. We turn now to prove the uniqueness of x which is special for the case n = 3.  2g gfgg ggcgggg3 g=gg gfgggggggH  0޽h ? ̙33___PPT10e.+Ë7D' = @B DP' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*1D' =%(DM' =%(D' =K@BBBBPB0B%(/%( D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*1D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*2D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*2++0+10 ++0+00 ++0+20 +AY 0L0 --|{-(  B  HDjJ?" L 2 ,$D  0B  HDjJ?"  ,$D   0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"  B 6@ BD?"  ,$D   0    l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"   c $0<$D 0   6(x) = a - uniqueness: n=3eeeewwwg B  TDo?" B  TDo?"     H ?">C8 A3$(2g   Hd ?"K  E  A2$(2g  H ?"  A1$(2g  < ?"   ?a" 2g  Hh ?"G : ,$D  0 A.$(2g$   H ?"  ,$D 0 A.$(2g$  H ?" Y ,$D 0  (x1, x2, a3) 2gogoggwwwowwwg  Ht  ?" >h ,$D  0 .<f1(x1, x2, a3) =3(a1, x2, a3)(2gogogoggogoggogoggwwwowwwg 7 H` ?"  ,$D  0 A.$(2g$ 9 Z?"  ,$D 0  (a1, x2, a3) (2ggogoggwwwowwwg : H% ?" @ U ,$D 0 A.$(2g$ m N,o?"9 ,$D  0 Reminder: Looking for (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) 2; 2, g"ggogoggwwwowwwggg ggogoggwwwowwwggwwwowwwgoggwwwowwwgoggwwwowwwg q  f,*ԔD[Ȝ8c?" ,$D 0 >   2e s HHK?" f v ,$D  0 f1 is increasing with x1 2eme eeemd u HW?" (,$D 0 PConsider the set D = {(x1, x2, a3) S}>) 2eeeememeewwwmwwwegea$ v Hf?" ,$D 0 jand the functions on D, f1(x1, x2, a3) = 3(a1, x2, a3) f2(x1, x2, a3) = 3(x1, a2, a3) Z 2eeeemememeememeememeemeefmfememeemememeemeemee z N3o?" ,$D  0, {  0e0e    B?CSDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %$?]k.P5lylTS @    S"}  | Hc?"F  @ ?D" 2gH  0޽h ? ̙33**___PPT10*.+D^(' = @B D(' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*q%(D' =-o6Bdissolve*<3<*qD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*s%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*sD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*sDn' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-o6Bdissolve*<3<*9DH' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%( D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bwipe(up)*<3<*6D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+0+0+0 ++0+0 ++0+ 0 ++0+0 ++0+0 ++0+70 ++0+90 ++0+:0 ++0+q0 ++0+s0 +Wa 0L0 ?:7:%OD9(  Dd GD Ht?" (,$D 0 PConsider the set D = {(x1, x2, a3) S}>) 2eeeememeewwwmwwwegea$ HD Ht?" ,$D 0 jand the functions on D, f1(x1, x2, a3) = 3(a1, x2, a3) f2(x1, x2, a3) = 3(x1, a2, a3) Z 2eeeemememeememeememeemeefmfememeemememeemeemeeB D@ BD?"},$D  0B D HDjJ?" ,$D  0B D HDjJ?"  ,$D  0B D HDjJ?" L 2 ,$D  0B  D HDjJ?"  ,$D  0  D  0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"  B  D@ BD?"  ,$D   0  D  l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S" v  D c $h0   6(x) = a - uniqueness: n=3eeeewwwg B D TDo?" B D TDo?"   D H( ?">C8 A3$(2g D H ?"K  E  A2$(2g D H ?"  A1$(2g D <8 ?"   ?a" 2g D H ?"G : ,$D 0 A.$(2g$ D H  ?"o 6 i ,$D  0  (x1, x2, a3) 2gogoggwwwowwwg D H ?"D_g ,$D  0 0>f1(x1, x2, a3) =3(a1, x2, a3)  (2gogogoggogoggogoggwwwowwwg D H ?"  ,$D  0 A.$(2g$ D Z#?"  ,$D  0  (a1, x2, a3) (2ggogoggwwwowwwg D H0 ?" @ U ,$D 0 A.$(2g$  D H/ ?" XK ,$D 0 A.$(2g$ !D H|3 ?"eN,$D 0 A.$(2g$ "D H7 ?"d,$D  0 A.$(2g$ =D N|?o?"9 ,$D  0 Reminder: Looking for (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) 2; 2, g"ggogoggwwwowwwggg ggogoggwwwowwwggwwwowwwgoggwwwowwwgoggwwwowwwg AD  f@ԔD[Ȝ8c?"  >   2e: DD  0e0e    BCLDEF Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||Ox@5L@  S"c ] ,$D   04 ED  0e0e    BCDEF Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||KP]:@  S"f ,$D   0 JD N3o?" ,$D  0 LD H _?" f v ,$D  0 f1 is increasing with x1 2eme eeem MD Hg ?"  ,$D 0 A.$(2g$, ND  0e0e    B?CSDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %$?]k.P5lylTS @    S"}  OD HXl?"F  @ ?D" 2g D Hpp ?" < V ,$D 0 A.$(2g$H D 0޽h ? ̙33&&___PPT10&+0VD$' r= @B D$' = @BA?%,( < +O%,( < +DL' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*DD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*DD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-s6Bwipe(down)*<3<*DD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*DD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*DDH' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DD#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*DD' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*DD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*DD' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!D%(D' =%( D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-o6Bwipe(up)*<3<*DD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"D%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-o6Bdissolve*<3<*DDz' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*DD%(D' =-o6Bwipe(up)*<3<*DDD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ED%(D' =-s6Bwipe(down)*<3<*ED++0+D0 ++0+D0 ++0+D0 ++0+ D0 ++0+!D0 ++0+"D0 ++0+D0 +R 0L0 //./(  B @ <DfjJ?"D P] A ,$D  0B  HDfjJ?"OO? ,$D   0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"    c $0<$D  0   6(x) = a - uniqueness: n=3eeeewwwg B  TDo?" B  TDo?"     l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H4 ?">C8 A3$(2g   Hġ ?"K  E  A2$(2g   Hp ?"  A1$(2g   < ?"   ?a" 2g   HȬ ?"Z ,$D  0 A.$(2g$f  Ht ?"b @\,$D 0  (x1, a2, a3) (2ggoggoggog  H ?" Y ,$D 0  (x1, x2, a3) 2gogoggwwwowwwg  H ?")  ,$D 0 A.$(2g$fB  BDf?"O,$D   0  HL ?"`,$D  0 A.$(2g$f  Z?")J,$D  0 0>f2(x1, x2, a3) = 3(x1, a2, a3) (2gfofgogoggogogoggwwwowwwggwwwowwwg,   0e0e    B?CSDE4F o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| %$?]k.P5lylTS @    S"}   H?"F  @ ?D" 2g $ No?"9 ,$D  0 Reminder: Looking for (x1, x2, a3) S such that 3(x1, a2, a3) = 3(a1, x2, a3) 2; 2, g"ggogoggwwwowwwggg ggogoggwwwowwwggwwwowwwgoggwwwowwwgoggwwwowwwg (  fԔD[Ȝ8c?" ,$D  0 >   2e " H?" b r,,$D  0 f2 is decreasing with x1 2efmfe eeem  H ?"  ,$D 0 A.$(2g$fd ) H?" (,$D 0 PConsider the set D = {(x1, x2, a3) S}>) 2eeeememeewwwmwwwegea$ * HH%?" ,$D 0 jand the functions on D, f1(x1, x2, a3) = 3(a1, x2, a3) f2(x1, x2, a3) = 3(x1, a2, a3) Z 2eeeemememeememeememeemeefmfememeemememeemeemee , N3o?" ,$D  0 . HL#?" f v ,$D  0 f1 is increasing with x1 2eme eeemH  0޽h ? ̙33""___PPT10".+$C D ' w= @B D ' = @BA?%,( < +O%,( < +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<*"Dn' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<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<*%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*DH' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(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' =%(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 +e 0L0 G>?>'==(  B @ <DfjJ?" ` ,$D  0B @ <DjJ?"D P] A ,$D  0B  HDjJ?"OO? ,$D  0   0e0e    BCMDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||M@S"  v  c $Tl0   6(x) = a - uniqueness: n=3eeeewwwg B  TDo?" B  TDo?"      l0e0e    BCH DEF>  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%%(cR|bj+hWONy2L#1 % jH v$obQ5( @            `S"    H`w ?">C8 A3$(2g   H{ ?"K  E  A2$(2g   H ?"  A1$(2g   <H ?"   ?a" 2g  H ?"Z ,$D 0 A.$(2g$  H ?"vp,$D 0  (x1, a2, a3)