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0.50 setcmykcolor}DC/Black{0 0 0 1 setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end %%EndProcSet %%BeginProcSet: hps.pro 0 0 %! /HPSdict 20 dict dup begin/braindeaddistill 50 def/rfch{dup length 1 sub 1 exch getinterval}bind def/splituri{dup(#)search{exch pop}{()exch} ifelse dup(file:)anchorsearch{pop exch pop 3 -1 roll pop false}{pop 3 -1 roll exch pop true}ifelse}bind def/lookuptarget{exch rfch dup /TargetAnchors where{pop TargetAnchors dup 3 -1 roll known{exch get true }{pop(target unknown:)print == false}ifelse}{pop pop (target dictionary unknown\012)print false}ifelse}bind def/savecount 0 def/stackstopped{count counttomark sub/savecount exch store stopped count savecount sub 1 sub dup 0 gt{{exch pop}repeat}{pop}ifelse}bind def /tempstring 128 string def/targetvalidate{1 index dup length 127 gt exch tempstring cvs dup(/)search{pop pop pop exch pop true exch}{pop}ifelse token{pop length 0 ne}{true}ifelse or not}bind def/targetdump-hook where {pop}{/targetdump-hook{dup mark exch gsave initmat setmatrix{{mark/Dest 4 2 roll targetvalidate{aload pop exch pop/Page 3 1 roll/View exch[exch /FitH exch]/DEST pdfmark}{cleartomark}ifelse}forall}stackstopped pop grestore}bind def}ifelse/baseurl{mark exch 1 dict dup 3 -1 roll/Base exch put/URI exch/DOCVIEW{pdfmark}stackstopped pop}bind def /externalhack systemdict/PDF known def/oldstyle true def/initmat matrix currentmatrix def/actiondict 2 dict dup/Subtype/URI put def /weblinkhandler{dup 3 1 roll mark 4 1 roll/Title 4 1 roll splituri 3 -1 roll dup length 0 gt{cvn/Dest exch 4 2 roll}{pop}ifelse{externalhack{ /HTTPFile exch}{actiondict dup 3 -1 roll/URI exch put/Action exch} ifelse}{externalhack{/HTTPFile exch}{/File exch/Action/GoToR}ifelse} ifelse counttomark 2 sub -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll /Color exch oldstyle{/LNK}{/Subtype/Link/ANN}ifelse gsave initmat setmatrix{pdfmark}stackstopped grestore}bind def/externalhandler where{ pop}{/externalhandler{2 copy{weblinkhandler}exec{/externalhack externalhack not store 2 copy{weblinkhandler}exec{/externalhack externalhack not store/oldstyle false store 2 copy{weblinkhandler}exec{ (WARNING: external refs disabled\012)print/externalhandler{pop pop}bind store externalhandler}{pop pop}ifelse}{pop pop/externalhack externalhack not store}ifelse}{pop pop/externalhandler{weblinkhandler pop}bind store} ifelse}bind def}ifelse/pdfmnew{dup type/stringtype eq{externalhandler}{ exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll/Color exch/LNK gsave initmat setmatrix pdfmark grestore}{pop pop}ifelse}ifelse}bind def/pdfmold{dup type/stringtype eq{externalhandler}{exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop pop 0 3 getinterval /Rect 3 1 roll/Border exch/LNK gsave initmat setmatrix pdfmark grestore} {pop pop}ifelse}ifelse}bind def/pdfm where{pop}{/pdfm /currentdistillerparams where{pop currentdistillerparams dup /CoreDistVersion known{/CoreDistVersion get}{0}ifelse dup braindeaddistill le{(WARNING: switching to old pdfm because version =) print ==/pdfmold}{pop/pdfmnew}ifelse load}{/pdfmark where{pop{dup type /stringtype eq{externalhandler}{2 copy mark 3 1 roll{pdfmnew} stackstopped{2 copy mark 3 1 roll{pdfmold}stackstopped{ (WARNING: pdfm disabled\012)print/pdfm{pop pop}store}{ (WARNING: new pdfm failed, switching to old pdfm\012)print/pdfm/pdfmold load store}ifelse}{/pdfm/pdfmnew load store}ifelse pop pop}ifelse}}{{ pop pop}}ifelse}ifelse bind def}ifelse end def %%EndProcSet TeXDict begin @defspecial systemdict /pdfmark known{userdict /?pdfmark systemdict /exec get put}{userdict /?pdfmark systemdict /pop get put userdict /pdfmark systemdict /cleartomark get put}ifelse /DvipsToPDF{72.27 mul Resolution div} def/PDFToDvips{72.27 div Resolution mul} def/BPToDvips{72 div Resolution mul}def product (Ghostscript) search {pop pop pop revision 927 gt}{pop false} ifelse{/BorderArrayPatch{} def}{/BorderArrayPatch{[exch{dup dup type/integertype eq exch type/realtype eq or{BPToDvips}if}forall]}def} ifelse /HyperBorder {1 PDFToDvips} def/H.V {pdf@hoff pdf@voff null} def/H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def/H.S {currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF 72 add /pdf@hoff exch def HyperBorder sub /pdf@llx exch def} def/H.L {2 sub dup/HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def/pdf@urx exch def} def/H.A {H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def} def/H.R {currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def} def /pgfHrgb{/pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfArgb} def /pgfdir { dup 0 moveto dup 5 index lineto } bind def} bind def /pgfVrgb{/pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfArgb} def /pgfdir { dup 0 exch moveto dup 5 index exch lineto } bind def} bind def /pgfArgb{ /pgfdiff 8 index round cvi 8 index round cvi sub 2 mul 1 add def 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div pgfheight 9 index 9 index 9 index 14 index pgfdiff { 3 index 3 index 3 index setrgbcolor pgfdir stroke 4 -1 roll 7 index add 4 -1 roll 6 index add 4 -1 roll 5 index add 4 -1 roll .5 sub } repeat mark 15 1 roll cleartomark exch pop }bind def /pgfR1rgb{ newpath dup dup dup 0 360 arc clip newpath dup /pgfendx exch def /pgfendy exch def 0.875 setlinewidth [] 0 setdash /pgfshade {pgfRrgb} def /pgfstartx exch def /pgfstarty exch def /pgfdiffx pgfendx pgfstartx sub def /pgfdiffy pgfendy pgfstarty sub def dup /pgfdomb exch def }bind def /pgfR2rgb{ newpath 0.5 add pgfcircx pgfcircy 3 2 roll 0 360 arc setrgbcolor fill pop}bind def /pgfRrgb{ /pgfdiff 8 index round cvi 8 index round cvi sub 4 mul 1 add def /pgfcircx pgfstartx 9 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 9 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 8 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 8 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 8 index 8 index 8 index 13 index pgfdiff { 3 index 3 index 3 index setrgbcolor pgfcircx pgfcircy 2 index 0 360 arc closepath stroke 4 -1 roll 6 index add 4 -1 roll 5 index add 4 -1 roll 4 index add 4 -1 roll .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 14 1 roll cleartomark exch pop }bind def /pgfHcmyk{ /pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfAcmyk} def /pgfdir { dup 0 moveto dup 6 index lineto } bind def} bind def /pgfVcmyk{ /pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfAcmyk} def /pgfdir { dup 0 exch moveto dup 6 index exch lineto } bind def} bind def /pgfAcmyk{ /pgfdiff 10 index round cvi 10 index round cvi sub 2 mul 1 add def 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div pgfheight 12 index 12 index 12 index 12 index 18 index pgfdiff { 4 index 4 index 4 index 4 index setcmykcolor pgfdir stroke 5 -1 roll 9 index add 5 -1 roll 8 index add 5 -1 roll 7 index add 5 -1 roll 6 index add 5 -1 roll .5 sub } repeat mark 19 1 roll cleartomark exch pop }bind def /pgfR1cmyk{ newpath dup dup dup 0 360 arc clip newpath dup /pgfendx exch def /pgfendy exch def 0.875 setlinewidth [] 0 setdash /pgfshade {pgfRcmyk} def /pgfstartx exch def /pgfstarty exch def /pgfdiffx pgfendx pgfstartx sub def /pgfdiffy pgfendy pgfstarty sub def dup /pgfdomb exch def }bind def /pgfR2cmyk{ newpath 0.5 add pgfcircx pgfcircy 3 2 roll 0 360 arc setcmykcolor fill pop}bind def /pgfRcmyk{ /pgfdiff 10 index round cvi 10 index round cvi sub 4 mul 1 add def /pgfcircx pgfstartx 11 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 11 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 10 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 10 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 11 index 11 index 11 index 11 index 17 index pgfdiff { 4 index 4 index 4 index 4 index setcmykcolor pgfcircx pgfcircy 2 index 0 360 arc closepath stroke 5 -1 roll 8 index add 5 -1 roll 7 index add 5 -1 roll 6 index add 5 -1 roll 5 index add 5 -1 roll .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 18 1 roll cleartomark exch pop }bind def /pgfHgray{ /pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfAgray} def /pgfdir { dup 0 moveto dup 3 index lineto } bind def} bind def /pgfVgray{ /pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfAgray} def /pgfdir { dup 0 exch moveto dup 3 index exch lineto } bind def} bind def /pgfAgray{ /pgfdiff 4 index round cvi 4 index round cvi sub 2 mul 1 add def dup 2 index sub pgfdiff div pgfheight 3 index 6 index pgfdiff { 1 index setgray pgfdir stroke exch 3 index add exch .5 sub } repeat mark 7 1 roll cleartomark exch pop }bind def /pgfR1gray{ newpath dup dup dup 0 360 arc clip newpath dup /pgfendx exch def /pgfendy exch def 0.875 setlinewidth [] 0 setdash /pgfshade {pgfRgray} def /pgfstartx exch def /pgfstarty exch def /pgfdiffx pgfendx pgfstartx sub def /pgfdiffy pgfendy pgfstarty sub def dup /pgfdomb exch def }bind def /pgfR2gray{ newpath 0.5 add pgfcircx pgfcircy 3 2 roll 0 360 arc setgray fill pop}bind def /pgfRgray{ /pgfdiff 4 index round cvi 4 index round cvi sub 4 mul 1 add def /pgfcircx pgfstartx 5 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 5 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 4 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 4 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def dup 2 index sub pgfdiff div 2 index 5 index pgfdiff { 1 index setgray pgfcircx pgfcircy 2 index 0 360 arc closepath stroke exch 2 index add exch .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 6 1 roll cleartomark exch pop }bind def /pgfsc{}bind def/pgffc{}bind def/pgfstr{stroke}bind def/pgffill{fill}bind def/pgfeofill{eofill}bind def/pgfe{a dup 0 rlineto exch 0 exch rlineto neg 0 rlineto closepath}bind def/pgfw{setlinewidth}bind def/pgfs{save pgfpd 72 Resolution div 72 VResolution div neg scale magscale{1 DVImag div dup scale}if pgfx neg pgfy neg translate pgffoa .setopacityalpha}bind def/pgfr{pgfsd restore}bind def userdict begin/pgfo{pgfsd /pgfx currentpoint /pgfy exch def def @beginspecial}bind def /pgfc{newpath @endspecial pgfpd}bind def /pgfsd{globaldict /pgfdelta /delta where {pop delta} {0} ifelse put}bind def/pgfpd{/delta globaldict /pgfdelta get def}bind def /.setblendmode where {pop} {/.setblendmode{pop}def} ifelse /.setfillconstantalpha where {pop /.setopacityalpha {.setfillconstantalpha} def} {/.setopacityalpha where {pop} {/.setopacityalpha {pop} def} ifelse} ifelse /.pgfsetfillopacityalpha{/pgffoa exch def /.setfillconstantalpha where {pop pgffoa .setfillconstantalpha} {/pgffill{gsave pgffoa .setopacityalpha fill 1 .setopacityalpha newpath fill grestore newpath}bind def /pgfeofill{gsave pgffoa .setopacityalpha eofill 1 .setopacityalpha newpath eofill grestore newpath}bind def} ifelse} bind def /.pgfsetstrokeopacityalpha{/pgfsoa exch def /.setstrokeconstantalpha where {pop pgfsoa .setstrokeconstantalpha} {/pgfstr{gsave pgfsoa .setopacityalpha stroke grestore newpath}bind def} ifelse}bind def /pgffoa 1 def /pgfsoa 1 def /.pushpdf14devicefilter where {pop [userdict /bop-hook known {userdict /bop-hook get aload pop} if {0 .pushpdf14devicefilter} aload pop] cvx userdict exch /bop-hook exch put [userdict /eop-hook known {userdict /eop-hook get aload pop} if {.poppdf14devicefilter} aload pop] cvx userdict exch /eop-hook exch put} if systemdict /pdfmark known not {userdict /pdfmark systemdict /cleartomark get put} if end /pgfwritesamplecmyk { 4 index 0 5 index pgfcheckcolorrange 255 mul round cvi put 4 index 1 4 index pgfcheckcolorrange 255 mul round cvi put 4 index 2 3 index pgfcheckcolorrange 255 mul round cvi put 4 index 3 2 index pgfcheckcolorrange 255 mul round cvi put pop pop pop pop } bind def /pgfwritesamplergb { 3 index 0 4 index pgfcheckcolorrange 255 mul round cvi put 3 index 1 3 index pgfcheckcolorrange 255 mul round cvi put 3 index 2 2 index pgfcheckcolorrange 255 mul round cvi put pop pop pop } bind def /pgfwritesamplegray { pgfcheckcolorrange 16777215 mul round cvi 1 index 0 2 index -16 bitshift put 1 index 1 2 index 65535 and -8 bitshift put 1 index 2 2 index 255 and put pop } bind def /pgfcheckcolorrange { dup 0.0 lt {pop 0.0} if dup 1.0 gt {pop 1.0} if } bind def /pgfchanneldepthcmyk 8 def /pgfchanneldepthrgb 8 def /pgfchanneldepthgray 24 def /pgfcolorsamplecmyk 4 string def /pgfcolorsamplergb 3 string def /pgfcolorsamplegray 3 string def /pgfrangecmyk [0 1 0 1 0 1 0 1] def /pgfrangergb [0 1 0 1 0 1] def /pgfrangegray [0 1] def /pgf1{gsave exec 1.0 pgfw 2.00002 0.0 moveto -6.00006 4.00005 lineto -3.00003 0.0 lineto -6.00006 -4.00005 lineto pgffill grestore} bind def /pgf2{gsave exec 1.0 pgfw 0.8 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -3.00003 4.00005 moveto -2.75002 2.50002 0.0 0.24998 0.75 0.0 curveto 0.0 -0.24998 -2.75002 -2.50002 -3.00003 -4.00005 curveto pgfstr grestore} bind def /pgf3{gsave exec 1.0 pgfw [ ] 0.0 setdash 0.0 -5.00005 moveto 0.0 5.00005 lineto pgfstr grestore} bind def /pgf4{gsave exec 1.0 pgfw [ ] 0.0 setdash -3.00003 -5.00005 moveto 0.0 -5.00005 lineto 0.0 5.00005 lineto -3.00003 5.00005 lineto pgfstr grestore} bind def /pgf5{gsave exec 1.0 pgfw [ ] 0.0 setdash -2.00002 -5.00005 moveto 1.0 -3.00003 1.0 3.00003 -2.00002 5.00005 curveto pgfstr grestore} bind def /pgf6{gsave exec 1.0 pgfw [ ] 0.0 setdash -4.50003 -5.00005 moveto 0.49998 0.0 lineto -4.50003 5.00005 lineto pgfstr grestore} bind def /pgf7{gsave exec 1.0 pgfw -2.50002 0.0 translate [ ] 0.0 setdash 3.00003 0.0 moveto 3.00003 1.65689 1.65689 3.00003 0.0 3.00003 curveto -1.65689 3.00003 -3.00003 1.65689 -3.00003 0.0 curveto -3.00003 -1.65689 -1.65689 -3.00003 0.0 -3.00003 curveto 1.65689 -3.00003 3.00003 -1.65689 3.00003 0.0 curveto closepath gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath grestore} bind def /pgf8{gsave exec 1.0 pgfw [ ] 0.0 setdash 1.0 0.0 moveto -5.00005 3.00003 lineto -11.00012 0.0 lineto -5.00005 -3.00003 lineto closepath gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath grestore} bind def @fedspecial end %%BeginFont: URWPalladioL-Roma %!PS-AdobeFont-1.0: URWPalladioL-Roma 1.05 %%CreationDate: Wed Dec 22 1999 % Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development % (URW)++,Copyright 1999 by (URW)++ Design & Development % See the file COPYING (GNU General Public License) for license conditions. % As a special exception, permission is granted to include this font % program in a Postscript or PDF file that consists of a document that % contains text to be displayed or printed using this font, regardless % of the conditions or license applying to the document itself. 12 dict begin /FontInfo 10 dict dup begin /version (1.05) readonly def /Notice ((URW)++,Copyright 1999 by (URW)++ Design & Development. See the file COPYING (GNU General Public License) for license conditions. As a special exception, permission is granted to include this font program in a Postscript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /Copyright (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development) readonly def /FullName (URW Palladio L Roman) readonly def /FamilyName (URW Palladio L) readonly def /Weight (Roman) readonly def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /URWPalladioL-Roma def /PaintType 0 def /WMode 0 def /FontBBox {-166 -283 1021 943} readonly def /FontType 1 def /FontMatrix [0.001 0.0 0.0 0.001 0.0 0.0] readonly def /Encoding StandardEncoding def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BC2C03103C68570A7B354A4A280AE 6FBF7F9888E039AB60FCAF852EB4CE3AFEB979D5EA70FDE44A2AE5C8C0166C27 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A9CC193BD7D33F78E823C7321F814472D5BA349BB93BDF92 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: URWPalladioL-Bold %!PS-AdobeFont-1.0: URWPalladioL-Bold 1.05 %%CreationDate: Wed Dec 22 1999 % Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development % (URW)++,Copyright 1999 by (URW)++ Design & Development % See the file COPYING (GNU General Public License) for license conditions. % As a special exception, permission is granted to include this font % program in a Postscript or PDF file that consists of a document that % contains text to be displayed or printed using this font, regardless % of the conditions or license applying to the document itself. 12 dict begin /FontInfo 10 dict dup begin /version (1.05) readonly def /Notice ((URW)++,Copyright 1999 by (URW)++ Design & Development. See the file COPYING (GNU General Public License) for license conditions. As a special exception, permission is granted to include this font program in a Postscript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /Copyright (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development) readonly def /FullName (URW Palladio L Bold) readonly def /FamilyName (URW Palladio L) readonly def /Weight (Bold) readonly def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /URWPalladioL-Bold def /PaintType 0 def /WMode 0 def /FontBBox {-152 -301 1000 935} readonly def /FontType 1 def /FontMatrix [0.001 0.0 0.0 0.001 0.0 0.0] readonly def /Encoding StandardEncoding def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BC2C03103C68570A7B354A4A280AE 6FBF7F9888E039AB60FCAF852EB4CE3AFEB979D5EA70FDE44A2AE5C8C0166C27 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cleartomark %%EndFont %%BeginFont: URWPalladioL-Ital %!PS-AdobeFont-1.0: URWPalladioL-Ital 1.05 %%CreationDate: Wed Dec 22 1999 % Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development % (URW)++,Copyright 1999 by (URW)++ Design & Development % See the file COPYING (GNU General Public License) for license conditions. % As a special exception, permission is granted to include this font % program in a Postscript or PDF file that consists of a document that % contains text to be displayed or printed using this font, regardless % of the conditions or license applying to the document itself. 12 dict begin /FontInfo 10 dict dup begin /version (1.05) readonly def /Notice ((URW)++,Copyright 1999 by (URW)++ Design & Development. See the file COPYING (GNU General Public License) for license conditions. As a special exception, permission is granted to include this font program in a Postscript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /Copyright (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development) readonly def /FullName (URW Palladio L Italic) readonly def /FamilyName (URW Palladio L) readonly def /Weight (Regular) readonly def /ItalicAngle -9.5 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /URWPalladioL-Ital def /PaintType 0 def /WMode 0 def /FontBBox {-170 -305 1010 941} readonly def /FontType 1 def /FontMatrix [0.001 0.0 0.0 0.001 0.0 0.0] readonly def /Encoding StandardEncoding def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BC2C03103C68570A7B354A4A280AE 6FBF7F9888E039AB60FCAF852EB4CE3AFEB979D5EA70FDE44A2AE5C8C0166C27 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cleartomark %%EndFont %%BeginFont: PazoMath-Italic %!PS-AdobeFont-1.0: PazoMath-Italic 001.003 %%CreationDate: Fri May 17 11:17:28 2002 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.003) readonly def /Copyright ((c) Diego Puga, 2000, 2002.) readonly def /Notice (Copyright (c) Diego Puga, 2000, 2002. Distributed under the GNU General Public License (http://www.gnu.org/copyleft/gpl.txt). As a special exception, permission is granted to include this font program in a PostScript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /FullName (Pazo Math Italic) readonly def /FamilyName (PazoMath) readonly def /ItalicAngle -9.50 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) readonly def end readonly def /FontName /PazoMath-Italic def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /alpha put dup 101 /epsilon1 put dup 112 /pi put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.00100 0 0 0.00100 0 0] readonly def /FontBBox {-70 -277 902 733} readonly def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BCF7C3C3333173232E3FDBFF43949 1DB866C39088C203DC22FDC758584860EC7BB67FDA28CC6208249060E18FAB32 204779B5C03C0493BBBBC95CF02692CC4DEAA8D2EA90B5C2E64374E92BCB8501 429B8FAE4A76C0C6B76D6FF7CF9A7D5EDFBCA0E959541C59BD05B7DE43D25D53 FC3DDA6EF0C2743978A6D03E19CCED4A11F2EA4BCC3110BE8B8D9E2772361969 C19258EFAFDC276CB1ADE9208A941A36D18F9FB1C33DEF76AA315DDB0A7F85C4 9167E58A181840EC46CC379EAAE07071FBD381C7EDF50327298FF86A6A4F6812 2553C2CAF67371991CB73DD3F08A7BE3D12E5D0B885AA74F37D1940EEE91E918 45D5847CDBAAB475B142774C14C38D20D16027AAB426CFC43043B69E67862958 8D309D463F6EC259BC9BAB2689D1380DCBB972BC5D1C6C0E70024BB94743B0D3 0E708C018C6FC49FB3675CB4742FEDFE96626AA9EA7D3C796CF48BB4AA39B8A6 F6CE05666C879D0C0A32D94DD56B8F9D70D13AEB18E9CE60CFA59B7C09A1315C F5CA850E22F7A7D830CD703ABA72AF0809353FBF829BFEA6DC6BDBF71868E50F 4FB5CF3E74205306F4A0483E4392234521E9BEE73DCC872013FA0C00CEF2FCE2 56118ECB17B68E6CFE34CA1ACB4ECAFE88D23CEAA9A342E6F99DF3E6D5C384B9 73331941B9FCDC23DE031D356228C697FBA2B4B23D1F6666CD4B46786097125B BB9405A45FAAF63F5368C50330336F4B21261921A08C5A26FD946FC76DAA4D31 DE30F498228E6805F19356249B0CB284D0C92F91C19B50408B9A1CBCACE3A2CD 13642CAC2B58B5C098805B8D26695ABFAC108093696675CFFB67877D51E33527 8A9A4191C6A0F004CB20C4A34BC4446593A38EC5CF591C6CA42E6F2687735A3A 91BA805B7E95C85D8B19F58845029CC7F63FE860355B39504ABBE922C21E84CF 0223052B9600CBD2D7F195ADF94FB5BCFB53393ABA60795FA7875C01CBE21F84 3DE9937E9DFA1FA1FC62E3D51CC8F9BD7B033B181C2E360D4778644FE826B56E 67BBD6442947964BA6234EFFE258C04969109B86BF934FDBDC3A2EDB5A22A727 47C69DE261E264E6D19CE391C9812D7953777F081C42F90C9A7A72B39A1F0DF4 B294926C617885AFF629567C9F7AF2CABC182BEFD7EDD2AE8AA82BF9D60F2FA6 1048F1FCA7F17ECB7CDB1E29C436D1A232F19E436B8B6E503C2E481755CBA7A9 8BC9F176A27F776F07A7CA94474DE1277A94E6D20255339FF569DF7E17EDDE5D 39DA2DB64A52F694A638F304D28922426D16A7C257214DB8EA73693EDC9C5A7E 4720DBBAACD9F118DA418D31EE78E18377AAB2DF1A89C7765C6CD1645E14915E F2B4A6C813B5DBF32AFA9DB2C058F102F5CAF03077885A830C9A73A18C4F4D08 74215B49AC3C1C2C327A2EE088B956510184F34D8EC2BAF04A0C93DED7CA3743 1DFD9E94287E8D28C5366066E8AE5C80BE7FD8E766F80543E21DC3DFC0F28859 61670BE2D2CE87F11B7F4250A50DD561FFF7E3669A392CC6F47981F4D01BAEDF 6D983D1E4A85C06190EF3A270698F6B9230D200E42B424AC335873F8D910852C 291F926FCC512B2D5F4DC3E857BACF340C8C4E164007A384477ABC4A7F281319 7CF8F595F1555AAA669C28B477CE8019F106BF81D08D68F7920A1938932638DE 190F01064558C6272165A2DBA4978322F3B706D42300C96FED233F800A39AC56 1D5ED025A2448EDA8A35B556426602CDC757B0FD0A10AC1127B4E7D18ADDEF5F FE63A9BBE0499676CD65BFD9F684CE70EF2873CFBFD2C695707D172CF5DCFABF B73E4152C23C1D4F3AE318FCF4DF3990EAEB73B9CEAB98A5E5B24B1B627B7A8F AEC8F49E518EBC05643DA5F9D1A2030F95C95E9A39346F11BB95071304715032 04710CCBA0E0138D450960AE56D745749D6FB26B8B04890816F5651CFFDBCBBA BBA5C913ADC92551B57F617D13D190A8AC6287644374D8F219F90D165EF1C2A0 6A9B7500C6AD185B4115777FF5C249D0492D0BFC40D8DF6B9D4B8299B2399045 A14879EE9FF01BB0A1B8C24DB58F7FAAF67B92282C58FCE20277695CCC2E54A2 5041A959AC738AB6CDEA63D4F21BA9C0D030F2F68C45ED7A97E0AACDAD46C964 B24846294C55F890EC1805347D10189DDBEFFC14D7543280E59E14C1F64C508C 3C81BC1871607DB02675BC3C8C802A1517D9C6FDA3474591F19D573EB051F069 2F3C6FE22E584B1197B3CEF5DD0B1376678D381A64842187A9B2C04618D40E0C FDD1B23AA85A323260482525EE8072950A606224285C7B6382E516245ABBA1C3 E0FE565DDBD45AE467A782725D8B3E30CF079565393F94B97579D351D72148A9 5277622125B11C5840EB67966E9B6F104FDDFF62E82B16CBE3D401AAF8EE7443 5A6FCF0C7E4C041BB3ABCC68CEAEE80B1BD4A027782BA27C44FD991761CA10FB FBE802FE062AD8CDDDA1D8C89CF7B69B3CAC447A2D2CBE22433B74955C34B978 482F9041012AD781066C37459250D8608EC19D436A0B5C8448D0420E8062916A 646F58084A79CA3D7ABA56B8ED4E7AF0F4D8740C71D0E6305207F349BCA8EDBF B3CC9869F5BEFFCB86768D0C654D71C2B81897E7F8734C7573B0B3286806EC59 743E296828E9C3DBC27F7E781713D5DE3E2EF43D7BBF9757FD5623538AB64664 92608DE79700A2696D228555A71ECE576F131B71830D84172E7CD7C05B2585A5 3C7CB0DF58C23358698AB2A4E7319B9FF1A7480F80EC4A178BCDC9DCFE5DB374 3D73F39A7785A0B2F313EA4AC6875727DF6EC29E90C08D7617119BDC68150065 FDEA0A4CEE94D1CEDF1E2D29BB43864AB5B770A14C98A431FE892002C19B3221 F02826617596B5855B28202F696E0DF3AF3420D94859E53AEFC9D1DD75BF4A02 A4E2C12F0DBAA9DC2F6628F61497ADF417AABEAD37062788829B757D4563DFA8 FDC2A66E45C96C8FFFBC31898A8FCE588E1E18810BCB001AFB8F065FCBF8EF6B E2025AE75FD5A8955F09EB5CFEAF9C0DB75F5239E01332E7FE4B7CB450A9C45F CC1397C6B0A86284D5B10037E9C9E5B4450986908EC3E769861BB34154B0E15A 2475CE6183A9BCF859B0BEE2E561F8EBCB4590430C4FFEB4739742646360A8CD 13FA687EA4AF4032AB2A1D003273383CA1D9E65D5326ED55853C746DCD8017D9 50A70713AE1C88E377FEE874AEEFA995DBB2F234A83E36D2113C305DFA544AF4 A65B248D54C82E7BD925148EF431540D265DE6290F0C26C56F8AD7A81F41B574 C8CE8FE1CD3174E521774848AE2DE238E7E6F223332A76E62FFAC1BB2C6B401A 1FE8C8175B6E9E729853C94305B6A96D480F5557D393D39EAB82895969C04206 6C2DEEF60EB2881C70A896EB6102E5F14DA205678AE28E303522AD488CB9B8E4 F9CCA83FDEE875952C2B87FA5BC3CB57D7026FBC38C159BCFA8E6D6152BFE64B 051E97DC339583CB1C3F1708E45E0C475023EA119AA38575C6D19736A4FE9228 F1950932C2152B6A823AFC00A7C928E33ECF12B4DEB5549E314B20AA82DC68A9 39E1719A132D61EB6322C29488237F433D899F1638EAD62FFC30FBEE19967237 07A741A6F1AD9B51BC0F86A82CEB81DE1BE0ACC6ABE68A361F8CF9CCF71008F0 ACF9A9131A2725B366A1D8DBEB2624CCEBC46875049BCDB62CC85485192B1702 EC386EDA24DF5FF55855F237F3CBBC08723B8270250C1B765159C4F392DC6AE3 80CE613F137AFD7514107AA7E94FE3B5D806A965ECDD8BB96C694F7EB2DF0E69 773AC8C71D5BD7ADA788D81D3733574155208551EE6714BCC56F491BD61BCF43 240C1462551D49BF847659169E7DCB194784E4886A2071600FBCC90C3EE6069F 6F1A6E2923C46DA2947211D47F744C36F4034B317C561DC4DA79AB1AEE96A42C 54E5C688D16378056863A751FA776B66519ABD568E1F8494B20CED7DC98F5EE5 824F98187FD10AB93E17B12750F1FEC1659E7113C584312EE49616225CB39EC8 E4162D05AD8231837B8BC2E116C386FF8249F2A4201159C1D6616A001A48E1DF 1D2A6FBBA5635172346EF9C610BD086D4C8F1CC41F70797D5B2ECC33CD92A64B 0C6366DC7840EB8CD15DD83A6160A38616D3B012630731827DBA58A4D1B12F31 54D0E1943B0293EAB8D163AC5BB304761FA358A7571064CACEA76773597140F8 D9AA2B8C1CC1BBC82818E593FFEEC260C365CA98377B28999B3A68795309F2CF 5ECEEF0D2FB56EA74F36250241AF16FF571D04C4E22DE7D39BF7FC0F138CCE61 0A5D283F64A028796AEE8A18731ED4D59984F08E0FECDA5001ED377E85414882 64143D2ADD512C8BF2BEFE8D3C8DD8E3E9CEF65A59C21C2C7653E3415B9DA885 82AB95A728FA76FEDC1ADB5562BDBB4ADCA1FC700DD5CF179DBD1B6C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: URWPalladioL-BoldItal %!PS-AdobeFont-1.0: URWPalladioL-BoldItal 1.05 %%CreationDate: Wed Dec 22 1999 % Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development % (URW)++,Copyright 1999 by (URW)++ Design & Development % See the file COPYING (GNU General Public License) for license conditions. % As a special exception, permission is granted to include this font % program in a Postscript or PDF file that consists of a document that % contains text to be displayed or printed using this font, regardless % of the conditions or license applying to the document itself. 12 dict begin /FontInfo 10 dict dup begin /version (1.05) readonly def /Notice ((URW)++,Copyright 1999 by (URW)++ Design & Development. See the file COPYING (GNU General Public License) for license conditions. As a special exception, permission is granted to include this font program in a Postscript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /Copyright (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development) readonly def /FullName (URW Palladio L Bold Italic) readonly def /FamilyName (URW Palladio L) readonly def /Weight (Bold) readonly def /ItalicAngle -9.9 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /URWPalladioL-BoldItal def /PaintType 0 def /WMode 0 def /FontBBox {-170 -300 1073 935} readonly def /FontType 1 def /FontMatrix [0.001 0.0 0.0 0.001 0.0 0.0] readonly def /Encoding StandardEncoding def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BC2C03103C68570A7B354A4A280AE 6FBF7F9888E039AB60FCAF852EB4CE3AFEB979D5EA70FDE44A2AE5C8C0166C27 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: PazoMathBlackboardBold %!PS-AdobeFont-1.0: PazoMathBlackboardBold 001.003 %%CreationDate: Fri May 17 11:17:28 2002 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.003) readonly def /Copyright ((c) Diego Puga, 2000, 2002.) readonly def /Notice (Copyright (c) Diego Puga, 2000, 2002. Distributed under the GNU General Public License (http://www.gnu.org/copyleft/gpl.txt). As a special exception, permission is granted to include this font program in a PostScript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /FullName (Pazo Math Blackboard Bold) readonly def /FamilyName (PazoMathBlackboardBold) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (BlackboardBold) readonly def end readonly def /FontName /PazoMathBlackboardBold def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.00100 0 0 0.00100 0 0] readonly def /FontBBox {-15 -195 1118 709} readonly def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BCF7C3C3333173232E3FDBFF43949 1DB866C39088C203DC22FDC758584860EC7BB67FDA28CC6208249060E18FAB32 204779B5C03C0493BBBBC95CF02692CC4DEAA8D2EA90B5C2E64374E92BCB8501 429B8FAE4A76C0C6B76D6FF7CF9A7D5EDFBCA0E959541C59BD05B7DE43D25D53 FC3DDA6EF0C2743978A6D03E19CCED4A11F2EA4BCC3110BE8B8D9E2772361969 C19258EFAFDC276CB1ADE9208A941A36D18F9FB1C33DEF76AA315DD8F6FC13F6 956DF14F0BC043FC51433EF506A869D4DC4F4CC8551813BD4FB32A549ACA63E0 54D96D2899DF1C91C0DF800BE3BA3183ED98550A1B6E8341018E5958D6FE55D9 31F13205A8D7E8E8D272D572FA455E4F79A422E23BF4446DD2E07B074F30D20B 7B327FC652BF42CA69B3AF4EB4EDCC4DC3FDD9F6E97566FB964CB4941B14186F 90ACF11EB96EF5DBE23EE591471C6CBB1E5FEBED31B6233CF057945C928EEF4A BA00C461E2BD02E6845FABF2469862C1868AC67E9820944DF750735283EE93DB C5AC907F4DB0E3FFAA4879F572B8A31E14282100F00A93CC110AB9DB3D83E5A1 F4A3B69A6847A30060728BF2FE4ECFA2CC78D6D332E81D449B54D4C5ED4B0743 4B7F2FC9ACE090F2E35DB9BCD03AE87894485BCCD3DB72684636A4E42E58436C A6A21BA4F014B9C6BF83C7FDC71368E1710460D3FEBFCD506C3C37CFC202E0FD E93B06F345D460E8B09FB4BD3E3C8F6298DD646E421A6213EAB9714E2F770AD8 F73B72AF134C745D08A1B27D8524D8A3B26602FC08FE91B6EA727A66370A25B0 26278282F06DC022F2B5B73313D4CAE299EA9D6E460C9053DED9BEFC5D4AFA4A D2F3A52F7C51CA75C57255288CEBCAC17E5FF702FF96EBA92A735856224DC0AB F42956ABF57816DF995F73554C9731F708C2ACC44A8445A70674C80F2C802851 F9B56D4BCE72BA21D204950920F6693ECDFA6045CEA1E3ECEC4557F79DCCBAE8 A6449BCC4612C44B46B6C3F868B90CE48D0EB31D50D450AF1B73F38BF5C1183F 3D03D74C5C90E114F70FF1C5A57C6B65905F08E67DB1D2AA4B56D9871EF9D678 1F9E0973C62CA6023BE0867F8CB0D212ED40902165CF886FD6E41E5DF3F3D491 18A16198EF898EE374D37409110FA3A9431A57BD918BCD631490291AB39A8DBF 29A3A36A015163334C085CB8025AE64294CD7BF78C4017107490975F8FE6E41D AA2F3D6837DED80360D2EECBCC926002321A47C10DE63B298E2F88F669B2F977 A348E28994C87CCA39AE1F17BCA3ABD09FADFEF46F124CF880B36162E8838DB2 EEF382D5AD4BC17091EA476EB9DA01703BE04C1B27FB8E6ABD97BCDCA95F11BA 428CD8721484547F144B689B02E2D40285011696FB42CE221D35EF6D3A4D1711 1A0D28C1AC9D19A6F9ED2ED65CE6DCF6213C4F8D9EE3A45217B2259C4A076BF5 8F0B89D6115BF34E660434E53077206B06696878E08D696F299A0821E42C97CC 50E6C4D6C1ABDAA045D3402FD0AB800B022A9B59AC93D44CA1463EC3CF8726A3 FD7C28917CFB4D3B4587CBF59A38C4418DC5B1639D45D5A0ABBF1FA44FA6A5D7 779A5031AB41EA020AF7B697A616F07AA95FC25807B9C3C1786622618A58440D 82194CDB01A2CFC41133C69AD75946DBF8881B61D9AF69A47DF33A789FF79255 ECF2AF644ED9C88D07A5D43B48D40D3DE3B9062DEBFA016CF35B3296A4D2331A 8ACFD51DD8F95023A5763651EEF6D328079912F6012B974D2061E7E1BDAAF50B 9DA03EA412A45C4F2DDA808D365A7468BFCFBF7E680446F888A04685A6D16D46 9A20CE3D0401A7B165F5271B585DF664199E9CA3CF34E140D7DBDF0CD2C2808B F33DBF18EAF9E90B484FC8C8D637F6F6C0136E0E1F0FB0A3B6845B26DEE73E36 0BA24603A6B42398895F1F95D5F0D035FAE9DC24349F29614784C3339DBF4C4D 4D95243483D723FB557CBC51ED0027A70B19A0039DEF9FD4E7BAEE0B3AF17AA5 0A1DA4B0CFEF053F44EFDFEA9519D114360B4D3ED785EC763635391ADC754BC4 4144C5544715156A99395CC93B6189189FDFD3AB62C759498BE62E4A4C69618F E8EE03001B4A00839DB50767293D6F873666DE01A7354BDBD3B73F695138F3AB 72016B33B5418F210BACAA8CCC7D60AF299F7A6109C2682642104D9BB6DABACC 4AA136286FC74158421FBA6568591BD979D642FD47D7677CAD5B82B4548BCFEA FC7C74304E863B1CAB3DC4679896560ED8BD50E7E141A38ED6285646AFB0D163 B7F08BA3D9C9446A52615F03468042550ED9739076EB30477D455C7DF2258DD2 F13A33FE49FED460A102B225FF51C3CAD527A8BF9CB710BC75E0106F5F94BDCD 5EC3826DA19468F81C96DEDFCE3088529A5C2BFE6B271EFE69C67B6E0BE067C9 3285487121FB3EBFE72A4B9268E1F3D065E8BCD98D19BF27F59CB0616C73AC6D 0D113E9D918049692DAA676AF73508E9704D6E01E3FC943AA32D2E5C8DA3642A 91F1947AAB4A780D034BC00B16D7C4D77CA434A078A4716A24CDF975CC7EA03D 80D1B3F78C5A26817EF500183246F0377048C48AB0D05E92A03FCD2E6CD1B748 90B2029F4CDAF8B95A33FFD5B823CD6A47154F9AE5827A5B89299EC09650AEAD 8210F7AC185DD0F5F92CB77F7843C0D4BCC174D4FBD61DE7DDB8C5B82AC7D6D5 21E9D0A70DB9A2BCA8A15A617ACEDDE0A068E3977C4C730F2A37988D9D03CA30 5BED5E3FA22E8D0C64F81BABC9E046AEE8866049379032547DE1BBCEC7D01555 1B07AAD8C85C6196DC28076CF4C4A1C12A54ACA688D325B8FB6FD35D8B2A9420 802B29FDE12CA630B01A0EF0761F13888A4306667786F6967E002C318363FF20 B5A0762A4A9A546A212DB60D5241CFAAA85225DFF931D0829C9385C6D5F48BC1 5D9DD07B2F6C52952E53E988DE37F3AE41F517DA87CCC5DA3E4C99DCACF80286 44105038841988EEE3BC2351F0834216B411CD0127DFC19CEFDD5752FE117E17 5F1D4CDA471FBD381152783F047D62B2BEE984B17FD89C314D618E29B9620271 F7C8E0F5882A10A8F5449AA859831FF2B40D41EB2FD9C97C134944DFCAB36B89 3DEAF6514F9F1200759948846896C7ADB1FF1960B85AD009DEA2B826266D5C9F F99CE2D3EE2119DB141979F9F0A08C568A14BF67D0DDD15B7692928512B33B4C 579DE37C2F4E685A06F9202EE899AD9C341351AD59E7345098946C002CA22064 26BF4187816FA0B96CD49C0F860EB6DA1BD42A065562095B06DC28C2669F21DE A96DAD51F9605E33F07025D5C099A0CEC012CC13D5F5822F0FD4573D799B25E3 0A6B14826FB19236868E2AAD5651E0066EE46DAFAEA884C45DF208F8440809D2 DE5153874F44BBD533B0F7F060F661B70ECA742B19220A7102F03B2B9688D2F2 BF484FB09C7E789AE7873B80DA3346B580E302A5D0979538EE40BBE8B069DC43 D5D918ECC4120B9BAE603204B5AAA57C1143F50A87057B0EC0C8DB61258440A3 B9D4E052F86FAC5D267B524F67F643B6B833BE7609292B1FED6F854EBE19FBB6 6761E041B6871421FE2A3313187091D09ACECCF37CF6B155DA3846AB1EB9C7CA 97B5295A088A41851CE4EBADC9B42C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: PazoMath %!PS-AdobeFont-1.0: PazoMath 001.003 %%CreationDate: Fri May 17 11:17:28 2002 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.003) readonly def /Copyright ((c) Diego Puga, 2000, 2002.) readonly def /Notice (Copyright (c) Diego Puga, 2000, 2002. Distributed under the GNU General Public License (http://www.gnu.org/copyleft/gpl.txt). As a special exception, permission is granted to include this font program in a PostScript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /FullName (Pazo Math) readonly def /FamilyName (PazoMath) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) readonly def end readonly def /FontName /PazoMath def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 71 /Gamma put dup 198 /emptyset put dup 229 /summation put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.00100 0 0 0.00100 0 0] readonly def /FontBBox {-40 -283 878 946} readonly def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BCF7C3C3333173232E3FDBFF43949 1DB866C39088C203DC22FDC758584860EC7BB67FDA28CC6208249060E18FAB32 204779B5C03C0493BBBBC95CF02692CC4DEAA8D2EA90B5C2E64374E92BCB8501 429B8FAE4A76C0C6B76D6FF7CF9A7D5EDFBCA0E959541C59BD05B7DE43D25D53 FC3DDA6EF0C2743978A6D03E19CCED4A11F2EA4BCC3110BE8B8D9E2772361969 C19258EFAFDC276CB1ADE9208A941A36D18F9FB1C33DEF76AA315DD8F6FC13F6 956DF14F0BC043FC51433EF506A869D4DC4F4CC8551813BD4FB32A549ACA63E0 54D96D2899DF1C91C0DF800BE3BA3183ED98550A1B6E8341018E5958D6FE55D9 31F13205A8D7E8E8D272D572FA455E4F79A422E23BF4446DD2E07B074F30D20B 7B327FC652BF42CA69B3AF4EB4EDCC4DC3FDD9F6E97566FB964CB4941B14186F 90ACF11EB96EF5DBE23EE591471C6CBB1E5FEBED31B6233CF057945C928EEF4A BA00C461E2BD02E6845FABF2469862C1868AC67E9820944DF750735283EE93DB C5AC907F4DB0E3FFAA4879F572B8A31E142821042905BCE7CDD20581B4566DEF 6C7308188CA240DD953378AE17C8F146670B99CE786C72D24D538A986424AC14 F9616D0DF97D915C8E27F2F99C3BC0F97A7F58D0DA960A5241DB44CCC3AAD3EB B33F0BAFA3A307D3A12D586EBD694D666DEC2DE5F336192F0FC8CA12F2873A97 3A5233DB87054ABC74E097E919D8AFFE50E496365F3FF99CA21C0E7914253F12 97E11974DB6201F01987F082178ED90876E981DA2BC21E8A3E1BC06B64C0496B 6168202BDF877EE0627B3A422374E4B5F1AE5FB419EAF4012354EECFC0E86647 F34168C09224513607BB45FBBE45184DBE99D6276A99A79E5F72A0C22BC630F2 1C80258BFCB5CF958AA75E68D9A2B7BAE95BCBABBEA8CDD6D173A8B2829A91A8 202146B6BDBA12756472D83845580B897B0670DBA5093DE4BFF241556278867B FA4BD8F6B16BA3DB087744731F1B354F905CE1EC277700E17B3873714FF586D8 D8805E82BEEFFFD7E1A5BBC8064FCA6873AF4963E1AEB4556E9E50DEC15E6C86 9F6F483D23EBB55678821D43F68B570B0C2254239D644A205940CFCBC5D893A7 C6180C939327BD93B8DDAF770335D9255CDB6FBE2AFF74CAAE8D1CB13A12FA4F DBE9ECAD5A374C948FC10EF8049044A4BE109CA21B610D75453D75D618EF81DB BBD55B4C46DFC11592954E10D29A3C9F823445AEA45158E2946EBB55E87CD195 1AA77B2B2A73039B66EA4D2CAFA649D1AC291B0C7A3A82B7D9D7A60E0D3323A0 B709F40562D01C19C91E637A132BC8D6E023FED6817B1AECBA92325205A358F5 0893E39108F4A2128D046C3861AF304CD788C83E960F452C7FE3C53FD30C838C 3019AD019D719EF487B22532E8976800D6F7BD0FCCFF837E1CD0A59D88ABE3C8 FF2C7F99930FE769403B46E25CFC2041759BBC27CBA7D3F3F15E77B3FF03CA99 C867D2C9933AE39EC903795A6CB08AF08E8E864A163A4B146C17049270201098 006F532E9227C04A548A84E8652AE73A8A984BA235A8448523E4FB61522B5153 6AD69F860DA404DD8C5FF89A73B433863DEBC330EFA6104EE532580A7D087ABB 152C60E40979EC387233F4596B941EDFA080E35F9330B4DBA656A3ADC5513117 574C43A8A8D071B9F28835B537A70FA57D4B5105A1EA77879EC0A95B2D8EE4A1 518E2BCDF099DEB2C473C510CA28AB90147613F0EBC94C3EA0001040ECB1C333 F104AE66CE447123DF5C4349561CBC7B16A8BFC44F1D2767BD0D84ACBEDFDCF3 BC90CBA818881707232339F3FCD771914933CABF613F0F9FA3E201263F0ED9A6 A38BB39CA02A3504FEB9ECF4787CF3CB1F50F2589FE68B29179A3E97A3F0B35A 3FE8DF5B25820AA9A46B3F762BD799AED8E8A2C89898F7DFE12656D272E07AEA B1F2D0C9BFC8600FC97EEB7775397289A3E04E291EE8E9E19ED316C9751654A3 CD885FDF9CA95FB6020B097082E73509B70F0C64694B73BFF6E9007AB89F1C77 E0F1D1BE46155FA2C45930779B7BE808F27E984ADFC4099E1610A8F1ACBC0DFE 56DEF39B9F77173E5FBD8530DC9CDE44DC931AE06231CEEBA97739C6F3D04C9A 8B1801E807227D29389161E744FD4CAE8A227CAB65E8F6DAFF1885F4D4AC55BF B53444C98BB16FD06B69192A2146790B67AA56FADF3E60144C391EE9AADC6D26 F1D3B0B0E9797836CA0B1D5845A638E474A98F1EC24FFE4A5B5B06C74BA977CB FB4C1BD05563EC9E6754600AAC5AC9DFA7381C5F099A739E924266ABB0F5AF20 02DD17FB09401108241414A900D70C6060927EC46753645F880BD138BCBADAA1 55B45925E649AC3F7C555829C9CAF71C5318B590F535072495E724E633F2433A 5ED9413970F91DBF8F65A2A956BDAF4392A5C844DF9DFD50B9BEA18D947C780E 1D37D1EC32C188F311A107CC79FE5120FF3149420DA789CBE20DEC1DA560FCE2 87B86E393C824960DAE409001F68B04377E9DAD87BCD6759B3DDBACBD0BCBF65 BF856CC55D2B72AC6766790C81ECE3210433F8CB0B1528BD444DD1B0EFDAF667 A918171169D86699CB07512ECB44D71092A3CFC7C91EA6FD21F1AADFF04257A8 3A6CB987E1C0442D9FBB38539AE5B64C3EE39776D8D365CDD8FF305059C2E913 B7978671AC30C86917148D20F8B239B690400EF95A204F74A5EB0BCFEAF8F047 62557FD0CFD23B1E1537EB5C1C96FC8E13E483E22AAD36199164159504EE78B8 43D655B78B2CCB9241F5F460C546781735AB163091D8B993F1888658BF8073C2 6DA39D8F3C9900F826E51945DE6356BA976B0B490A04896A823C3529050AA960 B9D932833D83FCB09071FF52D9B99EFF6E2AC36895C02494F1D1E7FD184DB05C ACF3296865532E0597C8B7410E7E4445AF07FFCFF0488D22883A4EFFF3493935 C05BD6232B0B778BCAA304F10AA59C7F68FF91DB97DCDE55F701A63A7D1FA6F2 34C5C1E6AB854152E675C1476AD160D15D1AA6C439B5A799005E94BD1FAA0B18 4B920E187FF792FDEDB2321F0A36B1B1C0E935AC84696C5AC192C44C4A7E4982 5D19519C5EB55715F8DAE292731C7375DD80EF7BD18D688A1DB92A2B11E02BBA EE2FA914C82A303C6F314EC6B11679E916AABD5C467828ECBA2D6E89AD428BF4 E1035E92E55FB21E639AF66E6A3A504CED0223734E5A7F68B96A1184091D122A BB1E38913B4347FB11B98441AC98E80F6F1927389F1E876DE480507FB9BE8940 73228575A9E4DEB8CD7FAB42099157516030841C4C77FA2857264BB1E487B89A D6CD7F966FE947DBF44B6AAA3CA60E7F033C4AAC0F877B14B81469A4103BE305 AC460691CBD49F4CA398D01F40060FCADF610AFF655F504C0F7AAC8F68C95D04 6DD47CE6505D1BE740F26EEFF704859D4502D4C10758ED19AE1970E28D0FA3D6 E666B4AE09D4195593C86505EE89434400BAF6C60A928F5F238B8956064CEEC6 93C16B072F7C6D6CE676E526B14DA57A4FE629AC482DA81D616D8F71BE9C8608 A17B0020B7D3EAF3AE61C6BAE4B70717C6FDA3E24F184343FC691A6E03B7C69C 2D044042B4475A240F86C5E79812B9A536D0011C623776F871093E9CC4DF08E0 EDA044E1D4EFB0A55286F8749F78D8F90149FDEB7FBAFC72F146412C7F230C72 B3BE7623D35CC2B7E8E73E2B0327AC057B6FA5F160BFD5B43D0E41A9A27625AB C03EABEBF80581B416EDFE344C6D5CD0251E9EB85A17EFC722D8DD6B938FB5CE 1EE7761DE6AA3A6553CD40E8C1CD2493495BC5CF9926C3EAA6557E6DAA80FB2C CDF48A9E1B6A435CBE5BF71AEFE9B5D6F907CAD5BDAF3F570A2F1AED3712E129 26D3B31DE2E644B283CE45CE1473FF7028AD307A8D2EC30EDDBB195F7423ADC6 0419A29452EA165063A1C7E64301A9B0CF2A10EA6C86EF909D2C3B11BF30D80A E73DE2562E651584F620C02870DE71DB8684FFCCEDB3E97A24D84398DB80108C 0596956849CD43F56557EE5DF1F0CDC3E8E2D31E355CB3A38CE5A0B1DA1DDF9E B97B8DAAC7C5A61652DEA1D6DB61390D872202DB7D78F2A77F449A28E815496A E77590C304EACBBAB45022C61B0E47A522FE963349861C7AA3085866F2B22F21 E89D9447DE71A3C3CB470644FF8359D5F75DF58F13160516B086928DAA3D28EC 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h(pr)n(ocedur)n(e)f(can)h(also)h(ensur)n(e)d(a)j(str)n(onger)d(ex-post) g(fairness)h(pr)n(operty)f(called)0 3098 y FK(envy-fr)n(eeness)k(up)c (to)h(any)g(indivisible)h(good)g(for)g(mixed)f(goods)h(\(EFXM\))e FL([)p 0 0.58038 0.45097 TeXcolorrgb 2400 3098 a SDict begin H.S end 2400 3098 a FL(Bei)22 b(et)g(al.)2738 3098 y SDict begin 13.6 H.L end 2738 3098 a 2738 3098 a SDict begin [/Subtype /Link/Dest (cite.BeiLiLi21) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 2738 3098 a 0 TeXcolorgray 1 w FL(,)p 0 0.58038 0.45097 TeXcolorrgb 2779 3098 a SDict begin H.S end 2779 3098 a FL(2021a)3004 3098 y SDict begin 13.6 H.L end 3004 3098 a 3004 3098 a SDict begin [/Subtype /Link/Dest (cite.BeiLiLi21) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3004 3098 a 0 TeXcolorgray 2 w FL(;)p 0 0.58038 0.45097 TeXcolorrgb 3046 3098 a SDict begin H.S end 3046 3098 a FL(Nishimura)h(and)g(Sumita)3966 3098 y SDict begin 13.6 H.L end 3966 3098 a 3966 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/Link/Dest (equation.4.2) cvn H.B /ANN pdfmark end 1537 2558 a 0 TeXcolorgray FL(\))g(may)h(fail.)30 b(In)21 b(mor)n(e)i(detail,)f(bundle)28 b FK(A)2958 2574 y Fw(j)2981 2558 y FL(,)22 b(wher)n(e)j FK(j)j Fz(2)g(I)i FL(may)23 b(con-)0 2701 y(tain)h(mor)n(e)f(than)g (one)f(indivisible)j(good.)j(Hence,)22 b(it)h(may)h(be)f(that)2368 2708 y FJ(\345)2442 2722 y Fw(j)r Fv(2)r(I)2573 2701 y FK(u)2626 2717 y Fw(i)2650 2701 y FI(\()7 b FK(A)2764 2717 y Fw(j)2789 2701 y FI(\))28 b FG(>)2956 2708 y FJ(\345)3028 2653 y Fv(j)r(I)7 b(j)3030 2731 y Fw(j)r Fp(=)q FC(1)3158 2701 y FK(u)3211 2717 y Fw(i)3235 2701 y FI(\()t FK(B)3334 2668 y Fw(i)3334 2730 y(j)3358 2701 y FI(\))q FL(.)30 b(As)22 b(a)i(r)n(esult,)0 2838 y(Equation)c(\()p Plum 416 2838 a SDict begin H.S end 416 2838 a Plum FL(2)p Plum 462 2775 a SDict begin H.R end 462 2775 a 462 2838 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.2) cvn H.B /ANN pdfmark end 462 2838 a 0 TeXcolorgray 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b(a)i(good)d(at)h(this)h(r)n(ound)f FK(virtually)j FL(\(lines)p Plum 1783 4590 a SDict begin H.S end 1783 4590 a Plum -2 x FL(20)p Plum 1873 4525 a SDict begin H.R end 1873 4525 a 1873 4588 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.20) cvn H.B /ANN pdfmark end 1873 4588 a 0 TeXcolorgray FL(-)p Plum 1903 4588 a SDict begin H.S end 1903 4588 a Plum FL(21)p Plum 1995 4525 a SDict begin H.R end 1995 4525 a 1995 4588 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.21) cvn H.B /ANN pdfmark end 1995 4588 a 0 TeXcolorgray FL(\).)141 4701 y(If)h(ther)n(e)f(ar)n(e)h(not)f(enough)g(goods)e(to)j (allocate)h(one)e(to)g(each)h(unfr)n(ozen)g(agent,)f(the)g Fj(while)p FL(-loop)h(in)g(line)p Plum 3855 4701 a SDict begin H.S end 3855 4701 a Plum FL(4)p Plum 3900 4638 a SDict begin H.R end 3900 4638 a 3900 4701 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest 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FL(to)c(another)g(agent)k FK(j)e FL(under)f FH(p)j FL(can)d(be)g(eliminated)g(by)f(the)g(gain)h(under)f (another)0 3583 y(permutation)23 b FH(p)593 3550 y Fv(0)639 3583 y FL(which)h(just)e(exchanges)f FK(i)k FL(and)i FK(j)d FL(in)g FH(p)s FL(.)0 3608 y SDict begin H.S end 0 3608 a 0 3608 a SDict begin 13.6 H.A end 0 3608 a 0 3608 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.4) cvn /DEST pdfmark end 0 3608 a 0 TeXcolorgray 152 x FA(Lemma)f(5.4.)p 0 TeXcolorgray 42 w FK(Algorithm)p Plum 915 3761 a SDict begin H.S end 915 3761 a Plum -1 x FK(2)p Plum 961 3697 a SDict begin H.R end 961 3697 a 961 3760 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algocf.2) cvn H.B /ANN pdfmark end 961 3760 a 0 TeXcolorgray 22 w FK(r)n(eturns)h(an)f(ex-ante)h (EF)e(allocation.)0 3922 y SDict begin H.S end 0 3922 a 0 3922 a SDict begin 13.6 H.A end 0 3922 a 0 3922 a SDict begin [/View [/XYZ H.V]/Dest (subsection.5.3) cvn /DEST pdfmark end 0 3922 a 91 x FD(5.3)99 b(Ef)n(\002ciency)0 4190 y FL(W)-8 b(e)24 b(leverage)g(the)f(Fisher)h(market)g(\(r)n (elevant)h(concepts)e(in)i(Appendix\))f(to)g(pr)n(ovide)g(our)g(ef)n (\002ciency)h(guaran-)0 4303 y(tee.)0 4305 y SDict begin H.S end 0 4305 a 0 4305 a SDict begin 13.6 H.A end 0 4305 a 0 4305 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.5) cvn /DEST pdfmark end 0 4305 a 0 TeXcolorgray 175 x FA(Lemma)e(5.5.)p 0 TeXcolorgray 42 w FK(Every)h(integral)g (allocation)h(r)n(eturned)f(by)g(Algorithm)p Plum 2293 4481 a SDict begin H.S end 2293 4481 a Plum -1 x FK(2)p Plum 2339 4417 a SDict begin H.R end 2339 4417 a 2339 4480 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algocf.2) cvn H.B /ANN pdfmark end 2339 4480 a 0 TeXcolorgray 22 w FK(is)g(fPO.)p 0 TeXcolorgray 0 4658 a(Pr)n(oof.)p 0 TeXcolorgray 46 w FL(T)-8 b(o)19 b(leverage)g(Theor)n(em)p Plum 1120 4660 a SDict begin H.S end 1120 4660 a Plum -2 x FL(C.1)p Plum 1252 4595 a SDict begin H.R end 1252 4595 a 1252 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closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 127 91 a FL(\225)p 0 TeXcolorgray 45 w(for)33 b(each)h(agent)e FK(i)48 b Fz(2)i FK(N)1107 58 y Fw(r)1173 91 y FL(who)32 b(r)n(eceives)h(both)g(lar)n(ge)g(and)g (small)h(goods,)g(set)k FK(p)3127 105 y Fw(g)3211 91 y FI(=)47 b FK(u)3385 107 y Fw(i)3409 91 y FI(\()t FK(g)r FI(\))35 b FL(for)e(each)230 204 y FK(g)28 b Fz(2)33 b FK(A)467 220 y Fw(i)490 204 y FL(.)0 392 y(For)h(the)f(prices)h(of)h (the)f(r)n(emaining)h(goods,)g(we)f(will)i(set)d(them)h(iteratively:)52 b(if)35 b(ther)n(e)f(exists)f(one)h(agent)f FK(i)0 505 y FL(without)d(pricing)i(her)e(goods)f(such)i(that)g(some)f(agent)g 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FK(A)2803 972 y Fw(i)2847 956 y FL(is)21 b(exactly)h(1.)28 b(For)20 b(each)i(good)3 1069 y FK(g)31 b FL(which)e(is)g(allocated)h(to)e(some)g(agent)k FK(j)f FL(in)e(a)i FK(Z)1747 1086 y Fw(t)1771 1067 y Fb(0)1791 1086 y Fv(\024)r Fw(t)1873 1069 y FL(,)g(if)k FK(p)2066 1083 y Fw(g)2143 1069 y FI(=)k FK(u)2311 1085 y Fw(j)2335 1069 y FI(\()t FK(g)r FI(\))h(=)g FK(a)q FL(,)31 b(this)e(good)e(is)i FK(actually)h FL(allocated)0 1182 y(at)25 b(line)p Plum 274 1184 a SDict begin H.S end 274 1184 a Plum -2 x FL(26)p Plum 365 1119 a SDict begin H.R end 365 1119 a 365 1182 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.26) cvn H.B /ANN pdfmark end 365 1182 a 0 TeXcolorgray 25 w FL(and)j FK(g)f FL(is)e(in)30 b FK(M)931 1149 y Fw(r)959 1162 y Fn(i)1000 1182 y Fz(n)21 b FJ(G)r FI(\()s FK(Z)1222 1196 y Fw(t)1250 1182 y FI(\))q FL(,)k(which)h(is)f(valued)i FK(a)g FL(by)e FK(i)i FL(fr)n(om)e(the)f(second)g(term)g(in)i (Observation)p Plum 3764 1184 a SDict begin H.S end 3764 1184 a Plum -2 x FL(5.2)p Plum 3877 1119 a SDict begin H.R end 3877 1119 a 3877 1182 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.2) cvn H.B /ANN pdfmark end 3877 1182 a 0 TeXcolorgray FL(,)0 1295 y(whose)e(ratio)i FK(u)552 1311 y Fw(i)576 1295 y FI(\()t FK(g)r FI(\))r FL(/)7 b FK(p)813 1309 y Fw(g)879 1295 y FL(is)25 b(at)g(most)f(1.)36 b(For)24 b(each)i(good)g FK(g)h FL(which)f(is)f(allocated)h(to)e(some)h (agent)j FK(j)f FL(in)i FK(N)3649 1262 y Fw(r)3677 1275 y Fn(i)3720 1295 y Fz(n)22 b FK(Z)3850 1309 y Fw(t)3877 1295 y FL(,)0 1408 y(we)j(also)i(have)g FK(u)605 1424 y Fw(i)629 1408 y FI(\()t FK(g)r FI(\))34 b(=)h FK(a)28 b FL(fr)n(om)e(Observation)p Plum 1707 1410 a SDict begin H.S end 1707 1410 a Plum -2 x FL(5.2)p Plum 1820 1345 a SDict begin H.R end 1820 1345 a 1820 1408 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.2) cvn H.B /ANN pdfmark end 1820 1408 a 0 TeXcolorgray FL(.)38 b(Thus,)26 b(each)g(agent)g FK(i)i FL(in)f(some)g FK(Z)3050 1422 y Fw(t)3103 1408 y FL(only)e(r)n(eceives)h(goods)0 1521 y(in)d(MBB)299 1535 y FC(i)323 1521 y FL(.)141 1634 y(For)h(each)h (low-price)g(agent,)f(fr)n(om)h(the)f(thir)n(d)h(term)f(in)h (Observation)p Plum 2619 1636 a SDict begin H.S end 2619 1636 a Plum -2 x FL(5.2)p Plum 2732 1571 a SDict begin H.R end 2732 1571 a 2732 1634 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.2) cvn H.B /ANN pdfmark end 2732 1634 a 0 TeXcolorgray FL(,)g(all)i(goods)22 b(allocated)j(to)g(her)f (in)0 1747 y(the)j(multi-r)n(ound)i(pr)n(ocedur)n(e)d(must)i(be)g(lar)n (ge.)44 b(Because)27 b(this)h(agent)f(is)i(not)e(priced)h(befor)n(e)f (the)h(end)f(of)h(the)0 1860 y(iteration,)20 b(she)e(r)n(eceives)h(no)g (small)i(goods.)k(Since)19 b(the)g(goods)e(she)i(owns)f(ar)n(e)i (priced)f(at)j FK(a)f FL(which)f(is)g(the)e(lowest)0 1973 y(price)k(we)g(set)f(and)i(all)g(goods)e(in)i(her)e(bundle)h(must) g(be)g(valued)h(at)h FK(b)f FL(under)e(her)h(valuation,)i(all)f(goods)e (in)i(her)0 2086 y(bundle)f(must)g(be)h(in)g(MBB)948 2100 y FC(i)971 2086 y FL(.)141 2198 y(W)-8 b(e)25 b(then)g(come)h(to)g (some)f(agent)g FK(i)35 b Fz(2)i FK(N)1564 2165 y Fw(r)1623 2198 y FL(who)25 b(sets)31 b FK(p)2058 2212 y Fw(g)2129 2198 y FI(=)j FK(u)2290 2214 y Fw(i)2314 2198 y FI(\()t FK(g)r FI(\))28 b FL(for)e(each)j FK(g)34 b Fz(2)39 b FK(A)3069 2214 y Fw(i)3092 2198 y FL(.)f(The)25 b(ratio)j FK(u)3599 2214 y Fw(i)3623 2198 y FI(\()t FK(g)r FI(\))r FL(/)7 b FK(p)3860 2212 y Fw(g)0 2311 y FL(for)24 b(each)g(good)h FK(g)30 b Fz(2)35 b FK(A)819 2327 y Fw(i)865 2311 y FL(is)24 b(exactly)g(1.)32 b(If)24 b(she)e(values)i(a)h(good)g FK(g)h FL(in)e(some)29 b FK(A)2676 2327 y Fw(j)2699 2311 y FL(,)19 b FK(j)30 b Fz(2)i FK(N)c FL(at)d FK(b)g FL(\(thus)j FK(j)d FL(cannot)f(be)g(a)0 2424 y(low-price)g(agent)f(otherwise)f(it)i (will)h(be)f(priced)f(befor)n(e)g(the)g(end)g(fr)n(om)i(the)e (iteration\))h(and)g FK(u)3327 2440 y Fw(j)3352 2424 y FI(\()t FK(g)r FI(\))30 b(=)k FK(p)3663 2438 y Fw(g)3731 2424 y FI(=)c FK(a)q FL(,)0 2537 y(we)21 b(can)g(backwar)n(d)h(the)e (iteration)h(fr)n(om)h(agent)e FK(i)k FL(and)d(\002nally)h(achieve)f (an)h(agent)e FK(i)2903 2504 y Fv(0)2954 2537 y Fz(2)30 b FK(N)3121 2504 y Fw(r)3175 2537 y FL(who)21 b(r)n(eceives)g(both)0 2650 y(lar)n(ge)27 b(and)f(small)h(goods)e(fr)n(om)h(her)g (perspective.)37 b(This)26 b(backwar)n(d)h(path)f(along)h(with)f(the)f (good)g(pr)n(esented)0 2763 y(in)30 b(the)g(condition)f(of)h(the)g (iteration)f(is)i(a)f(feasible)h(augmenting)e(path)h(at)g(line)p Plum 2836 2763 a SDict begin H.S end 2836 2763 a Plum FL(24)p Plum 2926 2700 a SDict begin H.R end 2926 2700 a 2926 2763 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.24) cvn H.B /ANN pdfmark end 2926 2763 a 0 TeXcolorgray 31 w FL(and)g(agent)f FK(i)3421 2730 y Fv(0)3475 2763 y FL(would)h(not)0 2876 y(r)n(eceive)c(a)g(small)h(good,)e(which)h (leads)g(to)f(a)h(contradiction.)37 b(This)26 b(completes)e(the)h(pr)n (oof)g(for)h(the)f(case)h(when)1 2989 y FK(m)h Fz(\025)g FK(n)q FL(.)p 3834 2989 4 62 v 3838 2931 55 4 v 3838 2989 V 3892 2989 4 62 v 141 3176 a(W)-8 b(e)22 b(ar)n(e)h(now)f(r)n (eady)g(to)g(pr)n(ove)g(our)g(main)i(r)n(esult)e(in)h(this)g(section.)p 0 TeXcolorgray 0 3364 a FK(Pr)n(oof)h(of)f(Theor)n(em)p Plum 646 3365 a SDict begin H.S end 646 3365 a Plum -1 x FK(5.1)p Plum 759 3301 a SDict begin H.R end 759 3301 a 759 3364 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.1) cvn H.B /ANN pdfmark end 759 3364 a 0 TeXcolorgray FK(.)p 0 TeXcolorgray 45 w FL(Fr)n(om)k(Lemmas)p Plum 1440 3366 a SDict begin H.S end 1440 3366 a Plum -2 x FL(5.3)p Plum 1554 3301 a SDict begin H.R end 1554 3301 a 1554 3364 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.3) cvn H.B /ANN pdfmark end 1554 3364 a 0 TeXcolorgray FL(,)p Plum 1577 3366 a SDict begin H.S end 1577 3366 a Plum -2 x FL(5.4)p Plum 1690 3301 a SDict begin H.R end 1690 3301 a 1690 3364 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.4) cvn H.B /ANN pdfmark end 1690 3364 a 0 TeXcolorgray 28 w FL(and)p Plum 1899 3366 a SDict begin H.S end 1899 3366 a Plum -2 x FL(5.5)p Plum 2012 3301 a SDict begin H.R end 2012 3301 a 2012 3364 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.5) cvn H.B /ANN pdfmark end 2012 3364 a 0 TeXcolorgray FL(,)h(it)g(suf)n(\002ces)e(to)h (show)f(Algorithm)p Plum 3274 3364 a SDict begin H.S end 3274 3364 a Plum FL(2)p Plum 3319 3301 a SDict begin H.R end 3319 3301 a 3319 3364 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algocf.2) cvn H.B /ANN pdfmark end 3319 3364 a 0 TeXcolorgray 28 w FL(can)i(terminate)0 3477 y(in)33 b(polynomial)h(time.)59 b(Fr)n(om)33 b(the)f(description)g(of)h (our)f(algorithm,)k(since)e Fz(j)6 b FK(M)s Fz(j)34 b FL(keeps)d(decr)n(easing)i(which)0 3590 y(bounds)26 b(the)g(total)h (number)g(of)g(r)n(ounds)f(by)i FK(m)h FL(and)e(the)f(only)h(thing)f (we)h(need)f(is)h(to)g(show)f(lines)p Plum 3492 3592 a SDict begin H.S end 3492 3592 a Plum -2 x FL(6)p Plum 3537 3527 a SDict begin H.R end 3537 3527 a 3537 3590 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.6) cvn H.B /ANN pdfmark end 3537 3590 a 0 TeXcolorgray FL(-)p Plum 3567 3592 a SDict begin H.S end 3567 3592 a Plum -2 x FL(9)p Plum 3613 3527 a SDict begin H.R end 3613 3527 a 3613 3590 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.9) cvn H.B /ANN pdfmark end 3613 3590 a 0 TeXcolorgray 27 w FL(can)i(be)0 3703 y(terminated)23 b(in)h(polynomial)h(time.)31 b(Fr)n(om)24 b(the)f(constr)o(uction)g(of)j FK(Z)2371 3717 y Fw(t)2422 3703 y FL(accor)n(ding)e(to)g(Algorithm)p Plum 3391 3703 a SDict begin H.S end 3391 3703 a Plum FL(4)p Plum 3437 3640 a SDict begin H.R end 3437 3640 a 3437 3703 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algocf.4) cvn H.B /ANN pdfmark end 3437 3703 a 0 TeXcolorgray FL(,)g(if)g(we)g(keep)0 3816 y(the)i(perfect)f(matching)j(between)h FK(N)24 b Fz(n)e FK(Z)1458 3830 y Fw(t)1512 3816 y FL(and)32 b FK(M)21 b Fz(n)h FJ(G)r FI(\()s FK(Z)2028 3830 y Fw(t)2056 3816 y FI(\))28 b FL(by)e(just)g(r)n(eplacing)h(the)f(good)i FK(g)3255 3783 y Fv(0)3306 3816 y FL(by)i FK(g)q FL(,)e(one)e(edge)0 3929 y(fr)n(om)h FK(j)h Fz(2)h FK(Z)427 3943 y Fw(t)476 3929 y FL(to)c FK(g)628 3896 y Fv(0)675 3929 y FL(is)e(added,)e(this)h (leads)g(to)g(one)g(of)g(the)g(following)h(cases:)k(either)21 b(\(1\))j(one)e(augmenting)f(path)0 4042 y(can)k(be)f(performed)f(and)h (the)g(size)g(of)h(the)e(maximum)j(matching)f(between)i FK(N)h FL(and)i FK(M)c 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FK(u)r FI(\()t FK(B)r FI(\))21 b Fz(\000)i FK(u)r FI(\()7 b FK(A)r FI(\))39 b FG(>)h FK(u)r FI(\()t FK(g)r FI(\))31 b FL(and)g FK(u)r FI(\()7 b FK(A)r FI(\))22 b(+)g FK(u)r FI(\()t FK(g)r FI(\))40 b FG(>)h FK(u)r FI(\()t FK(B)r FI(\))21 b Fz(\000)h FK(u)r FI(\()t FK(g)r FI(\))32 b FL(fr)n(om)e(the)0 4744 y(conditions)h(in)g (lines)p Plum 784 4744 a SDict begin H.S end 784 4744 a Plum FL(14)p Plum 875 4681 a SDict begin H.R end 875 4681 a 875 4744 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.14) cvn H.B /ANN pdfmark end 875 4744 a 0 TeXcolorgray 31 w FL(and)p Plum 1092 4744 a SDict begin H.S end 1092 4744 a Plum FL(17)p Plum 1183 4681 a SDict begin H.R end 1183 4681 a 1183 4744 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.17) cvn H.B /ANN pdfmark end 1183 4744 a 0 TeXcolorgray FL(,)i(so)e(the)f(utility)i(dif)n(fer)n(ence)f(between)f (the)h(two)f(bundles)h(decr)n(eases)f(fr)n(om)0 4857 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FL(\),)h(so)f(an)h(item)g(can)h(be)e(moved)g(between)f(the)h(two)g (bundles)g(at)h(most)e(once.)40 b(The)0 5309 y(overall)22 b(time)e(complexity)h(is)f(bounded)f(by)h FK(O)r FI(\()r FK(m)c FL(log)g FK(m)r FI(\))q FL(,)21 b(wher)n(e)f(we)g(may)h(\002rst) f(sort)f(the)h(items)g(accor)n(ding)i(to)p 0 TeXcolorgray 1905 5649 a(19)p 0 TeXcolorgray eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.20) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 1 91 a FK(u)q FL(,)22 b(and)h(then)f(perform)g(the)f 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1025 a 2297 1088 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.6) cvn H.B /ANN pdfmark end 2297 1088 a 0 TeXcolorgray 27 w FL(fail\),)j(fr)n(om)d(Lemma)p Plum 3088 1090 a SDict begin H.S end 3088 1090 a Plum -2 x FL(3.1)p Plum 3201 1025 a SDict begin H.R end 3201 1025 a 3201 1088 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 3201 1088 a 0 TeXcolorgray FL(,)h(both)g Fz(A)3540 1055 y FC(1)3604 1088 y FL(and)g Fz(A)3863 1055 y FC(2)0 1200 y FL(ar)n(e)d(EFX)e(if)i(for)f(each)h FK(i)31 b Fz(2)g FI([)q FL(2)q FI(])q FL(,)26 b(agent)d(3)d Fz(\000)f FK(i)27 b FL(picks)d(her)g(pr)n(eferr)n(ed)f(bundle)h (\002rst)f(in)j Fz(A)2988 1167 y Fw(i)3011 1200 y FL(.)33 b(Because)23 b(condition)h(in)0 1313 y(line)p Plum 178 1313 a SDict begin H.S end 178 1313 a Plum FL(4)p Plum 223 1250 a SDict begin H.R end 223 1250 a 223 1313 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 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Fw(i)1782 1436 y FL(is)g(upper)f(bounded)g(by)i(the)e (gain)i(of)g FK(u)3160 1452 y Fw(i)3185 1436 y FI(\()7 b FK(A)3296 1399 y FC(3)q Fv(\000)r Fw(i)3296 1465 y FC(2)3409 1436 y FI(\))20 b Fz(\000)h FK(u)3614 1452 y Fw(i)3638 1436 y FI(\()7 b FK(A)3749 1399 y FC(3)q Fv(\000)r Fw(i)3749 1466 y FC(1)3862 1436 y FI(\))0 1549 y FL(under)27 b Fz(A)348 1516 y FC(3)q Fv(\000)r Fw(i)461 1549 y FL(.)43 b(Thus,)28 b(this)f(randomized)h(allocation)h(is)e (ex-ante)g(EF)g(and)h(ex-post)e(EFX.)g(It)h(suf)n(\002ces)g(to)g(show)0 1662 y(that)22 b(this)h(algorithm)g(can)g(terminate)f(in)h(polynomial)g (steps,)e(wher)n(e)h(we)g(adopt)f(a)j(similar)g(analysis.)141 1775 y(Fr)n(om)35 b(the)g(condition)g(at)g(line)p Plum 1267 1775 a SDict begin H.S end 1267 1775 a Plum FL(4)p Plum 1313 1712 a SDict begin H.R end 1313 1712 a 1313 1775 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.4) cvn H.B /ANN pdfmark end 1313 1775 a 0 TeXcolorgray 35 w FL(and)g(the)g(monotonic)g(pr)n(operty)e(in)j(the)e(pr)n(oof)h(of)h (Lemma)p Plum 3540 1777 a SDict begin H.S end 3540 1777 a Plum -2 x FL(3.1)p Plum 3654 1712 a SDict begin H.R end 3654 1712 a 3654 1775 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 3654 1775 a 0 TeXcolorgray FL(,)i(after)0 1888 y(executing)26 b(line)p Plum 593 1890 a SDict begin H.S end 593 1890 a Plum -2 x FL(5)p Plum 638 1825 a SDict begin H.R end 638 1825 a 638 1888 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.5) cvn H.B /ANN pdfmark end 638 1888 a 0 TeXcolorgray FL(,)i(the)f(utility)g(dif)n(fer)n(ence)g(between)f(the)g(two)h (bundles)f(in)j Fz(A)2777 1855 y FC(3)q Fv(\000)r Fw(i)2917 1888 y FL(under)e FK(u)3240 1904 y FC(3)q Fv(\000)r Fw(i)3380 1888 y FL(decr)n(eases)e(\(to)0 2000 y(see)30 b(this,)j(Lemma)p Plum 698 2002 a SDict begin H.S end 698 2002 a Plum -2 x FL(3.1)p Plum 811 1937 a SDict begin H.R end 811 1937 a 811 2000 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 811 2000 a 0 TeXcolorgray 32 w FL(implies)f(that)f(the)g(utility)g(dif)n(fer)n(ence)h(between)d (the)i(two)g(bundles)f(in)i(the)f(updated)1 2113 y Fz(A)78 2080 y FC(3)q Fv(\000)r Fw(i)222 2113 y FL(is)g(weakly)f(smaller)i (than)f(the)f(utility)h(dif)n(fer)n(ence)g(of)g(the)f(two)g(bundles)g (in)j Fz(A)3040 2080 y Fw(i)3094 2113 y FL(under)d FK(u)3420 2129 y FC(3)q Fv(\000)r Fw(i)3533 2113 y FL(,)j(and)e(the)0 2226 y(condition)k(at)h(line)p Plum 715 2226 a SDict begin H.S end 715 2226 a Plum FL(4)p Plum 760 2163 a SDict begin H.R end 760 2163 a 760 2226 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.4) cvn H.B /ANN pdfmark end 760 2226 a 0 TeXcolorgray 36 w FL(implies)g(that)f(the)f(utility)i(dif)n (fer)n(ence)f(between)f(the)h(two)g(bundles)f(in)j Fz(A)3474 2193 y FC(3)q Fv(\000)r Fw(i)3622 2226 y FL(strictly)0 2339 y(decr)n(eases)24 b(after)i(executing)e(line)p Plum 1209 2341 a SDict begin H.S end 1209 2341 a Plum -2 x FL(5)p Plum 1255 2276 a SDict begin H.R end 1255 2276 a 1255 2339 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.5) cvn H.B /ANN pdfmark end 1255 2339 a 0 TeXcolorgray FL(\).)37 b(W)-8 b(e)25 b(\002rst)g(notice)g(that)h(line)p Plum 2309 2341 a SDict begin H.S end 2309 2341 a Plum -2 x FL(13)p Plum 2400 2276 a SDict begin H.R end 2400 2276 a 2400 2339 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.13) cvn H.B /ANN pdfmark end 2400 2339 a 0 TeXcolorgray 25 w FL(cannot)g(be)g(executed)d(in)j(this)g(call)h(of)f(the)0 2452 y(subr)n(outine)i(because)g(of)g(the)g(failur)n(e)i(of)f(the)f (conditions)f(at)i(line)p Plum 2358 2452 a SDict begin H.S end 2358 2452 a Plum FL(2)p Plum 2404 2389 a SDict begin H.R end 2404 2389 a 2404 2452 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.2) cvn H.B /ANN pdfmark end 2404 2452 a 0 TeXcolorgray 28 w FL(or)f(the)g(pr)n(eviously)g (executed)f(line)p Plum 3725 2454 a SDict begin H.S end 3725 2454 a Plum -2 x FL(6)p Plum 3770 2389 a SDict begin H.R end 3770 2389 a 3770 2452 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.6) cvn H.B /ANN pdfmark end 3770 2452 a 0 TeXcolorgray FL(.)46 b(If)0 2565 y(ther)n(e)24 b(is)h(no)f(swap)h(step)e(\(line)p Plum 1082 2565 a SDict begin H.S end 1082 2565 a Plum FL(17)p Plum 1173 2502 a SDict begin H.R end 1173 2502 a 1173 2565 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.17) cvn H.B /ANN pdfmark end 1173 2565 a 0 TeXcolorgray FL(\))i(executed)e(in)i(this)g(call)i (of)e(the)f(subr)n(outine,)g(since)h(the)f(original)i(allocation)1 2678 y FI(\()7 b FK(A)113 2645 y Fw(i)112 2706 y FC(1)149 2678 y FL(,)21 b FK(A)261 2645 y Fw(i)260 2705 y FC(2)298 2678 y FI(\))31 b FL(is)g(EFX)e(under)i FK(u)987 2694 y Fw(i)1010 2678 y FL(,)h(either)f FK(u)1382 2694 y Fw(i)1406 2678 y FI(\()7 b FK(A)1517 2641 y FC(3)q Fv(\000)r Fw(i)1517 2708 y FC(1)1631 2678 y FI(\))42 b Fz(\025)g FK(u)1879 2694 y Fw(i)1903 2678 y FI(\()7 b FK(A)2014 2641 y FC(3)q Fv(\000)r Fw(i)2014 2707 y FC(2)2127 2678 y FI(\))32 b FL(and)f(then)e(meets)h(the)f(condition)i(at)f(line)p Plum 3714 2680 a SDict begin H.S end 3714 2680 a Plum -2 x FL(6)p Plum 3759 2615 a SDict begin H.R end 3759 2615 a 3759 2678 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.3.6) cvn H.B /ANN pdfmark end 3759 2678 a 0 TeXcolorgray FL(,)i(or)0 2791 y(the)23 b(allocation)j Fz(A)644 2758 y FC(3)q Fv(\000)r Fw(i)781 2791 y FL(is)e(also)f(EFX)g(for)h(agent)f FK(i)r FL(.)31 b(In)24 b(both)f(cases,)g(the)g(algorithm)h(terminates.) 30 b(Thus,)23 b(we)g(then)0 2904 y(assume)c(some)g(swap)h(step)e(is)i 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SDict begin H.S end 1742 1744 a Plum -2 x FL(B.9)p Plum 1866 1679 a SDict begin H.R end 1866 1679 a 1866 1742 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.9) cvn H.B /ANN pdfmark end 1866 1742 a 0 TeXcolorgray 19 w FL(separately)f(and)g(r)n(espectively)-10 b(.)26 b(W)-8 b(e)18 b(start)g(with)h(a)g(lemma)0 1855 y(which)k(will)h(be)f(used)e(in)i(the)f(pr)n(oof)g(of)h(Lemma)p Plum 1703 1857 a SDict begin H.S end 1703 1857 a Plum -2 x FL(B.8)p Plum 1827 1792 a SDict begin H.R end 1827 1792 a 1827 1855 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.8) cvn H.B /ANN pdfmark end 1827 1855 a 0 TeXcolorgray FL(.)0 1880 y SDict begin H.S end 0 1880 a 0 1880 a SDict begin 13.6 H.A end 0 1880 a 0 1880 a SDict begin [/View [/XYZ H.V]/Dest (theorem.B.7) cvn /DEST pdfmark end 0 1880 a 0 TeXcolorgray 153 x FA(Lemma)g(B.7.)p 0 TeXcolorgray 41 w FK(The)g(indivisible)i(goods)g(in)d(Y)j(ar)n(e)f(fully)g 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FL(;)d(otherwise,)e(ther)n(e)h(is)h(an)h (augmenting)e(path,)g(and)h(the)g(allocation)0 2663 y(should)d(have)h (been)f(updated)f(during)h(the)g(\002rst)g(phase)g(of)g(Step)g(3.)28 b(Every)19 b(agent)h(in)j FK(T)e Fz(n)e FK(T)3152 2678 y FC(1)3211 2663 y FL(values)i(each)g(good)0 2776 y(matched)k(to)g (agents)i FK(T)817 2791 y FC(1)880 2776 y FL(at)h FK(a)f FL(as)f(we)f(have)h(alr)n(eady)g(found)f(all)i(the)e(alternating)g (paths)g(starting)g(fr)n(om)k FK(g)q FL(.)37 b(Let)1 2889 y FK(S)31 b Fa(\()h FK(T)239 2904 y FC(1)301 2889 y FL(and)27 b FK(S)32 b Fz(6)r FI(=)g FJ(\306)q FL(.)k(Then,)26 b FK(S)g FL(is)f(an)h(evidence)e(demonstrating)g(that)j FK(T)h FL(is)d(not)g(the)f(minimal)k(unmatchable)0 3001 y(gr)n(oup)22 b(\(see)f(De\002nition)p Plum 875 3001 a SDict begin H.S end 875 3001 a Plum FL(B.4)p Plum 999 2938 a SDict begin H.R end 999 2938 a 999 3001 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.4) cvn H.B /ANN pdfmark end 999 3001 a 0 TeXcolorgray FL(\),)h(a)i(contradiction.)p 3834 3001 4 62 v 3838 2943 55 4 v 3838 3001 V 3892 3001 4 62 v 0 3100 a SDict begin H.S end 0 3100 a 0 3100 a SDict begin 13.6 H.A end 0 3100 a 0 3100 a SDict begin [/View [/XYZ H.V]/Dest (theorem.B.8) cvn /DEST pdfmark end 0 3100 a 0 TeXcolorgray 87 x FA(Lemma)f(B.8.)p 0 TeXcolorgray 41 w FK(The)g(randomized)i(allocation)h Fz(R)e FK(is)f(ex-ante)i(PROP)c (for)j(agents)g(in)g(T)s(.)p 0 TeXcolorgray 0 3365 a(Pr)n(oof.)p 0 TeXcolorgray 46 w FL(W)-8 b(e)26 b(de\002ne)f(baseline)i(allocation)h (as)f(in)g(De\002nition)p Plum 2116 3365 a SDict begin H.S end 2116 3365 a Plum FL(4.4)p Plum 2230 3302 a SDict begin H.R end 2230 3302 a 2230 3365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 2230 3365 a 0 TeXcolorgray FL(,)g(and)g(adopt)e(a)i(similar)h (analysis)f(as)g(in)g(Theo-)0 3478 y(r)n(em)p Plum 179 3480 a SDict begin H.S end 179 3480 a Plum -2 x FL(4.3)p Plum 293 3415 a SDict begin H.R end 293 3415 a 293 3478 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.4.3) cvn H.B /ANN pdfmark end 293 3478 a 0 TeXcolorgray FL(.)g(For)20 b(any)h(agent)f FK(i)29 b Fz(2)g FK(T)s FL(,)21 b(as)g(ther)n(e)f(is)h (no)f(edge)f(fr)n(om)24 b FK(T)g FL(to)h FK(M)19 b Fz(n)e FK(Y)24 b FL(and)d Fz(j)p FK(Y)t Fz(j)29 b FG(<)e Fz(j)s FK(T)t Fz(j)h(\024)f FK(n)q FL(,)21 b(the)f(number)h(of)0 3591 y(goods)e(valued)i(at)h FK(b)h FL(by)e(agent)f FK(i)j FL(is)e(at)h(most)f FK(n)q FL(.)27 b(Hence,)20 b(each)h(bundle)j FK(B)2512 3558 y Fw(i)2510 3622 y(k)2565 3591 y FL(wher)n(e)d FK(k)28 b Fz(2)g FI([)r FK(n)r FI(])23 b FL(in)e(the)g(baseline)g (allo-)0 3704 y(cation)j Fz(B)335 3671 y Fw(i)381 3704 y FL(contains)e(at)g(most)f(one)h(indivisible)h(good)e(of)h(value)i FK(b)g FL(and)e(the)f(number)h(of)g(bundles)g(that)g(contain)0 3817 y(two)h(indivisible)i(goods)d(with)i(values)h FK(b)g FL(and)g FK(a)h FL(r)n(espectively)d(is)h(maximized)h(accor)n(ding)f (to)g(the)e(pr)n(operty)g(of)0 3930 y(Round-Robin.)141 4043 y(W)-8 b(e)26 b(use)32 b FK(A)r FI(\()r FH(p)s FL(,)16 b FK(k)r FI(\))28 b FL(to)f(denote)d(the)i(bundle)h(allocated)g(to)f (the)g(agent)g(ranked)h FK(k)q FL(-th)h(in)f(the)f(permutation)h FH(p)0 4156 y FL(in)g(the)f(allocation)i(output)d(by)h(our)g (algorithm.)40 b(Consider)26 b(an)h(arbitrary)g(permutation)f FH(p)k FL(wher)n(e)c(agent)g FK(i)j FL(is)0 4269 y(ranked)d FK(k)q FL(-th)g(\(i.e.,)h FH(p)t FI(\()r FK(k)r FI(\))34 b(=)e FK(i)r FL(\),)27 b(we)e(will)i(show)e(that)h FK(u)1924 4285 y Fw(i)1948 4269 y FI(\()7 b FK(A)r FI(\()r FH(p)s FL(,)16 b FK(k)r FI(\))r(\))33 b Fz(\025)g FK(u)2514 4285 y Fw(i)2538 4269 y FI(\()t FK(B)2637 4236 y Fw(i)2635 4299 y(k)2670 4269 y FI(\))q FL(.)k(Since)26 b(an)g(agent)f(in)i FH(p)t FI([)r FL(:)33 b FK(m)21 b Fz(\000)g FK(n)r FI(])0 4382 y FL(r)n(eceives)31 b(two)g(goods)f(and)h(otherwise)f(one)h(good)f (in)i(our)f(algorithm,)j(the)d(number)g(of)h(indivisible)h(goods)0 4495 y(r)n(eceived)c(by)g(agent)g FK(i)j FL(is)d(the)g(same)g(as)h (that)f(in)h(the)e(baseline)i(allocation)h(befor)n(e)e(the)f(water)n (-\002lling)i(pr)n(ocess)0 4607 y(with)f(whichever)g(index)f(in)j FH(p)s FL(.)47 b(In)29 b(addition,)h(she)e(will)i(r)n(eceive)g(a)f (good)f(with)h(value)h FK(b)h FL(if)i FK(B)3328 4574 y Fw(i)3326 4638 y(k)3389 4607 y FL(contains)c(one)0 4720 y(good.)d(Hence,)d FK(u)612 4736 y Fw(i)636 4720 y FI(\()7 b FK(A)r FI(\()r FH(p)s FL(,)16 b FK(k)r FI(\))r(\))28 b Fz(\025)f FK(u)1191 4736 y Fw(i)1215 4720 y FI(\()t FK(B)1314 4687 y Fw(i)1312 4751 y(k)1347 4720 y FI(\))d FL(if)i FK(B)1548 4687 y Fw(i)1546 4751 y(k)1603 4720 y FL(contains)c(only)h(indivisible)h(goods.)141 4833 y(If)37 b FK(B)297 4800 y Fw(i)295 4864 y(k)363 4833 y FL(contains)d(a)g(fraction)g(of)g(the)f(divisible)i(good)g FK(D)r FL(,)h(let)h FK(x)49 b FI(=)f FK(u)2589 4849 y Fw(i)2613 4833 y FI(\()t FK(B)2712 4800 y Fw(i)2710 4864 y(k)2745 4833 y FI(\))q FL(.)61 b(Denote)33 b(the)g(actual)i(output)0 4946 y(allocation)25 b(by)f Fz(A)29 b FI(=)g(\()7 b FK(A)860 4961 y FC(1)896 4946 y FL(,)16 b(.)h(.)g(.)g(,)k FK(A)1167 4960 y Fw(n)1209 4946 y FI(\))q FL(,)i(and)h(denote)d(the)i(set)f(of)h (bundles)g(that)g(only)g(contain)h(indivisible)h(goods)0 5059 y(in)30 b(the)f(actual)i(allocation)h Fz(A)f FL(by)f Fz(S)48 b(\022)40 b(A)q FL(.)49 b(By)28 b(Lemma)p Plum 2036 5059 a SDict begin H.S end 2036 5059 a Plum FL(B.7)p Plum 2160 4996 a SDict begin H.R end 2160 4996 a 2160 5059 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.7) cvn H.B /ANN pdfmark end 2160 5059 a 0 TeXcolorgray 30 w FL(and)h(accor)n(ding)i(to)e(the)f(\002rst)h(phase)g(of)h(Step)e (3,)0 5172 y(each)f(agent)e(in)k FK(T)h FL(r)n(eceives)c(at)h(most)e (one)h(good)f(fr)n(om)h FK(Y)s FL(,)h(so)f(no)g(bundle)g(in)h(the)e (actual)j(allocation)g(contains)0 5285 y(two)e(indivisible)i(goods)c (both)i(valued)h(at)h FK(b)g FL(to)e(agent)f FK(i)r FL(.)40 b(As)26 b(the)g(number)g(of)h(bundles)e(with)i(the)f(maximum)0 5398 y(possible)g(value)i(of)f(indivisible)h(goods)d(\(which)j(is)g FK(b)21 b FI(+)h FK(a)q FL(\))29 b(is)e(maximized)h(in)g Fz(B)2834 5365 y Fw(i)2858 5398 y FL(,)g(we)e(have)3276 5405 y FJ(\345)3351 5419 y Fw(A)3403 5432 y Fn(j)3422 5419 y Fv(2)r(S)3538 5398 y FK(u)3591 5414 y Fw(i)3615 5398 y FI(\()7 b FK(A)3729 5414 y Fw(j)3753 5398 y FI(\))35 b Fz(\024)p 0 TeXcolorgray 1905 5649 a FL(24)p 0 TeXcolorgray eop end end %%Page: 25 25 TeXDict begin HPSdict begin 25 24 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin 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Fp(+)q FC(1)2736 760 y FK(u)2789 776 y Fw(i)2813 760 y FI(\()t FK(B)2912 723 y Fw(i)2912 785 y(j)2936 760 y FI(\))27 b(=)g FK(u)3154 776 y Fw(i)3178 760 y FI(\()6 b FK(M)21 b Fz([)h FK(D)s FI(\))q FL(,)0 1047 y(wher)n(e)j(the)g(\002rst)g(inequality)i(holds)e(as)h(each)g (bundle)32 b FK(A)2000 1063 y Fw(j)2062 1047 y FL(/)-62 b Fz(2)34 b(S)h FL(that)25 b(contains)h(divisible)i(goods)c(is)i (valued)g(at)0 1170 y(most)h FK(x)274 1137 y Fv(0)324 1170 y FL(to)d(agent)h FK(i)j FL(guaranteed)23 b(by)i(EFM,)f(and)i(the) e(thir)n(d)h(inequality)h(holds)e(as)i FK(u)2976 1186 y Fw(i)3000 1170 y FI(\()t FK(B)3099 1137 y Fw(i)3099 1200 y(j)3123 1170 y FI(\))32 b Fz(\025)i FK(x)28 b FL(if)h FK(B)3516 1137 y Fw(i)3516 1200 y(j)3564 1170 y FL(contains)0 1292 y(no)22 b(divisible)i(good.)141 1405 y(As)33 b(the)h(permutation)f (is)h(generated)d(uniformly)k(at)f(random,)i(the)d(lemma)i(concludes)f (following)g(the)0 1518 y(same)23 b(analysis)g(for)f(ex-ante)g(PROP)h (as)g(in)g(Theor)n(em)p Plum 1887 1520 a SDict begin H.S end 1887 1520 a Plum -2 x FL(4.3)p Plum 2001 1455 a SDict begin H.R end 2001 1455 a 2001 1518 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.4.3) cvn H.B /ANN pdfmark end 2001 1518 a 0 TeXcolorgray FL(:)345 1782 y FK(u)398 1798 y Fw(i)422 1782 y FI(\()r Fz(R)r FI(\))k(=)749 1684 y Fw(n)719 1801 y Fr(\345)709 1881 y Fw(k)r Fp(=)q FC(1)933 1801 y Fr(\345)844 1884 y Fy(p)t FC(:)r Fy(p)t Fp(\()r Fw(k)r Fp(\))r(=)r Fw(i)1148 1721 y FK(u)1201 1737 y Fw(i)1225 1721 y FI(\()7 b FK(A)r FI(\()r FH(p)s FL(,)16 b FK(k)r FI(\))r(\))p 1147 1762 453 4 v 1334 1845 a FK(n)r FL(!)1636 1782 y Fz(\025)1778 1684 y Fw(n)1747 1801 y Fr(\345)1737 1881 y Fw(k)r Fp(=)q FC(1)1962 1801 y Fr(\345)1872 1884 y Fy(p)t FC(:)r Fy(p)t Fp(\()r Fw(k)r Fp(\))r(=)r Fw(i)2176 1716 y FK(u)2229 1732 y Fw(i)2254 1716 y FI(\()t FK(B)2353 1683 y Fw(i)2351 1746 y(k)2386 1716 y FI(\))p 2175 1762 249 4 v 2260 1845 a FK(n)r FL(!)2460 1782 y FI(=)2602 1684 y Fw(n)2571 1801 y Fr(\345)2561 1881 y Fw(k)r Fp(=)q FC(1)2706 1716 y FK(u)2759 1732 y Fw(i)2783 1716 y FI(\()t FK(B)2882 1683 y Fw(i)2880 1746 y(k)2915 1716 y FI(\))p 2705 1762 V 2804 1845 a FK(n)2990 1782 y FI(=)3101 1721 y FK(u)3154 1737 y Fw(i)3178 1721 y FI(\()6 b FK(M)20 b Fz([)i FK(D)s FI(\))p 3100 1762 425 4 v 3286 1845 a FK(n)3534 1782 y FL(.)p 3834 1782 4 62 v 3838 1724 55 4 v 3838 1782 V 3892 1782 4 62 v 0 1994 a SDict begin H.S end 0 1994 a 0 1994 a SDict begin 13.6 H.A end 0 1994 a 0 1994 a SDict begin [/View [/XYZ H.V]/Dest (theorem.B.9) cvn /DEST pdfmark end 0 1994 a 0 TeXcolorgray 80 x FA(Lemma)h(B.9.)p 0 TeXcolorgray 41 w FK(The)g(randomized)i (allocation)h Fz(R)e FK(is)f(ex-ante)i(PROP)c(for)j(agents)g(in)i(N)d Fz(n)e FK(T)s(.)p 0 TeXcolorgray 0 2255 a(Pr)n(oof.)p 0 TeXcolorgray 46 w FL(Dif)n(fer)n(ent)37 b(fr)n(om)g(the)e(pr)n(oof)g (in)h(Lemma)p Plum 1763 2257 a SDict begin H.S end 1763 2257 a Plum -2 x FL(B.8)p Plum 1886 2192 a SDict begin H.R end 1886 2192 a 1886 2255 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.8) cvn H.B /ANN pdfmark end 1886 2255 a 0 TeXcolorgray 36 w FL(above,)k(a)c(bundle)g(in)g(the)f(actual)i (allocation)i Fz(A)e FL(may)0 2368 y(contain)27 b(two)e(indivisible)j (goods)c(both)h(valued)i(at)g FK(b)h FL(to)d(agent)g FK(i)k FL(for)d(some)f FK(i)36 b Fz(2)h FK(N)24 b Fz(n)e FK(T)s FL(.)39 b(W)-8 b(e)25 b(thus)g(de\002ne)g(the)0 2481 y(baseline)e(allocation)h(for)f(agent)e FK(i)26 b FL(in)d(a)g(dif)n(fer)n(ent)g(way)f(below)-8 b(.)0 2506 y SDict begin H.S end 0 2506 a 0 2506 a SDict begin 13.6 H.A end 0 2506 a 0 2506 a SDict begin [/View [/XYZ H.V]/Dest (theorem.B.10) cvn /DEST pdfmark end 0 2506 a 0 TeXcolorgray 137 x FA(De\002nition)24 b(B.10)h FL(\(Baseline)g (allocation)i Fz(B)1526 2610 y Fw(i)1575 2643 y FL(for)e(agent)f FK(i)32 b Fz(2)j FK(N)24 b Fz(n)e FK(T)s FL(\))p FA(.)p 0 TeXcolorgray 43 w FL(The)i FK(baseline)i(allocation)i Fz(B)3376 2610 y Fw(i)3425 2643 y FK(for)e(agent)g(i)33 b Fz(2)4 2756 y FK(N)23 b Fz(n)e FK(T)26 b FL(is)d(obtained)f(by)0 2814 y SDict begin H.S end 0 2814 a 0 2814 a SDict begin 13.6 H.A end 0 2814 a 0 2814 a SDict begin [/View [/XYZ H.V]/Dest (Item.5) cvn /DEST pdfmark end 0 2814 a 0 TeXcolorgray 114 2938 a FL(1.)p 0 TeXcolorgray 45 w(letting)36 b(each)h(bundle)f(contain)i(one)d(distinct)i(good)e(that)h(is)h (matched)f(during)g(the)g(\002rst)g(phases)f(of)227 3050 y(Steps)24 b(2)i(and)f(3)h(\(note)e(that)h(ther)n(e)f(ar)n(e)j FK(n)f FL(goods)d(matched)j(during)e(the)h(\002rst)f(phases)g(of)h (Steps)f(2)i(and)f(3,)227 3163 y(so)d(each)h(bundle)f(can)i(contain)f (exactly)g(one)e(good\);)0 3225 y SDict begin H.S end 0 3225 a 0 3225 a SDict begin 13.6 H.A end 0 3225 a 0 3225 a SDict begin [/View [/XYZ H.V]/Dest (Item.6) cvn /DEST pdfmark end 0 3225 a 0 TeXcolorgray 114 3348 a FL(2.)p 0 TeXcolorgray 45 w(arranging)i(the)f(bundles)g(in)h(descending)d(or)n (der)i(accor)n(ding)h(to)f(agent)g FK(i)r FL('s)h(utility;)0 3410 y SDict begin H.S end 0 3410 a 0 3410 a SDict begin 13.6 H.A end 0 3410 a 0 3410 a SDict begin [/View [/XYZ H.V]/Dest (Item.7) cvn /DEST pdfmark end 0 3410 a 0 TeXcolorgray 114 3533 a FL(3.)p 0 TeXcolorgray 45 w(letting)f(agent)g FK(i)j FL(allocate)f(r)n(emaining)g(indivisible)g(goods)c(by)j (Round-Robin;)e(and)0 3595 y SDict begin H.S end 0 3595 a 0 3595 a SDict begin 13.6 H.A end 0 3595 a 0 3595 a SDict begin [/View [/XYZ H.V]/Dest (Item.8) cvn /DEST pdfmark end 0 3595 a 0 TeXcolorgray 114 3719 a FL(4.)p 0 TeXcolorgray 45 w(executing)h(the)g(water)n(-\002lling)h(pr)n(ocess)e (for)i(the)f(divisible)i(goods)c(accor)n(ding)j(to)f(agent)g FK(i)r FL('s)h(utility)-10 b(.)141 3900 y(Assume)23 b(that)h(agent)f FK(i)k FL(is)d(ranked)g FK(k)q FL(-th)i(in)e(the)g(permutation)g FH(p)s FL(,)g(and)h(denote)d(the)h(actual)i(output)e(alloca-)0 4013 y(tion)d(by)h Fz(A)27 b FI(=)g Fz(f)7 b FK(A)626 4028 y FC(1)664 4013 y FL(,)16 b(.)h(.)g(.)f(,)21 b FK(A)934 4027 y Fw(n)976 4013 y Fz(g)q FL(,)g(wher)n(e)26 b FK(A)1413 4029 y Fw(i)1456 4013 y FL(is)20 b(also)g(denoted)e(by)26 b FK(A)r FI(\()r FH(p)s FL(,)16 b FK(k)r FI(\))22 b FL(similar)g(to)e (Lemma)p Plum 3236 4015 a SDict begin H.S end 3236 4015 a Plum -2 x FL(B.8)p Plum 3360 3950 a SDict begin H.R end 3360 3950 a 3360 4013 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.8) cvn H.B /ANN pdfmark end 3360 4013 a 0 TeXcolorgray FL(.)27 b(The)19 b(number)0 4126 y(of)25 b(the)f(indivisible)i(goods)d(in)28 b FK(B)1139 4093 y Fw(i)1137 4156 y(k)1196 4126 y FL(equals)d(to)f(that)g(she)g(r)n (eceives)h(in)g(the)f(actual)i(allocation)h(befor)n(e)d(the)g(water)n (-)0 4238 y(\002lling)29 b(pr)n(ocess.)43 b(In)28 b(the)f(\002rst)h (phase)f(of)h(Step)f(2,)j(agent)d FK(i)k FL(will)e(r)n(eceive)g(a)f (good)f(with)h(value)i FK(b)q FL(,)g(which)e(is)g(no)0 4351 y(less)h(than)h(the)g(value)h(of)f(the)f(\002rst)g(good)g(added)g (to)j FK(B)1944 4318 y Fw(i)1942 4382 y(k)2007 4351 y FL(\(as)e(the)g(value)g(is)h(at)f(most)g 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y(water)n(-\002lling)j(pr)n(ocess.)63 b(Denoted)34 b(by)1469 5007 y(\257)1442 5029 y FK(A)1512 5045 y Fw(j)1570 5029 y FL(and)1780 5009 y(\257)1762 5029 y FK(B)1820 4996 y Fw(i)1820 5058 y(j)1878 5029 y FL(the)k FK(j)q FL(-th)d(bundles)g(that)f(contain)i(only)e(the)h(indivisible)0 5165 y(goods)27 b(befor)n(e)i(the)f(water)n(-\002lling)i(pr)n(ocess)e (for)33 b FK(j)40 b Fz(\024)f FK(n)q FL(.)48 b(The)28 b(\002rst)h(good)e(added)h(to)3045 5144 y(\257)3018 5165 y FK(A)3088 5181 y Fw(j)3140 5165 y FL(and)3344 5145 y(\257)3326 5165 y FK(B)3384 5132 y Fw(i)3384 5195 y(j)3436 5165 y FL(is)h(identical.)0 5287 y(Guaranteed)23 b(by)h(the)f(second)f (and)i(thir)n(d)f(steps)f(when)h(constr)o(ucting)g(the)g(baseline)h (allocation,)h(the)e(possible)0 5400 y(number)32 b(of)g(bundles)g(with) g(two)f(goods)f(both)i(with)g(value)i FK(b)g FL(to)e(agent)f FK(i)k FL(is)d(maximized,)k(i.e.,)e(ther)n(e)d(is)h(no)p 0 TeXcolorgray 1905 5649 a(25)p 0 TeXcolorgray eop end end %%Page: 26 26 TeXDict begin HPSdict 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Fw(i)2912 1410 y(j)2936 1385 y FI(\))27 b(=)g FK(u)3154 1401 y Fw(i)3178 1385 y FI(\()6 b FK(M)21 b Fz([)h FK(D)s FI(\))q FL(,)141 1675 y(W)-8 b(e)27 b(have)i(shown)d(that)j FK(u)1054 1691 y Fw(i)1078 1675 y FI(\()7 b FK(A)r FI(\()r FH(p)s FL(,)16 b FK(k)r FI(\))r(\))38 b Fz(\025)f FK(u)1653 1691 y Fw(i)1677 1675 y FI(\()t FK(B)1776 1642 y Fw(i)1774 1706 y(k)1809 1675 y FI(\))29 b FL(holds)e(for)h(each)h FH(p)i FL(and)d FK(i)j FL(such)c(that)i FH(p)t FI(\()r FK(k)r FI(\))38 b(=)e FK(i)r FL(.)45 b(Ther)n(e-)0 1788 y(for)n(e,)26 b(for)g(each)h FH(p)s FL(,)f(each)g(agent)f(will)i(r)n (eceive)f(no)f(less)g(value)h(in)g(the)f(actual)i(allocation)g(than)f (in)g(the)f(baseline)0 1901 y(allocation.)k(Following)23 b(the)e(same)i(analysis)g(in)g(Lemma)p Plum 2031 1903 a SDict begin H.S end 2031 1903 a Plum -2 x FL(B.8)p Plum 2155 1838 a SDict begin H.R end 2155 1838 a 2155 1901 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.8) cvn H.B /ANN pdfmark end 2155 1901 a 0 TeXcolorgray FL(,)f(ex-ante)g(PROP)h (is)g(guaranteed.)p 3834 1901 4 62 v 3838 1843 55 4 v 3838 1901 V 3892 1901 4 62 v 141 2086 a(W)-8 b(e)22 b(ar)n(e)h(now)f(r) n(eady)g(to)g(establish)h(Theor)n(em)p Plum 1748 2086 a SDict begin H.S end 1748 2086 a Plum FL(4.1)p Plum 1862 2023 a SDict begin H.R end 1862 2023 a 1862 2086 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.4.1) cvn H.B /ANN pdfmark end 1862 2086 a 0 TeXcolorgray FL(.)p 0 TeXcolorgray 0 2270 a FK(Pr)n(oof)h(of)f(Theor)n(em)p Plum 646 2271 a SDict begin H.S end 646 2271 a Plum -1 x FK(4.1)p Plum 759 2207 a SDict begin H.R end 759 2207 a 759 2270 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.4.1) cvn H.B /ANN pdfmark end 759 2270 a 0 TeXcolorgray FK(.)p 0 TeXcolorgray 45 w FL(Lemmas)p Plum 1188 2272 a SDict begin H.S end 1188 2272 a Plum -2 x FL(B.8)p Plum 1312 2207 a SDict begin H.R end 1312 2207 a 1312 2270 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.8) cvn H.B /ANN pdfmark end 1312 2270 a 0 TeXcolorgray 17 w FL(and)p Plum 1501 2272 a SDict begin H.S end 1501 2272 a Plum -2 x FL(B.9)p Plum 1624 2207 a SDict begin H.R end 1624 2207 a 1624 2270 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.B.9) cvn H.B /ANN pdfmark end 1624 2270 a 0 TeXcolorgray 18 w FL(together)14 b(show)i(the)h(randomized)g(allocation)h(is)g(ex-ante)e(PROP)-12 b(,)0 2383 y(and)22 b(it)g(is)h(also)f(ex-post)e(EFM)h(guaranteed)g(by) h(the)f(fact)i(that)f(each)g(partial)i(allocation)f(containing)g(only)e (indi-)0 2496 y(visible)e(goods)c(is)j(EF1)f(and)g(by)h(Lemma)p Plum 1406 2496 a SDict begin H.S end 1406 2496 a Plum FL(4.2)p Plum 1520 2433 a SDict begin H.R end 1520 2433 a 1520 2496 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.4.2) cvn H.B /ANN pdfmark end 1520 2496 a 0 TeXcolorgray FL(.)26 b(Finally)-10 b(,)19 b(it)f(is)f(straightforwar)n(d)g(that)g (each)h(part)f(of)h(our)f(algorithm)0 2609 y(r)o(uns)22 b(in)h(polynomial)g(time.)28 b(This)23 b(concludes)f(Theor)n(em)p Plum 2026 2609 a SDict begin H.S end 2026 2609 a Plum FL(4.1)p Plum 2140 2546 a SDict begin H.R end 2140 2546 a 2140 2609 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.4.1) cvn H.B /ANN pdfmark end 2140 2609 a 0 TeXcolorgray FL(.)p 3834 2609 4 62 v 3838 2551 55 4 v 3838 2609 V 3892 2609 4 62 v 0 TeXcolorgray 0 2794 a FA(Remark.)p 0 TeXcolorgray 42 w FL(The)i(above)g(algorithm)g(can)h(be)f (generalized)f(to)h(the)f(case)h(wher)n(e)f(agents)g(have)h (personalized)f(bi-)0 2907 y(valued)c(utilities)f(\(the)g(value)j FK(a)f FL(and)f FK(b)h FL(of)e(each)h(agent)e(ar)n(e)i(dif)n(fer)n (ent\))g(to)e(indivisible)j(goods)c(and)j(no)f(constraint)0 3020 y(to)k(divisible)i(goods.)0 3191 y SDict begin H.S end 0 3191 a 0 3191 a SDict begin 13.6 H.A end 0 3191 a 0 3191 a SDict begin [/View [/XYZ H.V]/Dest (appendix.C) cvn /DEST pdfmark end 0 3191 a 124 x FM(C)120 b(Omitted)29 b(Details)f(in)i(Section)p Plum 1693 3316 a SDict begin H.S end 1693 3316 a Plum -1 x FM(5)p Plum 1753 3235 a SDict begin H.R end 1753 3235 a 1753 3315 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 1753 3315 a 0 TeXcolorgray 0 3415 a SDict begin H.S end 0 3415 a 0 3415 a SDict begin 13.6 H.A end 0 3415 a 0 3415 a SDict begin [/View [/XYZ H.V]/Dest (subsection.C.1) cvn /DEST pdfmark end 0 3415 a 114 x FD(C.1)99 b(Proof)25 b(of)g(Lemma)p Plum 1002 3530 a SDict begin H.S end 1002 3530 a Plum -1 x FD(5.3)p Plum 1126 3463 a SDict begin H.R end 1126 3463 a 1126 3529 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.3) cvn H.B /ANN pdfmark end 1126 3529 a 0 TeXcolorgray 0 3706 a FL(W)-8 b(e)32 b(can)j(verify)e(the)g(r) n(equir)n(ed)f(pr)n(operty)f(after)j(each)f(r)n(ound)g(and)g(pr)n(ove)g (this)g(by)g(induction.)60 b(The)32 b(initial)0 3819 y(empty)19 b(allocation)j(is)e(trivially)i(EFX.)c(\(Recall)k(that)e(we) f(use)24 b FK(N)2172 3786 y Fw(r)2224 3819 y FL(and)i FK(M)2492 3786 y Fw(r)2543 3819 y FL(to)20 b(denote)e(the)h(corr)n (esponding)f(set)23 b FK(N)0 3932 y FL(and)35 b FK(M)e FL(encounter)n(ed)28 b(in)j(Algorithm)p Plum 1402 3932 a SDict begin H.S end 1402 3932 a Plum FL(2)p Plum 1447 3869 a SDict begin H.R end 1447 3869 a 1447 3932 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algocf.2) cvn H.B /ANN pdfmark end 1447 3932 a 0 TeXcolorgray 31 w FL(at)f(the)g(beginning)g (of)g(its)g(r)n(ound)g FK(r)r FL(.\))52 b(At)30 b(the)f(next)h(r)n (ound)g FK(r)r FL(,)i(if)g(we)0 4045 y(cannot)26 b(\002nd)f(a)i (perfect)e(matching)h(between)j FK(N)1710 4012 y Fw(r)1768 4045 y FL(and)d(goods)j FK(M)2311 4012 y Fw(r)2368 4045 y FL(\(lines)p Plum 2613 4047 a SDict begin H.S end 2613 4047 a Plum -2 x FL(13)p Plum 2704 3982 a SDict begin H.R end 2704 3982 a 2704 4045 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.13) cvn H.B /ANN pdfmark end 2704 4045 a 0 TeXcolorgray FL(-)p Plum 2734 4047 a SDict begin H.S end 2734 4047 a Plum -2 x FL(19)p Plum 2825 3982 a SDict begin H.R end 2825 3982 a 2825 4045 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.19) cvn H.B /ANN pdfmark end 2825 4045 a 0 TeXcolorgray FL(\),)e(ther)n(e)e(exists)f(a)j (non-empty)0 4158 y(minimal)c(unmatchable)f(gr)n(oup)g FK(Z)1230 4172 y Fw(t)1256 4158 y FL(.)28 b(For)20 b(each)h(agent)e FK(i)29 b Fz(2)i FK(N)2131 4125 y Fw(r)2180 4158 y Fz(n)20 b FK(Z)2308 4172 y Fw(t)2356 4158 y FL(at)g(this)h(r)n(ound,)f(since)g (she)g(can)i(still)f(r)n(eceive)0 4271 y(a)d(lar)n(ge)h(good)d(and)i (every)f(agent)g(can)i(r)n(eceive)f(at)g(most)f(one)g(good)f(per)i(r)n (ound,)f(ther)n(e)g(is)h(no)g(envy)f(fr)n(om)i(agent)e FK(i)r FL(.)141 4384 y(For)28 b(each)i(agent)e FK(i)41 b Fz(2)g FK(Z)1000 4398 y Fw(t)1027 4384 y FL(,)30 b(if)g(she)f(is)g (fr)n(ozen)g(in)h(this)f(loop,)h(since)f(she)f(will)i(be)f(only)g(fr)n (ozen)h(for)f(the)f(next)1 4497 y Fz(b)r FK(b)r FL(/)s FK(a)c Fz(\000)c FL(1)q Fz(c)29 b FL(r)n(ounds,)e(the)g(total)g (utility)g(accumulated)i(by)e(any)g(other)f(agent's)h(bundle)g(during)f (these)g(fr)n(ozen)0 4610 y(r)n(ounds)34 b(is)g(upper)g(bounded)f(by)k FK(a)25 b FI(+)f Fz(b)r FK(b)r FL(/)s FK(a)i Fz(\000)d FL(1)q Fz(c)i(\001)h FK(a)51 b Fz(\024)e FK(b)q FL(,)38 b(fr)n(om)e(the)e(second)f(term)h(in)i(Observation)p Plum 3764 4612 a SDict begin H.S end 3764 4612 a Plum -2 x FL(5.2)p Plum 3877 4547 a SDict begin H.R end 3877 4547 a 3877 4610 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.2) cvn H.B /ANN pdfmark end 3877 4610 a 0 TeXcolorgray FL(.)0 4723 y(Thus,)27 b(each)h(fr)n(ozen)g(agent)e(will)j(not)d(envy)h (others)f(during)g(the)h(fr)n(ozen)h(r)n(ounds.)40 b(If)28 b(an)g(agent)e FK(i)38 b Fz(2)f FK(Z)3621 4737 y Fw(t)3675 4723 y FL(is)28 b(not)0 4835 y(fr)n(ozen)f(\(in)g(other)f(wor)n(ds,)g (becomes)h FK(quiet)p FL(\),)h(she)e(will)i(not)e(envy)g(the)g(agents)g (not)g(in)j FK(Z)3097 4849 y Fw(t)3151 4835 y FL(or)d(the)g(agents)g (in)j FK(Z)3873 4849 y Fw(t)0 4948 y FL(without)g(fr)n(eezing)h(since)g (these)e(agents)h(can)i(only)f(r)n(eceive)g(a)h(good)d(in)36 b FK(M)2673 4915 y Fw(r)2725 4948 y Fz(n)23 b FJ(G)r FI(\()s FK(Z)2949 4962 y Fw(t)2977 4948 y FI(\))q FL(,)32 b(which)e(is)g(valued)h(at)h FK(a)0 5061 y FL(under)c FK(u)325 5077 y Fw(i)348 5061 y FL(.)44 b(For)27 b(some)g(agent)k FK(j)38 b Fz(2)g FK(Z)1298 5075 y Fw(t)1353 5061 y FL(is)28 b(fr)n(ozen)g(in)h(this)e(r)n(ound,)h(if)h(all)h(goods)25 b(allocated)k(to)i FK(j)f FL(until)e(now)g(ar)n(e)0 5174 y(valued)f(at)g FK(b)h FL(by)e(agent)f FK(i)r FL(,)j(agent)d FK(i)k FL(will)f(not)d(envy)30 b FK(j)e FL(after)e(r)n(emoving)h(the)e (last)i(good)d(fr)n(om)j(the)f(second)f(term)0 5287 y(in)i(Observation) p Plum 627 5289 a SDict begin H.S end 627 5289 a Plum -2 x FL(5.2)p Plum 741 5224 a SDict begin H.R end 741 5224 a 741 5287 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.2) cvn H.B /ANN pdfmark end 741 5287 a 0 TeXcolorgray FL(.)39 b(Otherwise,)26 b(agent)f FK(i)k FL(will)f(not)e(envy)j FK(j)g FL(since)d(ther)n(e)f(is)i(exactly)f(one)f(small)j(good)d(in)i FK(i)r FL('s)0 5400 y(bundle)22 b(and)h(ther)n(e)f(is)g(at)h(least)f (one)g(small)i(good)d(in)27 b FK(j)q FL('s)d(bundle)e(under)g FK(u)2583 5416 y Fw(i)2606 5400 y FL(.)p 0 TeXcolorgray 1905 5649 a(26)p 0 TeXcolorgray eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.27) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 141 91 a FL(For)24 b(each)i(agent)e FK(i)k FL(not)c(in)30 b FK(N)1150 58 y Fw(r)1183 91 y FL(,)25 b(i.e.,)g(a)h(quiet)f(agent)f(in)h(a)h(pr)n(evious)h FK(Z)2561 108 y Fw(t)2585 89 y Fb(0)2607 91 y FL(,)f(it)f(does)f(not)g (envy)h(the)f(agents)g(not)0 204 y(in)30 b FK(Z)170 221 y Fw(t)194 202 y Fb(0)243 204 y FL(and)e(the)e(agents)g(in)j FK(Z)1035 221 y Fw(t)1059 202 y Fb(0)1109 204 y FL(without)d(fr)n (eezing)i(befor)n(e)e(because)h(all)i(agents)d(r)n(eceive)h(a)h(small)g (good)e(fr)n(om)0 317 y FK(i)r FL('s)32 b(perspective.)51 b(Compar)n(ed)31 b(to)g(some)f(other)g(quiet)h(agent)j FK(j)f FL(in)h FK(Z)2443 334 y Fw(t)2467 315 y Fb(0)2521 317 y FL(who)c(has)i(been)e(fr)n(ozen)h(befor)n(e,)i(after)0 430 y(eliminating)26 b(one)e(small)i(good)d(in)29 b FK(j)q FL('s)c(bundle,)g(we)f(have)i FK(b)32 b Fz(\024)g FK(a)21 b FI(+)g Fz(b)r FK(b)r FL(/)s FK(a)h Fz(\000)d FL(1)q Fz(c)j(\001)g FK(a)f FI(+)g FK(a)q FL(,)26 b(which)g(can)f(eliminate)0 543 y(the)d(envy)-10 b(.)28 b(The)22 b(term)i FK(a)d FI(+)f Fz(b)r FK(b)r FL(/)s FK(a)i Fz(\000)c FL(1)q Fz(c)j(\001)g FK(a)k FL(r)n(epr)n(esents)c(the)h(utility)h(during)f(the)g(fr)n(ozen)h (r)n(ounds)f(of)27 b FK(j)d FL(earned)e(by)0 656 y FK(i)j FL(and)e(the)f(last)j FK(a)f FL(r)n(epr)n(esents)c(the)i(utility)h(at)g (this)f(r)n(ound.)141 769 y(W)-8 b(e)21 b(then)f(come)h(to)g(the)g (second)f(case)h(wher)n(e)g(we)f(can)j(\002nd)e(a)h(perfect)e(matching) i(between)i FK(N)3427 736 y Fw(r)3481 769 y FL(and)d(goods)5 882 y FK(M)93 849 y Fw(r)144 882 y FL(\(Lines)p Plum 411 884 a SDict begin H.S end 411 884 a Plum -2 x FL(10)p Plum 502 819 a SDict begin H.R end 502 819 a 502 882 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.10) cvn H.B /ANN pdfmark end 502 882 a 0 TeXcolorgray FL(-)p Plum 532 882 a SDict begin H.S end 532 882 a Plum FL(11)p Plum 624 819 a SDict begin H.R end 624 819 a 624 882 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.11) cvn H.B /ANN pdfmark end 624 882 a 0 TeXcolorgray FL(\).)27 b(For)19 b(each)h(agent)f FK(i)k FL(in)h FK(N)1529 849 y Fw(r)1561 882 y FL(,)d(since)e(it)h(r)n(eceives)g(lar)n(ge)g(goods)e(at)i(all)h (r)n(ounds,)e(ther)n(e)g(is)h(no)g(envy)0 995 y(fr)n(om)j(her)-7 b(.)28 b(For)22 b(each)h(agent)e FK(i)26 b FL(in)d(some)e(pr)n(evious)j FK(Z)1828 1009 y Fw(t)1855 995 y FL(,)f(the)f(case)g(is)h(similar)h(to) e(the)g(case)h(above.)141 1108 y(Fr)n(om)17 b(the)g(induction,)h(it)g (suf)n(\002ces)f(to)g(show)g(we)g(can)h(still)g(maintain)h(EFX)e(when)f (allocating)j(the)e(r)n(emaining)0 1220 y(goods)26 b(at)h(Lines)p Plum 619 1220 a SDict begin H.S end 619 1220 a Plum FL(22)p Plum 710 1157 a SDict begin H.R end 710 1157 a 710 1220 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.22) cvn H.B /ANN pdfmark end 710 1220 a 0 TeXcolorgray FL(-)p Plum 740 1222 a SDict begin H.S end 740 1222 a Plum -2 x FL(25)p Plum 831 1157 a SDict begin H.R end 831 1157 a 831 1220 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (AlgoLine.2.25) cvn H.B /ANN pdfmark end 831 1220 a 0 TeXcolorgray FL(.)43 b(Because)27 b(every)f(agent)h FK(i)j FL(does)c(not)h(envy)g (agent)k FK(j)f FL(except)c(for)h(one)g(special)h(case)g(as)0 1333 y(stated)19 b(in)i(the)f(pr)n(operty)f(we)h(maintain,)j(the)c (envy)i(between)e(these)g(agents)g(can)j(be)e(eliminated)h(by)g(r)n (emoving)0 1446 y(the)28 b(good)g(allocated)i(at)f(this)g(\002nal)h (step.)46 b(The)28 b(r)n(emaining)i(is)f(to)g(show)f(the)g(case)i(wher) n(e)e FK(i)k FL(and)h FK(j)e FL(ar)n(e)e(in)h(the)0 1559 y(same)23 b FK(Z)292 1573 y Fw(t)340 1559 y FL(and)f(agent)e FK(i)k FL(has)d(never)g(been)f(fr)n(ozen)i(but)f(agent)j FK(j)f FL(has)e(been)g(fr)n(ozen)g(befor)n(e.)27 b(If)21 b(the)g(value)h(of)f(agent)0 1672 y FK(i)29 b FL(over)e(the)f(good)i FK(g)g FL(allocated)g(to)e(agent)k FK(j)e FL(at)f(r)n(ound)f FK(r)1940 1688 y Fw(j)1991 1672 y FL(is)j FK(a)q FL(,)g(the)d(envy)g (can)i(be)e(eliminated)h(after)g(r)n(emoving)0 1785 y(the)22 b(good)f(allocated)i(at)f(this)g(\002nal)i(step)d(since)h(ther)n(e)g (is)g(no)g(envy)g(fr)n(om)h FK(i)i FL(to)h FK(j)f FL(during)c(the)h (loop.)27 b(Otherwise,)e FK(j)0 1898 y FL(must)f(be)g(befor)n(e)g FK(i)j FL(in)e(the)e(r)n(eversed)g(or)n(der)h(of)h FH(p)s FL(.)33 b(This)24 b(is)h(because)f(if)h FK(i)i FL(is)d(befor)n(e)k FK(j)f FL(in)d(the)g(r)n(eversed)f(or)n(der)g(of)1 2011 y FH(p)s FL(,)h(ther)n(e)e(must)h(exist)g(an)h(augmenting)e(path)i(fr)n (om)g FK(i)i FL(to)d(the)f(good)j FK(g)g FL(\(fr)n(om)g(the)d(fact)j (that)e FK(i)j FL(can)e(dir)n(ectly)g(take)0 2124 y(the)e(good)h FK(g)i FL(and)d(we)g(keep)e(the)i(r)n(emaining)h(allocation)h(to)e(r)n (each)h(a)g(matching)g(with)f(a)h(lar)n(ger)f(size)h(befor)n(e)f(the)0 2237 y(position)e(of)25 b FK(j)q FL(\))d(and)f(then)f FK(i)k FL(would)c(be)h(fr)n(ozen.)28 b(Thus,)20 b FK(i)j FL(will)g(pick)e(a)g(good)f(befor)n(e)k FK(j)f FL(at)f(this)e(\002nal)i (step)e(fr)n(om)i FH(p)s FL(,)0 2350 y(which)j(means)g(the)f(envy)g(fr) n(om)i FK(i)h FL(to)h FK(j)f FL(can)f(be)e(eliminated)i(after)e(r)n (emoving)h(the)f(good)g(r)n(eceived)g(by)29 b FK(j)d FL(at)f(this)0 2462 y(\002nal)f(step)d(\(if)i(it)g(exists\).)0 2627 y SDict begin H.S end 0 2627 a 0 2627 a SDict begin 13.6 H.A end 0 2627 a 0 2627 a SDict begin [/View [/XYZ H.V]/Dest (subsection.C.2) cvn /DEST pdfmark end 0 2627 a 90 x FD(C.2)99 b(Proof)25 b(of)g(Lemma)p Plum 1002 2718 a SDict begin H.S end 1002 2718 a Plum -1 x FD(5.4)p Plum 1126 2650 a SDict begin H.R end 1126 2650 a 1126 2717 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.4) cvn H.B /ANN pdfmark end 1126 2717 a 0 TeXcolorgray 0 2894 a FL(For)g(each)i(pair)f(of)g(two)g(agents)f FK(i)j FL(and)i FK(j)e FL(except)d(for)h(the)f(case)i(that)e(agent)h FK(i)i FL(and)f(agent)i FK(j)f FL(ar)n(e)e(in)h(the)e(same)j FK(Z)3873 2908 y Fw(t)0 3007 y FL(and)23 b(agent)e FK(i)26 b FL(has)c(never)g(been)g(fr)n(ozen)h(while)g(agent)j FK(j)e FL(has)f(been)f(fr)n(ozen,)g(fr)n(om)h(the)f(maintained)h(pr)n (operty)e(as)0 3120 y(stated)e(befor)n(e)h(Lemma)p Plum 861 3122 a SDict begin H.S end 861 3122 a Plum -2 x FL(5.3)p Plum 975 3057 a SDict begin H.R end 975 3057 a 975 3120 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5.3) cvn H.B /ANN pdfmark end 975 3120 a 0 TeXcolorgray FL(,)g(it)h(suf)n (\002ces)f(to)g(show)f(the)h(allocation)i(at)f(the)f(\002nal)h(step)e (\(Lines)p Plum 3210 3120 a SDict begin H.S end 3210 3120 a Plum FL(22)p Plum 3301 3057 a SDict begin H.R end 3301 3057 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