Proof that if Voting is Perfect in One Dimension, then the First. Eigenvector Extracted from the Double-Centered Transformed

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1 Proof tht f Votng s Perfect n One Dmenson, then the Frst Egenvector Extrcted from the Doule-Centered Trnsformed Agreement Score Mtrx hs the Sme Rn Orderng s the True Dt Keth T Poole Unversty of Houston 4 Ferury 00 Notton nd Defntons Let the true del onts of the legsltors e denoted s,,, Wthout loss of generlty let the orderng of the true del onts of the legsltors on the dmenson from left to rght e: 3 Let e the numer of non-unnmous roll cll votes wth > 0 nd let the cutont for the jth roll cll e Z j Votng s erfect Tht s, ll legsltors re sncere voters nd ll legsltors to the left of cutont vote for the sme lterntve nd ll legsltors to rght of cutont vote for the oostve lterntve For exmle, f ll legsltors to the left of Z j vote Ny, then ll legsltors to the rght of Z j vote Ye Wthout loss of generlty we cn ssume tht every legsltor to the left of Z j votes Ye nd every legsltor to the rght of Z j votes Ny Tht s, the olrty of the roll cll does not ffect the nlyss elow

2 Let e the numer of cutonts etween legsltors nd, e the numer of cutonts etween legsltors nd 3, nd so on, wth - eng the numer of cutonts etween legsltors - nd Hence - () = = > 0 The greement score etween two legsltors s the smle roorton of roll clls tht they vote for the sme outcome Hence, the greement score etween legsltors - nd s smly ecuse nd gree on ll roll clls excet for those wth cutonts etween them Smlrly, the greement score etween legsltors nd 3 s nd the greement score etween legsltors nd 3 s In generl, for two legsltors nd where, the greement score s: A = - - = () These defntons llow me to stte the followng theorem: Theorem: If Votng s Perfect n One Dmenson, then the Frst Egenvector Extrcted From the Doule Centered y Mtrx of Sured Dstnces from Euton (3) hs t Lest the Sme We Monotone Rn Orderng s the Legsltors Proof: The greement scores cn e treted s Euclden dstnces y smly sutrctng them from Tht s:

3 - - - = d = - A = - = = (3) The d s comuted from euton (3) stsfy the three xoms of dstnce: they re nonnegtve ecuse y () 0 A so tht 0 d ; they re symmetrc, d = d ; nd they stsfy the trngle neulty To see ths, consder ny trle of onts < < c The dstnces re: d = - = nd d = c c- = nd d = c c- = Hence d c = d + d c (4) Becuse ll the trngle neultes re eultes, n Euclden geometry ths mles tht,, nd c ll le on strght lne (Borg nd Groenen, 997, ch 8) Becuse ll the trngle neultes re eultes nd ll trles of onts le on strght lne, the dstnces comuted from () cn e drectly wrtten s dstnces etween onts: - d = = = (5) where d = 0 The y mtrx of sured dstnces s: D = 0 ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 (6) 3

4 To recover the s, smly doule-center D nd erform n egenvlueegenvector decomoston The frst egenvector s the soluton To see ths: Where Let the men of the jth column of D e Let the men of the th row of D e Let the men of the mtrx D e = = s the men of the dj = = j j j d = = - + dj j j= j= d = = - + dj j = j= j= = d = = - + The mtrx D s doule-centered s follows: from ech element sutrct the row men, sutrct the column men, dd the mtrx men, nd dvde y ; tht s, (d j - d j - d + d ) y j = = ( )( j ) Ths roduces the y symmetrc ostve semdefnte mtrx Y: Y = (7) Becuse Y s symmetrc wth rn of one, ts egenvlue-egenvector decomoston s smly: 4

5 u u Y = λ u u u u (8) Hence, the soluton s u u = λ u Becuse, wthout loss of generlty, the orgn cn e lced t zero, tht s, = 0, the soluton cn lso e wrtten s: u u = λ u (9) The onts from (9) exctly reroduce the dstnces n (4), the greement scores n (), nd the orgnl roll cll votes In ddton, note tht the mrror mge of the onts n (9) ( multlcton y mnus one) lso exctly reroduces the orgnl roll cll votes Furthermore, for ny r of true legsltor del onts nd wth one or more mdonts etween them, < Zj <, the recovered legsltor del onts must hve the sme orderng, < If there re no mdonts etween nd -- tht s, 5

6 ther roll cll votng ttern s dentcl -- then the recovered legsltor del onts re dentcl; = Hence, f there re cuttng onts etween every r of djcent legsltors, tht s, for =,, -, then the rn orderng of the recovered del onts s the sme s the true rn orderng If some of the = 0, then the recovered del onts hve we monotone trnsformton of the true rn orderng (n other words there re tes, some legsltors hve the sme recovered del onts) Ths comletes the roof QED Dscusson Note tht n ntervl level set of onts s recovered However, ths s n rtfct of the dstruton of cuttng onts For exmle, f >, ths hs the effect of mng d > d 3 even f the true coordntes,, 3 were evenly sced Wth erfect one dmensonl votng, the legsltor confgurton s only dentfed u to we monotone trnsformton of the true rn orderng The rn orderng cn lso e recovered from the mtrx Y gven n (7) wthout erformng n egenvlue-egenvector decomoston Note tht, wth the orgn t zero, the dgonl elements of Y re smly the legsltor coordntes sured The rn orderng cn e recovered y tng the sure root of the frst dgonl element nd then dvdng through the frst row of the mtrx Note tht ths sets > 0 nd the remnng onts re dentfed vs vs 6