Algorithmic verification of feasibility for generalized median voter schemes on compact ranges Korgin N.*

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1 Preprs of he 8h IFAC orld Cogress Mlao (Ialy) Augus 8 - Sepeber 0 Algorhc verfcao of feasbly for geeralzed eda voer schees o copac rages Korg N* *Isue of Corol Sceces of Russa Acadey of Sceces Mosco Russa Federao (Tel ; e-al org@puru) Absrac Barberá Massó ad Serzaa (998) provded full characerzao for class of sraegy-proof socal choce fucos for socees here he se of aleraves s ay full desoal copac subse of a Eucldea space ad all voers have geeralzed sgle-peaed prefereces They proved ha hs class s coposed by geeralzed eda voer schees sasfyg a addoal codo called he erseco propery Bu accordg o her resuls order o udersad heher ay geeralzed eda voer schee sasfes erseco propery for gve se of aleraves or o as ecessary o chec all he aleraves fro he se of ufeasble aleraves - addo of he se of feasble aleraves o al Caresa produc rage coag hs se So he uber of aleraves o be checed as fe I hs paper s proved ha s eough o chec fe uber of aleraves fro he se of ufeasble aleraves ad cosrucve algorh o deere aleraves ha should be checed s provded INTRODUCTION For socees h ages facg a se Z of aleraves a socal choce fuco deeres ha alerave o choose for each possble profle of prefereces Oe of he pora properes of socal choce fucos s sraegy-proofess he he bes sraegy for each age s o repor s prefereces ruhfully Bu geeral hs propery s hard o oba due o he Gbbard Saerhae Theore [see Gbbard (97) ad Saerhae (975)] all socal choce fucos hose rage coas ore ha o aleraves are eher dcaoral or apulable f all possble prefereces over aleraves are adssble for all ages Bu applyg soe resrcos for he doa of adssble prefereces oe ca acheve exsece of odcaoral sraegy-proof socal choce fucos I hs paper he seg s cosdered he he se of feasble aleraves Z s a full desoal copac se -desoal Eucldea space ad age s prefereces - uldesoal sglepeaed h he added requree ha he ucosraed axal elee of hese prefereces (age s op) belogs o Z Due o Barberá Massó ad Serzaa (998) ay socal choce fuco hs seg s sraegy-proof f ad oly f s a geeralzed eda voer schee (GMVS) sasfyg he erseco propery for Z Ths resul ally proved by Barberá Massó ad Nee (997) for fe ses of aleraves follos hose obaed by Border ad Jorda (98) for hole as he se of aleraves ad Moul s (980) al aalyss of he oe-desoal case Srucure of GMVS leads o he fac ha he resul each deso s deered depedely Thus here s exs possbly ha he fal oucoe ll be ousde Z bu he al Caresa produc rage BZ ˆ( ) coag hs se eve f ops of all ages belog o Z I oher ords a ufeasble alerave ca be chose as resul of geeralzed eda voer schee applcao Ierseco propery as offered as a ool for checg heher he GMVS respecs feasbly for he se of feasble aleraves or o Bu order o ae sure ha ay GMVS had sasfes erseco propery for ay se of aleraves Z urs ou ha all he ufeasble aleraves fro se BZ ˆ( )\ Z us be speced Ths s que covee he oe ors h couous seg because here s fe uber of ufeasble aleraves o be checed Accordg o Bosser ad eyar (006) ad Barberá (006) hs resuls ca be reaed as soe d of frole soluo of he proble cosdered For fe seg of aleraves Nehrg ad Puppe (007) provde alerave defo of erseco propery ha s ore sple ha orgal oe bu sll all he ufeasble aleraves fro se BZ ˆ( )\ Z us be speced accordg o her defo I hs paper s proved ha due o srucure of a GMVS s eough o spec he fe uber of ufeasble aleraves fro se BZ ˆ( )\ Z order o udersad heher hs GMVS sasfes erseco propery or o Moreover sgh of ha he aleraves should be speced s provde These resuls allo o characerze cosrucve algorh of feasbly verfcao for geeralzed eda voer schees o copac rages The paper s orgazed as follos Seco coas oaos defos ad soe prelary resuls Seco roduces he oao of brcs Caresa produc rages hch ay GMVS dvdes BZ ˆ( ) I Seco 4 oao of dreco fro a ufeasble alerave o a se of feasble Copyrgh by he Ieraoal Federao of Auoac Corol (IFAC) 84

2 Preprs of he 8h IFAC orld Cogress Mlao (Ialy) Augus 8 - Sepeber 0 aleraves s roduced ad erseco propery s reforulaed usg hs oao I Seco 5 a heore s proved algorh s provded llusraed by he exaple of s applcao Seco 6 PRELIMINARIES GENERALIZED MEDIAN VOTER SCHEMES AND INTERSECTION PROPERTY e or h he seg cosdered by Barberá Massó ad Serzaa (998) There are a se of ages N = { } ad a se of coordaes M = { } Z - a se of feasble aleraves hch s full-desoal copac Gve M deoe projeco of Z o -h coordae as Z he al box coag Z BZ ˆ( ) = [ Z ax Z ] M Aleraves fro BZ ˆ( )\ Z ll be called ufeasble Preferece u of each age N s couous coplee preorder o aleraves ad uldesoal sgle-peaed has uque axal elee τ( u ) Z - he op of u for ay z z [ ˆ({ ( z B z τ u )}) ad z z ] [ u ( z ) > u ( z) ] Le U be doa of all prefereces cosdered The socal choce fuco (SCF) F s appg fro U o Z Le SCF F U Z o be called apulable o U f here s exs u = ( u u ) U N ad u U such ha ( ( u F u u )) u ( F( u)) here ages excep A SCF F U o apulable o U Z u - uly profle of all s sraegy-proof f s Accordg o Barberá Massó ad Serzaa (998) se of all sraegy-proof SCF for he seg cosdered ay be characerzed follog ay A socal choce fuco o he doa of uldesoal sgle-pea prefereces s sraegy-proof ff s a geeralzed eda voer schee sasfyg erseco propery I hs saee here are o ey oaos ha e us expla - geeralzed eda voer schee ad erseco propery Noao of geeralzed eda voer schee (GMVS) as ally roduced by Moul (980) for oe-desoal seg Here e use defo of GMVS ers of fales of rgh (lef) -coalo syses roduced by Barberá Massó ad Nee (997) A rgh (or lef) -coalo syse o Z [ a b] s a correspodece ha assgs o every z Z a colleco ( z ) of coalos of ages sasfyg N N ( z) ad ( a ) = \ ( ( b ) = \ ) ) Coalo ooocy f ( z ) ad ( z) ) Oucoe ooocy f z < (> ) z ad ( z) ( z ) 4) Upper secouy for ay N ay z Z ad ay sequece { z} Z such ha l z = z [ ( z )] [ ( z )] A faly R of rgh-coalo syses o BZ ˆ( ) s a colleco { R } = here each R s a rgh-coalo syse o Z Slarly a faly L of lef-coalo syses o BZ ˆ( ) s a colleco { L } = here each L s a lef coalo syse o Z For each deso le ( τ τ ) be he vecor of ops projeced o hs deso The ay GMVS s a fuco F U Bˆ ( Z) duced by ( ZR ) or ( ZL ) follog ay u U M F ( u) = ax{ z Z { N τ z } R ( z )} or F ( u) = { z Z { N τ z } L ( z )} Gve a rgh-coalo syse R correspodg lefcoalo syse L * s * N L ( z ) = { z > z R ( z ) = } Coalo syses * L = L R ad L duce sae GMVS ff Due o fac ha ay GMVS s defed o BZ ˆ( ) sead of Z s possble ha resul reured by soe GMVS for soe profle of prefereces u U ll be ufeasble - Fu ( ) BZ ˆ( )\ Z A GMVS respecs feasbly f for ay u U reurs feasble resul I as proved ha ay GMVS duced by ( ZR ) respecs feasbly (or sasfes erseco propery) ff faly of rgh-coalo syses R has erseco propery for Z A faly R = { R } = of rgh-coalo syses o BZ ˆ( ) has he erseco propery for Z f for ay y B ˆ( Z)\ Z T ad ay fe subse { z z } Z T l( y) r( y) () = M ( y z ) M ( y z ) T for every r( y) R( y) h M ( y z ) ad for = T every l( y) L( y) h M ( y z ) here = ) Voer soveregy z ( a b ] ([ a b )) ( z ) 85

3 Preprs of he 8h IFAC orld Cogress Mlao (Ialy) Augus 8 - Sepeber 0 M ( y x) = { M x > y} M ( y x) = { M x < y } hle hs defo s ecessary o chec () for ay fe subse { T z z } Z as sho by Barberá Massó ad Nee (997) ha for ay ufeasble alerave y B ˆ( Z)\ Z here s s o uque crucal se such ha for ay faly of rgh-coalo syses s eough o chec () every ufeasble alerave oly for s crucal se Here e provde o he orgal defo of crucal se ally offered by Barberá Massó ad Nee (997) bu s esseal for us purposes properes A fe subse S Z s crucal for y B ˆ( Z)\ Z ff xz S x z eher M ( y x) M ( y z) or M ( y x) M ( y z) z Z \ T x S such ha M ( y x) M ( y z) ad M ( y x) M ( y z) Forally hese resuls provde full characerzao of class of sraegy-proof SCFs for he seg cosdered Bu hey are o cosrucve follog sese defo of erseco propery deads o chec each ufeasble alerave Ad here s fe uber of ufeasble aleraves because BZ ˆ( )\ Z - couous se BRICKS The defo of GMVS ers of fales of rgh ad lef coalo syses leads o follog each deso M here s exs fe se { z z z } Z z = Z z = Z of cardaly such ha for every < z ( z z ) ( ) ( R z = R z ) L ( ) ( z = L z ); z < z ( ) ( R z R z ) L ( ) ( z L z ); z > z ( ) ( R z R z ) Lz ( ) ( Lz ) Elees of hese ses ay be reaed as oe-desoal ops of phao voers accordg o al defo of GMVS by Moul (980) These ses deered copleely by defo of rgh ad lef-coalo syses correspodg deso Le us deoe = { = ( ) M < } The Caresa produc of such rages fors a brc B = [ z z ] = Ay GMVS defed o BZ ˆ( ) duces o a se of a brcs B = { B} such ha B = Bˆ( Z ) A brc B B s he border brc for Z f B Z ad B Bˆ( Z )\ Z Le us deoe he se of all border brcs for Z va B ( cl( Z )) e ll say ha GMVS sasfes erseco propery for Z brc B B f sasfes erseco propery for Z ay ufeasble alerave y B Bˆ( Z )\ Z Usefuless of brcs for erseco propery verfcao s based o hs sple ye very pora resul Lea A faly R = { R } = of rgh-coalo syses spls BZ ˆ( ) o se of brcs B The f for ay B B ad ay z B here are exs fe se T M M M T r( y) R( y) M T l( y) L( y) M such ha T l( y) r( y) = T M M he z B here are exs r( z) R( z) M l( z) L( z) M such ha T l( z) r( z) = T M M T Proof I s obvous fro he fac ha yz B Ry () = Rz () L () y = L () z ad x clb Ry () Rx () L () y L () x QED Lea resuls fac ha s eough o spec fe uber of aleraves (oe fro each brc) order o chec heher a GMVS sasfes erseco propery for a se of feasble aleraves or o The proble ha e should solve s hch cobao of lef ad rgh coalos should be explored for each bloc I orgal defo of erseco propery hs cobaos as deered by crucal ses for each alerave o be explored Bu s o que clear ho crucal ses for ufeasble aleraves fro oe bloc correspods o each oher I order o solve hs proble e apply oao of a dreco ally roduced Korg 00a 4 INTERSECTION PROPERTY IN TERMS OF DIRECTIONS Gve -desoal Eucldea space ca be roduced several oaos Dreco -uple d M here d { 0} M Usg hs oao s aural o prese a dreco fro oe alerave o aoher ers o he lef (-) o he rgh () ad cocdes (0) Dreco dyz ( ) fro y here d( y z ) = f y z d ( y z ) = 0f y = z o z - s dreco < d( y z ) = f y > z 86

4 Preprs of he 8h IFAC orld Cogress Mlao (Ialy) Augus 8 - Sepeber 0 Soees ll be useful o use oo d ( y z) - dreco fro y o z all desos excep M The dea of drecos ca be expaded order o deere relave posos of a se of feasble aleraves ad sgle ufeasble alerave sae ers Le us deoe clz() y = { z clz Bˆ ( z y) Z = z} Defo Dreco p s dreco fro y \ Z o Z f () () z clz y d y z = p ad f M d ( y z) = 0 he x clz() y d () y z = d ( y x) d ( y x) 0 I urs ou ha accordg o hs defo fro a ufeasble alerave here ay be ore he oe dreco o Z Tha s hy le us deoe se of drecos fro M y \ Z o Z - DyZ ( ) = { p p s dreco fro y \ Z o Z } Shape of Z defes heher here s y B ˆ( Z)\ Z such ha # DyZ ( ) > or o Se of feasble aleraves Z s brc covex f zz Z { zz } Bˆ ({ zz }) Z I s obvous ha ay covex copac se s also brc covex I s que easy o sho ha y B ˆ( Z)\ Z # DyZ ( ) = ff Z s brc covex Gve ay brc B B such ha B Bˆ( Z )\ Z le us deoe he se of drecos fro B o Z DB ( Z) = { DyZ ( )} y B Bˆ( Z )\ Z e ll cosder a brc B B such ha B Bˆ( Z )\ Z o be a bad brc f y B Bˆ( Z )\ Z such ha # DyZ ( ) > Le us deoe he se of all bad brcs va BB ( Z ) Usg all hs oaos defo of he erseco propery ay be reforulaed ( spr of Nehrg ad Puppe (007) usg oly erseco ad o uos) accordg o follog lea Lea A faly R = { R } = of rgh coalo syses o BZ ˆ( ) has he erseco propery for Z ff y B ˆ( Z)\ Z r ( y ) R ( y ) M ( y Z) l ( y ) L ( y ) M ( y Z) M ( y Z) M ( y Z) l () y r () y () here M ( y Z) = { M d D( y Z) d = } ad M ( y Z) = { M d D( y Z) d = } Proof See Korg 00(a) Ths forulao of erseco propery allos fdg ou soe addoal regulares hch helps o reduce coplexy of verfcao of he erseco propery Lea (Ierseco propery ooosy) y B ˆ( Z)\ Z # DyZ ( ) = f a faly R = { R } = of rgh coalo syses o BZ ˆ( ) has he erseco propery for Z y he ll have erseco propery ay x B ˆ( Z)\ Z dxy ( ) DyZ ( ) Proof See Korg 00(a) Lea resuls fac ha for all brc covex rages s eough o spec aleraves close o cl( Z ) - jus fro border brcs 5 THE CONSTRUCTIVE ALGORITHM OF FEASIBILITY VERIFICATION All he resuls above allo forulag a heore of hs paper Theore A faly R = { R } = of rgh coalo syses o BZ ˆ( ) has he erseco propery for Z ff B B( cl( Z)) BB( Z) for ay oe arbrary chose alerave y B holds ha D D( B Z) r ( y ) R ( y ) M ( D) l ( y ) L ( y ) M ( D) l () y r () y () M ( D) M ( D) here M ( D) = { M d D d = } ad M ( D) = { M d D d = } Proof See he exeded verso of hs arcle o hp//asru/upload/lbrary/korg_cafvgvspdf Ths heore resuls fac ha hle uber of ufeasble aleraves s fe s eough o chec fe uber of aleraves for fe uber of drecos order o udersad heher a GMVS sasfes erseco propery for a se of feasble aleraves or o Ad he cosrucve algorh of feasbly verfcao based o hs heore s follog Sep Gve GVMS o verfy feasbly for se Z s ecessary o defe he se of brcs B produced by hs GMVS o BZ ˆ( ) Sep Defe he se of border brcs B ( cl( Z )) Sep For each brc fro he se of border brcs chec heher gve GMVS sasfes erseco propery for Z hs brc or o accordg o heore If here s exs B B( cl( Z)) such ha () s o rue he he algorh sops he GMVS uder cosderao does o sasfy erseco propery for Z I oher case algorh goes o he ex sep 87

5 Preprs of he 8h IFAC orld Cogress Mlao (Ialy) Augus 8 - Sepeber 0 Sep 4 Defe he se of bad brcs BB ( Z ) If s epy he algorh sops hs GMVS sasfes erseco propery for Z I oher case algorh goes o he ex sep Sep 5 For each brc fro he se of bad brcs chec heher gve GMVS sasfes erseco propery for Z hs brc or o accordg o heore If B BB( Z) () s rue ha hs GMVS sasfes erseco propery for Z For all brc-covex ses of feasble aleraves se of bad brcs ll be epy so he algorh ll sop o sep 4 6 EXAMPLE OF THE ALGORITHM APPLICATION Le us llusrae algorh s applcao h follog exaple hch ca be characerzed as Proble of resource allocao by vog There are hree projecs ( = ) ad hree ages ( = ) ha voe abou ho he led aou C of resources should be allocaed aog hs hree projecs hle o ore ha C /5aou of he resources ca be allocaed o each projec ad s o ecessary o sped all he resources avalable Top of each age s he os preferable resource allocao fro hs po of ve The se of feasble aleraves s (see fg ) Z = { z = { z z z} z z z [0 C] M z [0 C /5]} Mal box for hs se ll be BZ ˆ( ) = [0 C /5] = Graphcal represeao of Z (hached area) ad BZ ˆ( ) s preseed o fg BZ ˆ( ) pleeao (see Korg 00b) for SCF ha calculae ea value of age s ops z = f( τ) M f = j ( τ) τ j= Le us apply he algorh developed for hs exaple Sep The GMVS (4) decoposes BZ ˆ( ) o he se of brcs B = { B} here = { = ( ) M < } B = [ z z ] M z = C = 5 For hs GMVS here s follog correspodece beee ay brc s dex (deoed as ) ad srucure of coalos ha sasfes fales of rgh ad lef-coalo syses (ha geerae hs GMVS) for ay z B z B r( z) R( z)# r( z) M (5) l( z) L( z)# l( z) For case of ages ad projecs se of brcs B geeraed by GMVS (4) cosss fro 7 brcs ad 4 M z = 0 z = C /5 z = C /5 z = C /5 Sep Se of border brcs B ( cl( Z )) our exaple cosss fro brcs B B z C ad = z C = I ca be easly sho ha B B( cl( Z)) {67} The oal uber of brcs B ( cl( Z)) s Graphcal represeao of B ( cl( Z )) (shaded brcs) s preseed o fg z z z = C BZ ˆ( ) Z Fg The se of feasble aleravesz for exaple cosdered ad s al box ˆ( ) BZ Le us see f he GMVS x = πτ ( ) C M π( τ) = ax{ z [0 ] 5 (4) 5z #{ N τ z} } C respecs feasbly or o for gve Z I case he GMVS (4) s feasble urs ou ha s he bes vrual ruhful B ( ( )) cl Z Fg The se of border brcs Sep I our exaple dreco fro ay ufeasble alerave o Z ll be o he lef each deso y B ˆ( Z )\ Z DyZ ( ) = {( )} Thus B B( cl( Z)) DB ( Z ) = {( )} Tha s ea ha he GMVS cosdered sasfes erseco 88

6 Preprs of he 8h IFAC orld Cogress Mlao (Ialy) Augus 8 - Sepeber 0 propery for Z B B( cl( Z)) ff for ay oe arbrary chose alerave y B \ Z holds ha r( y) R( y) M r () y (6) M I ca be easly sho ha f a border brc has dex such ha = 7 ha eas ha y B \ Z r ( y ) R ( y ) M r( y) r( y) r( y) 7 resulg fac ha (6) holds for hs border brcs (for deals see Korg 00a) If a border brc has dex such ha = 6 (6) s o rue because y clb M r ( y ) R ( y ) such ha # r ( z ) = ad r( y) r( y) r( y) = 6 I s obvous ha hs case ay be har ( y ) r ( y ) r ( y ) = For exaple le us cosder ufeasble alerave(7 C / 0 7 C / 0 7 C / 0) Ths alerave belogs o B () C < 7 C < C Ths alerave ay be chose by GMVS (4) f a leas o ages voe for each projec Le he ops of age be τ = (7 C / C / 0) τ = (7 C / 0 7 C / 0 0) τ = (07 C /07 C /0) All hs ops are feasble ad M #{ N τ 7 C /0} = Tha s hy he resul accordg o GMVS (4) s πτ ( ) = (7 C /07 C /07 C /0) Tha s hy he GMVS (4) does o sasfy erseco propery for he se of feasble aleraves Z cosdered hs exaple Tha s hy he decso rule based o sraegy-proof ulcera vog rule (4) s o applcable for he proble cosdered Aalyzg srucure of he se of brcs geeraed by he GMVS (4) ca easly be see ha f he brc srucure s such ha M z = 0 z = C /5 z = C / z 4 = C /5 he he se of border brcs ll be B( cl( Z)) = { B B 7} If he correspodece beee brc s dexes ad codos for lef ad rgh coalos sde each brc reas he sae as for GMVS (4) (see (5)) he (6) ll be sasfed for all he brcs fro se of border brcs The GMVS ha geeraes such se of brcs s x = πτ ( ) C M π ( τ) = ax{ z [0 ] 5 5z #{ N τ z} } 4C 4 Sep ll reur ha all of border brcs sasfes (5) for hs GMVS The algorh ll sop a sep 4 hs exaple because of brc covexy of Z Tha s hy hs GMVS geeraes sraegy-proof decso ag echas for he proble cosdered For exaple for ages h ops τ = (7 C / C / 0) τ = (7 C / 0 7 C / 0 0) τ = (07 C /07 C /0) resource allocao ll be πτ ( ) = ( C / C / C /) 7 CONCLUSIONS The a resul of hs paper ay be shorly ouled as follos Frs e sho ha case of fe uber of feasble aleraves s eough o spec fe uber of ufeasble aleraves order o udersad heher ay GMVS respec feasbly or o for cocree se of feasble aleraves Secod e provde algorh for udersadg ha aleraves should e spec hch s good sese ha zes uber of calculao o be perfored order o udersad heher ay GMVS respec feasbly or o for cocree se of feasble aleraves Also al proble solved hs paper s o so crucal case of fe uber of feasble aleraves (because uber of ufeasble aleraves s also fe) algorh provded ll be useful hs case oo because allos reducg uber of aleraves o be speced REFERENCES Barberá S (006) Sraegy-proof socal choce I Hadboo of Socal Choce ad elfare (Arro K J Se A K Suzuura K (Ed)) vol Elsever/Norh- Hollad Aserda Barberá S Masso J Nee A (997) Vog uder cosras J Eco Theory vol76 pp 98- Barberá S Masso J Serzaa S (998) Sraegy-proof vog o copac rages Gaes Eco Behav vol5 pp Border K Jorda J(98) Sraghforard elecos uay ad phao voers Rev Eco Sudes vol50 pp Bosser eyar JA (006) Socal choce Rece developes Caher de recherche # pages Déparee de sceces écooques Uversé de Moréal 6 Korg N (00a) Use of Ierseco Propery for aalyss of feasbly of Mulcrera Experse Resuls Auoao ad Reoe corol Vol 7 N6 pp Korg N (00b) Vrual Ipleeao of Socal Choce Fuco of Lear Aggregao I Colleced absracs of papers preseed o Fourh Ieraoal Cofrece Gae Theory ad Maagee (Perosya L Zeevch N (Ed)) 0-04 SPb Graduae School of Maagee SPbU 8 Moul H(980) O sraegy-proofess ad sglepeaedess Publc Choce vol 5 pp Nehrg K Puppe C (007) Effce ad sraegyproof vog rules A characerzao Gaes Eco Behav vol 59 Issue pp -5 89