A New Single Transferable Vote Method and its Axiomatic Justification

Transcription

1 A New Sngle Tranferable ote Metho an t Axomatc utfcaton Fua Alekerov ab* Alexaner Karpov a a Natonal Reearch Unverty Hgher School of conomc 2 Myantkaya tr. Mocow Rua; b Ruan Acaemy of Scence Inttute of Control Scence 65 Profoyuznaya Str Mocow Rua. *Correponng author. Tel Fax: mal aree: alek@he.ru (F. Alekerov) akarpov@he.ru (A. Karpov) We propoe a new metho for conuctng Sngle Tranferable ote (ST) electon an prove a unfe metho for ecrbng clac ST proceure (the Gregory Metho the Incluve Gregory Metho an the Weghte Incluve Gregory Metho) a an teratve proceure. We alo propoe a mofcaton for uota efnton that mprove the theoretcal properte of the proceure. The metho utfe by utlng a new et of axom. We how that th metho exten the Weghte Incluve Gregory Metho wth the mofe efnton of uota an ranom euprobable electon of a wnnng coalton n each teraton. The reult are extene to the metho allowng fractonal number of vote. Keywor. Sngle Tranferable ote ornal proportonal repreentaton ytem Gregory Metho

2 Introucton The Sngle Tranferable ote (ST) a preferental votng ytem. oter rank canate accorng to ther preference though ome canate can reman unranke. More than one canate electe from each electoral trct. For a canate to be electe the number of frt place vote the canate receve houl excee a uota. The uota et at the begnnng of the proceure bae on the number of voter an the number of eat. The vote countng proce run teratvely. In each teraton of the proceure the frt place vote for each canate are counte; the total then compare wth the uota. ach canate who acheve a vote count euallng or exceeng the uota electe. If electe canate have urplu vote thoe vote are tranferre to other canate. If all canate have le than the et uota n frt place vote an the number of canate excee the number of eat whch are not flle then the canate wth the leat number of vote elmnate an her vote are tranferre to other canate. If the number of canate eual the number of eat that are not flle then all canate are eclare electe. Although ST ue n proportonal repreentaton ytem t retan a key feature of maortaran electoral ytem - each trct repreente by t own canate. Keen nteret n Sngle Tranferable ote ytem ha been aren n recent year. In Brth Columba two referena were hel n 25 an 29 (lecton BC 25 29). In both referena voter were ake f they whe to change the votng ytem to ST. Both referena were efeate. A referenum on the votng ytem for UK parlamentary electon took place n May 2 (The lectoral Common 2). The ueton propoe aree the alternatve vote whch the one-wnner counterpart of ST. At the en of 2 voter n New Zealan wll chooe an electoral ytem wth ST a one of the alternatve (lecton New Zealan 2). There are everal metho of mplementng ST. The man fference among the metho n how a urplu vote tranferre. The man metho are ecrbe n Farrel an McAllter (23). We prove here hort ecrpton of fferent veron of the Gregory metho. Gregory metho (orgnal veron). Intally ST wa mplemente through the ranom electon of tranferre vote. The proce of manual countng n an ST electon ecrbe n Hoag an Hallett (926). The ballot are orte nto bunle n accorance wth the trbuton of frt place vote. Quota urplu vote are ranomly remove from a bunle of a canate who met the uota an tranferre to the other canate bunle. A oon a another canate reache the uota only lat parcel ballot are tranferre to ubeuent canate. xample. There are 4 canate 3 eat to be flle an the followng voter preference The uota efne a 2 vote a b c 2 vote a vote b vote c vote. Total 54 vote. vote number eat number () where ncate rounng own. In our example 2

3 Therefore 4 vote the uota.e. the mnmum number of vote that ecure a eat for a canate. Canate a receve 24 vote a total that excee the uota. Therefore a eclare electe. Canate a urplu vote are then trbute. ach vote ha a probablty of /24=4.67% of beng tranferre. Let u examne the mot lkely outcome. Fve vote are tranferre to canate b an 5 vote are move to non-tranferable vote becaue thee voter o not ncate ubeuent preference. On the next teraton we obtan 7 vote a b c 7 vote a 5 vote b c (tranfer to a) vote b vote c vote 5 non-tranferable vote. Canate b reache the uota. Accorng to the Gregory metho urplu vote are tranferre from the lat parcel.e. from canate a urplu. In other wor canate b bunle goe up becaue canate a urplu tranferre untl a vote total meet the uota. Reual ballot are tranferre to other canate. In our example one vote tranferre to canate c. Canate c n the lat teraton ha more vote than canate o he electe. In the fnal tally canate a b an c are eclare the wnner. Wth nonzero probablty vote are move from the econ group of vote to nontranferable vote. 2 vote a b c 2 vote a vote b vote c vote non-tranferable vote. Canate {b c } have eual chance of beng elmnate. Three poble electon reult can be {a b c} {a c } an {a b }. Ranomne n the choce of the wnnng coalton lea to fferent reult. Gregory metho (moern veron). In th veron there no ranom electon. Ballot are electe proportonally to the econ place vote. The lat parcel tranfer prncple reman unchange. In our example /24=4.67% of each group of vote tranferre n the frt teraton. From 2 vote wth the ame econ place vote /24*2=5 vote are tranferre. The econ teraton urplu homogenou (t contan only b c vote). Th urplu tranferre to canate c. In th veron canate a b an c are the wnner. Incluve Gregory metho. Th veron ffer from the moern Gregory veron n the metho of tranferrng ubeuent urplue. The frt teraton the ame (proportonal veron). On the econ teraton one vote from canate b houl be tranferre n our example. The Incluve Gregory metho count all vote for canate b (22 vote): 2 vote a b c vote b. Depte the fact that the maorty of the 2 vote are ue for canate a to wn th metho take nto account an ntal preference profle. Thu 2/22=.55 vote houl be 3

4 tranferre from the frt group of vote an /22=.45 from the econ group. We coner the metho that tranfer a non-fractonal number of vote;.e. one ballot goe to canate c. In th cae canate a b an c are electe. Weghte ncluve Gregory metho. Face wth a urplu tranference problem th metho count only uch vote that were tranferre to the canate before the current teraton. The frt teraton the ame a n the Gregory metho. On the econ teraton when one vote from canate b houl be tranferre all 5 vote of all group of voter are counte: The group of vote 5 (=/24*2) vote a b c vote b. b contan more voter o canate a b an are electe. In thee metho only whole (non-fractonal) number of vote are tranferre. Th retrcton come from ol manual countng practce; later electoral law fxe th feature. However moern computatonal metho allowng fractonal vote cannot be mplemente mmeately becaue a ere to preerve tranparency n votng proceure an confence n electoral reult lea to retance to new metho. Moreover torton n real electon that are from rounng are ngnfcant ue to the enormou number of vote cat. There are metho that take nto account fractonal number of vote. A an example we can preent a weghte ncluve Gregory Metho wthout correponng rounng. We wll coner a veron of the propoe moel wth fractonal vote n Secton 4. ental fference among thee metho are notceable only when euental urplue are trbute an proceure become more an more complcate. Htorcal example o not prove the bet metho for uncoverng thee fference. In actual electon the number of teraton can approach everal hunre whch make any nterpretaton of the avantage of fferent metho rather ffcult. In aton the metho ue often change uner the nfluence of fferent poltcal force when uch force beleve that the reult of electon obtane by a metho are unfar. We ue an axomatc approach to tuy the Sngle Tranferable ote. Th approach allow u to compare fferent metho on the ba of the axom we ntrouce. There are a number of publcaton emontratng that ST volate varou ratonal choce axom. Doron an Kronck (977) howe that ST volate monotoncty an Nurm (996) emontrate that ST ncorporate the No-Show Paraox (f ome voter who prefer canate x partcpate n electon x loe) volate the Conorcet Prncple an oe not atfy the Contency Property (f a choce on two group of ballot conce then the choce on the one group houl be the ame). Mller (27) contructe an example calle a butterfly effect whch reveal chaotc behavour on the part of ST (ee alo the cuon n Hll 28)). Moreover th example alo emontrate that ST manpulable;.e. a mall change n the preference profle can lea to a ubtantal change n the reult. Laler an an er Straeten (peronal communcaton) have alo elaborate on the manpulaton problem n the cae of alternatve vote proceure. Wooall (987) prove the axom for ST an prove an mpoblty theorem howng that there no proceure that atfe all thoe axom. The proof wa mae not only for ST but alo for aggregaton proceure n general. Thee axom o not allow for the electon of any partcular proceure. A formalaton of ST To analye the metho mplementng ST let u formale the proceure an formulate the properte that thee metho houl atfy. 4

5 Let u enote v... - a et of voter nexe by k v n C c... c m - a et of canate nexe by the number of eat to be flle (aume that <m<n) nex of teraton of countng of vote the et of voter conere on the th teraton C C the et of canate on the th teraton the et of canate electe before the th teraton P preference profle of voter P a preference profle correponng to the et of voter an the et of canate on the th teraton c C a coalton of voter for whch a canate ranke frt n ther preference c C the mal coalton of th type.e. all other voter vote for other canate n the uota.e. the mnmum number of vote neceary for a canate to be electe. For an nteger number of vote the uota efne a If c C n n n (2) then th coalton a wnnng one becaue uch a coalton can prove vctory for any canate c. Let u efne formally a ST proceure that performe teratvely untl all eat are flle. Th algorthm ecrbe a cla of teratve metho. ach metho an example of an ST proceure. At the begnnng of the proceure the uota efne accorng to formula (2). Moreover on the ntal teraton we put : : Ø. The teraton : Ø. a) f there ext a wnnng coalton c C upporte by th coalton eclare electe. Then for ome canate : c. c then the canate c f then : \ c C C \ c : C an a new teraton begn. Otherwe the proceure top. : b) If there no wnnng coalton then the algorthm procee a follow. If C all canate are electe.e. : C C an the proceure top. Otherwe f 5

6 we contruct a partton of the et of voter n the coalton c C canate c C wth the leat carnalty wth mal coalton c C the new teraton then begn. : : C \ c : C : ; for all c C The loe an Note that n ome teraton a voter may not have a preference on the et of canate that have yet to be electe. Th mean that her ballot move to the category of non-tranferable vote;.e. her vote are not nclue n the wnnng coalton of thoe canate for whch he vote an thee vote o not have any mpact on the votng reult after her preferre canate are exclue from the lt. Formally however uch a voter reman n the lt of voter untl the en of the proceure. In certan teraton everal canate can acheve the uota multaneouly. ach metho houl fx the rule by whch the proceure chooe the partcular canate from the et of canate who have receve more than the uota of frt place vote. xample 2. Let ecrbe the orgnal Gregory metho n new term. The mal coalton for canate c efne a follow c C c C C \ C C f there wa no wnnng coalton on the prevou teraton an c C c C C \ C C \ C \ C C where C C C n \ f the lat teraton ene by ome canate electon. In Gregory metho only the lat parcel ballot are tranferre. Therefore the wnnng coalton houl entrely nclue the mal coalton from prevou teraton c C c C where c C n. xample 3. Coner a cae wth 3 canate an 5 voter that llutrate how the et of voter an canate ecreae. The ntal preference profle hown n Table. Table. The preference profle Frt preference Secon preference Thr preference oter a a a b c c b c c b b c b a a Wth =2 to wn t neceary to get 2 vote (=2). Canate a ha a wnnng coalton from the voter { 2} (hown n Table unerlne); thu he electe. The ballot of thee voter an canate a are exclue from the profle. In Table 2 the change preference profle hown ung the proceure n the next teraton. 6

7 Table 2. The change preference profle Frt preference Secon preference oter c b c b c b The remanng et of voter {3 4 5} an the remanng et of canate {b c}. Due to the excluon of the canate a ballot 3 move from canate a to canate c. There a wnnng coalton for canate c (voter {3 5}). The proceure termnate an canate a an c are electe. Way of mplementng ST ffer on the metho of etermnng how the wnnng coalton choen. Th fference n turn efne whch vote reman n favour of the canate an whch are tranferre. Thu the metho of etermnng how the wnnng coalton efne etermne all ubeuent path of vote tranfer. That why the metho of choong the wnnng coalton n each teraton houl be etermne n avance. Some natural contrant allow u to retrct metho to ome cla of proceure. Axom an the repreentaton theorem In th ecton we gve new et of axom how ther nepenence an prove the characteraton theorem. Axom for the wnnng coalton choce metho are the followng:. The nepenence of the prevou teraton. For any teraton f one begn the proceure a f tartng from the begnnng whle keepng the current trbuton of vote unchange the current choce of the wnnng coalton houl alo reman unchange. 2. The nepenence of coneuent preference. The preference of voter that were not taken nto account n the proceure (.e. thoe preference that were not frt preference n the current teraton) houl not nfluence the choce of the wnnng coalton. 3. Th axom can be conere a a counterpart of the Inepenence of Irrelevant Alternatve by K.Arrow (963). Inee IIA tate that n the contructon of ocal choce between alternatve a an b the preference among other alternatve a an c b an c an etc. o not matter. Axom 2 tate that n fllng the eat on the -th tep we take nto account only frt level of the nvual preference on that teraton. 4. The orgnal Gregory metho wth manual count naturally oe not take nto coneraton the coneuent preference an atfe th axom. 5. Anonymty. The name of voter o not matter. 6. Neutralty. The name of the canate o not matter. The neceary conton for Axom to hol the re-evaluaton of the uota n each teraton accorng to the formula. (3) If the uota not re-evaluate then the ntal uota an the uota evaluate on the ba of vote an eat wll not be eual n ome teraton whch naturally volate Axom. The et of voter an the number of remanng eat are change only at the moment of the electon of the next canate. In each teraton when a canate not electe the uota not change. The non-tranferable vote are taken nto account when the uota re-evaluate. It turn out that 7

8 8 the re-evaluaton of the uota oe not change the proceure crucally becaue the uota can ecreae only by over all teraton. Lemma. The uota evaluate accorng to formula (3) oe not ncreae on any teraton. Proof. By efnton of the uota the number of canate who can attan the uota mut be at leat eual to the number of the eat avalable n that teraton. (4) Aume that after the electon on ome teraton of a current canate the uota ncreae. Th mean that the ol uota lea to the electon of a number of canate larger than.e. contractng (4). Thu n each teraton the uota ecreae or reman unchange.e.. Theorem. For the uota evaluate n the fnal teraton l of the proceure.e. l l l the followng neualty l hol. Proof. By Lemma n each teraton the uota can ecreae or oe not change.e.. From the frt teraton the uota can change a follow. In each teraton n whch a canate electe the et of voter ecreae. Then where a et of teraton untl whch reult n the electon of a canate.e.. Wthout rounng own th expreon can ncreae to le than. Thu the nteger part of the fference oe not excee the fference of nteger part.e.

9 9. Ung the efnton of we obtan where x x x Denote the lat teraton when hol a teraton then begnnng wth + the expreon oe not ecreae. Coner the cae when the ntal teraton. In th cae eat are trbute an. Becaue t contnue the prevou teraton the uota o not change an the value reman the ame. For th teraton t true that In the um n the nomnator only the frt component negatve an Thu

10 We obtan whch true for the fnal teraton a well Theorem 2. The only metho atfyng Axom -4 the metho wth ranom euprobable electon of the wnnng coalton n each teraton wth a re-evaluaton of the uota accorng to formula (3) Proof. Becaue the choce oe not epen on the ubeuent alternatve (Axom 2) an the choce n each teraton mae mlarly to the choce on the ntal teraton (Axom ) whch oe not have a htory then the voter ffer only by ther name an by ther frt preferable alternatve n the th teraton. Anonymty (Axom 3) mple that each voter hence each coalton a pror (before the proceure begn) ha eual chance of wnnng. By Axom the ntal an ubeuent teraton o not ffer;.e. the eual nepenent chance are preerve n each teraton. The only metho that gve eual chance the euprobable metho n each teraton of coalton electon. For the ze of wnnng coalton not to epen on the teraton telf t neceary an uffcent to re-evaluate the uota accorng to formula (3). Thu the only metho atfyng Axom -3 th metho. By contructon t alo neutral;.e. Axom 4 alo obeye. The metho ntrouce can be calle the probabltc veron of the weghte ncluve Gregory metho wth the re-evaluaton of the uota n each teraton. Obvouly the probabltc metho of electng the wnnng coalton can lea to fferent electon reult a emontrate n the example n Secton. Two remark are n orer. Remark. Theorem 2 llutrate an mportant feature of the rule uner tuy. If only etermntc rule are allowe then no rule can atfy Axom -4. Remark 2. Becaue there may be many canate who are upporte by fferent wnnng coalton t eem reaonable to etermne how thee canate are electe. Below we prove two more axom howng how th electon can be axomate. Axom for the wnnng canate choce rule: 7. Herety. If canate x choen from the et of canate whoe frt place vote excee the uota then canate x houl be choen from any ubet that nclue hm. 8. Inepenence from other canate. The rule to chooe the canate from the et of canate whoe frt place vote excee the uota houl epen only on nformaton from voter whoe frt alternatve from th et. The metho etermne by Theorem 2 can be tnguhe by the wnnng canate choce rule. There are many way of choong a canate from the et of canate whoe frt place vote excee the uota whle alo atfyng Axom -6. For example one mght chooe the canate wth the hghet number of vote or the lowet number of vote. An analy of Axom -3 We wll emontrate next that Axom -3 are nepenent. To o th we wll contruct example that volate only one axom of the three. Metho. In the ntal teraton the voter are ranomly ranke. Accorng to th orerng lexcographc orer mpoe on the et of coalton. We chooe the coalton wth the mnmum number. Axom 2-4 are atfe but Axom volate.

11 Metho 2. In each teraton the uota re-evaluate an the canate are ranomly ranke. The coalton electe from thoe voter for whch the econ bet canate cloet to the choen canate. Among coalton that are not comparable wth repect to th crteron we elect one canate by mean of a ranom mechanm. Obvouly Axom 3 4 are obeye but Axom 2 volate. Metho 3. Rank coalton lexcographcally wth repect to the name of voter. In each teraton re-evaluate the uota an chooe the coalton wth the lowet rank. Axom 2 an 4 are atfe but Axom 3 volate. Theorem 2 mple that there no metho that atfe Axom -3 but alo volate Axom 4. We wll how now that the Gregory metho wth proportonal electon or wth the ranom electon of wnnng coalton oe not atfy Axom. To th en coner xample. After the trbuton of the urplu vote of canate a the followng tuaton oberve 5 vote b c (receve from a) voter b voter c voter. The uota eual to 4. Canate b electe. Accorng to the Gregory Metho one vote tranferre to canate c. If uch a tuaton oberve n the frt teraton then vote have been electe wth eual probablty among all ballot. Thu wth the probablty of 5/5 the vote of the frt group have been re-trbute an wth the probablty of /5 the vote of the econ group have been re-trbute;.e. the electon of vote to tranfer to the next canate ha been change. Th mean that the Gregory Metho oe not atfy Axom the nepenence of the prevou teraton. In the proportonal re-trbuton of ballot a can be een from th example the Gregory Metho volate Axom. The ncluve Gregory Metho alo volate Axom but the weghte ncluve Gregory Metho complete wth the re-evaluaton of the uota n each teraton atfe Axom. The Gregory Metho wth the ranom electon of ballot atfe Axom 2 the nepenence of coneuent preference becaue any change of the preference profle oe not change the probablte of the electon of wnnng coalton n any prevou teraton. Smlarly all extene veron of the Gregory Metho atfy Axom 2. However the Gregory Metho wth proportonal electon of ballot oe not atfy Axom 2. Inee coner the followng example: xample 4. Let the number of voter an canate be uch that the uota eual to 6. In Table 3 one can ee the tuaton n the frt teraton wth repect to the preference of voter votng for canate a. Other vote are uch that canate a receve the mum number of vote. Table 3. The electon of the wnnng coalton ote for a Other vote Frt preference a a a a a a a a Secon preference c c c c The proceure chooe a wnnng coalton (vote {23678} - unerlne n the Table 3) proportonally to the econ preference. Becaue the number of the ballot n whch c an take econ preference are eual then the repectve voter houl be repreente n eual part

12 n the coalton. ote 4 an 5 are tranferre to canate c an. However f we change the econ preference n th profle then the trbuton of the econ preference n the prevou coalton wll not be proportonal whch can be een from Table 4. Table 4. A change of econ preference ote for a Other vote Frt preference a a a a a a a a Secon preference c c c c After the change of econ preference the wnnng coalton mut be change whch hown n Table 5. The poble et of wnnng coalton wa change after the change n the econ preference. The metho epen on ubeuent (econ) preference. Th outcome volate Axom 2. Table 5. A electon of the wnnng coalton ote for a Other vote Frt preference a a a a a a a a Secon preference c c c c The ame reaonng can be apple to the extene Gregory Metho becaue they o not ffer n the ntal teraton. Fractonal vote cae Coner an example. There are 4 vote for wnnng canate an one of them urplu vote whch houl be tranferre. We can o th exactly proportonally tranferrng a uarter of each vote. Another poble oluton to tranfer half of ballot an half of ballot 2. There nfnte number of poble oluton. The fractonal vote approach generale prevou moel. v v... v v. The remanng vote n teraton wll be preente by a vector n k Let v... where the unt () enote a complete vote. w w... w w A charactertc vector of a coalton enote a n k. If n the teraton the k th voter belong to the wnnng coalton for the canate then w. In aton we put w f v. Let k k w voter vote for other canate. Note that v k an be a mal coalton n favour of ;.e. all other w k can be fractonal. A coalton wnnng f the number of vote (ot prouct of thee vector) eual to the uota v w n. (5) Next we ecrbe the proceure. At the ntal teraton the uota efne. The uota can be efne n the cae of nteger n n vote by but t can be efne a wth beng an arbtrary mall real number. A before n the ntal teraton we put : : Ø. k An teraton. 2

13 a) If there a wnnng coalton w for ome canate c then the canate c upporte by th coalton electe. Then : c. If then v k : vk wk C \ c : C : an the new teraton begn; otherwe the proceure top. b) If uch a wnnng coalton oe not ext the proceure contnue a follow. If C then all canate are electe : C an the proceure termnate. In other cae.e. f C the canate c C wth mnmal value v w erae from the lt an v : : v C \ c : C then the new teraton begn. : In aton to the euprobable Gregory Metho the weghte ncluve Gregory Metho atfe Axom -4 f the uota re-evaluate. If the uota efne a n the cae wth nteger vote wth t change to not more than. If t efne wth t change to not more than. The weghte ncluve Gregory Metho trbute eual part of each vote; namely from each complete or fractonal vote t gve to the wnnng coalton that fracton of the k th vote that efne the rato of the uota n all wnnng coalton w k w. (6) v Th efnton mple that the contructon of a coalton oe not epen on further preference (Axom 2) an oe not epen on the number of teraton (Axom ). Becaue all voter are taken nto account n eual part the proceure atfe anonymty (Axom 3). Theorem 3. In a fractonal vote framework there ext a non-neutral metho atfyng Axom -3. Proof. We contruct th metho. The canate x electe at the begnnng of the proceure. All wnnng coalton can be ecrbe by (5) ubect to w. If canate x k w k 3

14 wn then elect the wnnng coalton euprobably from the et of all wnnng coalton. If ome other canate wn then efne the wnnng coalton by (6) tranferrng eual fracton of vote. Obvouly Axom -3 are obeye but not Axom 4. Axom -4 efne the cla of metho whch combne euprobable trbuton an a tranfer of eual part of vote ether completely nepenently or epenent on the frt vote n the voter preference. Meek Metho (Meek ) a well a other metho contructe on t ba cannot be formale by the moel ecrbe above becaue ther pecfc feature a permanent re-contructon of wnnng coalton of canate alreay electe. Meek Metho now ue only n electon n New Zealan (To 23). Conclung remark We contruct an extenon of fferent metho bae on ST a an teratve proceure. The extng metho can be conere a partcular cae of th proceure. We prove a et of axom for th proceure that favour none of the component of the efnton of wnner; the et oe not favour the name of canate the name of voter the name of a coalton or the number of teraton. Th feature mmeately lea to the ranom electon of a wnnng coalton n each teraton a concluon that proven n Theorem 2. We have contructe a new metho of ST that we call ST wth re-evaluaton of the uota n each teraton wth vote tranferre ether euprobably or n eual fracton f fractonal vote are allowe. We prove an axomatc utfcaton of th metho. The re-evaluaton of the uota not gnfcant becaue we prove that the uota cannot ncreae an can only ecreae to no more than. If the uota not re-evaluate th acton mple a fference between the ntal teraton of the proceure an the ubeuent teraton. Thu the choce of the proceure epen on the number of teraton. Th reaon le to Axom (the nepenence of prevou teraton). In actual electon th pecfc aton wll not play an mportant role but the re-evaluaton of the uota wll ncreae the tranparency of the proceure becaue t ecreae the fference between the frt an ubeuent teraton. Depte of the local nature of the axom The nepenence of coneuent preference t turn out that the re-evaluaton of the uota on each teraton oe not lea to large fference n the uota value t can change only on no more than. Acknowlegement The partal upport of the Acaemc Founaton of the Natonal Reearch Unverty Hgher School of conomc (proect # -4-3) an the Internatonal Laboratory of Decon Choce an Analy n the framework of the Programme of Funamental Stue of the NRU HS (proect 53. an 55.) Mocow RF to BM acknowlege. We alo thank the partcpant of the emnar on Publc Choce Reearch Centre (Turku Unverty Fnlan May 2) an the Workhop on conomc Degn (Blg Unverty Itanbul Turkey May 4-5 2) an Profeor Hannu Nurm for ueful comment. 4

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