From: Subject: Kennislink - Is democratie wiskundig onmogelijk? Date: Fri, 15 Dec 2006 09:58:32 +0100 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C7202F.940D2670" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2962 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C7202F.940D2670 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.kennislink.nl/web/show?id=133140&vensterid=811&cat=60360 Kennislink - Is democratie = wiskundig onmogelijk?
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TERUG|MAI= L|SLA OP 3D""=20
16/6/2005 Is=20 democratie wiskundig onmogelijk?=20
SAMENVATTING
Bij de Britse lagerhuisverkiezingen op = 5 mei won=20 Labour 55% van de zetels met slechts 35% van de stemmen. In = 2000 won=20 Bush de Amerikaanse presidentsverkiezingen met 500.000 = stemmen=20 minder dan Gore. Bij veel referenda is het voor = =91voor=92-stemmers=20 beter om thuis te blijven dan om te gaan stemmen. Fraude? = Omkoperij?=20 Intimidatie? Nee, nog erger: simpele wiskunde.=20

Rare verschijnselen = zoals=20 hierboven worden in de wiskunde verkiezingsparadoxen = genoemd. Door=20 de manier waarop de verkiezingen georganiseerd zijn, kan de = uitkomst=20 van de verkiezingen anders zijn dan je logischerwijze zou=20 verwachten. In het geval van Groot Brittanie en Amerika is = de=20 oorzaak het districtenstelsel (zie kader). Het is geen groot = nieuws=20 dat dit soort stelsels tot verkiezingsparadoxen kunnen = leiden.=20 Schokkender is het dat in ieder kiessysteem dit soort = paradoxen=20 kunnen voorkomen. In 1951 bewees Kenneth Arrow dat geen = enkel=20 kiessysteem helemaal eerlijk is, en dat volmaakte democratie = dus=20 wiskundig onmogelijk is. Dit schokkende feit was goed voor = een=20 Nobelprijs economie.=20

Zeker met alle veranderingen de organisatie van de EU die = voor de=20 deur staan, is het interessant om te kijken wat voor = paradoxen er=20 zoal mogelijk zijn. Daarom in dit artikel een groot aantal=20 (vereenvoudigde) voorbeelden. Ook zal ik uitleggen wat = Kenneth Arrow=20 bedoelt met 'volmaakte democratie'. Wie meer wil weten over = het=20 bewijs dat dit een onhaalbare kaart is of over de = kiessystemen zoals=20 die in de Europese grondwet staan, verwijs ik naar de links = onderaan=20 dit artikel. Speciaal voor scholieren/docenten is er een = verkorte=20 versie van dit artikel beschikbaar met vragen en=20 opdrachten.

3D""=20 Kenneth=20 Arrow, mede-winnaar van de Nobelprijs voor de Economie in=20 1972.



Verschillend kiessysteem, verschillende = uitslag.=20
Drie vrienden (die zoals de meeste vrienden in=20 wiskunderaadsels Anna, Bob en Charles heten) hebben na jaren = zoeken=20 eindelijk een mooi huis in Amsterdam gevonden waar ze kunnen = gaan=20 wonen. Ze zijn nu bezig met de inrichting van hun = gemeenschappelijke=20 woonkamer en zoeken op de meubelboulevard in Beverwijk naar = een=20 stoel. Anna is helemaal weg van een rieten rotanstoel, Bob = gaat voor=20 een lederen fauteuil en Charles is zeer gecharmeerd van een=20 multifunctionele klapstoel. Ze zijn het erover eens dat het = een van=20 deze stoelen moet worden, want alle andere stoelen zijn of = te lelijk=20 of te duur. Bob wil echter absoluut de rieten rotanstoel = niet, omdat=20 hij als driejarige ooit zijn vinger aan zo'n soort stoel = heeft open=20 gehaald. Anna heeft ernstige bezwaren tegen de = multifunctionele=20 klapstoel, omdat dat helemaal geen sfeer in huis geeft, en = Charles=20 is fanatiek tegen de lederen fauteuil omdat die zou = betekenen dat=20 hij zijn kat de hele dag krampachtig in zijn eigen kleine = kamertje=20 opgesloten zou moeten houden.

Omdat ze het niet eens worden besluiten ze te stemmen. De = voorkeuren zijn dus als volgt verdeeld:

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Gewoon per stoel stemmen heeft geen zin: = voor iedere=20 stoel is er precies een stem. Daarom stelt Anna voor eerst = tussen de=20 Lederen Fauteuil en de Multifunctionele Klapstoel te = stemmen, en dan=20 tussen de winnaar en de Rieten Rotanstoel. Bij iedere = stemronde moet=20 de uitslag 2 tegen 1 of 3 tegen 0 zijn, dus is er in elk = geval een=20 uitslag.

Wat Anna er niet bij vertelt, is dat ze heel goed weet = wat die=20 uitslag zal zijn. Eerst verplettert de Fateuil de Klapstoel = met 2=20 stemmen (Bob en zijzelf) tegen 1, om vervolgens het loodje = te leggen=20 tegenover de Rotanstoel, die op zijn beurt ook op twee = stemmen kan=20 rekenen (dit keer van Charles en Anna) . Bob is hier om=20 begrijpelijke redenen fanatiek op tegen. Hij doet een ander=20 voorstel: eerst stemmen tussen de Multifunctionele Klapstoel = en de=20 Rieten Rotanstoel en dan tussen de winnaar en de Lederen = Fauteuil.=20 Bob kan niet wachten om de Rieten Rotanstoel roemloos ten = onder te=20 zien gaan in zijn verkiezingsstrijd tegen de Klapstoel. Dat = de=20 Lederen Fateuil vervolgens de Multifunctionele klapstoel ook = nog=20 eens voorbijstreeft in de tweede ronde is helemaal mooi = meegenomen.=20 Het wordt aan de lezer overgelaten te bedenken met wat voor=20 alternatief kiessysteem Charles op de proppen zal komen.

Uiteindelijk krijgt Anna haar zin. Er wordt eerst gestemd = tussen=20 de Fauteuil en de Klapstoel en dan tussen de winnaar en de = Rieten=20 Rotanstoel. Bob voelt de bui al hangen, en omdat zijn = rotantrauma zo=20 sterk is dat hij liever verhuist dan op een rotan stoel te = moeten=20 zitten, besluit hij in de eerste stemronde niet op zijn = eerste keus=20 (de Lederen Fauteuil) maar op zijn tweede keus (de = Multifuncionele=20 Klapstoel) te stemmen. Tot ontsteltenis van Anna wint de = Klapstoel=20 nu de eerste ronde, en alsof het allemaal niet erg genoeg = is, de=20 tweede ronde ook. Charles kan zijn geluk niet op, Bob is = blij dat=20 hij erger heeft weten te voorkomen en Anna, tsja, Anna zit = er maar=20 mooi mee.

Op zoek = naar het=20 perfecte kiessysteem
Nu we weten dat verschillende=20 kiessystemen tot verschillende uitslagen leiden, kunnen we = ons=20 afvragen: wat is het beste kiessysteem? Bij welk kiessysteem = komt de=20 uitslag het best overeen met de meningen van de kiezers? = Deze vraag=20 is nogal moeilijk te beantwoorden omdat "de mening van de = kiezers"=20 een begrip is waarvan we niet goed weten wat het is. Iedere = kiezer=20 heeft zo zijn eigen mening. Wat we bijvoorbeeld wel kunnen = zeggen=20 dat er mis is met het Nederlandse kiessysteem, is dat geen = rekening=20 houdt met wat de kiezers vinden van de partijen op wie ze = niet=20 stemmen. Als de meerderheid van de kiezers liever partij A = heeft dan=20 partij B, dan zouden we willen dat A ook meer zetels haalt = dan B.=20 Ook als een groot deel van die kiezers op partij C stemt. In = het=20 Nederlandse systeem is dit niet altijd het geval. Laten we = kijken=20 naar het volgende vereenvoudigde voorbeeld.=20

Een pasopgerichte politieke partij, moet een kleur = kiezen. rood,=20 blauw en groen zijn al geclaimd, paars is ook geen optie en = dus gaat=20 de keus tussen geel, oranje en turqoise. Op het congres dat = deze=20 belangrijke beslissing moet nemen zijn 31 leden aanwezig. De = voorkeuren zijn als volgt verdeeld:


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Net als bij de echte verkiezingen mogen de = leden hun=20 eerste keus op een briefje schrijven. De uitslag wordt dus:

1. Geel met 12 stemmen
2. Turquoise met 10 = stemmen
3.=20 Oranje met 9 stemmen

Als we echter de kleuren twee aan twee zouden vergelijken = krijgen=20 we een heel andere uitslag: meer dan de helft van de leden = (16=20 namelijk) heeft liever Oranje dan Turquoise, en ook meer dan = de=20 helft (ook 16) heeft liever Oranje dan Geel. Het lijkt er = dus op dat=20 Oranje veruit de populairste kleur is. Bovendien hebben 16 = leden=20 liever Turquoise dan Geel. Geel kunnen we dus wel aanmerken = als de=20 grote verliezer. Deze uitslag:

1. Oranje (Want populairder dan zowel Turquoise als = Geel)
2.=20 Turqoise (Want populairder dan Geel, maar minder populair = dan=20 Oranje)
3. Geel (Want minder populair dan zowel Oranje = als=20 Turquoise)

is echter precies tegengesteld aan de vorige. Welke is = nou=20 eerlijker?

Meeste = stemmen=20 gelden vs Twee aan twee vergelijken
Over het algemeen = wordt=20 'twee aan twee vergelijken' wat bij de tweede uitslag gedaan = eerlijker beschouwd dan 'de meeste stemmen gelden', wat tot = de=20 eerste uitslag leidde. De reden hiervoor is dat twee aan = twee=20 vergelijken aan een aantal redelijke eisen voldoet, waaraan = de=20 meeste stemmen helaas niet altijd kan voldoen. De eerste eis = is al=20 besproken: als een meerderheid van de kiezers liever A dan B = heeft,=20 dan willen we dat A ook hoger eindigt dan B. En in het = extreme geval=20 dat
ALLE kiezers liever partij A dan partij B hebben, = willen we=20 natuurlijk helemaal dat A meer zetels haalt dan B. Zelfs = hieraan is=20 in het Nederlandse kiessysteem niet altijd voldaan (bedenk = zelf=20 hoe).

De tweede eis wordt duidelijk uit het volgende voorbeeld. = Stel=20 dat de zeven kiezers die als volgorde
"1. Geel 2. Oranje = 3.=20 Turquoise" hadden, zich op het laatste moment bedenken en = besluiten=20 dat ze Turquoise eigenlijk mooier vinden dan Oranje (maar = Geel nog=20 steeds mooier dan de andere twee). Je zou redelijkerwijs = mogen=20 verwachten dat zo'n aardverschuiving in "de mening van de = kiezers"=20 van grote invloed is op de uitslag van de verkiezingen. Dit = is bij=20 de meeste stemmen gelden echter helemaal niet het geval!

Wat is met de nieuwe voorkeuren:

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de 'nieuwe' uitslag als ieder lid zijn 1e = keus op een=20 briefje schrijft? En wat is de 'nieuwe' uitslag bij twee aan = twee=20 vergelijken? 'Uw mening telt' geldt dus meer bij twee aan = twee=20 vergelijken dan bij 'De meeste stemmen gelden'.

Iets meer in het algemeen is het bezwaar tegen het = systeem dat in=20 Nederland gebruikt wordt, dat het geen rekening houdt met = wat de=20 kiezers vinden van de partijen waar ze niet op stemmen. Twee = aan=20 twee vergelijken is niet de enige manier om hier wat aan te = doen. Er=20 zijn talloze andere kiessystemen te bedenken en bedacht die = op dit=20 punt =91eerlijker=92 lijken dan het gewone Nederlandse = systeem. Ik roep=20 van harte op zo=92n systeem te bedenken. =

3D""=20 Verkiezingen=20 in Afghanistan. Naast de verschillende mankementen van = kiesstelsels=20 an sich had de bevolking hier ook nog eens te maken = met de=20 dreiging van Taliban-aanslagen...


Ik heb = getwijfeld=20 over Belgi=EB
Er is zelfs nog een derde punt waarop=20 twee-aan-twee vergelijken beter scoort dan de meeste stemmen = gelden,=20 genaamd Onafhankelijkheid van Irrelevante Alternatieven. = Bekijk het=20 volgende voorbeeld.

Een schoolklas gaat een weekje op = kamp en=20 mag kiezen uit drie bestemmingen: China, Japan en Belgi=EB. = De=20 meningen van de 20 leerlingen zijn als volgt=20 verdeeld:

3D""=20



Er wordt gestemd op de gebruikelijke manier = en de=20 uitslag is als volgt:

1. Belgi=EB (met 16 stemmen)
2. Japan (met 1 = stem)
3. China=20 (met 0 stemmen)

De klas besluit naar Belgi=EB te gaan. =92s Avonds is = echter op het=20 nieuws dat in Belgi=EB oorlog is uitgebroken. De ouderraad = acht het=20 niet verantwoordelijk om in deze omstandigheden naar = Belgi=EB te gaan,=20 en de klas kiest voor zijn (volgens de verkiezingen) tweede = keus:=20 Japan. Dit is natuurlijk niet helemaal eerlijk. Als de = oorlog een=20 dag eerder was uitgebroken en er meteen tussen alleen China = en Japan=20 gestemd was, had China overtuigend met 16 tegen 4 gewonnen. = Dit=20 mankement van het kiessysteem nodigt natuurlijk uit tot = vreselijke=20 fraude. Stel dat de keus oorspronkelijk alleen tussen China = en Japan=20 ging. Piet voelde al aankomen dat hij vreselijk ging = verliezen, en=20 zat danig met de handen in het haar. Toen hoorde hij opeens = een=20 piepje in zijn broekzak. Zijn sms-nieuwsdienst vertelde hem = als=20 eerste en als enige over de oorlog in Belgi=EB die zojuist = begonnen=20 was. Prompt lanceerde hij het "Irrelevante Alternatief" = Belgi=EB als=20 derde vakantiebestemming. Hij gaf hoog op van de goede = keuken en de=20 frisse boslucht en al zijn klasgenoten gingen voor de bijl. = Belgi=EB=20 won de verkiezingen als hierboven, met als goede tweede = Japan. s=20 Avonds zag iedereen het nieuws en de volgende dag besloot de = klas=20 naar Japan te gaan, in plaats van het logischer lijkende = China. We=20 zeggen dat dit de meeste stemmen gelden systeem niet = 'onafhankelijk=20 van irrelevante alternatieven' is.

Ga na dat twee aan twee vergelijken wel onafhankelijk van = irrelevante alternatieven is: het toevoegen en achteraf weer = weghalen van een extra keuzemogelijkheid verandert niks aan = de=20 verhoudingen tussen de andere = mogelijkheden.

Twee aan twee vergelijken in de = praktijk
Dit=20 alles roept twee vragen op: 'merken we nou ook wat van in = het echt?'=20 en 'als twee aan twee vergelijken dan zoveel beter is dan de = meeste=20 stemmen gelden, waarom doen we dat dan niet?

Het antwoord op de eerste vraag is waarschijnlijk ja. Op = grond=20 van opiniepeilingen is dit berekend voor een aantal = Nederlandse=20 tweede kamer verkiezingen (zie de links onderaan het = artikel). In de=20 verkiezingen van 1994 won D66 bij twee aan twee vergelijken = van de=20 PvdA, van het CDA en van de VVD, maar al deze partijen = haalden meer=20 zetels dan D66.

Dan de vraag waarom twee aan twee vergelijken niet wordt=20 toegepast. De reden is vrij eenvoudig: er is niet altijd een = uitslag. Kijk nog eens naar het voorbeeld van de stoelen. = Bij twee=20 aan twee vergelijken wint de Rieten Rotanstoel van de = Lederen=20 Fauteuil, de Lederen Fauteuil op zijn beurt wint weer van de = Multifunctionele Klapstoel, maar de Multifunctionele = Klapstoel wint=20 van de Rieten Rotanstoel. Twee aan twee vergelijken geeft = misschien=20 het best de "mening van de samenleving als geheel" weer, = maar de=20 samenleving als geheel denkt niet altijd logisch na.

We zouden natuurlijk (zoals Anna voorstelt in het=20 stoelenvoorbeeld) een van de drie vergelijkingen niet kunnen = houden,=20 maar in dat geval worden de stoelen niet gelijkwaardig = behandeld.=20 Als extra eis aan volmaakte democratie zouden we willen = toevoegen=20 dat alle partijen of dingen waartussen gekozen moet worden, = gelijk=20 worden behandeld.

Soms zijn er gewoon maar twee mogelijkheden om uit te = kiezen. In=20 dat geval kan het probleem dat =93de samenleving als = geheel=94=20 mogelijkheid A boven B verkiest, mogelijkheid B boven C maar = mogelijkheid C boven A niet optreden en kunnen we met een = gerust=20 hart =91twee aan twee vergelijken=92, bijvoorbeeld in een = referendum.=20 Ook hierbij kunnen vreemde verschijnselen optreden zoals uit = de=20 volgende twee voorbeelden blijkt.

Verkiezingen versus referenda
In = de Gemeente=20 Zwaanhoven zijn twee partijen: Zwaanhoven Belangen en = Leefbaar=20 Zwaanhoven. Bovendien draaien de verkiezingen geheel om drie = thema's: de bouw van een parkeergarage in een park, een = fusie met de=20 nabijgelegen gemeente Oest en de privatisering van het = Zwaanhovens=20 Vervoerbedrijf. Op al deze drie punten zijn de partijen het = met=20 elkaar oneens. Als de een voor is, is de ander tegen en = andersom.

De meningen van de kiezers zijn verdeeld. Ze zijn in te = delen in=20 4 groepen Hieronder staat aangegeven hoe deze groepen het = per=20 vraagstuk met de verschillende partijen eens zijn en hoe = groot ze=20 zijn. We gaan er vanuit dat alle kiezers de drie thema=92s = even=20 belangrijk vinden. De kiezers uit groep A zijn het op twee = punten=20 (de parkeergarage en de fusie met Oest) eens met Leefbaar = Zwaanhoven=20 en slechts op een punt met Zwaanhoven Belangen. Deze kiezers = stemmen=20 dus op Leefbaar Zwaanhoven. Op dezelfde manier kiezen de = andere=20 kiezers voor de partij met wie ze het op 2 van de 3 punten = eens=20 zijn.

3D""=20



Bij de verkiezingen wint Leefbaar Zwaanhoven = met 60%=20 van de stemmen tegen 40% voor Zwaanhoven Belangen kan dus in = op alle=20 drie de punten zijn zin doordrukken. Maar als er referenda = gehouden=20 zouden worden zou juist Zwaanhoven Belangen op alle drie de = punten=20 zijn zin krijgen (steeds met een meerderheid van 60% tegen = 40%) en=20 zou Leefbaar Zwaanhoven achter het net = vissen.

O was ik maar bij moeder = thuisgebleven.
In=20 de nabijgelegen gemeente Oest is de situatie een beetje = anders dan=20 in Zwaanhoven. Hier draait alles om de aanleg van een snelle = metroverbinding tussen Oest-Oost en Oest-West. Het = gemeentebestuur=20 is voor, evenals 60% van de inwoners van Oest (om precies te = zijn de=20 inwoners Oest-West en Oest-Oost, ieder goed voor 30% van de=20 bevolking). De inwoners van het oude centrum van Oest zijn = tegen,=20 omdat ze geen zin hebben in alle graafwerkzaamheden onder = hun=20 antieke dorpskern. Deze mensen vormen de overige 40% van de = Oestse=20 bevolking. (Oest-Noord bestaat geheel uit industrieterrein = en=20 Oest-Zuid bestaat om onduidelijke redenen niet.)

Het gemeentebestuur van Oest besluit een referendum over = de=20 kwestie uitteschrijven. =91Zo hebben de burgers niet het = gevoel dat ze=20 niks te zeggen hebben=92, redeneert het gemeentebestuur, = =91en bovendien=20 hebben we niets te verliezen met 60% voorstanders.=92 Als de = opkomst=20 bij het referendum 50% of meer is, is het referendum bindend = en zal=20 de gemeente zich houden aan de uitslag. Als de opkomst lager = is dan=20 50% is het referendum ongeldig en mag het gemeentebestuur = doen wat=20 ze wil, in dit geval dus de metrolijn aanleggen.

Hier zien we een mooi voorbeeld van de =91Paradox van de=20 thuisblijver=92. Door een lokale regenbui regent het in = Oest-Oost en=20 in het oostelijke deel van Oest-centrum. De 30% voorstemmers = in=20 Oest-Oost denken =91ja, zeg ik ga niet helemaal door de = regen lopen=20 voor een metrolijn die er pas in 2030 ligt=92 en blijven = thuis. De 40%=20 tegenstemmers in Oest-Centrum denken: =91Een verkoudheid = duurt=20 misschien een week, maar die graafwerkzaamheden zullen als = we niet=20 uitkijken 25 jaar in beslag nemen=92 en trotseren dus weer = en wind om=20 tegen te stemmen.

De vraag is nu wat de inwoners van het zonovergoten = Oest-West=20 moeten doen. Als rechtgeaarde democraten snellen ze = natuurlijk=20 massaal naar de stembus om hun voorstem te laten klinken, = maar=20 eigenlijk is dat niet in hun belang. Als de 30% inwoners van = Oest-West v=F3=F3r stemmen is het referendum rechtsgeldig = met een=20 opkomst van 70%, maar de tegenstanders winnen met 4/7 = oftewel 57%=20 van de stemmen. Als de Oest-Westenaren daarentegen massaal = waren=20 thuisgebleven hadden ze hun zin gekregen: met een opkomst = van=20 slechts 40% hoeft het gemeentebestuur geen rekening met de = uitslag=20 van het referendum te houden, ook al is in dat geval 100% = van de=20 stemmen tegen. De voorstemmers moeten dus eigenlijk zorgen = dat ze =F3f=20 allemaal wel =F3f allemaal niet gaan stemmen Maar ja, zie = maar eens=20 met zoveel mensen tot een handige afspraak te=20 komen....

3D""=20 De Tweede=20 Kamer: 150 vertegenwoordigers van de Nederlandse kiezer. = Moeten we=20 natuurlijk wel zorgen dat ze zo eerlijk mogelijk worden=20 gekozen...



Eisen aan je kiessysteem en de stelling = van=20 Arrow
Na deze voorbeelden van hoe een kiessysteem tot = een=20 vreemde of zelfs =91oneerlijke=92 uitslag kan leiden, kunnen = we bekijken=20 welke eisen we aan ons kiessysteem zouden willen opleggen, = om hem=20 maar zo eerlijk mogelijk te maken. We zijn in de tekst tot = nu toe 5=20 eisen tegengekomen waaraan een volmaakte democratie zou = moeten=20 voldoen:

Neutraliteit: alle partijen worden gelijk = behandeld (Dit=20 gebeurt niet in het voorbeeld van de stoelen, maar ook niet = bij het=20 referendum in Oest)

Meerderheidsprincipe: als de meerderheid partij A = boven=20 partij B verkiest moet A ook boven B eindigen (Dit gebeurt = niet in=20 het voorbeeld van de kleuren)

Monotonie: als iemand van gedachten verandert en = partij A=20 hoger waardeert dan eerst, dan moet A er ook in de uitslag = op=20 vooruit gaan. (Dit gebeurt niet bij het referendum in Oest: = als een=20 voorstander besluit t=F3ch maar te gaan stemmen, kan hij = daarmee juist=20 net voorkomen dat de metro er komt)

Onafhankelijkheid van Irrelevante Alternatieven: = toevoegen=20 en weer verwijderen van partijen heeft geen invloed (Dit = ging mis in=20 het voorbeeld van de schoolklas)

Transitiviteit: als A eindigt boven B en B boven C = dan=20 eindigt A automatisch ook boven C. (Dit is niet haalbaar bij = twee=20 aan twee vergelijken zoals blijkt uit het voorbeeld van de = stoelen.

De uitdaging is nu: kunnen we een kiessysteem bedenken = dat aan=20 alle vijf deze eisen voldoet.

Het antwoord is nee. Dit neemt echter niet weg dat je het = niet=20 kunt proberen. Er zijn een hoop verschillende kiessystemen = mogelijk=20 die elk hun eigen paradoxen hebben. Je zult moeten kiezen = welke=20 eisen je belangrijker vindt dan andere. Je zou bijvoorbeeld = kunnen=20 zeggen dat het meerderheidsprincipe en monotonie gewoon te = veel=20 gevraagd zijn. Laten we monotonie gewoon afschaffen als eis = en het=20 meerderheidsprincipe vervangen door (onbetwistbaar = redelijke)=20 =91Pareto eis=92:

Als IEDEREEN partij A boven B verkiest moet A ook boven B = eindigen in de verkiezingen.=20

Is er een systeem dat aan deze eis voldoet, transitief = is,=20 onafhankelijk van irrelevante alternatieven en alle partijen = gelijk=20 behandelt? Het antwoord is ja, en dit systeem heet = dictatuur. Met=20 dictatuur wordt in dit geval bedoeld dat alle kiezers een = mening=20 hebben over hoe de uitslag eruit zou moeten zien, maar deze = uitslag=20 per definitie altijd gelijk is aan de mening van =E9=E9n van = hen, de=20 dictator. Het is duidelijk dat dit systeem aan alle vier = deze eisen=20 voldoet.

Dit wil niet zeggen dat dictatuur wiskundig gezien het = meest=20 democratische systeem is. We kunnen namelijk gewoon een eis = aan onze=20 volmaakte democratie toevoegen, bijvoorbeeld dat alle = stemmen even=20 zwaar moeten meetellen. Ga na dat in een districtensysteem = zoals in=20 Engeland ook niet aan deze eis voldoet.

Wat nu in 1950 door Keneth Arrow bewezen is, is het = volgende: er=20 is geen systeem dat:
* niet dictatoriaal is (niet = iedereen hoeft=20 gelijk behandeld te worden, maar je moet het niet te bont=20 maken)
* neutraal is (alle partijen worden gelijk = behandeld)
*=20 aan de Pareto eis voldoet (als iedereen A beter vindt dan B = moet A=20 ook beter uit de bus komen dan B)
* transitief is (Er is = wel een=20 uitslag).

Dit klinkt nogal schokkend. De (nog niet eens zo heel) = Volmaakte=20 Democratie is wiskundig bewezen (niet te verwarren met = statistisch=20 bewezen) onhaalbaar.

Voor wie=20 meer wil lezen
Meer leuke voorbeelden van=20 verkiezingsparadoxen (en een hoop andere leuke wiskunde) is = te lezen=20 in: De wraak van Archimedes (Archimedes=92 revenge) van Paul = Hoffman.
Nog veel en veel meer informatie over=20 verkiezingsparadoxen met wiskundige achtergronden en = voorbeelden uit=20 de =91echte=92 politiek is te vinden in Verkiezingen, een = web van=20 paradoxen door H. de Swart, A van Deemen, E van der Hout en = P. Klop,=20 deel 8 uit de Zebrareeks van Epsilon Uitgaven.

Verkiezingsparadoxen bij echte Nederlandse verkiezingen = zijn te=20 vinden in het artikel Emperical evidence of paradoxes of = voting in=20 Dutch elections, door A. van Deemen en N. Vergunst, in = Public=20 Choice, 97, blz 475-490, 1998.

Keneth Arrow beschreef zijn stelling in zijn boek: Social = Choice=20 and Individual Values uit 1951.

Een=20 verkorte versie van dit artikel met vragen en = opdrachten.=20

Meer=20 weten:
EU = Grondwet=20 oneerlijk? (Kennislink artikel)
Uitgebreide=20 Engelstalige website over kiessystemen en=20 verkiezingsparadoxen
Three brief=20 proofs of Arrow=92s theorem, door John = GeanakoplosUitslag=20 Britse verkiezingen 2005
Ui= tslag=20 Amerikaanse presidentsverkiezingen 2000


Dossiers
Terug = naar het=20 dossier Verkiezingen
Naar = het dossier=20 Kiezersonderzoek



PARTNERS= =20 BOEKENPLEIN=20
Kennislink wordt in opdracht van het = ministerie=20 van Onderwijs, Cultuur en Wetenschap uitgevoerd door Stichting Nationaal Centrum = voor=20 Wetenschap en Technologie.
copyright 1999-2005 = Kennislink colofon /disclaimer=20 =
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SCROLLBAR-3DLIGHT-COLOR: white; SCROLLBAR-ARROW-COLOR: white; = SCROLLBAR-TRACK-COLOR: #e4e4e4; SCROLLBAR-DARKSHADOW-COLOR: white; = SCROLLBAR-BASE-COLOR: #960604; BACKGROUND-COLOR: #e4e4e4 } BODY { FONT-FAMILY: Verdana, Arial, Helvetica, Geneva, sans-serif } TD { FONT-FAMILY: Verdana, Arial, Helvetica, Geneva, sans-serif } TH { FONT-FAMILY: Verdana, Arial, Helvetica, Geneva, sans-serif } DIV { FONT-FAMILY: Verdana, Arial, Helvetica, Geneva, sans-serif } SPAN { FONT-FAMILY: Verdana, Arial, Helvetica, Geneva, sans-serif } TEXTAREA { BORDER-RIGHT: #bd0300 1pt solid; PADDING-RIGHT: 1px; BORDER-TOP: = #bd0300 1pt solid; PADDING-LEFT: 1px; FONT-SIZE: 9px; PADDING-BOTTOM: = 1px; BORDER-LEFT: #bd0300 1pt solid; PADDING-TOP: 1px; BORDER-BOTTOM: = #bd0300 1pt solid; FONT-FAMILY: Verdana; BACKGROUND-COLOR: white } SELECT { BORDER-RIGHT: #bd0300 1pt solid; PADDING-RIGHT: 1px; BORDER-TOP: = #bd0300 1pt solid; PADDING-LEFT: 1px; FONT-SIZE: 10px; PADDING-BOTTOM: = 1px; BORDER-LEFT: #bd0300 1pt solid; PADDING-TOP: 1px; BORDER-BOTTOM: = #bd0300 1pt solid; FONT-FAMILY: Verdana; BACKGROUND-COLOR: white } INPUT { BORDER-RIGHT: #bd0300 1px solid; BORDER-TOP: #bd0300 1px solid; = FONT-SIZE: 9px; MARGIN: 3px; BORDER-LEFT: #bd0300 1px solid; COLOR: = black; BORDER-BOTTOM: #bd0300 1px solid; FONT-FAMILY: Verdana, Arial; = BACKGROUND-COLOR: white } INPUT.VINK { FONT-SIZE: 9px; BORDER-LEFT-COLOR: #bd0300; BORDER-BOTTOM-COLOR: = #bd0300; COLOR: black; BORDER-TOP-STYLE: none; BORDER-TOP-COLOR: = #bd0300; FONT-FAMILY: Verdana, Arial; BORDER-RIGHT-STYLE: none; = BORDER-LEFT-STYLE: none; BORDER-RIGHT-COLOR: #bd0300; = BORDER-BOTTOM-STYLE: none; outline-style: solid; outline-width: thin } A { COLOR: #960604; TEXT-DECORATION: none } A:hover { COLOR: #960604; TEXT-DECORATION: underline } .tiny { FONT-SIZE: 9px; COLOR: black } .small { FONT-SIZE: 10px; COLOR: black } .normal { FONT-SIZE: 12px; COLOR: black } .main { PADDING-RIGHT: 0px; PADDING-LEFT: 0px; PADDING-BOTTOM: 0px; MARGIN: = 0px; PADDING-TOP: 0px; BACKGROUND-COLOR: #063c6e } .tiny_txt { MARGIN-TOP: 5px; FONT-SIZE: 9px; MARGIN-BOTTOM: 5px; COLOR: black; = LINE-HEIGHT: 14px; HEIGHT: 45px } .linkjes { BACKGROUND-POSITION: left center; PADDING-LEFT: 20px; FONT-SIZE: 10px; = FLOAT: left; BACKGROUND-IMAGE: = url(/upload/556_1201_1116845413339-pijltje_rechts.gif); = BACKGROUND-REPEAT: no-repeat; HEIGHT: 14px } .linkjes A { FONT-SIZE: 10px; COLOR: #960604; LINE-HEIGHT: 13px; TEXT-DECORATION: = none } .linkjes A:hover { FONT-SIZE: 10px; COLOR: #960604; LINE-HEIGHT: 13px; TEXT-DECORATION: = underline } .item_center_text { FONT-SIZE: 11px; LINE-HEIGHT: 16px } .item_center_title { FONT-WEIGHT: bold; FONT-SIZE: 12px; LINE-HEIGHT: 16px } .item_center_title A { FONT-WEIGHT: bold; FONT-SIZE: 12px; COLOR: black; LINE-HEIGHT: 16px; = TEXT-DECORATION: none } .item_center_title A:hover { FONT-WEIGHT: bold; FONT-SIZE: 12px; COLOR: #960604; LINE-HEIGHT: 16px; = TEXT-DECORATION: underline } .item_center_title_date { FONT-SIZE: 9px; COLOR: #960604; LINE-HEIGHT: 16px } .lees_meer_extra { PADDING-RIGHT: 20px; BACKGROUND-POSITION: right center; FONT-SIZE: = 10px; FLOAT: right; BACKGROUND-IMAGE: = url(/upload/556_1201_1116845413339-pijltje_rechts.gif); MARGIN-RIGHT: = 5px; BACKGROUND-REPEAT: no-repeat; HEIGHT: 20px } .lees_meer_extra A { FONT-WEIGHT: bold; FONT-SIZE: 10px; COLOR: #960604; TEXT-DECORATION: = none } .lees_meer_extra A:hover { FONT-WEIGHT: bold; FONT-SIZE: 10px; COLOR: #bd0300; TEXT-DECORATION: = underline } .lees_meer { PADDING-RIGHT: 20px; BACKGROUND-POSITION: right center; FONT-SIZE: = 10px; FLOAT: right; BACKGROUND-IMAGE: = url(/upload/556_1201_1116845413339-pijltje_rechts.gif); MARGIN-RIGHT: = 5px; BACKGROUND-REPEAT: no-repeat; HEIGHT: 16px } .lees_meer A { FONT-WEIGHT: bold; FONT-SIZE: 10px; COLOR: #960604; TEXT-DECORATION: = none } .lees_meer A:hover { FONT-WEIGHT: bold; FONT-SIZE: 10px; COLOR: #bd0300; TEXT-DECORATION: = underline } .nieuwsflits { DISPLAY: block; FONT-WEIGHT: bold; FONT-SIZE: 11px; MARGIN-LEFT: 5px; = COLOR: #960604; LINE-HEIGHT: 13px } .nieuwsflits A { FONT-WEIGHT: bold; FONT-SIZE: 11px; COLOR: #960604; LINE-HEIGHT: 13px; = TEXT-DECORATION: none } .nieuwsflits A:hover { FONT-WEIGHT: bold; FONT-SIZE: 11px; COLOR: #bd0300; LINE-HEIGHT: 13px; = TEXT-DECORATION: underline } .archief { BACKGROUND-POSITION: left center; PADDING-LEFT: 20px; FONT-SIZE: 11px; = FLOAT: left; BACKGROUND-IMAGE: = url(/upload/556_1205_1116845413329-pijltje_onder.gif); = BACKGROUND-REPEAT: no-repeat; HEIGHT: 14px } .archief A { FONT-SIZE: 10px; COLOR: #960604; LINE-HEIGHT: 13px; TEXT-DECORATION: = none } .archief A:hover { FONT-SIZE: 10px; COLOR: #960604; LINE-HEIGHT: 13px; TEXT-DECORATION: = underline } .footer { BACKGROUND-POSITION: left bottom; BACKGROUND-IMAGE: = url(/upload/556_1206_1116845413259-footer_bg.jpg); BACKGROUND-REPEAT: = no-repeat; BACKGROUND-COLOR: black } .footer_big A { FONT-WEIGHT: bold; FONT-SIZE: 12px; MARGIN-LEFT: 5px; COLOR: white; = TEXT-DECORATION: none } .footer_big A:hover { FONT-WEIGHT: bold; FONT-SIZE: 12px; COLOR: white; BORDER-BOTTOM: white = 1px dotted } .footer_small { FONT-SIZE: 9px; COLOR: white; LINE-HEIGHT: 14px } .footer_small A { FONT-SIZE: 9px; COLOR: white; TEXT-DECORATION: none } .footer_small A:hover { FONT-SIZE: 9px; COLOR: white; BORDER-BOTTOM: white 1px dotted } .kop { FONT-WEIGHT: bold; FONT-SIZE: 11px; COLOR: white; MARGIN-RIGHT: 5px; = HEIGHT: 15px; TEXT-DECORATION: none } .kop A { FONT-WEIGHT: bold; FONT-SIZE: 11px; COLOR: white; MARGIN-RIGHT: 5px; = TEXT-DECORATION: none } .kop A:hover { FONT-WEIGHT: bold; FONT-SIZE: 11px; COLOR: white; BORDER-BOTTOM: white = 1px dotted } .artikel_date { MARGIN-TOP: 5px; FONT-SIZE: 9px; FLOAT: right; WIDTH: 100px; COLOR: = #960604; LINE-HEIGHT: 16px; HEIGHT: 50px; BACKGROUND-COLOR: #f5f5f5 } .artikel_title_vervolg { MARGIN-TOP: 5px; FONT-WEIGHT: bold; FONT-SIZE: 16px; FLOAT: left; = WIDTH: 400px; LINE-HEIGHT: 16px; HEIGHT: 50px } .artikel_title { MARGIN-TOP: 5px; FONT-WEIGHT: bold; FONT-SIZE: 16px; FLOAT: left; = WIDTH: 400px; LINE-HEIGHT: 16px; HEIGHT: 50px; BACKGROUND-COLOR: #f5f5f5 } #inleiding_container { CLEAR: both; PADDING-RIGHT: 5px; PADDING-LEFT: 5px; PADDING-BOTTOM: = 5px; WIDTH: auto; PADDING-TOP: 5px; BACKGROUND-COLOR: #f4d5d4 } .inleiding_title { FONT-WEIGHT: bold; FONT-SIZE: 9px; LINE-HEIGHT: 16px } .inleiding { FONT-SIZE: 9px; LINE-HEIGHT: 16px } .artikel_bold { FONT-WEIGHT: bold; FONT-SIZE: 12px; LINE-HEIGHT: 18px } .artikel { FONT-SIZE: 12px; LINE-HEIGHT: 18px } #pic_container { BORDER-TOP: #960604 1px solid; MARGIN-TOP: 5px; FLOAT: left; = MARGIN-BOTTOM: 5px; PADDING-BOTTOM: 5px; WIDTH: auto; PADDING-TOP: 5px; = BORDER-BOTTOM: #960604 1px solid; HEIGHT: auto } .pic_txt { FONT-SIZE: 9px } .metadata { FONT-WEIGHT: bold; FONT-SIZE: 9px; FLOAT: left; MARGIN-LEFT: 5px; = TEXT-TRANSFORM: uppercase; COLOR: #960604; TEXT-ALIGN: left } .metadata_content { FONT-SIZE: 9px; FLOAT: right; COLOR: black; MARGIN-RIGHT: 5px } .metadata_bg { BACKGROUND-POSITION: 0px top; BACKGROUND-IMAGE: = url(/upload/556_1209_1116845413010-bg_metadata.jpg); BACKGROUND-REPEAT: = no-repeat } .roodblok { PADDING-RIGHT: 3px; DISPLAY: block; PADDING-LEFT: 6px; FONT-WEIGHT: = bold; FONT-SIZE: 10px; PADDING-BOTTOM: 3px; TEXT-TRANSFORM: uppercase; = WIDTH: auto; PADDING-TOP: 3px; HEIGHT: auto; BACKGROUND-COLOR: #f4d5d4 } UL { MARGIN-TOP: 3px; MARGIN-BOTTOM: 5px } LI { FONT-SIZE: 10px; LIST-STYLE-IMAGE: = url(/upload/556_1213_1116845413317-list.gif) } .startlijst { PADDING-RIGHT: 2px; DISPLAY: block; PADDING-LEFT: 2px; FONT-SIZE: 10px; = PADDING-BOTTOM: 2px; WIDTH: auto; PADDING-TOP: 2px; HEIGHT: auto; = BACKGROUND-COLOR: #f4d5d4 } .startlijst A { PADDING-RIGHT: 2px; DISPLAY: block; PADDING-LEFT: 2px; FONT-SIZE: 10px; = PADDING-BOTTOM: 2px; WIDTH: auto; PADDING-TOP: 2px; HEIGHT: auto; = BACKGROUND-COLOR: #f4d5d4; TEXT-DECORATION: none } .startlijst A:hover { PADDING-RIGHT: 2px; DISPLAY: block; PADDING-LEFT: 2px; FONT-SIZE: 10px; = PADDING-BOTTOM: 2px; WIDTH: auto; PADDING-TOP: 2px; HEIGHT: auto; = BACKGROUND-COLOR: #f4d5d4; TEXT-DECORATION: underline } .inhoud_kop { MARGIN-TOP: 8px; PADDING-LEFT: 20px; FONT-WEIGHT: bold; FONT-SIZE: = 11px; FLOAT: left; BACKGROUND-IMAGE: = url(/upload/556_1201_1116845413339-pijltje_rechts.gif); MARGIN-LEFT: = 8px; BACKGROUND-REPEAT: no-repeat } SPAN.inhoud_kop { MIN-HEIGHT: 16px; HEIGHT: 16px } SPAN[class].inhoud_kop { HEIGHT: auto } .inhoud_kop A { FONT-WEIGHT: bold; FONT-SIZE: 11px; COLOR: #960604; LINE-HEIGHT: 13px; = TEXT-DECORATION: none } .inhoud_kop A:hover { FONT-WEIGHT: bold; FONT-SIZE: 11px; COLOR: #960604; LINE-HEIGHT: 13px; = TEXT-DECORATION: underline } .inhoud_date { MARGIN-TOP: 5px; FONT-SIZE: 9px; FLOAT: right; COLOR: #7b7b7b } .inhoud_txt { CLEAR: both; DISPLAY: block; FONT-SIZE: 10px; PADDING-BOTTOM: 4px; = MARGIN-LEFT: 8px; LINE-HEIGHT: 13px; PADDING-TOP: 4px } .inhoud_meta { CLEAR: both; FONT-WEIGHT: bold; FONT-SIZE: 9px; MARGIN-LEFT: 8px; = TEXT-TRANSFORM: uppercase; WIDTH: 80px } .inhoud_metadata { CLEAR: both; DISPLAY: block; PADDING-LEFT: 15px; FONT-SIZE: 9px; = BACKGROUND-IMAGE: url(/upload/556_1212_1116845413227-blokje.gif); WIDTH: = 290px; COLOR: #960604; BACKGROUND-REPEAT: no-repeat } .nav_start { FONT-WEIGHT: bold; FONT-SIZE: 10px; MARGIN-LEFT: 8px; COLOR: #960604; = HEIGHT: 16px } .nav_start A { FONT-WEIGHT: bold; FONT-SIZE: 10px; COLOR: #960604; TEXT-DECORATION: = none } .nav_start A:hover { FONT-WEIGHT: bold; FONT-SIZE: 10px; COLOR: #960604; TEXT-DECORATION: = underline } .nav { PADDING-LEFT: 18px; FONT-WEIGHT: bold; FONT-SIZE: 9px; = BACKGROUND-IMAGE: = url(/upload/556_1201_1116845413339-pijltje_rechts.gif); MARGIN-LEFT: = 8px; BACKGROUND-REPEAT: no-repeat; HEIGHT: 16px } .nav A { FONT-WEIGHT: bold; FONT-SIZE: 9px; COLOR: #960604; TEXT-DECORATION: = none } .nav A:hover { FONT-WEIGHT: bold; FONT-SIZE: 9px; COLOR: #960604; TEXT-DECORATION: = underline } .wittxt { FONT-SIZE: 9px; COLOR: white } .wittxt A { FONT-SIZE: 9px; COLOR: white; TEXT-DECORATION: underline } .wittxt A:hover { FONT-SIZE: 9px; COLOR: #f2d0d0; TEXT-DECORATION: underline } #align_right { FLOAT: right; HEIGHT: 9px } #align_left { FLOAT: left; HEIGHT: 9px } .zoek_meta { CLEAR: both; FONT-WEIGHT: bold; FONT-SIZE: 9px; FLOAT: left; = MARGIN-LEFT: 8px; TEXT-TRANSFORM: uppercase; WIDTH: 80px } .zoek_metadata { CLEAR: both; DISPLAY: block; PADDING-LEFT: 15px; FONT-SIZE: 9px; FLOAT: = right; BACKGROUND-IMAGE: url(/upload/556_1212_1116845413227-blokje.gif); = WIDTH: 400px; COLOR: #960604; BACKGROUND-REPEAT: no-repeat } ------=_NextPart_000_0000_01C7202F.940D2670 Content-Type: application/octet-stream Content-Transfer-Encoding: 7bit Content-Location: http://www.kennislink.nl/web/show?id=100538 function openPage(pageId, qs) { loc = '/web/show?id='+pageId + qs; window.location = loc; } function openURL(url) { window.open(url); } // (C) 2000 www.CodeLifter.com // http://www.codelifter.com // Free for all users, but leave in this header // NS4-6,IE4-6 // Fade effect only in IE; degrades gracefully // ======================================= // set the following variables // ======================================= // Set slideShowSpeed (milliseconds) var slideShowSpeed = 5000 // Duration of crossfade (seconds) var crossFadeDuration = 10 // Specify the image files var Pic = new Array() // don't touch this // to add more images, just continue // the pattern, adding to the array below Pic[0] = '/upload/header_bg_aardweten.jpg' Pic[1] = '/upload/header_bg_bio.jpg' Pic[2] = '/upload/header_bg_gezondheid.jpg' Pic[3] = '/upload/header_bg_meteo.jpg' Pic[4] = '/upload/header_bg_natuurkunde.jpg' Pic[5] = '/upload/header_bg_ruimte.jpg' Pic[6] = '/upload/header_bg_scheikunde.jpg' Pic[7] = '/upload/header_bg_sterrenkunde.jpg' Pic[8] = '/upload/header_bg_techniek.jpg' Pic[9] = '/upload/header_bg_wiskunde.jpg' // ======================================= // do not edit anything below this line // ======================================= var t var j = 0 var p = Pic.length var preLoad = new Array() for (i = 0; i < p; i++){ preLoad[i] = new Image() preLoad[i].src = Pic[i] } function runSlideShow(){ if (document.all){ document.images.SlideShow.style.filter="blendTrans(duration=3 n)" document.images.SlideShow.filters.blendTrans.Apply() } document.images.SlideShow.src = preLoad[j].src if (document.all){ document.images.SlideShow.filters.blendTrans.Play() } j = j + 1 if (j > (p-1)) j=0 t = setTimeout('runSlideShow()', slideShowSpeed) } function newImage(arg) { if (document.images) { rslt = new Image(); rslt.src = arg; return rslt; } } function changeImagesArray(array) { if (preloadFlag == true) { var d = document; var img; for (var i=0; i