Graduate AI Lecture 20: Social Choice I. Teachers: Martial Hebert Ariel Procaccia (this time)

Transcription

1 Graduate AI Lecture 20: Scial Chice I Teachers: Martial Hebert Ariel Prcaccia (this time)

2 Scial chice thery A mathematical thery that deals with aggregatin f individual preferences Origins in ancient Greece Frmal fundatins: 18 th Century (Cndrcet and Brda) 19 th Century: Charles Ddgsn 20 th Century: Nbel prizes t Kenneth Arrw and Amartya Sen 2

3 Cmputatinal scial chice Tw-way interactin with AI AI scial chice Algrithms and cmputatinal cmplexity Machine learning in scial chice Knwledge representatin Markv decisin prcesses 3

4 Cmputatinal scial chice Scial chice AI Multiagent systems: reducing cmmunicatin Human cmputatin: aggregating peples pinins 4

5 The vting mdel Set f vters N={1,...,n} Set f alternatives A, A =m Each vter has a ranking ver the candidates x > i y means that vter i prefers x t y Preference prfile = cllectin f all vters rankings a c b b a c c b a 5

6 Vting rules Vting rule = functin frm preference prfiles t alternatives that specifies the winner f the electin Plurality Each vter awards ne pint t tp alternative Alternative with mst pints wins Used in almst all plitical electins 6

7 Mre vting rules Brda cunt Each vter awards m-k pints t alternative ranked k th Alternative with mst pints wins Prpsed in the 18 th Century by the chevalier de Brda Used in the natinal assembly f Slvenia Similar t rule used in the Eurvisin sng cntest Lrdi, Eurvisin 2006 winners 7

8 Mre vting rules Vet Each vter vetes his least preferred alternative Alternative with least vetes wins Psitinal scring rules Defined by a vectr (s 1,...,s m ) Each vter gives s k pints t k th psitin Plurality: (1,0,...,0); Brda: (m-1,m-2,...,0), Vet: (1,...,1,0) 8

9 Mre vting rules a beats b in a pairwise electin if the majrity f vters prefer a t b Plurality with runff First rund: tw alternatives with highest plurality scres survive Secnd rund: pairwise electin between these tw alternatives 9

10 Mre vting rules Single Transferable vte (STV) m-1 runds In each rund, alternative with least plurality vtes is eliminated Alternative left standing is the winner Used in Ireland, Malta, Australia, and New Zealand (and Cambridge, MA) 10

11 STV: example 2 vters 2 vters 1 vter a b c b a d c d b d c a 2 vters 2 vters 1 vter a b c b a b c c a 2 vters 2 vters 1 vter a b b b a a 2 vters 2 vters 1 vter b b b 11

12 Marquis de Cndrcet 18 th Century French Mathematician, philspher, plitical scientist One f the leaders f the French revlutin After the revlutin became a fugitive His cver was blwn and he died mysteriusly in prisn 12

13 Cndrcet winner Cndrcet winner = alternative that beats every ther alternative in pairwise electin Cndrcet paradx = Cndrcet winner may nt exist Cndrcet criterin = elect a Cndrcet winner if ne exists Des plurality satisfy criterin? Brda? a c b b a c c b a 13

14 Mre vting rules Cpeland Alternative s scre is #alternatives it beats in pairwise electins Why des Cpeland satisfy the Cndrcet criterin? Maximin Scre f x is min y {i N: x > i y} Why des Maximin satisfy the Cndrcet criterin? 14

15 Awesme example Plurality: a Brda: b Cndrcet winner: c STV: d Plurality with runff: e 33 vters 16 vters 3 vters 8 vters 18 vters 22 vters a b c c d e b d d e e c c c b b c b d e a d b d e a e a a a 15

16 Manipulatin Using Brda cunt Tp prfile: b wins Bttm prfile: a wins By changing his vte, vter 3 achieves a better utcme! b b a a a b c c c d d d b b a a a c c c d d d b 16

17 Strategyprfness A vting rule is strategyprf (SP) if a vter can never benefit frm lying abut his preferences: <, i N, < i, f(<) i f(< i,< -i ) If there are tw candidates then plurality is SP 17

18 Gibbard-Satterthwaite A vting rule is dictatrial if there is a vter wh always gets his mst preferred alternative A vting rule is nt if any alternative can win Therem (Gibbard-Satterthwaite): If m 3 then any vting rule that is SP and nt is dictatrial In ther wrds, any vting rule that is nt and nndictatrial is manipulable 18

19 Prf f G-S Therem We prve the fllwing statement n the bard If m 3 and n=2 then any vting rule that is SP and nt is dictatrial The prf als appears in: L.-G. Svenssn. The prf f the Gibbard- Satterthwaite Therem revisited, Therem 1 (available frm curse website) 19

20 Lemmas A vting rule satisfies mntnicity if: f < = a, i N, x A, [x a x a] implies that f(< ) = a Lemma: Any SP vting rule is mntnic A vting rule satisfies Paret ptimality (PO) if: i N, x > i y f(<) y Lemma: Any SP and nt vting rule is PO 20