Graduate AI Lecture 20: Scial Chice I Teachers: Martial Hebert Ariel Prcaccia (this time)
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
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
Cmputatinal scial chice Scial chice AI Multiagent systems: reducing cmmunicatin Human cmputatin: aggregating peples pinins 4
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 1 2 3 a c b b a c c b a 5
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
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
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
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
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
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
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
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? 1 2 3 a c b b a c c b a 13
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
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
Manipulatin Using Brda cunt Tp prfile: b wins Bttm prfile: a wins By changing his vte, vter 3 achieves a better utcme! 1 2 3 b b a a a b c c c d d d 1 2 3 b b a a a c c c d d d b 16
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
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
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
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