Choix Social Computationnel

Transcription

1 Choix Soil Computtionnel Niols Muet Université Pierre et Mrie Curie 05 Septemre 2011

2 Motivtion Quest for the est voting system Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 2 / 84 Figure: Referenum on Alterntive Vote (UK, 2011)

3 Motivtion Mnipulting the system : Gerrymnering Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 3 / 84 Figure: The slmner of Elrige Gerry (1812)

4 Motivtion Online voting systems Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 4 / 84 Figure: Choie of resturnt (mye open on Suny, to hek)

5 Motivtion Met-serh engines Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 5 / 84 Figure: Aggregting serh results

6 Pln u ours Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 6 / 84

7 Outline of the Tlk Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 7 / 84

8 Voting Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition 1. finite set of voters A = {1,..., n} ; 2. finite set of nites (lterntives) X ; 3. profile = preferene reltion (= liner orer) on X for eh voter P = (V 1,..., V n ) = ( 1,..., n ) V i (or i ) = vote expresse y voter i. 4. P n set of ll profiles. Inomplete Other Topis 8 / 84

9 Voting Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 8 / finite set of voters A = {1,..., n} ; 2. finite set of nites (lterntives) X ; 3. profile = preferene reltion (= liner orer) on X for eh voter P = (V 1,..., V n ) = ( 1,..., n ) V i (or i ) = vote expresse y voter i. 4. P n set of ll profiles. Voting rule F : P n X F (V 1,..., V n ) = soilly preferre (elete) nite Voting orresponene C : P n 2 X \ { } C (V 1,..., V n ) = set of soilly preferre nites. Soil welfre funtion H : P n P H (V 1,..., V n ) = soil preferene reltion ( P ) Note : Rules n e otine from orresponenes y tie-reking (usully y using preefine priority orer on nites).

10 Positionl soring rules Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete n voters, p nites fixe list of p integers s 1... s p voter i rnks nite x in position j sore i (x) = s j winner : nite mximizing s(x) = n i=1 sore i(x) Exmples : s 1 = 1, s 2 =... = s m = 0 plurlity ; s 1 = s 2 =... = s m 1 = 1, s m = 0 veto ; s 1 = m 1, s 2 = m 2,... s m = 0 Bor. Other Topis 9 / 84 2 voters 1 voter 1 voter plurlity winner Bor winner

11 Conoret winner Niols Muet 05 Septemre 2011 Bsis of Soil Choie N (x, y) = {i x i y} set of voters who prefer x to y. #N (x, y) numer of voters who prefer x to y. Conoret winner for P = 1,..., n : nite x suh tht y x, #N (x, y) > n 2 ( nite who ets ny other nite y mjority of votes). Mnipultion Communition Inomplete Mjority grph Other Topis 10 / 84 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 :

12 Conoret winner Niols Muet 05 Septemre 2011 Bsis of Soil Choie N (x, y) = {i x i y} set of voters who prefer x to y. #N (x, y) numer of voters who prefer x to y. Conoret winner for P = 1,..., n : nite x suh tht y x, #N (x, y) > n 2 ( nite who ets ny other nite y mjority of votes). Mnipultion Communition Inomplete Mjority grph Other Topis 10 / 84 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : No Conoret winner

13 Conoret winner Niols Muet 05 Septemre 2011 Bsis of Soil Choie N (x, y) = {i x i y} set of voters who prefer x to y. #N (x, y) numer of voters who prefer x to y. Conoret winner for P = 1,..., n : nite x suh tht y x, #N (x, y) > n 2 ( nite who ets ny other nite y mjority of votes). Mnipultion Communition Inomplete Mjority grph Other Topis 11 / 84 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 :

14 Conoret winner Niols Muet 05 Septemre 2011 Bsis of Soil Choie N (x, y) = {i x i y} set of voters who prefer x to y. #N (x, y) numer of voters who prefer x to y. Conoret winner for P = 1,..., n : nite x suh tht y x, #N (x, y) > n 2 ( nite who ets ny other nite y mjority of votes). Mnipultion Communition Inomplete Mjority grph Other Topis 11 / 84 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : 2 voters out of 3 : is the Conoret winner

15 Conoret-onsistent rules The Copeln rule Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 12 / 84 Consisteny with Conoret : the voting rule shoul elet the Conoret winner whenever there is one. Exmple : Copeln rule get 1 pt for eh pirwise win, voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : for tie, 0 otherwise Mjority grph

16 Conoret-onsistent rules The Copeln rule Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 12 / 84 Consisteny with Conoret : the voting rule shoul elet the Conoret winner whenever there is one. Exmple : Copeln rule get 1 pt for eh pirwise win, voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : for tie, 0 otherwise Mjority grph C() = 2 C() = 2 C() = 1 C() = 1

17 Conoret-onsistent rules The Simpson rule Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 13 / 84 Consisteny with Conoret : the voting rule shoul elet the Conoret winner whenever there is one. Exmple : Simpson rule pik the nite who minimizes the mx pirwise efet voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : (Weighte) Mjority grph

18 Conoret-onsistent rules The Simpson rule Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 13 / 84 Consisteny with Conoret : the voting rule shoul elet the Conoret winner whenever there is one. Exmple : Simpson rule pik the nite who minimizes the mx pirwise efet voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : 4 voters out of 5 : 3 voters out of 5 : (Weighte) Mjority grph S() = mx { 3 } S() = mx { 4 } S() = mx { 3, 3 } S() = mx { 4, 4 } 3

19 Sequentil Rules Simple trnsferle vote (STV) Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis if there exists nite rnke first y mjority of votes then wins else Repet let e the nite rnke first y the fewest voters ; eliminte from ll llots {votes for trnsferre to the next est remining nite} ; Until there exists nite rnke first y mjority of votes / 84

20 Sequentil Rules Simple trnsferle vote (STV) Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion if there exists nite rnke first y mjority of votes then wins else Repet let e the nite rnke first y the fewest voters ; eliminte from ll llots {votes for trnsferre to the next est remining nite} ; Until there exists nite rnke first y mjority of votes Communition Inomplete Other Topis / 84

21 Sequentil Rules Simple trnsferle vote (STV) Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion if there exists nite rnke first y mjority of votes then wins else Repet let e the nite rnke first y the fewest voters ; eliminte from ll llots {votes for trnsferre to the next est remining nite} ; Until there exists nite rnke first y mjority of votes Communition Inomplete Other Topis Winner : 14 / 84 with only 3 nites, STV oinies with plurlity with runoff. system use in Austrli, Ireln

22 Approvl Voting Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Here the input provie y the voters is ifferent. profile = suset of nites A i X for eh voter P = (A 1,..., A n ) S P (x) = numer of voters i suh tht x A i. winner : nite mximizing S P. Other Topis 15 / 84

23 Outline of the Tlk Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 16 / 84

24 A rief history (outesy of Jérôme Lng) Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1. en of 18th entury : Conoret n Bor Mrie Jen Antoine Niols e Critt (Mrquis e Conoret). Essi sur l pplition e l nlyse à l proilité es éisions renues à l plurlité es voix. Pris, Imprimerie Royle, : Arrow s theorem, irth of moern soil hoie theory Kenneth J. Arrow. Soil Choie n Iniviul Vlues, Yle University Press, from the lte 80 s on : omputer sientists (n espeilly AI reserhers) jump on or Brtholi, Tovey, Trik. Voting Shemes for whih It Cn Be Diffiult to Tell Who Won the Eletion. Soil Choie n Welfre, / 84

25 Proxes Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion The stuy of voting rules unveile mny proxes... Exmple (Sri, 1995) Communition Inomplete Other Topis Veto, Conoret n Bor gree on the rnking But plurlity inste sys Other results show striking istintions etween rules, eg : No positionl rule is Conoret-onsistent (Young) 18 / 84

26 An Axiomti Approh Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Most results in (lssil) soil hoie seek hrteriztions of voting rules in terms of xioms they fulfill. There re other wys to rtionlize the use of ertin voting rules : mximum likelihoo pproh (there is orret outome, n the votes re noisy/istore pereptions of this outome, for given moel of noise) istne-se rtionliztion (there is onsensus notion, n the winner is the winning nite in the losest onsensul profile, for given notion of istne) 19 / 84 Elkin, Fliszewski & Slinko. Distne Rtionliztion of., Conitzer & Snholm. Common s Mximum Likelihoo Estimtors. UAI, 2005.

27 An Axiomti Approh [Arrow] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Sometimes impossiility results stte tht no voting rule n stisfy given set of xioms. unnimity if x i y for every voter i, then x P y inepenene of irrelevnt lterntive the soil preferene mong x n y only epens on their reltive reltive rnking y every iniviul. N P (x, y) = N P (x, y) then x P y x P y ittorship voter i is ittor if the funtion mps ny profile to his vote, i.e. H : P n V i Theorem (Arrow, 1951) Any soil welfre funtion for 3 or more nites stisfying unnimity n inepenene must e ittorship. 20 / 84

28 An Axiomti Approh [My] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis An exmple of possiility result... nonymity oes not epen on the ientity of voters, i.e. F (V 1,..., V n ) = F (π(v 1 ),..., π(v n )) neutrlity oes not epen on the ientity of nites positive responsiveness if nite x is mong the winners, then it shoul eome the unique winner when some voters moify their preferene n put x t higher rnk (without moifying the rest). Theorem (My, 1952) A voting orresponene for extly 2 nites stisfies nonymity, neutrlity, n positive responsiveness iff it is the plurlity rule (simple mjority). 21 / 84

29 An Axiomti Approh [Gir-Stherwite] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Another importnt notion is tht of strtegy-proofness. A voting rule is strtegy-proof if no voter is etter-off (i.e. prefers the new otine winner) misrepresenting his vote (in ny profile). surjetivity no nite is isre (for ny nite x, there is profile P suh tht F (P) = x) Theorem (Gir-Stherwite, 1952) Any voting rule for 3 or more nites tht is surjetive n strtegy-proof must e ittorship. 22 / 84

30 Niols Muet Plurlity with runoff fils to meet positive responsiveness Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis st roun : eliminte 2n roun : elete (11/6) 23 / 84

31 Niols Muet Plurlity with runoff fils to meet positive responsiveness Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis st roun : eliminte 2n roun : elete (9/8) 24 / 84

32 Erly motivtions for omputtionl soil hoie Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete In ll these results, no onsiertion for omputtionl issues re some rules iffiult to ompute? how out the iffiulty of mnipulting the eletion? how o these rules ter in istriute environment? wht if the numer of nites is huge? Other Topis 25 / 84

33 Outline of the Tlk Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 26 / 84

34 voting rules Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 27 / 84 Most voting rules n e ompute in polynomil time Exmples : positionl soring rules, pprovl : O(np) Copeln, Simpson, STV : O(np 2 ) But some voting rules re NP-hr. Referene ppers Fliszewski, Hemspnr, Hemspnr & Rothe. A riher Unerstning of the of Eletion Systems. CoRR Brtholi, Tovey, & Trik. Voting Shemes for whih It Cn Be Diffiult to Tell Who Won the Eletion. Soil Choie n Welfre, Hury. Mein liner orers : heuristis n rnh n oun lgorithm. EJOR

35 Hr rules Kemeny Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Looking for rnkings tht re s lose s possile to the preferene profile n hooses the top-rnke nites in these rnkings. Kemeny istne : K (V, V ) = numer of (x, y) X 2 on whih V n V isgree K (V, V 1,..., V n ) = K (V, V i ) i=1,...,n Kemeny onsensus = liner orer P suh tht K ( P, V 1,..., V n ) minimum Kemeny winner = nite rnke first in Kemeny onsensus 28 / 84

36 Hr rules Kemeny Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 29 / 84 A hrteriztion of Kemeny With eh profile P ssoite the pirwise omprison mtrix (rell #N P (x, y) is the numer of voters who prefer x to y in P). Now given rnking R : K (R) = x R y #N (x, y) If x R y then this orrespons to #N (x, y) greements (n #N (y, x) isgreements) P is Kemeny onsensus iff K (P ) is mximum. 4 voters 3 voters 2 voters Fin the Kemeny winner(s).

37 Hr rules Kemeny Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Kemeny sores 4 voters 3 voters 2 voters N / 84 Kemeny onsensus : ; Kemeny winner : this nive pproh yiels O(p!p 2 n)

38 Hr rules Kemeny Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition erly results : Kemeny is NP-hr (Orlin, 81 ; Brtholi et l., 89 ; Hury, 89) eiing whether nite is Kemeny winner is not even in NP, ut higher up mny works on pproximtion Inomplete Other Topis 31 / 84

39 Hr rules Kemeny Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Tehnique of Lol Kemeniztion : 1. generte n initil rnking R [w.l.o.g., R = x 1... x m ] ; 2. for k := 2 to m o 3. for j := k 1 ownto 1 o 4. if x j +1 ets x j mjoritywise 5. then swp x j n x j +1 in R. 6. return R. omputle in polynomil time (provie the initil rnking is omputle in polynomil time) Dwork, Kumr, Nor & Sivkumr. Rnk ggregtion methos for the We. WWW / 84

40 Hr rules Kemeny Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 33 / 84 Use in met-serh engines (in tht se, rnkings re likely to e prtil, euse of limite size) The result my e ritrry fr from optiml The rnking is lolly optiml, ut is the onsensus rnking. the resulting rnking R stisfies property of generlize Conoret-onsisteny (whih turns out to e very pproprite for the prolem t hn, so omin-speifi riterion re lso to onsier)

41 Hr rules Dogson / Young Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Other exmples of rules iffiult to ompute : Dogson (= Lewis Crrol) rule for eh nite, ompute D() the numer of jent swps require to turn it into Conoret winner. Pik the nite minimizing D(). Deiing whether esignte nite x is Dogson winner is NP-hr, not in NP, ut higher up in the hierrhy. So even verifying is not esy. Communition Inomplete Other Topis 34 / 84

42 Hr rules Dogson / Young Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 34 / 84 Other exmples of rules iffiult to ompute : Dogson (= Lewis Crrol) rule for eh nite, ompute D() the numer of jent swps require to turn it into Conoret winner. Pik the nite minimizing D(). Deiing whether esignte nite x is Dogson winner is NP-hr, not in NP, ut higher up in the hierrhy. So even verifying is not esy. Young rule for eh nite, ompute Y () the smllest numer of voters tht we nee to remove to turn it into Conoret winner. Pik the nite minimizing Y (). Deiing whether esignte nite x is Young winner is NP-hr, not in NP, ut higher up in the hierrhy. Hemspnr, Hemspnr, & Rothe. Ext Anlysis of Dogson Eletions. J. of ACM, Rothe, Spkowski, & Vogel. Ext of the Winner Prolem for Young Eletions. Theory Comput. Syst

43 Outline of the Tlk Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 35 / 84

44 The mny fets of mnipultion Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Rememer Gerrymnering. Prouing utomtilly fir reistriting ws n erly motivtion to onsier lgorithmi issues in voting. (Grfinkel & Nemhuser, 70) : erly lgorithms. (Altmn, 1997) : fir reistriting is NP-hr. Survey pper Ri, Sozzri & Simeone. Politil istriting : from lssil moels to reent pprohes. 4OR, Inomplete Other Topis 36 / 84

45 The mny fets of mnipultion Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 36 / 84 Rememer Gerrymnering. Prouing utomtilly fir reistriting ws n erly motivtion to onsier lgorithmi issues in voting. (Grfinkel & Nemhuser, 70) : erly lgorithms. (Altmn, 1997) : fir reistriting is NP-hr. Survey pper Ri, Sozzri & Simeone. Politil istriting : from lssil moels to reent pprohes. 4OR, In our ontext, no istrits. But mny ifferent vrints of the prolem though : Who wnts to mnipulte? ( single voter, group/olition of voters, the hir of the eletion) Wht kin of mnipultion is llowe? (moifying only his own vote, uying to get the others to moify their votes)

46 Computtionl Brriers to Mnipultion Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Rell the Gir-Sttherwite Theorem... If mnipultion is omputtionlly prohiitive then this my e goo news. However lwys er in min tht this is only worst-se onept (so mnipultion my e esy on most instnes...) Inomplete Other Topis Survey pper Fliszewski & Proi. AI s Wr on Mnipultion : Are we Winning?. AI Mgzine, / 84

47 of mnipultion Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Two types of mnipultion n e istinguishe : onstrutive mnipultion existene : Given voting rule r, set of p nites X, nite x X, n the votes of voters 1,..., k < n Question is there wy for voters k + 1,..., n to st their votes suh tht x is elete? estrutive mnipultion existene : Given voting rule r, set of p nites X, nite x X, n the votes of voters 1,..., k < n Question is there wy for voters k + 1,..., n to st their votes suh tht x is not elete? 38 / 84

48 Mnipulting Bor The single voter se Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete e e e e Current Bor sores : : 10 : 10 : 8 : 7 e : 5 Is there onstrutive mnipultion for? for? for? for? for e? Other Topis 39 / 84

49 Mnipulting Bor The single voter se Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete e e e e Current Bor sores : : 10 : 10 : 8 : 7 e : 5 Is there onstrutive mnipultion for n for? Oviously yes. Other Topis 40 / 84

50 Mnipulting Bor The single voter se Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis e e e Is there onstrutive mnipultion for? e yes e e e e e Current Bor sores : : 10 : 10 : 8 : 7 e : 5 Bor sores : : 10+1 = 11 : 10+0 = 10 : 8+4 = 12 : 7+2 = 9 e : 5+3 = 8 41 / 84

51 Mnipulting Bor The single voter se Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis e e e Is there onstrutive mnipultion for? e e e e e e Current Bor sores : : 10 : 10 : 8 : 7 e : 5 Bor sores : : 10+1 = 11 : 10+0 = 10 : 8+2 = 10 : 7+4 = 11 e : 5+3 = 8 the nswer epens on the tie-reking priority of. 42 / 84

52 Mnipulting Bor The single voter se Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete e e e e Current Bor sores : : 10 : 10 : 8 : 7 e : 5 Is there onstrutive mnipultion for e? oviously not. Other Topis 43 / 84

53 of mnipultion Mnipulting the Bor rule y single voter Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 44 / 84 Without loss of generlity : P profile (without the mnipulting voter) x 1 nite tht the voter wnts to see winning x 2,..., x m other nites, rnke y eresing Bor sore w.r.t. the urrent profile Algorithm : ple x 1 on top, then x m in seon position, then x m 1,..., n finlly x 2 in the ottom position. If x 1 oes not eomes winner then there exists no mnipultion for x. thus for Bor, onstrutive mnipultion existene y one voter is in P. (Brtholi, Tovey & Trik, 89). mnipultion y olitions of more thn one voter : NP-hrness reently solve (Betzler et l., 2011) n (Dvies et l., 2011) some rules re hr to mnipulte even for single voter, for instne the STV rule (Brtholi & Orlin, 91) some empiril works on mnipultion s well (Wlsh et l. 2010)

54 of mnipultion Mnipulting the Bor rule y two voters Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Bor + tie-reking priority > > > > e. Current Bor sores : : 12, : 10, : 9, : 9, e : 4, f : 1 Is there onstrutive mnipultion y two voters for e? Inomplete Other Topis 45 / 84

55 of (unweighte) mnipultion Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis From Xi et l. (09) : Numer of mnipultors 1 t lest 2 Copeln P (1) NP-omplete (2) STV NP-omplete (3) NP-omplete (3) veto P (4) P (4) Simpson P (1) NP-omplete (6) Bor P (1) NP-omplete (7,8) (1) Brtholi et l. ; (2) Flisezwski et l. ; (3) Brtholi n Orlin ; (4) Zukermn et l. ; (7) Betzler et l. (8) Dvies et l. 46 / 84

56 Control : Mnipultion y the Chir Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis The min types of ontrol re ing/eleting voters/nites With respet to given type of ontrol, we sy tht voting rule is : immune if this ontrol n never turn non-winning nite into winning one ; resistnt if it is not immune ut it is iffiult (i.e. NP-hr) to eie whether the outome n e otine vulnerle if it not immune n, furthermore, esy. From (Brtholi et l., 92) n (Trik, 09) : Control y Plurlity Conoret Aing nites resistnt immune Deleting nites resistnt vulnerle Aing voters vulnerle resistnt Deleting voters vulnerle resistnt 47 / 84

57 Outline of the Tlk Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 48 / 84

58 Communition Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis importnt to know the mount of informtion tht nees to e exhnge to ompute the outome no onern regring the omputtionl power of gents here nive universl protool : 1. eh gent reports his own vote to the enter (n log p! its) 2. the enter sens k the result (nme of the winner) (n log p its) for speifi rules we my esign more lever protools speifi protools provie upper ouns on the ommunition omplexity of the voting rule Conitzer & Snholm. Communition of Common. EC / 84

59 Upper Bouns Plurlity with runoff Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis A possile protool : step 1 voters sen the nme of their most preferre nite to the entrl uthority n log p its step 2 the entrl uthority sens the nmes of the two finlists to the voters 2n log p its step 3 voters sen the nme of their preferre finlist to the entrl uthority n its totl n(3 log p + 1) its (in the worst se) 50 / 84 the ommunition omplexity of plurlity with runoff is in O(n. log p).

60 Upper Bouns Single Trnsferle Vote (STV) Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 51 / 84 A slightly more intrite protool... step 1 voters sen their most preferre nite to the entrl uthority (C ) n log p its step 2 let x e the nite to e eliminte. All voters who h x rnke first reeive messge from C sking them to sen the nme of their next preferre nite. There were t most n p suh voters log p its n p step 3 similrly with the new nite y to e eliminte. At n most p 1 voters vote for y log p its et. n p 1 totl n log p(1 + 1 p + 1 p ) = O(n.(log p)2 ).

61 Lower ouns : Communition Setting [Yo, Kushilevitz & Nisn] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Bsi ommunition omplexity setting set of n gents hve to ompute funtion f (x 1,..., x n ) given tht the input is istriute mong the gents (x 1 privtely known from gent 1, et.) protools : speify ommunition tion y the gents, given its (privte) input n the its exhnge so fr useful tree representtion where eh noe is lelle y either gent or gent (se of two gents), with funtion speifying whether to wlk left (0) or right (1) epening on its privte input. Kushilevitz & Nisn. Communition omplexity. Cmrige Univ. Press, / 84

62 Protools illustrte Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition (y 0 ) = 0 (y 1 ) = 0 (y 2 ) = 0 (y 2 ) = 1 (x 0 ) = 0 (x 1 ) = 1 (x 2 ) = 1 (x 3 ) = 0 0 y 0 y 1 y 2 y 3 x x x x Inomplete Other Topis (x 0 ) = 0 (x 1 ) = 0 (x 2 ) = 0 (x 3 ) = 1 (x 0 ) = 0 (x 1 ) = 1 (x 2 ) = 1 (x 3 ) = 1 53 /

63 Protools illustrte Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition (x 0 ) = 0 (x 1 ) = 0 (x 2 ) = 0 (x 3 ) = 1 (y 0 ) = 0 (y 1 ) = 0 (y 2 ) = 0 (y 3 ) = 1 (x 0 ) = 0 (x 1 ) = 1 (x 2 ) = 1 (x 3 ) = 1 y 0 y 1 y 2 y 3 x x x x Inomplete Other Topis / 84

64 Cost of protools Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis The ost of protool is the numer of its exhnge (in the worst se), i.e. the height of the tree. On our exmple, the est ost is the seon one (ost 2 vs. 3 for the first one) The ommunition omplexity of funtion f is the minimum ost of P mong ll protools P tht ompute f. But how o we know tht there is no etter protool? ommunition omplexity offers unh of tehniques to prove lower ouns one of them is the fooling set tehnique 55 / 84

65 Fooling sets Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 56 / 84 Oserve tht the protools, s esrie, in ft prtition the mtrix of inputs into monohromti (sme output) retngles y 0 y 1 y 2 y 3 x x x x monohromti retngles the numer of leves is the numer of retngles hene the ost of protool must e t lest the log(#retngles) if we fin lrge numer of inputs suh tht no two of them n e in the sme retngle, the numer of retngles must e lrge s well. when two input pirs (x 1, y 1 ) n (x 2, y 2 ) re in the sme monohromti retngle, so o (x 1, y 2 ) n (x 2, y 1 ) 0?? 0 Key result (Yo,1979) : CC is t lest log(#fooling set)

66 Communition omplexity of voting rules [Conitzer & Snholm, EC05] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis In our ontext, we hve : f is the voting rule x i is the llot of voter i we re intereste in istinguishe nite, so f returns 1 if wins, 0 otherwise A fooling set is then set of profiles P i suh tht : 1. there exists nite suh tht r(p i ) = 2. for ny pir (i, j ) (i j ), there exists (m 1, m 2,..., m n ) {i, j } n suh tht r(v m1 1, v m2 2,..., vn mn ) we n mix the profiles y piking votes either in P i or P j n fool the funtion 57 / 84

67 Exmple : Lower oun for the Bor rule [Conitzer & Snholm, EC05] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete We note p = p 2 n n = (n 2)/4, π n ritrry permuttion of nites X \ {, } n π the mirror permuttion n 1 n π π π... π π.. π. (p!) n suh profiles.. π π. π... π π π Other Topis 58 / 84

68 Exmple : Lower oun for the Bor rule [Conitzer & Snholm, EC05] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis We note p = p 2 n n = (n 2)/4, π n ritrry permuttion of nites X \ {, } n π the mirror permuttion n 1 n π π π... π π.. π... π π. π... π π π (p!) n suh profiles 1. Does wins in ny suh profile? Oserve tht is 1 point he of ny other nite (thnks to voter n) 58 / 84

69 Exmple : Lower oun for the Bor rule [Conitzer & Snholm, EC05] Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 58 / 84 We note p = p 2 n n = (n 2)/4, π n ritrry permuttion of nites X \ {, } n π the mirror permuttion n 1 n π π π... π π.. π... π π. π... π π π (p!) n suh profiles 1. Does wins in ny suh profile? Oserve tht is 1 point he of ny other nite (thnks to voter n) 2. Is it fooling? Tke two profiles P 1 n P 2, for t lest one voter i {1,..., n } the vote iffers. Thus t lest one nite {, } must e rnke higher in P 1 thn P 2. Mix profiles y piking votes 4i-3 n 4i -2 from P 1 n the rest from P 2. Now get 2 itionl points n wins.

70 Outline of the Tlk Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 59 / 84

71 Voting with Inomplete Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis There re mny ses where profiles n e inomplete (i) nnot ompre nites (intrinsi inompleteness) (ii) there re too mny nites to e rnke (iii) messges my e lost, elye, or fulty When profiles re inomplete, one my either rely on : further ommunition to eliitte the (relevnt) missing informtion omputtion of possile winner(s), i.e. nites who win in t lest one ompletion of the profile 60 / 84

72 Possile n neessry winners Niols Muet 05 Septemre 2011 Bsis of Soil Choie For eh voter : P i is prtil orer on the set of nites. P = P 1,..., P n inomplete profile Completion of P : full profile T = T 1,..., T n of P, where eh T i is liner rnking extening P i. r voting rule Mnipultion Communition Inomplete Other Topis is possile winner if there exists ompletion of P in whih is elete. is neessry winner if is elete in every ompletion of P. Konzk & Lng. Voting proeures with inomplete preferenes. AI-Pref, / 84

73 Possile n neessry winners Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis possile winners for, plurlity with tie-reking > > possile plurlity >> -winners : {, }. 62 / 84

74 Missing Voters n Missing Cnites Niols Muet 05 Septemre 2011 In his generl version, the prolem of voting uner inomplete preferenes mkes no ssumption on inompleteness : But two speifi su-ses of the prolem re nturl. Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Missing voters : Missing nites : Oserve tht the possile winner prolem with missing voters extly orrespon the olitionl mnipultion prolem. 63 / 84

75 Missing Voters Compiling Niols Muet 05 Septemre 2011 Bsis of Soil Choie Unknown numer of missing voters : how to store the urrent profile? Mnipultion Communition Inomplete Other Topis / 84 The winner is : x

76 Missing Voters Compiling Niols Muet 05 Septemre 2011 Bsis of Soil Choie Unknown numer of missing voters : how to store the urrent profile? ompiltion funtion Mnipultion Communition Inomplete Other Topis / 84 The winner is : x sme winner The winner is : x

77 Missing Voters Compiltion Funtions Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis We re fter the est ompiltion funtions for eh voting rule. To strt with, for ny nonymous voting rule, ompiling the profile into the orresponing voting sitution is possile : Profile : Voting sitution : Hene the ompiltion requires t most min(n log p!, p! log n). 65 / 84

78 Missing Voters Compiltion Funtions Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis We re fter the est ompiltion funtions for eh voting rule. To strt with, for ny nonymous voting rule, ompiling the profile into the orresponing voting sitution is possile : Profile : Voting sitution : Hene the ompiltion requires t most min(n log p!, p! log n). Very effiient when n p. Eg. n = 4703 n p = 4 we get min(4703 log 24, 24 log 4703) so 312 its vs its. 65 / 84

79 Missing Voters Compiltion Funtions for Speifi Rules Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Intuitively, for speifi voting rules one n get muh etter ompiltions, eg. for plurlity just ompile the sore yiels p log n. But these re upper ouns : how o we know tht no etter ompiltion is possile y very smrt guy? Lower ouns : gin, orrow notions from ommunition omplexity In ft, the prolem n e seen s one-roun ommunition omplexity prolem (the enter must sen the relevnt informtion in one single messge) 66 / 84

80 Missing Voters Compiling : Methoology Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis two profiles re equivlent for voting rule if they return the sme winner for ny possile ompletion. the key is to hrterize equivlene lsses for eh rules, n enumerte them (not lwys esy...). the ompiltion omplexity is given y tking the log of this numer. Voting rule Chrteriztion of equiv. Compiltion omplexity Any voting rule sme profiles O (np log p) Anonymous sme voting situtions O (p! log n) STV for ll Z C n x Z, Ω ( 2 p log n ) sore Pl (x, P Z ) = sore Pl (x, Q Z ) O ( p2 p log n ) ( Plurlity/runoff M P = M Q n Θ p 2 ) log n sore Pl (x, P) = sore Pl (x, Q) Con. WMG M P = M Q O ( p 2 ) log n Bor sore B (x, P) = sore B (x, Q) Θ (p log np) Plurlity sore Pl (x, P) = sore Pl (x, Q) Θ (p log(1 + n p ) + n log(1 + p ) n ) 67 / 84 Chevleyre et l. Compiling the votes of sueletorte. IJCAI, Xi & Conitzer. Compiltion of Common. AAAI, 2010.

81 Missing Cnites Possile Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Rell tht here prtil votes re simply liner orers on suset of nites (k missing nites) Exmple Jo ssignment eision with 4 vli pplitions n 2 pening verifitions. Given voting sitution π n voting rules r, x X is possile winner (wrt π n r) if there is ompletion, profile P extening P X, st. r(p) = x Note tht the neessry winner prolem is not very relevnt here, ny new nite eing (uner mil onitions) possile winner. Stuy the possile winner prolem with new nites fousing on soring rules s 1,..., s p (s i s i+1 n s 1 > s p ). 68 / 84 Chevleyre et l. Possile when New Cnites Are Ae : The Cse of Soring Rules. AAAI, 2010.

82 Missing Cnites An exmple with plurlity Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Exmple (Plurlity, 1 new nite) 1: 2: 3: 4: 5: 6: 7: 8: Tie-reking : > > > > y Plurlity sores : s() = 3, s() = 2, s() = 2, s() = 1 Who re the possile winners? ertinly is / 84

83 Exmple Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Exmple (Plurlity, 1 new nite) 1: y 2: y 3: y 4: y 5: y 6: y 7: y 8: y Tie-reking : > > > > y Plurlity sores : s() = 3, s() = 2, s() = 2, s() = 1 Who re the possile winners? is s well / 84

84 Exmple Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Exmple (Plurlity, 1 new nite) 1: y 2: y 3: y 4: y 5: y 6: y 7: y 8: y Tie-reking : > > > > y Plurlity sores : s() = 3, s() = 2, s() = 2, s() = 1 Who re the possile winners? is not. 71 / 84

85 Exmple Niols Muet 05 Septemre 2011 Exmple (Plurlity, 2 new nites) Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1: y 1 y 2 2: y 1 y 2 3: y 2 y 1 4: y 1 y 2 5: y 1 y 2 6: y 1 y 2 7: y 1 y 2 8: y 1 y 2 Tie-reking : > > > > y Plurlity sores : s() = 3, s() = 2, s() = 2, s() = 1 Who re the possile winners? now is. 72 / 84

86 Plurlity : n esy se Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis The generl onition is esy to stte. Intuitively : eh new nite n e ple t the top to erese the sore of nite ; for eh nite with higher sore thn x we must put the new nite on top numer of times equl to the ifferene of sores (+1 if tht nite hs priority in the tie-reking rule) ; the sore of the new nite must not e higher (or inee equl if the new nite hs priority) thn the urrent sore of x. Generlizes to k new nites. top(p X, x) 1 mx(0, top(p X, z) top(p X, x)) k z X 73 / 84

87 Bor Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Consier the soring vetor p 1, p 2,..., 0. For given nite x, the est sitution is tht the new nites y i re ple right fter x in the profile. 4, 3, 2, 1, 0 x y Communition Inomplete Other Topis 74 / 84

88 Bor Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Consier the soring vetor p 1, p 2,..., 0. For given nite x, the est sitution is tht the new nites y i re ple right fter x in the profile. 4, 3, 2, 1, 0 x y Hols in generl for rules where i : (s i s i+1 ) (s i+1 s i+2 ) Other Topis 74 / 84

89 Bor Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Consier the soring vetor p 1, p 2,..., 0. For given nite x, the est sitution is tht the new nites y i re ple right fter x in the profile. 4, 3, 2, 1, 0 x y Hols in generl for rules where i : (s i s i+1 ) (s i+1 s i+2 ) But the property oesn t hol if the sore vetor is onvex : 10, 3, 2, 1, 0 x y 74 / 84

90 Bor Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Consier the soring vetor p 1, p 2,..., 0. For given nite x, the est sitution is tht the new nites y i re ple right fter x in the profile. 4, 3, 2, 1, 0 x y Hols in generl for rules where i : (s i s i+1 ) (s i+1 s i+2 ) But the property oesn t hol if the sore vetor is onvex : 10, 3, 2, 1, 0 y x My e goo to put y ove x here (euse loses 7 points) / 84

91 Bor Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete This mens tht the onition is lso esy to stte. A nite n only gin points ginst nother nite when it is ove. Let N (P X, x, z) the numer of times x this hppens. k mx z X \{x} s(p X, z) s(p X, x) N (P X, x, z) Other Topis 76 / 84

92 Exmple Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Exmple (Bor) 1: 2: 3: 4: 5: 6: 7: 8: Bor sores : s() = 15, s() = 10, s() = 11, s() = 12 δ(, ) = (15 10)/3 = 5/3 δ(, ) = (11 10)/3 = 1/3 δ(, ) = (12 10)/4 = 1/2 Hene 2 new nites re require for. An the possile winners re... with 1 new nite with 2 new nites n 77 / 84

93 k-pprovl Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1,..., 1, 0,..., 0 For one nite, the onition n e esily stte. Intuitively : Only nites lying in the lst position of the pprovl set n e pushe wy ; Cnites with higher sore thn x must pper in the lst pprove position suffiiently mny times ; Overll, the sore of the new nite must not exee the urrent sore of x. 78 / 84

94 k-pprovl Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1,..., 1, 0,..., 0 For one nite, the onition n e esily stte. Intuitively : Only nites lying in the lst position of the pprovl set n e pushe wy ; Cnites with higher sore thn x must pper in the lst pprove position suffiiently mny times ; Overll, the sore of the new nite must not exee the urrent sore of x. Generlizes to more thn one new nite? 78 / 84

95 4-pprovl is hr Niols Muet 05 Septemre 2011 Bsis of Soil Choie Theorem Deiing if x is possile winner for 4-pprovl w.r.t. the ition of 3 nites is NP-omplete Proof (sketh) : Reution to the perfet 3D-mthing prolem. Mnipultion Communition Inomplete Other Topis Given : olletion of triples S 1... S n where S i = ( i, i, i ), fin perfet mthing if there exists one. 79 / 84

96 4-pprovl is hr Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 80 / 84 The voting profile is uil suh tht : sore(s i ) = 1, sore(x) = n sore( i ) = sore( i ) = sore( i ) = n + 1 S S n times x... + lots of fny votes To mke x win, 3 nites w 1, w 2, w 3 suh tht : These nites must pper t most n times (otherwise they win) They must lower the sore of eh i, i, i. e.g. S eomes S 1 w 1 w 2 w 3 The only wy to o this is to remove from top nites eh i, i, i extly one, whih is 3DM.

97 Outline of the Tlk Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis 1 2 Bsis of Soil Choie 3 4 Mnipultion 5 Communition 6 Voting with Inomplete 7 Other Topis 81 / 84

98 Other Topis Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Axiomtizing serh engine Applies the xiomti pproh to serh engines. Interestingly, in tht se, voters n nites oinie Jugement ggregtion The im is to ggregte jugments of gents on propositionl formule. The literture ws triggere y the so-lle otrinl prox : p q p q r r ? 82 / 84

99 Other Topis Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Resoure Allotion n Fir Division Prolems rise ue to the omintoril struture of the omin of lterntives (eg. lloting inivisile resoures), n prtiulr euse gents re typilly only onerne in resoures, not in full llotions. Autions or negoition-se pprohes well suite, with omin-relte onstrints (eg. kiney exhnges). Mthing In oule-sie mthing prolems, eh sie hs preferenes on the other sie n the ojetive is to mth them. One typilly seeks stle sttes (no pir of gents woul e etter off leving their mth to form new pir), eg. stle mrrige. 83 / 84

100 Thnks Niols Muet 05 Septemre 2011 Bsis of Soil Choie Mnipultion Communition Inomplete Other Topis Bse on joint work, slies, ppers, isussions, et. from/with (in prtiulr) : Ynn Chevleyre Ulle Enriss Jérôme Lng Jérôme Monnot More on these topis : Interntionl Workshop series Computtionl Soil Choie 84 / 84 Chevleyre, Enriss, Lng & Muet. A short introution to omputtionl soil hoie. SOFSEM, Enriss. Logi n Soil Choie Theory. ILLC Teh. Rep