Computational Social Choice

Transcription

1 Computtionl Soil Choie From Arrow's impossiility to Fishurn's mximl lotteries STACS 2015 Tutoril

2 Motivtion Wht is soil hoie theory? How to ggregte possily onfliting preferenes into olletive hoies in fir n stisftory wy? Origins: mthemtis, eonomis, n politil siene Essentil ingreients - Autonomous gents (e.g., humn or softwre gents) - A set of lterntives (epening on the pplition, lterntives n e politil nites, resoure llotions, olition strutures, et.) - Preferenes over lterntives - Aggregtion funtions The xiomti metho will ply ruil role in this tutoril. Whih forml properties shoul n ggregtion funtion stisfy? Whih of these properties n e stisfie simultneously? 2

3 Hnook of Computtionl Soil Choie (Cmrige University Press, forthoming in 2015) 1. Introution to Computtionl Soil Choie Prt 1: Voting 2. Introution to the Theory of Voting 3. Tournment Solutions 4. Weighte Tournment Solutions 5. Dogson s Rule n Young s Rule 6. Brriers to Mnipultion in Voting 7. Control n Briery in Voting 8. Rtionliztions of Voting Rules 9. Voting in Comintoril Domins 10. Inomplete Informtion n Communition in Voting Prt 2: Fir Allotion 11. Introution to the Theory of Fir Allotion 12. Fir Allotion of Inivisile Goos 13. Cke Cutting Algorithms Prt 3: Colition Formtion 14. Mthing uner Preferenes 15. Heoni Gmes 16. Weighte Voting Gmes Prt 4: Aitionl Topis 17. Jugment Aggregtion 18. The Axiomti Approh n the Internet 19. Knokout Tournments (B., Conitzer, Enriss, Lng, Proi) (Zwiker) (B., Brill, Hrrenstein) (Fisher, Hury, Nieermeier) (Crginnis, Hemspnr, Hemspnr) (Conitzer, Wlsh) (Fliszewski, Rothe) (Elkin, Slinko) (Lng, Xi) (Boutilier, Rosenshein) (Thomson) (Bouveret, Chevleyre, Muet) (Proi) (Klus, Mnlove, Rossi) (Aziz, Svni) (Chlkikis, Woolrige) (Enriss) (Tennenholtz, Zohr) (Vssilevsk-Willims)

4 Plurlity Why re there ifferent voting rules? Wht s wrong with plurlity (the most wiespre voting rule) where lterntives tht re rnke first y most voters win? Consier preferene profile with 21 voters, who rnk four lterntives s in the tle on the right. Alterntive is the unique plurlity winner espite - mjority of voters think is the worst lterntive, - loses ginst,, n in pirwise mjority omprisons, n - if the preferenes of ll voters re reverse, still wins In July 2010, 22 experts on soil hoie theory met in Frne n vote on whih voting rules shoul e use. Plurlity reeive no support t ll (mong 18 rules). 4

5 5 Common Voting Rules Plurlity Use in most emorti ountries, uiquitous Alterntives tht re rnke first y most voters Bor Use in Sloveni, emi institutions, Eurovision song ontest The most preferre lterntive of eh voter gets m-1 points, the seon most-preferre m-2 points, et. Alterntives with highest umulte sore win. Plurlity with runoff Use to elet the Presient of Frne The two lterntives tht re rnke first y most voters fe off in mjority runoff. 5

6 5 Common Voting Rules (t.) Instnt-runoff Use in Austrli, Ireln, Mlt, Aemy wrs Alterntives tht re rnke first y the lowest numer of voters re elete. Repet until no more lterntives n e elete. The remining lterntives win. In the UK 2011 lterntive vote referenum, people hose plurlity over instnt-runoff. Sequentil mjority omprisons Use y US ongress to pss lws (k menment proeure) n in mny ommittees Alterntives tht win fixe sequene of pirwise omprisons (e.g., (( vs. ) vs. ), et.). 6

7 A Curious Preferene Profile 33% 16% 3% 8% 18% 22% e e e e e e Plurlity: wins Exmple ue to Mihel Blinski Bor: wins Sequentil mjority omprisons (ny orer): wins Instnt-runoff: wins Plurlity with runoff: e wins 7

8 Rtionl Choie Theory A prerequisite for nlyzing olletive hoie is to unerstn iniviul hoie. Let U e finite universe of lterntives. A hoie funtion f mps fesile set A U to hoie set f(a) A. We require tht f(a)= only if A=. Not every hoie funtion omplies with our intuitive unerstning of rtionlity. Certin ptterns of hoie from vrying fesile sets my e eeme inonsistent, e.g., hoosing from {,,}, ut from {,}. A f(a) 8

9 Rtionlizle Choie Binry preferene reltion on U x y is interprete s x is t lest s goo s y. is ssume to e trnsitive n omplete. Best lterntives For inry reltion n fesile set A, Mx(,A)= {x A y A suh tht y x} f is rtionlizle if there exists preferene reltion on U suh tht f(a)=mx(,a) for ll A. The previously mentione hoie funtion f with f({,,})={} n f({,})={} nnot e rtionlize. 9

10 Consistent Choie It woul e nie if the non-existene of rtionlizing reltion oul e pointe out y fining inonsistenies. f stisfies onsisteny if for ll A,B with B A, f(a) B implies f(b)=f(a) B. A f(a) f(b) B Consequene: If x is hosen from fesile set, then it is lso hosen from ll susets tht ontin x. Exmple: Plurlity oes not stisfy onsisteny (when sores re ompute for eh fesile set). - f({,,}) = {} n f({,}) = {} Theorem (Smuelson, 1938; Arrow, 1959): A hoie funtion is rtionlizle iff it stisfies onsisteny

11 From Choie to Soil Choie N is finite set of t lest two voters. R(U) is the set of ll preferene reltions over U. Every R=( 1,..., N ) R(U) N is lle preferene profile. A soil hoie funtion (SCF) is funtion f tht ssigns hoie funtion to eh preferene profile. An SCF is rtionlizle (onsistent) if its unerlying hoie funtions re rtionlizle (onsistent) for ll preferene profiles. We will write f(r,a) s funtion of oth R n A. Let nxy = {i N x i y} n efine the mjority rule reltion s (x RM y) nxy > nyx. 11

12 Conoret s Prox Mrquis e Conoret Soil hoie from fesile sets of size two is esy. The mjority rule SCF is efine s f(r,{x,y}) = Mx(RM,{x,y}). Mjority rule n esily e hrterize using unontroversil xioms (e.g., My, 1952). Prolems rise whenever there re more thn two lterntives. Conoret prox (1785): RM n e intrnsitive. Alterntive x is Conoret winner in A if x RM y for ll y A\{x}. An SCF f is Conoret extension if f(r,a)={x} whenever x is Conoret winner in A

13 Arrow s Impossiility An SCF stisfies inepenene of infesile lterntives (IIA) if the hoie set only epens on preferenes over lterntives within the fesile set. An SCF stisfies Preto-optimlity if n lterntive will not e hosen if there exists nother lterntive suh tht ll voters prefer the ltter to the former. An SCF is ittoril if there exists voter whose most preferre lterntive is lwys uniquely hosen. Theorem (Arrow, 1951): Every rtionlizle SCF tht stisfies IIA n Preto-optimlity is ittoril when U 3. Nipkow (2009) hs verifie proof of Arrow s theorem using Iselle. Tng & Lin (2009) reue the sttement to finite se se tht ws solve y omputer. Kenneth J. Arrow 13

14 Wht now? Rtionlizility (or, equivlently, onsisteny) is inomptile with olletive hoie when U 3. Dropping IIA offers little relief (Bnks, 1995). Dropping Preto-optimlity offers little relief (Wilson, 1972). Dropping non-ittorship is uneptle. In this tutoril, we will onsier two espe routes from Arrow s impossiility: SCFs tht stisfy weker notions of onsisteny - Top yle, unovere set, Bnks set, tournment equilirium set Rnomize SCFs - Rnom ittorship, mximl lotteries 14

15 Wekly Consistent SCFs 15

16 Tournments For given preferene profile R, fesile set A n mjority rule RM efine irete grph (A,RM). We sy tht omintes if RM. Every symmetri irete grph is inue y some preferene profile (MGrvey, 1953). A mjoritrin SCF is n SCF whose output only epens on (A,RM). For simpliity, we will ssume tht iniviul preferenes re ntisymmetri n tht N is o. Hene, (A,RM) is tournment. SCF f is si to e finer thn SCF g if f g. Dominion D(x)={y A x RM y} Domintors D (x)={y A y RM x} 16

17 Mjoritrinness A =2 Non-ittorship IIA Preto-optimlity Rtionlizility/ Consisteny Arrow s Impossiility Expnsion Wek expnsion Strong retentiveness Retentiveness 17

18 The Top Cyle John I. Goo Consisteny n e wekene to expnsion: B A n f(a) B implies f(b) f(a). Theorem (Bores, 1976): There is unique finest mjoritrin SCF stisfying expnsion: the top yle. A ominnt set is nonempty set of lterntives B A suh tht for ll x B n y A\B, x RM y. The set of ominnt sets is totlly orere y set inlusion (Goo, 1971). Hene, every tournment ontins unique miniml ominnt set lle the top yle (TC). TC is Conoret extension. 18

19 Exmples f e TC(A,RM)={,,} TC(A,RM)={,,,} TC(A,RM)={,e,f} 19

20 Trnsitive Closure The essene of Conoret s prox n Arrow s impossiility is tht mjority rule fils to e trnsitive. Why not just tke the trnsitive (reflexive) losure RM*? Theorem (De, 1977): TC(A,RM) = Mx(RM*,A). Consequenes TC itself is yle. It is the soure omponent in the DAG (irete yli grph) of strongly onnete omponents. Liner-time lgorithms for omputing TC using Kosrju s or Trjn s lgorithm for fining strongly onnete omponents - Alterntively, one n initilize working set B with ll lterntives of mximl outegree n then itertively ll lterntives tht ominte n lterntive in B until no more suh lterntives n e foun. 20

21 Top Cyle n Preto-Optimlity The top yle is very lrge. In ft, it is so lrge tht it fils to e Preto-optiml when there re more thn three lterntives (Ferejohn & Grether, 1977) Sine Preto-optimlity is n essentil ingreient of Arrow s impossiility, this espe route is (so fr) not entirely onvining. Although, tehnilly, Arrow s theorem only requires Pretooptimlity for two-element sets (whih the top yle stisfies). 21

22 Mjoritrinness A =2 Non-ittorship IIA Preto-optimlity Rtionlizility/ Consisteny Arrow s Impossiility Expnsion Top Cyle (TC) Wek expnsion 22

23 The Unovere Set Peter C. Fishurn Nihols Miller Expnsion n e further wekene to wek expnsion: f(a) f(b) f(a B). Theorem (Moulin, 1986): There is unique finest mjoritrin SCF stisfying wek expnsion: the unovere set. Given tournment (A,RM), x overs y (x C y), if D(y) D(x). Propose inepenently y Fishurn (1977) n Miller (1980) Trnsitive sureltion of mjority rule The unovere set (UC) onsists of ll unovere lterntives, i.e., UC(A,PM) = Mx(C,A). 23

24 Exmples UC(A,RM)={,,} UC(A,RM)={,,} TC(A,RM)={,,,} 24

25 Properties of the Unovere Set Sine expnsion wek expnsion, UC TC. UC is Conoret extension. UC stisfies Preto-optimlity. Theorem (B. n Geist, 2014): UC is the lrgest mjoritrin SCF stisfying Preto-optimlity. How n the unovere set e effiiently ompute? Strightforwr O(n 3 ) lgorithm tht omputes the overing reltion for every pir of lterntives Cn we o etter thn tht? 25

26 Unovere Set Algorithm Equivlent hrteriztion of UC Theorem (Shepsle & Weingst, 1984): UC onsists preisely of ll lterntives tht reh every other lterntive in t most two steps. - Suh lterntives re lle kings in grph theory. Hene, UC n e ompute y squring the tournment s jeny mtrix. Fstest known mtrix multiplition lgorithm (Le Gll, 2014): O(n ) Just slightly fster thn Vssilevsk Willims, 2011: O(n ) Bse on Coppersmith & Winogr (1990): O(n ) Mtrix multiplition is elieve to e fesile in liner time (O(n 2 )). 26

27 Unovere Set Algorithm (Exmple) e B@ CA B@ CA B@ CA = B@ CA

28 Mjoritrinness A =2 Non-ittorship IIA Preto-optimlity Rtionlizility/ Consisteny Arrow s Impossiility Expnsion Top Cyle (TC) Wek expnsion Unovere Set (UC) Strong retentiveness 28

29 Bnks Set Wek expnsion n e wekene to strong retentiveness: f(d (x)) f(a) for ll x A. Theorem (B., 2011): There is unique finest mjoritrin SCF stisfying strong retentiveness: the Bnks set. A trnsitive suset of tournment (A,RM) is set of lterntives B A suh tht RM is trnsitive within B. Let Trns(A,RM) = {B A B is trnsitive}. The Bnks set (BA) onsists of the mximl elements of ll inlusion-mximl trnsitive susets (Bnks, 1985), i.e., BA(A,RM) = {Mx(RM,B) B Mx(,Trns(A,RM))} Jeffrey S. Bnks 29

30 Exmples (All missing eges re pointing ownwrs.) e f g UC(A,RM)={,,} BA(A,RM)={,,} TC(A,RM)={,,,,e,f,g} UC(A,RM)={,,,} BA(A,RM)={,,} 30

31 Properties of the Bnks Set Sine expnsion wek expnsion strong retentiveness, BA UC TC. As onsequene, BA is Conoret extension n stisfies Preto-optimlity. Rnom lterntives in BA n e foun in liner time y itertively onstruting mximl trnsitive sets. Yet, omputing the Bnks set is NP-hr (Woeginger, 2003) n remins NP-hr even for 5 voters (B. et l., 2013). Strong retentiveness n e further wekene to retentiveness: f(d (x)) f(a) for ll x f(a). 31

32 Mjoritrinness A =2 Non-ittorship IIA Preto-optimlity Rtionlizility/ Consisteny Arrow s Impossiility Expnsion Top Cyle (TC) Wek expnsion Unovere Set (UC) Strong retentiveness Bnks Set (BA) Retentiveness 32

33 Tournment Equilirium Set Let f e n ritrry hoie funtion. A non-empty set of lterntives B is f-retentive if f(d (x)) B for ll x B. Ie: No lterntive in the set shoul e properly ominte y n outsie lterntive. f is new hoie funtion tht yiels the union of ll inlusion-miniml f-retentive sets. f stisfies retentiveness. The tournment equilirium set (TEQ) of tournment is efine s TEQ=TEQ. Reursive efinition (unique fixe point of ring-opertor) Theorem (Shwrtz, 1990): TEQ BA. x Thoms Shwrtz B 33

34 Exmple {,,} is the unique miniml TEQ-retentive set. TEQ(D ()) = TEQ({}) = {} TEQ(D ()) = TEQ({,e}) = {} TEQ(D ()) = TEQ({,}) = {} TEQ(D ()) = TEQ({,}) = {} TEQ(D (e)) = TEQ({,,}) = {,,} e A thik ege from y to x enotes tht y TEQ(D (x)). 34

35 Properties of TEQ Computing TEQ is NP-hr (B. et l., 2010) n remins NP-hr even for 7 voters (Bhmeier et l., 2015). The est known upper oun is PSPACE! Theorem (Lffon et l., 1993; Houy 2009; B., 2011; B. n Hrrenstein, 2011): The following sttements re equivlent: Every tournment ontins unique miniml TEQ-retentive set. (Shwrtz Conjeture, 1990) TEQ is the unique finest mjoritrin SCF stisfying retentiveness. TEQ stisfies monotoniity (n mny other esirle properties). All or nothing: Either TEQ is most ppeling SCF or it is severely flwe. 35

36 Shwrtz s Conjeture There exists no ounterexmple with less thn 13 lterntives (154 illion tournments hve een heke). TEQ stisfies ll nie properties if A <13. No ounterexmple ws foun y serhing illions of rnom tournments with up to 50 lterntives. Cheking signifintly lrger tournments is intrtle. Mny non-trivil wekenings of Shwrtz s onjeture re known to hol (Goo, 1971; Dutt, 1988; B. et l., 2010; B., 2011). Theorem (B., Chunovsky, Kim, Liu, Norin, Sott, Seymour, n Thomssé, 2012): Shwrtz s onjeture is flse. 36

37 Aftermth Non-onstrutive proof relying on proilisti rgument y Erős n Moser (1964) Neither the ounter-exmple nor its size n e eue from proof. Smllest ounter-exmple of this type requires out lterntives. More reently, ounter-exmple with 24 lterntives ws foun with the help of omputer (B. & Seeig, 2013). In priniple, TEQ is severely flwe. However, ounter- exmples re so extremely rre tht this hs no prtil onsequenes. This sts out on the xiomti metho. 37

38 Wekly Consistent SCFs TC UC BA TEQ Top Cyle (1971) TC expnsion O(n 2 ) Unovere Set (1977) UC wek expnsion O(n 2.38 ) Bnks Set (1985) BA strong retentiveness 2 O(n) Tournment Equilirium Set (1990) TEQ (retentiveness) 2 O(n) 38

39 Rnomize SCFs 39

40 Rnom Dittorship Alln Gir A rnomize SCF mps preferene profile to lottery (proility istriution) over the lterntives. Perhps the most notorious rnomize SCF is rnom ittorship. One gent is pike uniformly t rnom n his most preferre lterntive is implemente s the soil hoie. Rnom ittorship is not s s it my soun. It stisfies most of the xioms tht re usully onsiere in soil hoie theory. Rnom ittorship is the only Preto-optiml rnomize SCF tht is strtegyproof, i.e., it nnot e mnipulte y lying out one s preferenes (Gir, 1977). 40

41 Mximl Lotteries Germin Krewers Peter C. Fishurn Krewers (1965) n Fishurn (1984) Reisovere y Lffon et l. (1993), Felsenthl n Mhover (1992), Fisher n Ryn (1995), Rivest n Shen (2010) Let g(x,y) = nxy - nyx e the mjority mrgin of x n y. Alterntive x is (wek) Conoret winner if g(x,y) 0 for ll y. Exten g to lotteries: g(p,q) = x,y p(x) q(y) g(x,y) Expete mjority mrgin p is mximl lottery if g(p,q) 0 for ll q. Rnomize (wek) Conoret winner Alwys exists ue to Minimx Theorem (v. Neumnn, 1928) 41

42 Exmples Two lterntives p(x) 1 0 Mximl lotteries 0 N nxy p(x) 1 0 Rnom ittorship 0 N nxy R n e trnsforme into symmetri zero-sum gme. Mximl lotteries re mixe minimx strtegies (or Nsh equiliri) The unique mximl lottery is 3/5 + 1/5 + 1/5. 42

43 Properties of Mximl Lotteries (ML) Mximl lotteries re lmost lwys unique. Alwys unique for o numer of voters (Lffon et l., 1997) ML oes not require symmetry, ompleteness, or even trnsitivity of preferenes. Rnom ittorship requires unique mximum. Cnonil generliztion (RSD) requires t lest one mximum. ML n e effiiently ompute vi liner progrmming. Computing RSD proilities, on the other hn, is #P-omplete (Aziz et l., 2013). In the ssignment omin, mximl lotteries re known s populr mixe mthings (Kvith et l., 2011). 43

44 Properties of Mximl Lotteries (t.) Preto-ominte lterntives lwys get zero proility in every mximl lottery. In ft, ML is even effiient with respet to stohsti ominne. No lottery gives more expete utility for ny utility representtion onsistent with the voters preferenes (Aziz et l., 2012). Violte y RSD (Bogomolni n Moulin, 2001). ML is wekly strtegyproof in well-efine sense (Aziz et l., 2013). ML n e uniquely hrterize using version of onsisteny for rnomize SCFs (Brnl et l., 2015). 44

45 Reommene Literture Books Allinghm: Choie Theory - A very short introution. Oxfor University Press, 2002 Austen-Smith n Bnks: Positive Politil Theory I, University of Mihign Press, 1999 Gärtner: A Primer in Soil Choie Theory, Oxfor University Press, 2009 Moulin: Axioms of Coopertive Deision Mking. Cmrige University Press, 1988 Nitzn: Colletive Choie n Preferene. Cmrige University Press, 2010 Introutory ook hpter B., Conitzer, n Enriss. Computtionl Soil Choie. In "Multigent Systems" (G. Weiss, e.), MIT Press,