Axiomatic social choice theory

Transcription

1 Axiomti soil hoie theory From Arrow's impossiility to Fishurn's mximl lotteries ACM EC 2014 Tutoril

2 Shedule : Tutoril prt : Coffee rek : Tutoril prt 2! In the interest of time, I nnot over strtegyproofness nd preferenes over lotteries. If your re interested in these topis, you re welome to ttend the poster session nd pper session 4 (oth on Tuesdy). 2

3 Motivtion Wht is soil hoie theory? How to ggregte possily onfliting preferenes into olletive hoies in fir nd stisftory wy? Origins: mthemtis, eonomis, nd politil siene Essentil ingredients - Autonomous gents (e.g., humn or softwre gents) - A set of lterntives (depending on the pplition, lterntives n e politil ndidtes, resoure llotions, olition strutures, et.) - Preferenes over lterntives - Aggregtion funtions The xiomti method will ply ruil role in this tutoril. Whih forml properties should n ggregtion funtion stisfy? Whih of these properties n e stisfied simultneously? 3

4 Plurlity Why re there different voting rules? Wht s wrong with plurlity (the most widespred voting rule) where lterntives tht re rnked first y most voters win? Consider preferene profile with 21 voters, who rnk four lterntives s in the tle elow.!!!! Alterntive is the unique plurlity winner despite the ft tht - mjority of voters think is the worst lterntive, d d - loses ginst,, nd d in pirwise mjority omprisons, nd - if the preferenes of ll voters re reversed, still wins. d d 4

5 5 Common Voting Rules Plurlity Used in most demorti ountries, uiquitous Alterntives tht re rnked first y most voters Bord Used in Sloveni, demi institutions, Eurovision song ontest The most preferred lterntive of eh voter gets m-1 points, the seond most-preferred m-2 points, et. Alterntives with highest umulted sore win. Plurlity with runoff Used to elet the President of Frne The two lterntives tht re rnked first y most voters fe off in mjority runoff. 5

6 5 Common Voting Rules (td.) Instnt-runoff Used in Austrli, Irelnd, Mlt, Ademy wrds Alterntives tht re rnked first y the lowest numer of voters re deleted. Repet until no more lterntives n e deleted. The remining lterntives win. In the UK 2011 lterntive vote referendum, people hose plurlity over instnt-runoff. Sequentil mjority omprisons Used y US ongress to pss lws (k mendment proedure) nd in mny ommittees Alterntives tht win fixed sequene of pirwise omprisons (e.g., (( vs. ) vs. ), et.). 6

7 A Curious Preferene Profile 33% 16% 3% 8% 18% 22% d e d e d e e d d e e d Plurlity: wins Exmple due to Mihel Blinski Bord: wins Sequentil mjority omprisons (,,,d,e): wins Instnt-runoff: d wins Plurlity with runoff: e wins 7

8 Desirle Properties (Axioms) Anonymity The voting rule trets voters eqully. Exhnging olumns in the preferene profile does not ffet the outome. Neutrlity The voting rule trets lterntives eqully. Renming the lterntives does not ffet the outome. Monotoniity A hosen lterntive will still e hosen when it rises in individul preferene rnkings (while leving everything else unhnged). Preto-optimlity An lterntive will not e hosen if there exists nother lterntive suh tht ll voters prefer the ltter to the former. Strtegyproofness No voter n otin more preferred lterntive y misrepresenting his preferenes. 8

9 Anonymity Neutrlity Monotoniity Preto Strtegyproofness Plurlity - Bord - Plurlity w/ runoff - - Instnt-runoff - - SMC d d d SMC fils neutrlity nd Preto-optimlity Runoff rules fil monotoniity

10 Outline Rtionl hoie theory My s Theorem, Condoret s Prdox, Arrow s Theorem Three espe routes: reple onsisteny with vrile-eletorte ondition - soring rules (e.g., plurlity, Bord) weken onsisteny - top yle, unovered set, Bnks set, tournment equilirium set rndomiztion - mximl lotteries 10

11 Choie Theory A prerequisite for nlyzing olletive hoie is to understnd individul hoie. Let U e finite universe of lterntives. A hoie funtion S mps fesile set A U to hoie set S(A) A. We require tht S(A)= only if A=. X S(X) For simpliity, we will fous on resolute (i.e., single-vlued) hoie funtions for now. Not every hoie funtion omplies with our intuitive understnding of rtionlity. Certin ptterns of hoie from vrying fesile sets my e deemed inonsistent, e.g., hoosing from {,,}, ut from {,}. 11

12 Preferene nd Mximlity Rtionl deision-mking proess (note the order) Wht is desirle? Wht is fesile? Choose the most desirle from mong the fesile. Binry preferene reltion R on U xry is interpreted s x is t lest s good s y For simpliity, we ssume tht R is symmetri nd omplete: for ll x y, either xry or yrx. Best lterntives For inry reltion R nd fesile set A, Mx(R,A)= {x A : there is no y suh tht yrx nd not xry} 12

13 Rtionlizle Choie S is rtionlizle if there exists inry reltion R on U suh tht S(A)=Mx(R,A) for ll A. A nturl ndidte for suh reltion is the se reltion RS: x RS y x S({x,y}) In ft, S n only e rtionlized y its se reltion RS, whih furthermore hs to e trnsitive when S is resolute (s otherwise Mx(R,A) my e empty). The previously mentioned hoie funtion S with S({,,})={} nd S({,})={} nnot e rtionlized. 13

14 Consisteny It would e nie if irrtionlity (i.e., the non-existene of rtionlizing reltion) ould e pointed out y finding inonsistenies. Consisteny onditions diretly relte hoies from vrile fesile sets with eh other. A resolute hoie funtion S stisfies onsisteny if for ll A,B with S(A) B A implies S(B)=S(A). If x is hosen in fesile set, then it is lso hosen in ll susets tht ontin x. - Exmple: Plurlity does not stisfy onsisteny (when sores re omputed for eh fesile set). - S({,,}) = {} nd S({,}) = {}

15 Rtionlizility nd Consisteny Amrty K. Sen Theorem (Sen, 1971): A resolute hoie funtion is rtionlizle iff it stisfies onsisteny. For resolute hoie funtions, onsisteny is equivlent to Smuelson s wek xiom of reveled preferene (WARP) nd the following ondition due to Shwrtz (1976): For ll A,B nd x A B, x S(A B) x S(A) S(B) A x B 15

16 From Choie to Soil Choie N is finite set of t lest two voters. For simpliity, we will ssume N is odd whenever possile. R(U) is the set of ll trnsitive, omplete, nd nti-symmetri reltions over U. Every RN=(R1,..., R N ) R(U) N will e lled preferene profile. A soil hoie funtion (SCF) is funtion f tht ssigns hoie funtion to eh preferene profile. We will write f(rn,a) s funtion of oth RN nd A. Rtionlizility nd onsisteny onditions rry over to SCFs. 16

17 My s Theorem Kenneth My We first restrit ttention to fesile sets of size two. Let nxy = {i N: x Ri y} nd define the mjority rule reltion s (x RM y) nxy > nyx. The mjority rule SCF is define s f(rn,{x,y}) = Mx(RM,{x,y}). Theorem (My, 1952): Mjority rule is the only resolute SCF on two lterntives tht stisfies nonymity, neutrlity, nd monotoniity. Mjority rule is very unontroversil. All voting rules mentioned erlier oinide with mjority rule on two lterntives. Mjority rule is strtegyproof. 17

18 The Condoret Prdox Theorem (Condoret, 1785; My, 1952): No nonymous, neutrl, nd monotoni resolute SCF is rtionlizle when U 3. Proof sketh: Let f e n SCF with the desired properties nd onsider the following preferene profile. My s theorem implies tht Rf=RM. RM is yli nd therefore nnot rtionlize f. Alterntive x is Condoret winner in A if x RM y for ll y A. Condoret winners my not exist, ut whenever they do they re unique. Arrow s theorem n e otined y signifintly wekening nonymity, neutrlity, nd monotoniity. Mrquis de Condoret

19 From Condoret to Arrow An SCF stisfies independene of infesile lterntives (IIA) if the hoie set only depends 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 f is dittoril if there exists voter whose most preferred lterntive is lwys uniquely hosen. These onditions n e formlly defined suh tht IIA is weker thn neutrlity, Preto-optimlity is weker thn monotoniity, nd non-dittorship is weker thn nonymity. 19

20 Arrow s Impossiility Kenneth J. Arrow Theorem (Arrow, 1951): There is no SCF tht stisfies IIA, Preto-optimlity, non-dittorship, nd rtionlizility when U 3. Arrow s theorem is usully presented in n lterntive formultion for soil welfre funtions, i.e., funtions tht ggregte individul preferene reltions into olletive preferene reltion. IIA, Preto-optimlity, nd non-dittorship n e ppropritely redefined for SWFs (y onsidering the se reltion). Theorem (Arrow, 1951): Every SWF tht stisfies IIA nd Pretooptimlity is dittoril when U 3. 20

21 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-dittorship is uneptle. A lssi espe from Arrow s impossiility is to onsider restrited domins of preferenes in whih mjority rule is trnsitive (suh s single-peked preferenes). In this tutoril, we will onsider three other espe routes: reple onsisteny with vrile-eletorte ondition weken onsisteny rndomiztion 21

22 Espe Route #1 Reple onsisteny with vrile-eletorte ondition 22

23 Bord vs. Condoret Jen-Chrles Chevlier de Bord ( ) Mthemtiin, physiist, nd silor Prtiipted in the onstrution of the stndrd-meter (1/10,000,000 of the distne etween the north pole nd the equtor) Mrie Jen Antoine Niols Critt, Mrquis de Condoret ( ) Philosopher nd mthemtiin Erly dvote of equl rights nd opponent of the deth penlty 23

24 Soring Rules nd Condoret Extensions Fix the fesile set A nd let A =m. A sore vetor is vetor s=(s1,..., sm) of rel numers. If voter rnks n lterntive t the ith position, it gets si points. A soring rule hooses those lterntives for whih the umulted sore is mximl. Exmples Bord s rule: s=(m-1, m-2,..., 0) plurlity rule: s=(1, 0,..., 0) An SCF f is Condoret extension if f(rn,a)={x} whenever x is Condoret winner in A ording to RN. Exmple (Copelnd s rule): Choose those lterntives tht win most pirwise mjority omprisons. 24

25 Soring Rules nd Condoret Extensions When U =2, mjority rule is the only monotoni resolute soring rule nd the only Condoret extension. Proposition (Condoret, 1785): Bord s rule is no Condoret extension when U 3. Theorem (Fishurn, 1973): No soring rule is Condoret extension when U 3. Theorem (Smith, 1973): A Condoret winner is never the lterntive with the lowest Bord sore. Bord s rule is the only soring rule for whih this is the se. 25

26 Vrile Eletortes One of the most remrkle results in soil hoie theory hrterizes soring rules in terms of vrile set of voters ( eletortes ). Reinforement All lterntives tht re hosen simultneously y two disjoint eletortes re preisely the lterntives hosen y the union of oth eletortes. 26

27 Chrteriztion of Soring Rules Reinforement is the equivlent of onsisteny for vrile eletorte! Consisteny: x f(rn,a) f(rn,a ) x f(rn,a A ) [x A A ] H. Peyton Young Reinforement: x f(rn,a) f(rn,a) x f(rn RN,A) [f(r,a) f(r,a) ] Theorem (Smith, 1973; Young, 1975): A neutrl nd nonymous SCF is soring rule iff it stisfies ontinuity nd reinforement. Loosely speking, n SCF stisfies ontinuity if negligile frtions of voters hve no influene on the hoie set. Continuity is tehnil xiom tht n e dropped when fixing n upper ound on the numer of voters. Reinforement is the defining property of soring rules. 27

28 The Dilemm of Soil Choie Theorem (Young nd Levenglik, 1978): No Condoret extension stisfies reinforement when U 3. Two enturies fter Bord nd Condoret, the rtionles etween oth ides were shown to e inomptile. Condoret extensions SCFs stisfying Reinforement When ggregting preferene reltions to sets of preferene reltions, the intersetion of these two sets ontins extly one neutrl funtion: Kemeny s rule! 28

29 Espe Route #2 Weken onsisteny 29

30 Mjoritrin SCFs An SCF is inry if the hoie set only depends on the pirwise hoies within the fesile set. Binriness is stronger thn IIA. A mjoritrin SCF is n SCF tht stisfies nonymity, neutrlity, monotoniity, nd inriness. The hoie set only depends on the se reltion, whih is furthermore fixed to e mjority rule (My s theorem). Mjoritrinness strengthens ll onditions from Arrow s theorem exept rtionlizility/onsisteny. Wekening onsisteny llows us to uniquely hrterize ppeling SCFs. 30

31 Tournments d For given preferene profile, mjority rule RM nd fesile set A define tournment (A,RM), n oriented omplete grph. We sy tht domintes if RM. Every tournment is indued y some preferene profile (MGrvey s, 1953). We will write mjoritrin SCFs s funtions of tournments (A,RM) rther thn funtions of (RN,A). SCF f is sid to e finer thn SCF g if f g. Dominion D(x)={y B x RM y} Domintors D (x)={y B y RM x} 31

32 The Top Cyle Consisteny n e wekened to expnsion: B A nd S(A) B implies S(B) S(A). John I. Good Theorem (Bordes, 1976): There is unique finest mjoritrin SCF stisfying expnsion: the top yle. A dominnt set is nonempty set of lterntives B A suh tht for ll x B nd y A\B, x RM y. The set of dominnt sets is totlly ordered y set inlusion (Good, 1971). Hene, every tournment ontins unique miniml dominnt set lled the top yle (TC). Also known s GETCHA (Shwrtz, 1986) or Smith set (Smith, 1973) TC is Condoret extension. 32

33 Exmples f d d e d TC(A,RM)={,,} TC(A,RM)={,,,d} TC(A,RM)={,e,f} 33

34 TC Liner-Time Algorithm Algorithm for omputing TCx, the miniml dominnt set ontining given lterntive x Initilize working set B with {x} nd then itertively dd ll lterntives tht dominte n lterntive in B until no more suh lterntives n e found. Computing TCx for every lterntive x nd then hoosing the smllest set yields n O(m 3 ) lgorithm where m= A. - A liner-time lgorithm is O(m 2 ) euse the input is qudrti in m. Alterntives with mximl degree re lwys ontined in TC (nd liner-time omputle). Hene, we only need to ompute TCx for some x with mximl degree. proedure TC(A, P M ) B C CO(A, P M ) loop C S C D A\B () if C = then return B end if B B C end loop 34

35 Trnsitive Closure The essene of Condoret s prdox nd Arrow s impossiility is tht the se reltion fils to e trnsitive. Why not just tke the trnsitive (reflexive) losure RM*? Theorem (De, 1977): TC(A,RM) = Mx(RM*,A). Consequenes RM* rtionlizes the top yle. TC itself is yle. It is the soure omponent in the DAG (direted yli grph) of strongly onneted omponents. Alterntive liner-time lgorithms using Kosrju s or Trjn s lgorithm for finding strongly onneted omponents 35

36 Top Cyle nd Preto- The top yle is very lrge. Optimlity In ft, it is so lrge tht it fils to e Preto-optiml when there re more thn three lterntives (Ferejohn & Grether, 1977) d Sine Preto-optimlity is n essentil ingredient of ll Arrovin impossiilities, this espe route is (so fr) not entirely onvining. d d 36

37 The Unovered Set Peter C. Fishurn Nihols Miller Expnsion n e further wekened to wek expnsion: S(A) S(B) S(A B). Theorem (Moulin, 1986): There is unique finest mjoritrin SCF stisfying wek expnsion: the unovered set. Given tournment (A,RM), x overs y (x C y), if D(y) D(x). Proposed independently y Fishurn (1977) nd Miller (1980) Trnsitive sureltion of mjority rule The unovered set (UC) onsists of ll unovered lterntives, i.e., UC(A,PM) = Mx(C,A). C rtionlizes the unovered set 37

38 Exmples d d UC(A,RM)={,,} UC(A,RM)={,,} TC(A,RM)={,,,d} 38

39 Properties of the Unovered Set Sine expnsion wek expnsion, UC TC. UC is Condoret extension. UC stisfies Preto-optimlity. Theorem (B. nd Geist, 2014): UC is the lrgest mjoritrin SCF stisfying Preto-optimlity. How n the unovered set e effiiently omputed? Strightforwrd O(m 3 ) lgorithm tht omputes the overing reltion for every pir of lterntives Cn we do etter thn tht? 39

40 Unovered 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 lled kings in grph theory. Algorithm vi mtrix multiplition - Fstest known mtrix multiplition lgorithm (Vssilevsk Willims, 2011): O(m ) - Strongly sed on previous lgorithm (Coppersmith & Winogrd, 1990): O(m ) - Mtrix multiplition is elieved to e fesile in liner time (O(m 2 )). proedure UC(A, P M ) for ll i, j 2 A do if ip M j then m ij 1 else m ij 0 end if end for M (m ij ) i, j2a U (u ij ) i, j2a M 2 + M + I B {i 2 A 8j2 A: u ij, 0} return B 40

41 Unovered Set Algorithm (Exmple) d e proedure UC(A, P M ) for ll i, j 2 A do if ip M j then m ij 1 else m ij 0 end if end for M (m ij ) i, j2a U (u ij ) i, j2a M 2 + M + I B {i 2 A 8j2 A: u ij, 0} return B B@ CA B@ CA B@ CA = B@ CA

42 Bnks Set Wek expnsion n e wekened to strong retentiveness: S(D (x)) S(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 42

43 Exmples (All missing edges re pointing downwrds.) d e f g d UC(A,RM)={,,} BA(A,RM)={,,} TC(A,RM)={,,,d,e,f,g} UC(A,RM)={,,,d} BA(A,RM)={,,} 43

44 Properties of the Bnks Set Sine expnsion wek expnsion strong retentiveness, BA UC TC. As onsequene, BA is Condoret extension nd stisfies Preto-optimlity. Rndom lterntives in BA n e found in liner time y itertively onstruting mximl trnsitive sets. Yet, omputing the Bnks set is NP-hrd (Woeginger, 2003) nd remins NP-hrd even for 7 voters (B. et l., 2013). Strong retentiveness n e further wekened to retentiveness: S(D (x)) S(A) for ll x S(A). 44

45 Tournment Equilirium Set Let S e n ritrry hoie funtion. A non-empty set of lterntives B is S-retentive if S(D (x)) B for ll x B. Ide: No lterntive in the set should e properly dominted y n outside lterntive. S is new hoie funtion tht yields the union of ll inlusion-miniml S-retentive sets. S stisfies retentiveness. The tournment equilirium set (TEQ) of tournment is defined s TEQ=TEQ. reursive definition (unique fixed point of ring-opertor) Theorem (Shwrtz, 1990): TEQ BA. x Thoms Shwrtz B 45

46 Properties of TEQ Computing TEQ is NP-hrd (B. et l., 2010) nd remins NP-hrd even for 9 voters (B. et l., 2013). The est known upper ound is PSPACE! Theorem (Lffond et l., 1993; Houy 2009; B., 2011; B. nd 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 (nd mny other desirle properties). All or nothing: Either TEQ is most ppeling SCF or it is severely flwed. 46

47 Shwrtz s Conjeture There exists no ounterexmple with less thn 13 lterntives (154 illion tournments hve een heked). TEQ stisfies ll nie properties if A <13. No ounterexmple ws found y serhing illions of rndom tournments with up to 50 lterntives. Cheking signifintly lrger tournments is intrtle. Mny non-trivil wekenings of Shwrtz s onjeture re known to hold (Good, 1971; Dutt, 1988; B. et l., 2010; B., 2011). Theorem (B., Chudnovsky, Kim, Liu, Norin, Sott, Seymour, nd Thomssé, 2012): Shwrtz s onjeture is flse. 47

48 Aftermth Non-onstrutive proof relying on proilisti rgument y Erdős nd Moser (1964) Neither the ounter-exmple nor its size n e dedued from proof. Smllest ounter-exmple of this type requires out lterntives. More reently, ounter-exmple with 24 lterntives ws found with the help of omputer (B. & Seedig, 2013). In priniple, TEQ is severely flwed. However, ounter- exmples re so extremely rre tht this hs no prtil onsequenes. This sts dout on the xiomti method. 48

49 Wekly Consistent SCFs TC UC BA TEQ Top Cyle (1971) TC expnsion O Unovered Set (1977) UC wek expnsion O Bnks Set (1985) BA strong retentiveness 2 Tournment Equilirium Set (1990) TEQ (retentiveness) 2 49

50 Espe Route #3 Rndomiztion 50

51 Soil Deision Shemes A soil deision sheme (SDS) mps preferene profile to lottery (proility distriution) p Δ(A) over the lterntives. Let g(x,y) = nxy - nyx e the mjority mrgin of x nd y. Alterntive x is Condoret winner if g(x,y) 0 for ll y A. g n e strightforwrdly extended to n expeted mjority mrgin g(p,q) = x,y A p(x) q(y) g(x,y). Lottery p is mximl if g(p,q) 0 for ll q Δ(A). Mximl lotteries re gurnteed to exist due to the minimx theorem nd re unique when N is odd (Lffond et l., 1997). 51

52 Mximl Lotteries Germine Krewers First studied y Krewers (1965) nd Fishurn (1984) redisovered y Lffond et l. (1993), Felsenthl nd Mhover (1992), Fisher nd Ryn (1995), Rivest nd Shen (2010) g n e seen s symmetri two-plyer zero-sum gme. Mximl lotteries re mixed minimx strtegies. Exmple The unique mximl lottery is 3/5 + 1/5 + 1/5. Peter C. Fishurn 52

53 Properties of Mximl Lotteries (ML) ML n e effiiently omputed vi LP. Preto-dominted lterntives lwys get zero proility in every mximl lottery. In ft, ML is even effiient with respet to stohsti dominne. - No lottery gives more expeted utility for ny utility representtion onsistent with the voters preferenes (Aziz et l., 2012). ML is wekly strtegyproof in well-defined sense (Aziz et l., 2012) ML n e uniquely hrterized using pproprite generliztions of onsisteny nd reinforement (Brndl et l., forthoming). The dilemm of soil hoie is resolved vi rndomiztion! 53

54 Tutoril Summry Rtionl hoie theory My s Theorem, Condoret s Prdox, Arrow s Theorem Three espe routes: reple onsisteny with vrile-eletorte ondition - Young s hrteriztion of soring rules (e.g., plurlity, Bord) weken onsisteny - top yle (expnsion) - unovered set (wek expnsion) - Bnks set (strong retentiveness) - tournment equilirium set (retentiveness) rndomiztion - mximl lotteries (rndomized Condoret winners) 54

55 Reommended Literture Introdutory ook hpter F. Brndt, V. Conitzer, nd U. Endriss. Computtionl Soil Choie. In "Multigent Systems" (G. Weiss, ed.), MIT Press, Books M. Allinghm: Choie Theory - A very short introdution. Oxford University Press, 2002 D. Austen-Smith nd J. Bnks: Positive Politil Theory I, University of Mihign Press, 1999 W. Gärtner: A Primer in Soil Choie Theory, Oxford University Press, 2009 J.-F. Lslier: Tournment Solutions nd Mjority Voting. Springer-Verlg, 1997 H. Moulin: Axioms of Coopertive Deision Mking. Cmridge University Press, 1988 S. Nitzn: Colletive Choie nd Preferene,