Elections and Strategic Voting: Condorcet and Borda. P. Dasgupta E. Maskin

Transcription

1 Electons and Strategc Votng: ondorcet and Borda P. Dasgupta E. Maskn

2 votng rule (socal choce functon) method for choosng socal alternatve (canddate) on bass of voters preferences (rankngs, utlt functons) promnent eamples Pluralt Rule (MPs n Brtan, members of ongress n U.S.) choose alternatve ranked frst b more voters than an other Majort Rule (ondorcet Method) choose alternatve preferred b majort to each other alternatve 2

3 Run-off Votng (presdental electons n France) choose alternatve ranked frst b more voters than an other, unless number of frst-place rankngs less than majort among top 2 alternatves, choose alternatve preferred b majort Rank-Order Votng (Borda ount) alternatve assgned 1 pont ever tme some voter ranks t frst, 2 ponts ever tme ranked second, etc. choose alternatve wth lowest pont total Utltaran Prncple choose alternatve that mamzes sum of voters utltes 3

4 Whch votng rule to adopt? Answer depends on what one wants n votng rule can specf crtera (aoms) votng rule should satsf see whch rules best satsf them One mportant crteron: nonmanpulablt voters shouldn t have ncentve to msrepresent preferences,.e., vote strategcall otherwse not mplementng ntended votng rule decson problem for voters ma be hard 4

5 But basc negatve result Gbbard-Satterthwate (GS) theorem f 3 or more alternatves, no votng rule s alwas nonmanpulable (ecept for dctatoral rules - - where one voter has all the power) Stll, GS overl pessmstc requres that votng rule never be manpulable but some crcumstances where manpulaton can occur ma be unlkel In an case, natural queston: Whch (reasonable) votng rule(s) nonmanpulable most often? Paper tres to answer queston 5

6 X = fnte set of socal alternatves socet conssts of a contnuum of voters [0,1] tpcal voter 0,1 reason for contnuum clear soon utlt functon for voter U : X R restrct attenton to strct utlt functons f, then U( ) U( ) = set of strct utlt functons U X profle U [ ] - - specfcaton of each ndvdual's utlt functon 6

7 votng rule (generalzed socal choce functon) F for all proflesu and all Y X, F U, Y Y ( ) F U, Y = optmal alternatve n Y f profle s U defnton sn t qute rght - - gnores tes wth pluralt rule, mght be two alternatves that are both ranked frst the most wth rank-order votng, mght be two alternatves that each get lowest number of ponts But eact tes unlkel wth man voters wth contnuum, tes are nongenerc so, correct defnton: for generc profle U and all Y X F U Y Y (, ) ( ) 7

8 pluralt rule: { ( ) = μ ( ) ( ) μ U a U b for all b for all a majort rule: rank-order votng: utltaran prncple: { } { ( ) ( ) } } P f U, Y a U a U b for all b ( ) μ ( ) ( ) { } U U ( ) = ( ) μ( ) ( ) μ( ) { } B f U, Y a r a d r b d for all b, ( ) = ( ) ( ) where r a # bu b U a U { } ( ) = ( ) μ() ( ) μ() U f U, Y a U a d U b d for all b 1 { { } 2 } f U, Y = a U a U b for all b 8

9 What propertes should reasonable votng rule satsf? ( ) ( ) for all Pareto Propert (P): f U > U and Y, then F( U, Y) f everbod prefers to, should not be chosen Anonmt (A): suppose [ ] [ ] π : 0,1 0,1 measure-preservng π permutaton. If U = U for all, then π () ( π ) = ( ) F U, Y F U, Y for all Y alternatve chosen depends onl on voters preferences and not who has those preferences voters treated smmetrcall 9

10 Neutralt (N): then Suppose ρ : Y Y permutaton. ρ Y ( ( )) ( ) alternatves treated smmetrcall ( ) ( ) ( ) ρ, Y, If U ρ > U ρ U > U for all,,, ρ, Y ( ) = ρ ( ) ( ) F U, Y F U, Y. All four votng rules pluralt, majort, rank-order, utltaran satsf P, A, N Net aom most controversal stll has qute compellng justfcaton nvoked b both Arrow (1951) and Nash (1950) 10

11 Independence of Irrelevant Alternatves (I): then ( ) f = F U, Y and Y Y (, ) F U Y = f chosen and some non-chosen alternatves removed, stll chosen Nash formulaton (rather than Arrow) no spolers (e.g. Nader n 2000 U.S. presdental electon, Le Pen n 2002 French presdental electon) 11

12 Majort rule and utltaransm satsf I, but others don t: pluralt rule.35 z.33 z.32 z (, {,, }) P f U z = (, {, }) P f U = rank-order votng.55 z.45 z (, {,, }) ( ) = B f U z B,{, } = f U 12

13 Fnal Aom: Nonmanpulablt (NM): then ( ) ( ) U = U j [ ] f = F U, Y and = F U, Y, where for all 0,1 j ( ) ( ) for some j U > U the members of coalton can t all gan from msrepresentng utlt functons as U 13

14 NM mples votng rule must be ordnal (no cardnal nformaton used) F s ordnal f whenever, for profles U and U, U > U U > U (*) F U, Y F U =, Y for all Y Lemma: If F satsfes NM and I, F ordnal ( ) ( ) ( ) ( ) for all,, ( ) ( ) ( ) ( ) ( { }) { } suppose = F U, Y = F U, Y, where U and U same ordnall F( U U { } ) = ( { }) = NM rules out utltaransm ( ) then = F U,, = F U,,, from I suppose f,,,, then wll manpulate f F U, U,,, then wll manpulate 14

15 But majort rule also volates NM F not even alwas defned.35 z.33 z.32 z (, {,, }) F U z = eample of ondorcet ccle F must be etended to ondorcet ccles one possblt F ( U, Y), f nonempt / B F ( U, Y) = B F U, Y, otherwse ( ) (Black's method) etensons make F vulnerable to manpulaton.35 z.33 z.32 z (, {,, }) / B F U z = z (, {,, }) = / B F U z z 15

16 Theorem: There ests no votng rule satsfng P,A,N,I and NM Proof: smlar to that of GS overl pessmstc - - man cases n whch some rankngs unlkel 16

17 Lemma: Majort rule satsfes all 5 propertes f and onl f preferences restrcted to doman wth no ondorcet ccles When can we rule out ondorcet ccles? preferences sngle-peaked 2000 US electon Nader Gore Bush unlkel that man had rankng Bush Nader or Nader Bush strongl-felt canddate Gore Gore n 2002 French electon, 3 man canddates: hrac, Jospn, Le Pen voters ddn t feel strongl about hrac and Jospn felt strongl about Le Pen (ranked hm frst or last) 17

18 Votng rule Fworkswellon doman U f satsfes P,A,N,I,NM when utlt functons restrcted to U e.g., F works well when preferences sngle-peaked 18

19 Theorem 1: Suppose F works well on doman U, then F works well on U too. onversel, suppose that F works well on U. Then f there ests profle on U such that U ( ) ( ) FU, Y F U, Y for some Y, there ests doman U on whch F works well but F does not Proof: From NM and I, f F works well on U, F must be ordnal Hence result follows from Dasgupta-Maskn (2008), JEEA shows that Theorem 1 holds when NM replaced b ordnalt 19

20 To show ths D-M uses Lemma: F works well on U f and onl f U has no ondorcet ccles Suppose F works well on U If F doesn't work well on U, Lemma mples U must contan ondorcet ccle z z z 20

21 onsder 1 2 n 1 U = z z z so 1 ( { }) (*) Suppose F U,, z = z 2 U = n z z z z ( ) ( { }) ( ) 2 2 { } ( 2 2 { }) { } F U,,, z = (from I) F U,, z =, contradcts (*) F U,,, z = (from I) F U,, =, contradcts (*) (A,N) ( 2, {,, }) { } F U z = z ( ) 2 so F U, so for, z = z (I) n 3 U = z z z z ( { }) F U 3,, z = z (N) 4 ontnung n the same wa, let U = ( { }) F U 4,, z = z, contradcts (*) 1 n 1 n z z z 21

22 So F can t work well on U wth ondorcet ccle onversel, suppose that F works well on U and ( ) ( ) F U, Y F U, Y for some U and Y Then there est α wth 1 α > α and U = 1 α such that α (, {, }) and, {, } ( ) = F U = F U But not hard to show that F unque votng rule satsfng P,A,N, and NM when X = contradcton 22

23 Let s drop I most controversal no votng rule satsfes P,A,N,NM on GS agan U X F works ncel on U f satsfes P,A,N,NM on U 23

24 Theorem 2: X = 3 Suppose F works ncel on U, then B F or F works ncel on U too. onversel suppose works ncel on, where B F U F = F or F. Then, f there ests profle U on U such that * there ests doman U on whch F works ncel but F does not Proof: F works ncel on an ondorcet-ccle-free doman B F ( ) ( ) B so F and F complement each other F U, Y F U, Y for some Y, works ncel onl when U s subset of ondorcet ccle f F works ncel on U and U doesn't contan ondorcet ccle, F works ncel too f F works ncel on U and U contans ondorcet ccle, then U can't contan an other rankng (otherwse no votng rule works ncel) B so F works ncel on U. 24

25 Strkng that the 2 longest-studed votng rules (ondorcet and Borda) are also onl two that work ncel on mamal domans 25