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

Transcription

1 Elections and Strategic Voting: ondorcet and Borda P. Dasgupta E. Maskin

2 voting rule (social choice function) method for choosing social alternative (candidate) on basis of voters preferences (rankings, utilit functions) prominent eamples Pluralit Rule (MPs in Britain, members of ongress in U.S.) choose alternative ranked first b more voters than an other Majorit Rule (ondorcet Method) choose alternative preferred b majorit to each other alternative 2

3 Run-off Voting (presidential elections in France) choose alternative ranked first b more voters than an other, unless number of first-place rankings less than majorit among top 2 alternatives, choose alternative preferred b majorit Rank-Order Voting (Borda ount) alternative assigned 1 point ever time some voter ranks it first, 2 points ever time ranked second, etc. choose alternative with lowest point total Utilitarian Principle choose alternative that maimizes sum of voters utilities 3

4 Which voting rule to adopt? Answer depends on what one wants in voting rule can specif criteria (aioms) voting rule should satisf see which rules best satisf them One important criterion: nonmanipulabilit voters shouldn t have incentive to misrepresent preferences, i.e., vote strategicall otherwise not implementing intended voting rule decision problem for voters ma be hard 4

5 But basic negative result Gibbard-Satterthwaite (GS) theorem if 3 or more alternatives, no voting rule is alwas nonmanipulable (ecept for dictatorial rules - - where one voter has all the power) Still, GS overl pessimistic requires that voting rule never be manipulable but some circumstances where manipulation can occur ma be unlikel In an case, natural question: Which (reasonable) voting rule(s) nonmanipulable most often? Paper tries to answer question 5

6 X = finite set of social alternatives societ consists of a continuum of voters [0,1] tpical reason for continuum clear soon utilit function for voter i Ui : X R restrict attention to strict utilit functions if U X profile [ ] voter i 0,1 ( ) ( ), then Ui Ui = set of strict utilit functions U - - specification of each individual's utilit function 6

7 voting rule (generalized social choice function) F for all profilesu and all Y X, F U, Y Y ( ) F U, Y = optimal alternative in Y if profile is U definition isn t quite right - - ignores ties with pluralit rule, might be two alternatives that are both ranked first the most with rank-order voting, might be two alternatives that each get lowest number of points But eact ties unlikel with man voters with continuum, ties are nongeneric so, correct definition: for generic profile U and all Y X F U Y Y (, ) ( ) 7

8 pluralit rule: { ( ) = µ ( ) ( ) µ iui a Ui b for all b for all a majorit rule: rank-order voting: utilitarian principle: { i i } { ( ) ( ) } } P f U, Y a iu a U b for all b ( ) µ ( ) ( ) { } U U ( ) = ( ) µ ( ) ( ) µ ( ) i { } B f U, Y a r a d i r b d i for all b, i ( ) = ( ) ( ) where r a # bu b U a U i i i { } i i ( ) = ( ) µ ( ) ( ) µ ( ) U f U, Y a U a d i U b d i for all b 1 { { i i } 2 } f U, Y = a iu a U b for all b 8

9 What properties should reasonable voting rule satisf? ( ) ( ) for all Pareto Propert (P): if Ui > Ui i and Y, then F( U, Y) if everbod prefers to, should not be chosen [ ] [ ] Anonmit (A): suppose π : 0,1 0,1 measure-preserving π permutation. If Ui = Uπ () i for all i, then π F U, Y = F U, Y for all Y ( ) ( ) alternative chosen depends onl on voters preferences and not who has those preferences voters treated smmetricall 9

10 Neutralit (N): then Suppose ρ : Y Y permutation. ρ Y ( ( )) ( ) alternatives treated smmetricall ( ) ( ) ( ) ρ, Y, If Ui ρ > Ui ρ Ui > Ui for all i,,, ρ, Y ( ) = ρ ( ) ( ) F U, Y F U, Y. All four voting rules pluralit, majorit, rank-order, utilitarian satisf P, A, N Net aiom most controversial still has quite compelling justification invoked b both Arrow (1951) and Nash (1950) 10

11 Independence of Irrelevant Alternatives (I): then ( ) if = F U, Y and Y Y (, ) F U Y = if chosen and some non-chosen alternatives removed, still chosen Nash formulation (rather than Arrow) no spoilers (e.g. Nader in 2000 U.S. presidential election, Le Pen in 2002 French presidential election) 11

12 Majorit rule and utilitarianism satisf I, but others don t: pluralit rule.35 z.33 z.32 z (, {,, }) P f U z = (, {, }) P f U = rank-order voting.55 z.45 z (, {,, }) ( ) = B f U z = B,{, } f U 12

13 Final Aiom: Nonmanipulabilit (NM): then ( ) ( ) U = U j [ ] if = F U, Y and = F U, Y, where for all 0,1 i j ( ) ( ) for some i j U > U i the members of coalition can t all gain from misrepresenting utilit functions as U i 13

14 NM implies voting rule must be ordinal (no cardinal information used) F is ordinal if whenever, U U U > U U > U i (*) F U, Y F U =, Y for all Y Lemma: If F satisfies NM and I, F ordinal ( ) ( ) ( ) ( ) for all,, ( ) ( ) i i i i ( ) ( ) { } FU ( U FU ( U { } ) = { } ) = NM rules out utilitarianism for profiles and, suppose = F U, Y = F U, Y, where U and U same ordinall ( ) ( { }) then = FU,, = FU,,, from I suppose if,,,, then will manipulate if,,,, then will manipulate 14

15 But majorit rule also violates NM F not even alwas defined.35 z.33 z.32 z (, {,, }) F U z = eample of ondorcet ccle F must be etended to ondorcet ccles one possibilit F / B ( U, Y) ( U Y) F,, if nonempt = B F,, otherwise ( U Y) (Black's method) etensions make.35 z.33 z.32 z F vulnerable to manipulation (, {,, }) / B F U z = z (, {,, }) = / B F U z z 15

16 Theorem: There eists no voting rule satisfing P,A,N,I and NM Proof: similar to that of GS overl pessimistic - - man cases in which some rankings unlikel 16

17 Lemma: Majorit rule satisfies all 5 properties if and onl if preferences restricted to domain with no ondorcet ccles When can we rule out ondorcet ccles? preferences single-peaked 2000 US election Nader Gore Bush unlikel that man had ranking Bush Nader or Nader Bush strongl-felt candidate Gore Gore in 2002 French election, 3 main candidates: hirac, Jospin, Le Pen voters didn t feel strongl about hirac and Jospin felt strongl about Le Pen (ranked him first or last) 17

18 Voting rule F works well on domain U if satisfies P,A,N,I,NM when utilit functions restricted to U e.g., F works well when preferences single-peaked 18

19 Theorem 1: Suppose F works well on domain U, then F works well on U too. onversel, suppose that F works well on U. Then if there eisits profile on U such that U ( ) ( ) F U, Y F U, Y for some Y, there eists domain U on which F works well but F does not Proof: From NM and I, if F works well on U, F must be ordinal Hence result follows from Dasgupta-Maskin (2008), JEEA shows that Theorem 1 holds when NM replaced b ordinalit 19

20 To show this D-M uses Lemma: F works well on U if and onl if U has no ondorcet ccles Suppose F works well on U If F doesn't work well on U, Lemma implies U must contain ondorcet ccle z z z 20

21 onsider 1 2n 1 U = z z z so 1 ( { }) (*) Suppose FU, z, = z 2 U = n z z z z ( 2 2 { }) ( { }) ( 2 2 { }) ( { }) FU ( { z} ) = z ( 2 { }) = F U,,, z = (from I) F U,, z =, contradicts (*) FU, z,, = (from I) FU,, =, contradicts (*) (A,N) so FU, z, z (I) so for n 3 U = z z z z ( { }) FU 3, z, = z (N) 4 ontinuing in the same wa, let U = ( { }) FU 4, z, = z, contradicts (*) 1n 1 n z z z 21

22 So F can t work well on U with ondorcet ccle onversel, suppose that F works well on U and ( ) ( ) F U, Y F U, Y for some U and Y Then there eist α with 1 α > α and U = 1 α such that α (, {, }) and, {, } But not hard to show that ( ) = F U = FU when F X = contradiction unique voting rule satisfing P,A,N, and NM 22

23 Let s drop I most controversial no voting rule satisfies P,A,N,NM on GS again U X F works nicel on U if satisfies P,A,N,NM on U 23

24 Theorem 2: X = 3 Suppose F works nicel on U, B then F or F works nicel on U too. onversel * there eists domain U on which F works nicel but F does not Proof: F works nicel on an ondorcet-ccle-free domain F B B so F and F complement each other suppose works nicel on, where B F U F = F or F. Then, if there eisits profile U on U such that works nicel onl when ( ) ( ) F U, Y F U, Y for some Y, U is subset of ondorcet ccle if F works nicel on U and U doesn't contain ondorcet ccle, F works nicel too if F works nicel on U and U contains ondorcet ccle, then U can't contain an other ranking (otherwise no voting rule works nicel) B so F works nicel on U. 24

25 Striking that the 2 longest-studied voting rules (ondorcet and Borda) are also onl two that work nicel on maimal domains 25