How Should Presidents Be Elected?

E. MaskinInstitute for Advanced Study

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Voting rule • method for choosing candidate on basis of

– ballot (set of available candidates)– voters’ preferences (rankings) over candidates

• prominent examples– plurality rule (MPs in Britain, members of Congress in

U.S.)choose candidate ranked first by more voters than any other

– majority rule (Condorcet Method)choose candidate preferred by majority to each other candidate

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− run-off voting (presidential elections in France)• choose candidate ranked first by more voters than any

other, unless gets less than majorityamong top 2 candidates, choose one preferred by majority

− rank-order voting (Borda Count)• candidate assigned 1 point every time voter ranks her

first, 2 points every time ranked second, etc.• choose candidate with lowest vote total

− utilitarianism• choose candidate that maximizes sum of voters’ utilities

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• Which voting rule to adopt?• Voting theorist’s answer:

– specify what one wants (axioms)– see which rules satisfy the axioms

• But basic negative results:Arrow Impossibility TheoremGibbard-Satterthwaite Theorem– if 3 or more candidates, no voting rule satisfies

set of compelling axioms

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• Prompts question:Which voting rule(s) satisfy axioms most often?

• try to answer question today– based on series of papers with P. Dasgupta

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• X = finite set of conceivable candidates• society = continuum of voters [0,1]

– – reason for continuum clear soon

• utility function for voter i– restrict attention to strict utility functions

• profile

typical voter 0,1i

:iU X

if , then i ix y U x U y = set of strict utility functionsXU

- - specification of each voter 's utility functionU i

, 0,1iU i

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• voting rule F

− winning candidates (could be ties)

• Example – plurality rule

, for all

for all

for all

Pluri i

i i

F U Y x i U x U y y

i U x U y y

x

for each profile and ballot ,

,U Y X

F U Y Y

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Axioms

– that is, there is clear-cut winner

• but that is too much to ask for• e.g., with plurality rule

– exact ties are possiblemight be 2 candidates who are both ranked first the most

• still, with large number of voters, tie very unlikely• similarly with majority rule, rank-order voting, etc.• continuum helps formalize this

(D): For all and , , is a Decisiveness U Y F U Y singleton

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•

– fraction of electorate who prefer x to y

•

•

– profile is regular if proportions of voters preferring one candidate to another fall outside exceptional set:

• Generic Decisiveness (GD): There exists finite S such that

•

Let be subset of - - S exceptional set

Let ,U i iq x y i U x U y

, ,U Uq x y q y x S

generically decisive with exceptional set 1PlurF S

for all and regular w.r.t. , , a singletonY U S F U Y

profile is with respect to if, for all U regular S x y

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• if everybody prefers x to y, then y should not win when x is on the ballot

• winner should not depend on which voter has which preferences– names don’t matter– all voters treated symmetrically

• all candidates treated symmetrically

(P): For all , , and , ,Pareto Property U Y x y Y

(A): Suppose : 0,1 0,1 (measure-preserving)Anonymity

If, for all , is profile with for all , then, for all ,iU U U U i Y

if for all , then ,i iU x U y i y F U Y

, ,F U Y F U Y

(N): For all , , and permutation of , letNeutrality Y U Y

, be profile such that for all , ,Y

i i i i

U i x y

U x U y U x U y

,Then , , .YF U Y F U Y

permutation of 0,1

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• only preference rankings matter– strength of preference not captured

• rules out utilitarianism• reflects standard idea that preference strength has no

operational meaning if only one “good” (no choice experiment)

• implied by later axiom• could be replaced by cardinality

– (as in von Neumann-Morgenstern preferences)

• what’s important: no interpersonal comparisons– such comparisons not operational

, , , where , 0,1i iF U Y F U Y U U i

(O): For all , , and increasing functions: , 0,1i

Ordinality U Yi

must be positive transformationsi affine

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• next axiom most controversialbut– has compelling rationale– invoked by Arrow (1951) and Nash (1950)

•

then

– if x elected and then some non-elected candidate removed from ballot, x still elected

– no “spoilers” (e.g. Nader in 2000 U.S. presidential election, Le Pen in 2002 French presidential election)

– Nash formulation

if , and x F U Y x Y Y

,x F U Y

(I): for all and Independence of Irrelevant Candidates U Y

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• plurality rule violates I

• so do rank-order voting and run-off voting• majority rule satisfies I but violates GD

, ,PlurF U x y y

.35xyz

, , ,PlurF U x y z x

.33yzx

.35xyz

.32zyx

.33yzx

Condorcet cycle.32zxy

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Theorem 1: No voting rule satisfies GD, P, A, N, O, and I– analog of Arrow Impossibility Theorem

• But natural follow-up question:Which voting rule (s) satisfies axioms as often as possible?– have to clarify “as often as possible”

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• F

•

– – –

2: Suppose works well on Theorem F U works well if consists of single-peaked preferencesMajF U

, , for some ,MajF U Y F U Y Y

Conversely, suppose works well on MajF U

if works well on domain, does tooMajF Fif there exists domain where works well and doesn'tMaj MajF F F F

on if it satisfiesXworks well restricted domain U UGD, P, A, N, O, I when profiles drawn from U U

there exists on which works well and doesn'tMajF FU

so works more often than MajF F

Then if on regular w.r.t. 's exceptional set andU F U

Then also works well on MajF U

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i.e.,

: works well on doesn't contain Condorcet cyclesMajLemma F U U

don't all belong to Ux y zy z xz x y

violates GD for Condorcet cycle

Condorcet cycle is only thing that can go wrong

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•

• Let S be exceptional for F. Because S finite, there exists n such that

− if divide population into n equal groups− assign everybody in a group same utility function

− resulting profile regular w.r.t. S.

If doesn't work well on , then there existsMajF U

2: Suppose works well on Proof of Theorem F U

x y zy z xz x y

U

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• Consider

• Consider

–

– then

regular w.r.t. S 1 2 nx z zz x x

1 2 3 nx y z zy z x xz x y y

then , , (I)F x z x

if , , ,F x y z y

1 2 3( )

nx y x x y Iy x y y

contradicts ( ) by (N)

assume , ,F x z z

if , , ,F x y z x

contradicts

1 2 3( )

ny x x x y Ax y y y

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–

Continuing iteratively

regular w.r.t. S 1 2 nx z zz x x

So doesn't work well on F U

assume , ,F x z z

So , , ,F x y z z

Contradicts

1 2 3 nx x z z z Nz z x x

1 2 3 ny y z z zz z y y

1 2 1n nx x x zz z z x

z

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Conversely, suppose

•

•

•

• F does not

works well on MajF UMajF F

To show: there exists whereU works wellMajF

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Consider

•

•

•

–

–

•

•

Fm k k yx yy x

x x yz y xy z z

U

then , , by IF y z y

if , , , , contradicts and IF U x y z x

So doesn't work well on F U

There exist , with -m k m k k

if , , , , contradicts PF U x y z z

contradicts by N

2k m k kUx x yz y xy z z

if , , ,F U x y z y

But work well on MajF does U

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Final Axiom:

• Nonmanipulability (NM):

then

– the members of coalition C can’t all gain from misrepresenting

if , and , ,

where for all 0,1j j

x F U Y x F U Y

U U j C

for some i iU x U x i C

utility functions as iU

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• Lemma: If F satisfies NM and I, F satisfies O– –

–

– –

• NM rules out utilitarianism

suppose , , , where and same ordinallyx F U Y y F U Y U U

then , , , , , from Ix F U x y y F U x y

Cyx

supposeCxy

if , , , , then will manipulateC CF U U x y y C

if , , , , then will manipulateC CF U U x y x C

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But majority rule also violates NM

• – – one possibility

– –

must be extended to Condorcet cyclesMajF

.35xyz

.33yzx

/ , , ,Maj BF U x y z z

/ , , ,Maj BF U x y z x

because not even always MajF defined

zyx

.32zxy

/

, , if nonempty,

, , otherwise

Maj

Maj B

B

F U YF U Y

F U Y

(Black's method)

extensions make vulnerable to manipulationMajF

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Theorem 3: There exists no voting rule satisfying GD, P,A,N,I and NM

Proof: similar to that of Gibbard-Satterthwaite Theorem

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•

–

Voting rule on domain if satisfies GD,P, A,N,I,NM when utility functions restricted to

F works very well UU

e.g., works very well when preferences single-peakedMajF

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•

• Conversely, suppose

Proof: From NM and I, if F works very well on U , F must be ordinal• Hence result follows from

– Theorem 2–

: Suppose works very well on domain then works very well on too.

MajTheorem 4 F FUU

that works very well on .Maj CF U

Then if there exisits regular profile on such thatCU U

, , for some ,MajF U Y F U Y Y

there exists domain on which works very well but does notMajF FU

lemma: if works well on , then works very well on MajF U U

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• Let’s drop I– most controversial

• – GS again

• F works nicely on U if satisfies GD,P,A,N,NM on U

voting rule satisfies GD,P,A,N,NM on Xno U

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Theorem 5: • Suppose F works nicely on U ,

•

Proof:•

•

• –

–

–

works nicely on any Condorcet-cycle-free domainMajF

Conversely suppose works nicely on , where or .Maj BF F F F UThen, if there exisits regular profile on such thatU

U , , for some ,F U Y F U Y Y

*there exists domain on which works nicely but does notF FU

works nicely only when is subset of Condorcet cycleBF U

then or works nicely on too.Maj BF F U

so and complement each otherMaj BF Fif works nicely on and doesn't contain Condorcet cycle, works nicely tooMajF FU U

if works nicely on and contains Condorcet cycle, then can't contain anyother ranking (otherwise voting rule works nicely)

Fno

U U U

so works nicely on .BF U

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Striking that the 2 longest-studied voting rules (Condorcet and Borda) are also

• only two that work nicely on maximal domains