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Voting Theory. Toby Walsh NICTA and UNSW. Motivation. Why voting? Consider multiple agents Each declares their preferences (order over outcomes) How do we make some collective decision? Use a voting rule!. Voting rule Social choice: mapping of a profile onto a winner(s)

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Voting theory l.jpg

Voting Theory

Toby Walsh

NICTA and UNSW


Motivation l.jpg
Motivation

  • Why voting?

    • Consider multiple agents

    • Each declares their preferences (order over outcomes)

    • How do we make some collective decision?

    • Use a voting rule!


Terminology l.jpg

Voting rule

Social choice: mapping of a profile onto a winner(s)

Social welfare: mapping of a profile onto a total ordering

Agent

Usually assume odd number of agents to reduce ties

Vote

Total order over outcomes

Profile

Vote for each agent

Terminology

Extensions include indifference,

incomparability, incompleteness


Voting rules plurality l.jpg
Voting rules: plurality

  • Otherwise known as “majority” or “first past the post”

    • Candidate with most votes wins

  • With just 2 candidates, this is a very good rule to use

    • (See May’s theorem)


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Voting rules: plurality

  • Some criticisms

    • Ignores preferences other than favourite

    • Similar candidates can “split” the vote

    • Encourages voters to vote tactically

      • “My candidate cannot win so I’ll vote for my second favourite”


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Two rounds

Eliminate all but the 2 candidates with most votes

Then hold a majority election between these 2 candidates

Consider

25 votes: A>B>C

24 votes: B>C>A

46 votes: C>A>B

1st round: B knocked out

2nd round: C>A by 70:25

C wins

Voting rules: plurality with runoff


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Voting rules: plurality with runoff

  • Some criticisms

    • Requires voters to list all preferences or to vote twice

    • Moving a candidate up your ballot may not help them (monotonicity)

    • It can even pay not to vote! (see next slide)


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Consider again

25 votes: A>B>C

24 votes: B>C>A

46 votes: C>A>B

C wins easily

Two voters don’t vote

23 votes: A>B>C

24 votes: B>C>A

46 votes: C>A>B

Different result

1st round: A knocked out

2nd round: B>C by 47:46

B wins

Voting rules: plurality with runoff


Voting rules single transferable vote l.jpg

STV

If one candidate has >50% vote then they are elected

Otherwise candidate with least votes is eliminated

Their votes transferred (2nd placed candidate becomes 1st, etc.)

Identical to plurality with runoff for 3 candidates

Example:

39 votes: A>B>C>D

20 votes: B>A>C>D

20 votes: B>C>A>D

11 votes: C>B>A>D

10 votes: D>A>B>C

Result: B wins!

Voting rules: single transferable vote


Voting rules borda l.jpg
Voting rules: Borda

  • Given m candidates

    • ith ranked candidate score m-i

    • Candidate with greatest sum of scores wins

  • Example

    • 42 votes: A>B>C>D

    • 26 votes: B>C>D>A

    • 15 votes: C>D>B>A

    • 17 votes: D>C>B>A

    • B wins

Jean Charles de Borda, 1733-1799


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Voting rules: positional rules

  • Given vector of weights, <s1,..,sm>

    • Candidate scores si for each vote in ith position

    • Candidate with greatest score wins

  • Generalizes number of rules

    • Borda is <m-1,m-2,..,0>

    • Plurality is <1,0,..,0>


Voting rules approval l.jpg
Voting rules: approval

  • Each voters approves between 1 and m-1 candidates

  • Candidate with most votes of approval wins

  • Some criticisms

    • Elects lowest common denominator?

    • Two similar candidates do not divide vote, but can introduce problems when we are electing multiple winners


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Voting rules: other

  • Cup (aka knockout)

    • Tree of pairwise majority elections

  • Copeland

    • Candidate that wins the most pairwise competitions

  • Bucklin

    • If one candidate has a majority, they win

    • Else 1st and 2nd choices are combined, and we repeat


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Voting rules: other

  • Coomb’s method

    • If one candidate has a majority, they win

    • Else candidate ranked last by most is eliminated, and we repeat

  • Range voting

    • Each voter gives a score in given range to each candidate

    • Candidate with highest sum of scores wins

    • Approval is range voting where range is {0,1}


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Voting rules: other

  • Maximin (Simpson)

    • Score = Number of voters who prefer candidate in worst pairwise election

    • Candidate with highest score wins

  • Veto rule

    • Each agent can veto up to m-1 candidates

    • Candidate with fewest vetoes wins

  • Inverse plurality

    • Each agent casts one vetor

    • Candidate with fewest vetoes wins


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Voting rules: other

  • Dodgson

    • Proposed by Lewis Carroll in 1876

    • Candidate who with the fewest swaps of adjacent preferences beats all other candidates in pairwise elections

    • NP-hard to compute winner!

  • Random

    • Winner is that of a random ballot


Voting rules l.jpg
Voting rules

  • So many voting rules to choose from ..

  • Which is best?

    • Social choice theory looks at the (desirable and undesirable) properties they possess

    • For instance, is the rule “monotonic”?

    • Bottom line: with more than 2 candidates, there is no best voting rule


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Axiomatic approach

  • Define desired properties

    • E.g. monotonicity: improving votes for a candidate can only help them win

  • Prove whether voting rule has this property

    • In some cases, as we shall see, we’ll be able to prove impossibility results (no voting rule has this combination of desirable properties)


May s theorem l.jpg
May’s theorem

  • Some desirable properties of voting rule

    • Anonymous: names of voters irrelevant

    • Neutral: name of candidates irrelevant


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May’s theorem

  • Another desirable property of a voting rule

    • Monotonic: if a particular candidate wins, and a voter improves their vote in favour of this candidate, then they still win

      • Non-monotonicity for plurality with runoff

        • 27 votes: A>B>C

        • 42 votes: C>A>B

        • 24 votes: B>C>A

      • Suppose 4 voters in 1st group move C up to top

        • 23 votes: A>B>C

        • 46 votes: C>A>B

        • 24 votes: B>C>A


May s theorem21 l.jpg
May’s theorem

  • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule

    • May, Kenneth. 1952. "A set of independent necessary and sufficient conditions for simple majority decisions", Econometrica, Vol. 20, pp. 680–68

    • Since these properties are uncontroversial, this about decides what to do with 2 candidates!


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May’s theorem

  • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule

    • Proof: Plurality rule is clearly anonymous, neutral and monotonic

    • Other direction is more interesting


May s theorem23 l.jpg
May’s theorem

  • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule

    • Proof: Anonymous and neutral implies only number of votes matters

    • Two cases:

      • N(A>B) = N(B>A)+1 and A wins.

        • By monotonicity, A wins whenever N(A>B) > N(B>A)


May s theorem24 l.jpg
May’s theorem

  • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule

    • Proof: Anonymous and neutral implies only number of votes matters

    • Two cases:

      • N(A>B) = N(B>A)+1 and A wins.

        • By monotonicity, A wins whenever N(A>B) > N(B>A)

      • N(A>B) = N(B>A)+1 and B wins

        • Swap one vote A>B to B>A. By monotonicity, B still wins. But now N(B>A) = N(A>B)+1. By neutrality, A wins. This is a contradiction.


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Condorcet’s paradox

  • Collective preference may be cyclic

    • Even when individual preferences are not

  • Consider 3 votes

    • A>B>C

    • B>C>A

    • C>A>B

    • Majority prefer A to B, and prefer B to C, and prefer C to A!

Marie Jean Antoine Nicolas de Caritat,

marquis de Condorcet (1743 – 1794)


Condorcet principle l.jpg
Condorcet principle

  • Turn this on its head

  • Condorcet winner

    • Candidate that beats every other in pairwise elections

    • In general, Condorcet winner may not exist

    • When they exist, must be unique

  • Condorcet consistent

    • Voting rule that elects Condorcet winner when they exist (e.g. Copeland rule)


Condorcet principle27 l.jpg
Condorcet principle

  • Plurality rule is not Condorcet consistent

    • 35 votes: A>B>C

    • 34 votes: C>B>A

    • 31 votes: B>C>A

    • B is easily the Condorcet winner, but plurality elects A


Condorcet principle28 l.jpg
Condorcet principle

  • Thm. No positional rule with strict ordering of weights is Condorcet consistent

    • Proof: Consider

      • 3 votes: A>B>C

      • 2 votes: B>C>A

      • 1 vote: B>A>C

      • 1 vote: C>A>B

    • A is Condorcet winner


Condorcet principle29 l.jpg
Condorcet principle

  • Thm. No positional rule with strict ordering of weights is Condorcet consistent

    • Proof: Consider

      • 3 votes: A>B>C

      • 2 votes: B>C>A

      • 1 vote: B>A>C

      • 1 vote: C>A>B

    • Scoring rule with s1 > s2 > s3

      • Score(B) = 3.s1+3.s2+1.s3

      • Score(A) = 3.s1+2.s2+2.s3

      • Score(C) = 1.s1+2.s2+4.s4

      • Hence: Score(B)>Score(A)>Score(C)


Arrow s theorem l.jpg
Arrow’s theorem

  • We have to break Condorcet cycles

    • How we do this, inevitably leads to trouble

  • A genius observation

    • Led to the Nobel prize in economics


Arrow s theorem31 l.jpg
Arrow’s theorem

  • Free

    • Every result is possible

  • Unanimous

    • If every votes for one candidate, they win

  • Independent to irrelevant alternatives

    • Result between A and B only depends on how agents preferences between A and B

  • Monotonic


Arrow s theorem32 l.jpg
Arrow’s theorem

  • Non-dictatorial

    • Dictator is voter whose vote is the result

    • Not generally considered to be desirable!


Arrow s theorem33 l.jpg
Arrow’s theorem

  • Thm: If there are at least two voters and three or more candidates, then it is impossible for any voting rule to be:

    • Free

    • Unanimous

    • Independent to irrelevant alternatives

    • Monotonic

    • Non-dictatorial


Arrow s theorem34 l.jpg
Arrow’s theorem

  • Can give a stronger result

    • Weaken conditions

  • Pareto

    • If everyone prefers A to B then A is preferred to B in the result

    • If free & monotonic & IIA then Pareto

    • If free & Pareto & IIA then not necessarily monotonic


Arrow s theorem35 l.jpg
Arrow’s theorem

  • Thm: If there are at least two voters and three or more candidates, then it is impossible for any voting rule to be:

    • Pareto

    • Independent to irrelevant alternatives

    • Non-dictatorial


Arrow s theorem36 l.jpg
Arrow’s theorem

  • With two candidates, majority rule is:

    • Pareto

    • Independent to irrelevant alternatives

    • Non-dictatorial

  • So, one way “around” Arrow’s theorem is to restrict to two candidates


Proof of arrow s theorem l.jpg
Proof of Arrow’s theorem

  • If all voters put B at top or bottom then result can only have B at top or bottom

    • Suppose not the case and result has A>B>C

    • By IIA, this would not change if every voter moved C above A:

      • B>A>C => B>C>A

      • B>C>A => B>C>A

      • A>C>B => C>A>B

      • C>A>B => C>A>B

      • Each AB and BC vote the same!


Proof of arrow s theorem38 l.jpg
Proof of Arrow’s theorem

  • If all voters put B at top or bottom then result can only have B at top or bottom

    • Suppose not the case and result has A>B>C

    • By IIA, this would not change if every voter moved C above A

    • By transitivity A>C in result

    • But by unanimity C>A

      • B>A>C => B>C>A

      • B>C>A => B>C>A

      • A>C>B => C>A>B

      • C>A>B => C>A>B


Proof of arrow s theorem39 l.jpg
Proof of Arrow’s theorem

  • If all voters put B at top or bottom then result can only have B at top or bottom

    • Suppose not the case and result has A>B>C

    • A>C and C>A in result

    • This is a contradiction

    • B can only be top or bottom in result


Proof of arrow s theorem40 l.jpg
Proof of Arrow’s theorem

  • If all voters put B at top or bottom then result can only have B at top or bottom

  • Suppose voters in turn move B from bottom to top

  • Exists pivotal voter from whom result changes from B at bottom to B at top


Proof of arrow s theorem41 l.jpg
Proof of Arrow’s theorem

  • If all voters put B at top or bottom then result can only have B at top or bottom

  • Suppose voters in turn move B from bottom to top

  • Exists pivotal voter from whom result changes from B at bottom to B at top

    • B all at bottom. By unanimity, B at bottom in result

    • B all at top. By unanimity, B at top in result

    • By monotonicity, B moves to top and stays there when some particular voter moves B up


Proof of arrow s theorem42 l.jpg
Proof of Arrow’s theorem

  • If all voters put B at top or bottom then result can only have B at top or bottom

  • Suppose voters in turn move B from bottom to top

  • Exists pivotal voter from whom result changes from B at bottom to B at top

  • Pivotal voter is dictator


Proof of arrow s theorem43 l.jpg
Proof of Arrow’s theorem

  • Pivotal voter is dictator

    • Consider profile when pivotal voter has just moved B to top (and B has moved to top of result)

    • For any AC, let pivotal voter have A>B>C

    • By IIA, A>B in result as AB votes are identical to profile just before pivotal vote moves B (and result has B at bottom)

    • By IIA, B>C in result as BC votes are unchanged

    • Hence, A>C by transitivity


Proof of arrow s theorem44 l.jpg
Proof of Arrow’s theorem

  • Pivotal voter is dictator

    • Consider profile when pivotal voter has just moved B to top (and B has moved to top of result)

    • For any AC, let pivotal voter have A>B>C

    • Then A>C in result

    • This continues to hold even if any other voters change their preferences for A and C

    • Hence pivotal voter is dicatator for AC

    • Similar argument for AB


Arrow s theorem45 l.jpg
Arrow’s theorem

  • How do we get “around” this impossibility

    • Limit domain

      • Only two candidates

    • Limit votes

      • Single peaked votes

    • Limit properties

      • Drop IIA


Single peaked votes l.jpg
Single peaked votes

  • In many domains, natural order

    • Preferences single peaked with respect to this order

  • Examples

    • Left-right in politics

    • Cost (not necessarily cheapest!)

    • Size


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Single peaked votes

  • There are never Condorcet cycles

  • Arrow’s theorem is “escaped”

    • There exists a rule that is Pareto

    • Independent to irrelevant alternatives

    • Non-dictatorial

    • Median rule: elect “median” candidate

      • Candidate for whom 50% of peaks are to left/right


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Conclusions

  • Many voting rules exist

    • Plurality, STV, approval, Copeland, …

  • For two candidates

    • “Best” rule is plurality

  • For more than two candidates

    • Arrow’s theorem proves there is no “best” rule

    • But there are limited ways around this (e.g. single peaked votes)


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