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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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Motivation

- Why voting?
- Consider multiple agents
- Each declares their preferences (order over outcomes)
- How do we make some collective decision?
- Use a 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

TerminologyExtensions include indifference,

incomparability, incompleteness

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)

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”

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 runoffVoting 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)

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 runoffIf 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 voteVoting 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

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

- 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

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

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}

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

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

- 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

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

- Some desirable properties of voting rule
- Anonymous: names of voters irrelevant
- Neutral: name of candidates irrelevant

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

- Non-monotonicity for plurality with runoff

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

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!

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 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 A wins.

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.

- N(A>B) = N(B>A)+1 and A wins.

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

- 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 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 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

- Proof: Consider

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)

- Proof: Consider

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 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 theorem

- Non-dictatorial
- Dictator is voter whose vote is the result
- Not generally considered to be desirable!

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 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 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 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

- 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 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 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 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 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 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 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 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 theorem

- How do we get “around” this impossibility
- Limit domain
- Only two candidates

- Limit votes
- Single peaked votes

- Limit properties
- Drop IIA
- …

- Limit domain

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
- …

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

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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