Voting Theory

1
Voting Theory

• Toby Walsh
• NICTA and UNSW

2
Motivation

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

3
Terminology

• 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

Extensions include indifference,
incomparability, incompleteness
4
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 Mays theorem)

5
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 Ill vote for my
  second favourite

6
Voting rules plurality with runoff

• Two rounds
• Eliminate all but the 2 candidates with most
  votes
• Then hold a majority election between these 2
  candidates

• Consider
• 25 votes AgtBgtC
• 24 votes BgtCgtA
• 46 votes CgtAgtB
• 1st round B knocked out
• 2nd round CgtA by 7025
• C wins

7
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)

8
Voting rules plurality with runoff

• Consider again
• 25 votes AgtBgtC
• 24 votes BgtCgtA
• 46 votes CgtAgtB
• C wins easily

• Two voters dont vote
• 23 votes AgtBgtC
• 24 votes BgtCgtA
• 46 votes CgtAgtB
• Different result
• 1st round A knocked out
• 2nd round BgtC by 4746
• B wins

9
Voting rules single transferable vote

• STV
• If one candidate has gt50 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 AgtBgtCgtD
• 20 votes BgtAgtCgtD
• 20 votes BgtCgtAgtD
• 11 votes CgtBgtAgtD
• 10 votes DgtAgtBgtC
• Result B wins!

10
Voting rules Borda

• Given m candidates
• ith ranked candidate score m-i
• Candidate with greatest sum of scores wins
• Example
• 42 votes AgtBgtCgtD
• 26 votes BgtCgtDgtA
• 15 votes CgtDgtBgtA
• 17 votes DgtCgtBgtA
• B wins

Jean Charles de Borda, 1733-1799
11
Voting rules positional rules

• Given vector of weights, lts1,..,smgt
• Candidate scores si for each vote in ith position
• Candidate with greatest score wins
• Generalizes number of rules
• Borda is ltm-1,m-2,..,0gt
• Plurality is lt1,0,..,0gt

12
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

13
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

14
Voting rules other

• Coombs 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

15
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

16
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

17
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

18
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, well be able to
  prove impossibility results (no voting rule has
  this combination of desirable properties)

19
Mays theorem

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

20
Mays 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 AgtBgtC
• 42 votes CgtAgtB
• 24 votes BgtCgtA
• Suppose 4 voters in 1st group move C up to top
• 23 votes AgtBgtC
• 46 votes CgtAgtB
• 24 votes BgtCgtA

21
Mays 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.
  68068
• Since these properties are uncontroversial, this
  about decides what to do with 2 candidates!

22
Mays 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

23
Mays 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(AgtB) N(BgtA)1 and A wins.
• By monotonicity, A wins whenever N(AgtB) gt N(BgtA)

24
Mays 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(AgtB) N(BgtA)1 and A wins.
• By monotonicity, A wins whenever N(AgtB) gt N(BgtA)
• N(AgtB) N(BgtA)1 and B wins
• Swap one vote AgtB to BgtA. By monotonicity, B
  still wins. But now N(BgtA) N(AgtB)1. By
  neutrality, A wins. This is a contradiction.

25
Condorcets paradox

• Collective preference may be cyclic
• Even when individual preferences are not
• Consider 3 votes
• AgtBgtC
• BgtCgtA
• CgtAgtB
• 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)
26
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)

27
Condorcet principle

• Plurality rule is not Condorcet consistent
• 35 votes AgtBgtC
• 34 votes CgtBgtA
• 31 votes BgtCgtA
• B is easily the Condorcet winner, but plurality
  elects A

28
Condorcet principle

• Thm. No positional rule with strict ordering of
  weights is Condorcet consistent
• Proof Consider
• 3 votes AgtBgtC
• 2 votes BgtCgtA
• 1 vote BgtAgtC
• 1 vote CgtAgtB
• A is Condorcet winner

29
Condorcet principle

• Thm. No positional rule with strict ordering of
  weights is Condorcet consistent
• Proof Consider
• 3 votes AgtBgtC
• 2 votes BgtCgtA
• 1 vote BgtAgtC
• 1 vote CgtAgtB
• Scoring rule with s1 gt s2 gt s3
• Score(B) 3.s13.s21.s3
• Score(A) 3.s12.s22.s3
• Score(C) 1.s12.s24.s4
• Hence Score(B)gtScore(A)gtScore(C)

30
Arrows 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

31
Arrows 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

32
Arrows theorem

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

33
Arrows 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

34
Arrows 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

35
Arrows 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

36
Arrows theorem

• With two candidates, majority rule is
• Pareto
• Independent to irrelevant alternatives
• Non-dictatorial
• So, one way around Arrows theorem is to
  restrict to two candidates

37
Proof of Arrows 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 AgtBgtC
• By IIA, this would not change if every voter
  moved C above A
• BgtAgtC gt BgtCgtA
• BgtCgtA gt BgtCgtA
• AgtCgtB gt CgtAgtB
• CgtAgtB gt CgtAgtB
• Each AB and BC vote the same!

38
Proof of Arrows 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 AgtBgtC
• By IIA, this would not change if every voter
  moved C above A
• By transitivity AgtC in result
• But by unanimity CgtA
• BgtAgtC gt BgtCgtA
• BgtCgtA gt BgtCgtA
• AgtCgtB gt CgtAgtB
• CgtAgtB gt CgtAgtB

39
Proof of Arrows 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 AgtBgtC
• AgtC and CgtA in result
• This is a contradiction
• B can only be top or bottom in result

40
Proof of Arrows 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

41
Proof of Arrows 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

42
Proof of Arrows 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

43
Proof of Arrows 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 AgtBgtC
• By IIA, AgtB in result as AB votes are identical
  to profile just before pivotal vote moves B (and
  result has B at bottom)
• By IIA, BgtC in result as BC votes are unchanged
• Hence, AgtC by transitivity

44
Proof of Arrows 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 AgtBgtC
• Then AgtC 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

45
Arrows theorem

• How do we get around this impossibility
• Limit domain
• Only two candidates
• Limit votes
• Single peaked votes
• Limit properties
• Drop IIA

46
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

47
Single peaked votes

• There are never Condorcet cycles
• Arrows 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

48
Conclusions

• Many voting rules exist
• Plurality, STV, approval, Copeland,
• For two candidates
• Best rule is plurality
• For more than two candidates
• Arrows theorem proves there is no best rule
• But there are limited ways around this (e.g.
  single peaked votes)