Transcript 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.

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!
Terminology
 Agent
Usually assume odd
number of agents to
reduce ties
 Vote
Total order over
outcomes
 Voting rule
Social choice: mapping
of a profile onto a
winner(s)
Social welfare: mapping
of a profile onto a total
ordering
 Profile
Vote for each agent
Extensions 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”
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: 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
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)
Voting rules: plurality with runoff
 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: single transferable vote
 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: 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
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)
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.
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
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
 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
…
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)