Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Social Choice Theory
Designing Election Mechanisms
January 28th, 2011
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Sources
The first part of this talk closely follows §9.2 in Algorithmic
Game Theory, available online through the following link:
http://www.cs.cornell.edu/courses/cs6822/2009sp/
The proof of Arrow’s Theorem is the first in the paper Three
Brief Proofs of Arrow’s Impossibility Theorem available here:
http://ideas.repec.org/p/cwl/cwldpp/1123r3.html
There are various supporting links for the second part of the
talk. They will be given in later slides.
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Introduction
Our goal is to design a voting system: a function that takes as
input individual voter preferences and outputs either a single
‘social’ preference (a full ranking of the candidates) or a single
candidate (a winner). We’ll also want our system to satisfy
some common sense criteria. For example, don’t elect a
candidate if he’s ranked last by every voter.
Before we delve into this topic, let’s see if we can’t guess our
way to a good voting system.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Majority Rule
When there are only two candidates, there is an obvious
solution: the winner is whichever of the two candidates is
preferred by a majority of voters (ignoring exact ties).
When there are ≥ 3 candidates, and a majority winner is not
assured, one option would be to just elect the person with the
most votes. This is called Plurality Voting, and we will see soon
that it is a poor voting system.
A possibly better option: Rank a above b in society’s preference
if a majority prefer a to b. This should produce a full ranking of
the candidates. Whoever is ranked first will be elected.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
The Condorcet Paradox
This idea immediately falls apart when we consider the
following example with three candidates and 24 voters with the
following preferences
Example 1
9 voters
8 voters
7 voters
a>b>c
b>c>a
c>a>b
13 votes constitutes a majority.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
The Condorcet Paradox
Working out who has majority support
9 voters
8 voters
7 voters
a>b>c
b>c>a
c>a>b
16 prefer a to b
17 prefer b to c
15 prefer c to a
So socially, we have a > b, b > c and c > a.
The ranking is intransitive!
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Mathematical Description of Voting
The following formal notation and definitions will help us
analyze voting systems:
Let A = {a, b, c, d, ...}, with |A| = k , be the set of
candidates and let N denote the number of voters.
A ballot for a voter is a total order ≺ on A:
∀a, b ∈ A, a ≺ b and b ≺ a ⇒ a = b (anti-symmetry)
∀a, b, c ∈ A, a ≺ b and b ≺ c ⇒ a ≺ c (transitivity)
∀a, b ∈ A, a ≺ b or b ≺ a (totality)
Let L be the set of total orders on A. (|L| = k !)
Voter i’s ballot will be denoted ≺i . Society’s ranking will be
denoted by ≺.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Mathematical Description of Voting
A Social Welfare Function is a function F : LN → L
F (≺1 , ≺2 , ... ≺N ) =≺
A Social Choice Function is a function f : LN → A
f (≺1 , ≺2 , ... ≺N ) = a
F produces a full (transitive) ranking of the candidates,
while f outputs only one candidate (the winner).
A set of ballots together with a SWF or SCF is an election.
We will start by studying SWFs.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
SWFs: Desirable Criteria
1
2
A SWF satisfies Unanimity if ∀ ≺∈ L, F (≺, ≺, ..., ≺, ≺) =≺.
That is, if every voter has exactly the same preference ≺,
society’s preference should be ≺.
i is a dictator if ∀(≺j )N
j=1 , F (≺1 , ..., ≺i , ..., ≺N ) =≺i . A SWF
is a non-dictatorship if there is no i that is a dictator.
3
Independence of Irrelevant Alternatives (IIA): Let
0
0 N
0
0
F (≺1 , ..., ≺N ) =≺, F (≺1 , ..., ≺N ) =≺ .∀(≺j )N
j=1 , (≺j )j=1 and
0
0
∀a, b ∈ A, if a ≺i b ⇔ a ≺i b ∀ i then a ≺ b ⇔ a ≺ b
In words: the relative social preference between a and b
depends only on voters relative preferences of a and b.
4
Together, Unanimity and IIA imply something which is also
sometimes called ‘unanimity’, but which we will call
Pairwise Unanimity : ∀a, b ∈ A, a ≺i b ∀i ⇒ a ≺ b
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Arrow’s Theorem
We will now show that there is no general Social Welfare
Function satisfying (1), (2) and (3) simultaneously. This result
was first proved, in a weaker form, by economist Kenneth Arrow
in 1951, and is usually called Arrow’s Impossibility Theorem.
Arrow shared the Nobel Memorial Prize in Economics in part
for this work:
Arrow’s Impossibility Theorem: Any Social Welfare Function
over a set A with |A| ≥ 3 that satisfies Unanimity and
Independence of Irrelevant Alternatives is a Dictatorship.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Proof of Arrow’s Theorem
There are many proofs of the Impossibility Theorem, some of
which are quite easy. We will prove it in three steps (link to
proof here)
1
2
3
Fix b ∈ A. If all voters put b either at the bottom or the top
of their ballots, then b must be ranked first or last in the
social preference.
Corollary: If we start with every voter ranking b last, and
sequentially, for each j = 1, 2, ....N, move b directly from the
bottom to the top of j’s ranking, then ∃i ∗ for which F will first
change from ranking b last to ranking b first, precisely when
i ∗ moves b from last to first.
i ∗ is a dictator over all ac pairs for a, c 6= b, i.e
∀ (≺j )N
j=1 , a, c 6= b, a ≺i c ⇔ a ≺ c.
∗
i is a dictator over all ab pairs as well, so F is a
Dictatorship. Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Strategic Voting
Obviously, we can’t abandon Unanimity or Non-Dictatorship, so
any Social Welfare Function we want to use will have to fail IIA.
What does this mean? Let’s consider a specific SWF. For any
election, denote by Rl (a) the number of voters which rank
candidate a in l th place. Formally,
Rl (a) = |{i|∃b1 ...bl−1 s.t. a ≺i b iff b ∈ {b1 , ..., bl−1 }}|
We’ll define F as follows: a ≺ b ⇔ R1 (a) < R1 (b) (ignore ties
for now). In other words, society ranks the candidates by the
number of first place votes they get . Consider the following 100
voter example:
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Strategic Voting
Example 2
Election 1
46 voters
49 voters
5 voters
a b c
b a c
c a b
Election 2
46 voters
49 voters
5 voters
a b c
b a c
a c b
In election 2, the five voters which voted for c switch their vote
to a. It should be clear that c ≺1 a ≺1 b and c ≺2 b ≺2 a, so
society’s relative preference between a and b changed. But
note that b ≺1i a ⇔ b ≺2i a, so IIA is violated.
The result of that violation is that the c voters can now change
the top ranked candidate from b (election 1) to a (election 2) by
dishonestly ranking a above c, a strategic move which gives
them a better result, since they prefer a to b.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Strategic Voting
So F is vulnerable to strategic manipulation. Specifically, there
are elections where we can change the highest ranked
candidate in the social preference through dishonest voting.
What we will do now is formally connect Arrow’s Theorem for
Social Welfare Functions to strategic manipulation of Social
Choice Functions.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Incentive Compatible SCFs
Definition: A Social Choice Function f can be strategically
0
manipulated by voter i if for some (≺j )N
j=1 and some ≺i we have
0
a ≺i a0 and f (≺1 , ..., ≺i , ..., ≺N ) = a 6= a0 = f (≺1 , ..., ≺i , ..., ≺N ).
i.e If i honestly prefers a0 to a, he can get a0 elected by casting
0
the dishonest ballot ≺i when every other voter j casts ≺j .
Definition: A Social Choice Function f is incentive compatible if
it cannot be strategically manipulated by any voter.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Monotonicity and Incentive Compatibility
Definition: A Social Choice Function f is (strong) monotone if
0
f (≺1 , ..., ≺i , ..., ≺N ) = a 6= a0 = f (≺1 , ..., ≺i , ..., ≺N ) ⇒
0
a0 ≺i a and a ≺i a0
i.e If the winner of the election changes from a to a0 and only i
0
changed his ballot (from ≺i to ≺i ), then i must have ranked a0
0
below a on ≺i and a0 above a on ≺i .
Looking at the definitions, it’s easy to see that f can be
strategically manipulated by some voter i ⇒ f is not monotone.
It is also easy to prove the other direction to show
Proposition: A SCF is incentive compatible iff it is monotone.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Monotonicity and Incentive Compatibility
Proof of Proposition:
Assume that monotonicity is violated for the choices
0
0
(≺j )N
j=1 , ≺i , a and a as in the definition. Then we must have
0
either a ≺i a0 or both a0 ≺i a and a0 ≺i a. In the first case,
when ≺i is i’s honest vote, i can dishonestly change his vote to
0
0
≺i to change the winner to a0 . In the second case, when ≺i is
i’s honest vote, i can dishonestly change his vote to ≺i to
change the winner to a (that is, we meet the definition of
0
strategically manipulable by voter i with the roles of ≺i and ≺i
switched and the roles of a and a0 switched). Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
The Gibbard-Satterthwaite Theorem
N
Definition: i is a dictator for a SCF f , if ∀(≺j )N
j=1 ∈ L
∀ b 6= a b ≺i a ⇒ f (≺1 , ..., ≺N ) = a
f is called a dictatorship if i is a dictator for f .
Ideally, we would like to find incentive compatible SCFs which
are not dictatorships. But, again, we will be able to show that
this is generally impossible. In fact, we’ll do this by constructing
a SWF from such a SCF that violates Arrow’s Theorem.
The Gibbard-Satterthwaite Theorem: Let f be an incentive
compatible Social Choice Function onto A with |A| ≥ 3. Then f
is a dictatorship.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
The Gibbard-Satterthwaite Theorem
Notes:
1
For |A| = 2, Majority Rule is incentive compatible (simple
exercise).
2
We need f to be onto, otherwise we could just restrict the
codomain of f to two candidates and use Majority Rule to
decide the winner.
To construct our SWF, we will introduce the following notation:
Let S ⊆ A, ≺ ∈ L. Define ≺S as the order which moves every
s ∈ S to the top of the ballot without changing the ordering
within S. Formally,
∀ a, b ∈ S a ≺S b ⇔ a ≺ b
∀ a, b ∈
/ S a ≺S b ⇔ a ≺ b
∀ b ∈ S, a ∈
/ S a ≺S b
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Proof of GS Theorem
Let f be an incentive compatible non-dictatorship onto A
(|A| ≥ 3) and define
a SWF as follows:
{a,b}
{a,b}
a ≺ b ⇐⇒ f ≺1 , ..., ≺N
= b.
We must show first of all, that this is indeed a SWF, i.e. that it is
transitive, total and anti-symmetric. Then we must show that it
satisfies Unanimity, IIA and Non-Dictatorship. In fact,
Non-Dictatorship of F is implied by Non-Dictatorship of f , so
we’ll only prove the first two. To do this we first need to prove a
lemma.
Lemma: If f is an incentive compatible set onto A, then we have
N
S
S
that ∀S ⊆ A, ∀(≺j )N
j=1 ∈ L , f (≺1 , ... ≺N ) ∈ S
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Proof of GS Theorem
Proof of Lemma:
0
We will prove it by induction. Since f is onto, ∃(≺j )N
j=1 such that
0
0
0
f (≺1 , ..., ≺N ) = a for some a ∈ S. Now sequentially change ≺j
to ≺Sj for j = 1, 2, ..., N.
Assume that at j = i we have
0
0
f (≺S1 , ..., ≺Si , ≺i+1 , ..., ≺N ) = a0 ∈ S.
Since ∀ b ∈
/ S, b ≺Si+1 a0 , we cannot have
0
0
f (≺S1 , ..., ≺Si+1 , ≺i+2 , ..., ≺N ) = b by monotonicity, so
0
0
f (≺S1 , ..., ≺Si+1 , ≺i+2 , ..., ≺N ) ∈ S.
Then, by induction, we are done. Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Proof of GS Theorem
This is all we need to finish off the theorem
F is a well-defined SWF.
{a,b}
{a,b}
Anti-symmetry and Totality: f ≺1
, ..., ≺N
∈ {a, b} by
the
totality. If a ≺ b and b ≺ a then
lemma. This proves
{a,b}
{a,b}
, ..., ≺N
= a = b.
f ≺1
Transitivity: Assume ≺ is intransitive. Then ∃a, b, c with
a ≺ b ≺ c ≺ a. Consider S = {a, b, c} and WLOG, let
f (≺S1 , ..., ≺SN ) = a. Now sequentially change each ≺Si to
{a,b}
≺i
. By monotonicity and induction,
{a,b}
{a,b}
f ≺1
, ..., ≺N
= a. But then b ≺ a. Contradiction.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Proof of GS Theorem
F satisfies Unanimity, and IIA.
{a}
{a,b}
{a,b}
Unanimity: If, ∀i, b ≺i a, then ≺i
= ≺i
. (i.e a is
first
on all ballots, bis second). But then, by the lemma,
{a,b}
{a,b}
f ≺1
, ..., ≺N
= a. So , b ≺i a ∀i ⇒ b ≺ a. This is
Pairwise Unanimity, which a fortiori implies Unanimity.
0
IIA:
If b ≺i a ⇔ b ≺i a and b ≺ a, then
{a,b}
f ≺1
0
{a,b}
, ..., ≺N
{a,b}
= a. Now sequentially change ≺i
{a,b}
to ≺i
. By monotonicity and induction, f will still select a
as the winner at every point since no voter changes his ab
preference and f must select from {a, b}.
Non-Dictatorship is obvious, so we have violated Arrow’s
Theorem. Contradiction. Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Extensions of GS
In a 1977 paper, Gibbard generalized his result to
functions which determine a winner through a combination
of ballot information and chance. He found that the GS
Theorem holds with only three exceptions
1
2
3
unilateral functions: only one voter can effect the outcome.
For example, f selects a voter at random and chooses his
top-ranked candidate as the winner.
duple functions: f restricts the outcome to only two
candidates. For example, f selects two candidates at
random and chooses the Majority Rule winner between
them.
Some probabilistic combination of unilateral and duple
functions. For example, use a unilateral function with
probability p and a duple function with probability 1 − p.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Extensions of GS
Note that these two exceptions are probabilistic
generalizations of dictatorships and SCFs with codomain
two, so this result is as good as we can get.
It can be shown that even if we allow ties on the ballot (that
is, we allow voters to rank a ≺ b, b ≺ a or a ' b), then the
GS Theorem still holds in that the winner will be chosen
from a (possible) tie at the top of the dictator’s ranking.
Lastly, we may want to consider the generalization of our
description to elections where instead of one winner, we
have many winners (elections to form a committee, for
instance). The generalization of GS is called the
Duggan-Schwartz Theorem. The above link has an
explanation and proof.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Real World Voting Systems
We’re now going to examine some actual SCFs used in real
world elections and see how strategic manipulation can lead to
undesirable and even perverse results. We’ll also look at some
natural criteria that we would like our SCFs to satisfy.
Before we begin, we will introduce an important piece of
notation:
Definition: arg max f (x) = {x ∈ X | ∀ x 0 6= x, f (x 0 ) ≤ f (x)}.
x∈X
‘argmax f(x)’ gives back the x ∈ X which maximize f (x).
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Plurality
The simplest example is one we have already seen.
f (≺1 , ..., ≺N ) = arg max R1 (a)
a∈A
This voting system is called Plurality, and it is widely used
throughout the world. It simply selects the candidate with the
highest number of first place votes. Now, we have already
shown (Example 2) that it’s possible to manipulate the winner of
such an election.
Consider an expanded example with five candidates from
different political parties, ordered alphabetically across the
political spectrum from most left-wing (a) to most right-wing (e),
and five groups of voters:
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Plurality - Example
Example 3
a supporters
b supporters
c supporters
d supporters
e supporters
9 voters
28 voters
20 voters
32 voters
11 voters
abcd e
bcad e
cd bae
d ceba
ed cba
We see that d wins the election. For the a supporters, this is a
pretty bad result, and their candidate doesn’t have much
popular support. So they will abandon him and switch their vote
to b.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Plurality - Example
a supporters’
b supporters
c supporters
d supporters
e supporters
9 voters
28 voters
20 voters
32 voters
11 voters
bacd e
bcad e
cd bae
d ceba
ed cba
So b now wins with 37 first place votes. But now the e
supporters will want to switch their vote.
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Plurality - Example
a supporters’
b supporters
c supporters
d supporters
e supporters’
9 voters
28 voters
20 voters
32 voters
11 voters
bacd e
bcad e
cd bae
d ceba
decba
So d goes back to being the winner with 43 first place votes.
Finall,y the a and b supporters move their support behind c.
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Plurality - Example
a supporters’
b supporters’
c supporters
d supporters
e supporters’
9 voters
28 voters
20 voters
32 voters
11 voters
cabd e
cbad e
cd bae
d ceba
decba
c wins with a majority, since no more voter manipulation can
occur.
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Duverger’s Law
What we see here is that the candidates belonging to the
smaller parties are abandoned in an attempt to make the
mainstream candidate closest to them win the election. So
under plurality, there’s a natural move away from many small
parties towards two large parties. This restriction of choice in
elections is well known to political scientists and has been
termed Duverger’s Law.
The complete dominance of the Democratic and Republican
parties in the United States is probably the most striking
example of this phenomenon.
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Weighted Positional Voting Systems
Let’s move on to something more interesting and hopefully
more promising. Let Bl be a decreasing sequence in l. Then
define
k
X
f (≺1 , ..., ≺N ) = arg max
Bl Rl (a)
a∈A
l=1
These systems are called Weighted Positional Voting
Systems. The Bl assign points for each rank on the ballot, and
the SCF adds up the total number of points a candidate earns
on all ballots. The candidate with the most points is the winner.
The special case of Bl = k − l is known as the Borda Count
after Jean-Charles de Borda, who invented it in 1770.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Is Borda Better?
Note
1
Since every candidate is at least ranked last, we can set
Bk = 0 without loss of generality.
2
The special case of B1 = 1, Bl = 0 ∀ l 6= 1 is just Plurality
voting.
The Borda Count, and WPVS’s in general, seem like an
improvement over Plurality in that we’re using the whole ranking
on each ballot to determine a winner, rather than just the top.
However, we can show by example that Borda is extremely
sensitive to strategic manipulation and the results can be even
worse than for Plurality.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Borda Count - Example
Example 4
a supporters
b supporters
c supporters
46 voters
44 voters
10 voters
abc
bac
cab
The Borda Scores are (46*2+44*1+10*1 =) 146 for a,
(46*1+44*2+10*0 =) 134 for b and (46*0+44*0+10*2 =) 20 for c,
making a the winner.
Of course, the dishonest strategy for the b supporters is
obvious.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Borda Count - Example
Example 4
a supporters
b supporters’
c supporters
46 voters
44 voters
10 voters
abc
bca
cab
The b supporters dishonestly move a to the bottom of their
ranking to avoid having to give her any points. As a result, b is
now the Borda winner with 134 pts. This tactic is known as vote
burying. In fact, we saw it in the Plurality example: voters buried
their favourite candidate in favour of a more electable one. Here
the b voters are burying their main opponent to help b.
To counter this tactic, a voters will try the same trick.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Borda Count - Example
Example 4
a supporters’
b supporters’
c supporters
46 voters
44 voters
10 voters
acb
bca
cab
Now who wins? c, the last choice of 90% of the population!
This is the worst possible outcome, but voters are are still
pushed towards it. The b supporters can only elect their
candidate by voting dishonestly, and if even 5/44 of them fail to
vote dishonestly, the a supporters can still win as long as they
vote a c b.
Note that almost any race where there are two top-tier
candidates can result in this disastrous outcome.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
The Majority Criterion
Another feature of the Borda Count, and Weighted Positional
Voting Systems in general, is the violation of the following
criteria:
Definition: A Social Choice function f satisfies the Majority
Criterion if R1 (a) > N/2 ⇒ f (≺1 , ..., ≺N ) = a.
In other words, if any candidate is the favourite of a majority of
voters, he must win the election. The following example shows
that Borda Count fails the Majority Criterion:
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
The Majority Criterion
Example 5:
60 voters
25 voters
10 voters
5 voters
abcd
cbd a
d bac
bd ca
a is the favourite of over 60% of the population, but b still wins
the election by a score of 205 to a’s 190.
By adding value to the lower ranks, Borda (and most Weighted
Positional Systems) allow ‘compromise’ candidates like b to win
elections, even in the extreme case where a different candidate
is preferred by a majority.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Instant Runoff Voting
Ever since their 2000 Presidential Election, there has been a
movement for Election Reform in the US. One of the most
popular reform options there is called Instant Runoff Voting
(also known as Single Transferable Vote or Preferential Voting).
IRV is used for electing the President of Ireland, the President
of India, and the Australian House of Representatives. Recently
IRV has been adopted to determine the winner of the Best
Picture ‘Oscar’.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
IRV - Defintion
IRV is defined recursively:
Let S := S((≺j )N
j=1 , A) = arg min R1 (a),
a∈A
and let there be some rule to select some s ∈ S when |S| > 1.
Then:
(
a
if R1 (a) > N/2,
f (≺1 , ..., ≺N ) =
A\{s}
A\{s}
f ≺1
, ..., ≺N
otherwise
If there is a candidate with a majority of first place votes, he
wins the election. Otherwise select a candidate with the
smallest number of first place votes, move him to the bottom of
every ballot, and repeat the election with the new ballots.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
IRV - Defintion
Since a candidate can only win with a majority of first place
votes, moving s to the bottom of the list is the same as
eliminating him from the set of candidates.
There can be no more than k − 2 steps before a winner is
found, since when we’ve eliminated all but two candidates,
one of them must have a majority of votes (ignoring exact
ties).
The outcome of this procedure is identical to that of a
‘runoff election’, where voters vote for their favourite
candidate over several rounds and the candidate with the
least support in each round cannot advance to the next
one. Hence the name "Instant Runoff".
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Does IRV encourage honest voting?
IRV is often said to be a better voting strategy because the
particular kind of vote burying we saw in Example 2 is no longer
necessary.
Election 1
46 voters
49 voters
5 voters
a b c
b a c
c a b
Election 2
46 voters
49 voters
5 voters
a b c
b a c
a c b
Now, under both elections, a is the winner. In election 1, c is
eliminated in the first round and his votes are transfered to a,
who wins with a majority in the second round. c no longer plays
the spoiler who gives the election to b.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Weak Monotonicity
Unfortunately, IRV trades this improvement over Plurality for a
much more serious defect.
Definition: A SCF f is weak monotone if
0
f (≺1 , ..., ≺N ) = a 6= a0 = f (≺1 , .., ≺i , .., ≺N ) and
0
{b ≺i c ⇒ b ≺i c ∀ b, c 6= a}
⇒
i must have lowered the ranking of a (not necessarily below a0 )
In words: If the winner of the election changes from a to a0
when a single voter i changes his ballot, and i changed only
preferences involving candidate a, then he must have moved a
down in his ranking.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Weak Monotonicity
There is another way of stating this criterion: If the winner
changes from a to a0 when only i changes his ballot and i
only changed preferences involving a0 , then he must have
moved a0 up in his ranking.
All Weighted Positional Systems (including Plurality) are
weak monotone: if a goes from being a winner to a loser,
then a must have lost points, which means he must have
moved down the rankings on at least one ballot.
Violation of weak monotonicity means that a candidate can lose
by getting too many votes!
We will see now that IRV violates weak monotonicity.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Weak Monotonicity - Example
To save space, we will shorten a b c to abc for this
example.
Example 6
Election 1
9 voters
8 voters
7 voters
abc
bca
cab
Election 2
10 voters
7 voters
7 voters
abc
bca
cab
Election 3
11 voters
6 voters
7 voters
abc
bca
cab
Moving from 1 to 2, one bca voter has changed his ballot to
abc. Moving from 2 to 3, one additional bca voter has changed
his ballot to abc.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Weak Monotonicity - Example
The winner in election 1 is a (c is eliminated first), in
election 3 it’s c (b is eliminated first) and in election 2 the
winner depends on the tie-breaker used.
However, b cannot win in election 2. Either he is eliminated
first and c wins, or c is eliminated and her supporters give
their votes to a.
The important point is that the bca voters who changed
their vote only moved a up the ranking. They changed their
vote to help a.
Yet, as a result, a goes from a winner to a loser as we move
from election 1 to 2 or 2 to 3, depending on the tie-breaker.
So regardless of the tie-breaker used, an increase in a’s
support has caused her to lose the election!
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
IRV - Conclusion
This alone should be enough to convince you that IRV cannot
work in practice.
Additionally the supposed advantages of IRV are also not very
convincing. IRV is not immune to vote burying and it doesn’t
satisfy a number of other important criteria, including the next
one we’ll discuss.
However, we’re going to end our discussion of IRV now and
move on to another class of systems.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
The Condorcet Criterion
Instead of defining a new voting system and then finding criteria
it violates, we’re going to try something new and use a desired
criterion to define a class of voting systems.
Definition: A SCF satisfies the Condorcet Criterion if
|{i|b ≺i a}| > |{i|a ≺i b}| ∀ b 6= a ⇒ f (≺1 , ..., ≺N ) = a
Any f satisfying this criterion is called a Condorcet Method and
any such a is called the Condorcet Winner.
This brings us back to our original discussion of Majority Rule.
In our first example, we saw that such an a did not always exist,
i.e. that it was possible to have a, b, c such that majorities
preferred a to b, b to c, and c to a.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
The Condorcet Criterion
But when there is an a which a majority of society prefers to
any other candidate, we should want him to win. Condorcet
Methods guarantee this.
Note that it doesn’t have to be the same group of voters
which prefer a for every b. In our Example 2, election 1,
the 46 a voters and 5 c voters preferred a to b, but it was
the 46 a voters and 49 b voters which preferred a to c.
Since we are only concerned with picking a winner, it
doesn’t matter if we have an intransitive cycle at the bottom
of our social ranking. We only care about top cycles.
A Condorcet Method is fully determined by how they
handle the case of a top cycle.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Minimax
The simplest idea for a Condorcet Method is to use |{i|b ≺i a}|
vs |{i|a ≺i b}| as a way to ‘measure’ the performance of a
relative to b. So define
Score(a, b) := |{i|b ≺i a}| − |{i|a ≺i b}|
Score(a, b) tells us how many more people prefer a to b than
the other way around. Obviously, if a is a Condorcet Winner,
then Score(a, b) > 0 for any b. If there is a top cycle, one
natural thing to do is select the winner as the candidate whose
worst score is the least negative.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Minimax
Or, since Score(a, b) = −Score(b, a),
f (≺1 , ..., ≺N ) = arg min max Score(b, a)
a∈A
b∈A
It should be clear that Minimax is a Condorcet Method. If there
is a Condorcet Winner c, c is the only candidate with negative
Score(b, c) for all other b.
Unfortunately, there are examples which show that if there is a
top cycle, the Minimax winner does not have to come from it. In
fact, a candidate b for which Score(a, b) < 0 ∀ a 6= b can still
win under Minimax. Such a candidate is called a Condorcet
Loser.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Ranked Pairs
Several Condorcet Methods that improve on Minimax have
been invented. We’ll look at just one, called Ranked Pairs.
Ranked Pairs can be described in four steps:
1
Compute w(a, b) = |{i|b ≺i a}| for all pairs a, b ∈ A.
Denote, for each pair, a as the ‘winner’ iff Score(a, b) > 0
2
Sort the w(a, b) from highest to lowest for the winners. If
there are ties, use Score as a tie-breaker.
3
Construct, edge by edge, a directed graph G with weighted
edges as follows: V (G) = A, E = {(a, b)|Score(a, b) > 0},
w[(a, b)] = w(a, b). Begin with the highest weighted
edges.
4
Stop construction when the next edge will create a directed
cycle. Then G is a Directed Acyclic Graph and must have a
source s. Declare s the winner.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Ranked Pairs - Example
Our example will be the one we saw at the beginning of this
talk:
9 voters
8 voters
7 voters
abc
bca
cab
16 prefer a to b
17 prefer b to c
15 prefer c to a
We have already worked out steps one and two. All three
candidates are winners in exactly one match-up, and the order
we add the edges in are (b, c), then (a, b), then (c, a).
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Ranked Pairs - Example
b
16
a
b
16
17
c
17
a
c
15
The first two edges give us the graph on the left. Adding the
next edge gives us a triangle with the cycle a → b → c → a
(right).
Since a has no incoming edge in the DAG, a is the winner of
the election. Note that Minimax would have also chosen a.
Beyond GS
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
What is the best Condorcet Method?
Though this is a somewhat subjective question, one particularly
good method, in terms of the positive criteria it satisfies, is
called the Schulze Method.
It is somewhat complicated and won’t be discussed here, but
you can find basic information on Wikipedia and a more
technical explanation and a proper analysis can be found in
this paper by Schulze.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Beyond Gibbard-Satterthwaite
The results of Arrow’s Theorem and Gibbard-Satterthwaite
leave us with mostly imperfect voting systems. But there are
several ways we can proceed theoretically or experimentally.
Relax IIA: It is possible have IIA hold only for a subset of
candidates in a given election. For more information on
this, look up the ‘Smith Set’ and Independence of Smith
Dominated Alternatives.
Model Elections: Consider which voter rankings are likely
and which are unlikely and determine the methods that
produce good results most often. For examples, look up
‘Single-Peaked Preferences’ or check out election
simulations here.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Approval Voting
Stop using preferences and start using numbers. We
saw that quite a few methods try and produce a numerical
score for candidates so that they can measure by how
much society prefers one candidate over another. We
assumed our ballots would be rankings, but they could just
as easily be functions bi : A → <. Much progress has been
made on this front.
The simplest such system has bi : A → {0, 1}. Voters can
either ‘approve’ (1) or ‘disapprove’ (0) of each candidate. The
candidate with the highest number of approvals wins. For more
information please see the following book on Approval Voting.
Introduction
Mathematical Description
Arrow’s Theorem
GS Theorem
Real World Voting Systems
Beyond GS
Conclusion - Mechanism Design
Arrow’s Theorem and the GS Theorem show the severe
problems that come with designing a Mechanism that can take
information about people’s preference, but can’t measure the
strength of those preferences.
However, in many economics problems, we can definitively
measure how much people value one outcome over another:
by how much they’re willing to pay for that outcome.
The addition of money to the problem of mechanism design,
we can get past the GS Theorem and design incentive
compatible mechanisms for a variety of problems. That will be
the subject of the Mechanism Design talk on the 11th.