Voting Geometry:A Mathematical Study of Voting Methods and Their Properties

Alan T. ShermanDept. of CSEE, UMBC

March 27, 2006

Reference Work

• Saari, Donald G., Basic Geometry of Voting, Springer (1995). 300 pages.

– Distinguished Professor: Mathematics and Economics (UC Irvine)

– National Science Foundation support– Former Chief Editor, Bulletin of the American

Mathematical Society– 103 hits on Google Scholar

Main Results

• Application of geometry to study voting systems

• New insights, simplified analyses, greater clarity of understanding

• Borda Count (BC) has many attractive properties, but all methods have limitations

Question:

• Does plurality always reflect the desires of the voters?

Example 1: Beer, Wine, Milk

Profile # Voters

M > W > B 6

B > W > M 5

W > B > M 4Total: 15

What beverage should be served?

Example 1: Plurality

Profile #

M > W > B 6

B > W > M 5

W > B > M 4

B W M

5 4 6

Example 1: Runoff

Profile #

M > W > B 6

B > W > M 5

W > B > M 4

B M

9 6

Example 1: Pairwise Comparison

Profile #

M > W > B 6

B > W > M 5

W > B > M 4

W > B > M

10 : 5 9 : 6

>

9 : 6

Example 1: Borda Count

Profile #

M > W > B 6

B > W > M 5

W > B > M 4

W B M

1 0 2

1 2 0

2 1 0

4 3 2

Example 1: Method Determines Outcome

Method Outcome

Plurality milk

Runoff beer

Pairwise wine

Borda Count wine

Outline

• Motivation• Why voting is hard to analyze• History• Modeling voting• Methods: pairwise, positional• Properties: Arrow’s Theorem• Other issues: manipulation, apportionment• Conclusion

Motivation

• Understand election results

• Understand properties of election methods

• Find effective methods for reasoning about election methods

• Identify desirable properties of election methods

• Help officials make informed decisions in choosing election methods

Why is Voting Difficult to Analyze?

• K candidates, N voters• K! possible rankings of candidates• Number of possible outcomes:

(k!)N - with ordering of votes cast

k! + N – 1 - without ordering of votes castN

(3!)15 = 615 = 470,184,984,576

History

• Aristotle (384-322 BC)– Politics 335-323 BC

• Jean-Charles Borda (1733-1799)– 1770, 1984

• M. Condorcet (1743-1794)

• Donald Saari – 1978

Modeling Voting

Profiles

(candidate rankings by each voter)

Election Outcome

Election

ProfilesFrequency counts of rankings by voters

P = (p1, p2, …, p6) (k = 3 candidates,

P = (6,5,4,0,0,0) N = 15 voters)

P = (6/15,5/15,4/15,0,0,0) normalized

M B

W

6 5

4

Election Mappings

f : Si(k!) → Si(k) (k = # candidates)

Si(k!) = normalized space of profiles;dimension k! – 1 (a simplex)

Si(k) = normalized space of outcomes;dimension k – 1 (a simplex)

f is linear

Voting Methods

• Pairwise methods– Agenda, Condorcet winner/loser

• Positional methods– Plurality, Borda Count (BC)

• Hybrid Rules– Runoff, Coomb’s runoff– Black’s procedure, Copeland method

Pairwise Methods:Outline

• Agenda

• Condorcet winner

• Arrow’s Theorem

Example 2: Agenda

• Bush > Kerry > Nader 5

• Kerry > Nader > Bush 5

• Nader > Bush > Kerry 5

• Who should win?

Example 2: Two Agendas

Agenda B,K,N

K 5

B 10

N 10

B 5

Agenda K,N,B

N 5

K 10

B 10

K 5

B > K > N 5

K > N > B 5

N > B > K 5

Condorcet Winner/Loser

• Condorcet Winner – wins all pairwise majority vote elections

• Condorcet Loser – loses all pairwise majority vote elections

Question:

• Does the Condorcet winner always reflect the first choice of the voters?

Problems with Condorcet Winners

• Condorcet winner does not always exist• Confused voters (non-transitive preferences)• Missing intensity of comparisons

election

Example 3: Condorcet Winner

M B

W

1 10

110

30 29

B

W

1 10

110

10 1

B

W

0 0

00

20 28

M

Moriginal

Condorcet

reduced

41-40 20-28

Remove confused voters!

Arrow’s Theorem: Hypotheses

• Universal Domain (UD)Each voter may rank candidates any way

• Independence of Irrelevant Alternatives (IIA)Relative rank x-y depends only on ranks x-y

• Involvement (Invl)candidates x,y, profiles p1,p2 p1 x>y and p2 y>x

• Responsiveness (Resp)Outcomes cannot always agree with some single voter

Arrow’s Theorem

Theorem (1963). For 3 voters, there is no voting procedure with strict rankings that satisfies UD, IIA, Invl, and Resp.

Corollary (Arrow). The only voting procedure that always gives strict rankings of 3 candidates, and that satisfies UD, IIA, and Invl, is dictatorship.

Borda Count

• “Appears to be optimal”

• Unique method to represent true wishes of voters

• Minimizes number and kind of paradoxes

• Minimizes manipulation

Additional Issues

• Manipulation / Strategic voting

• Apportionment

Gibbard-Satterthwaite

Theorem (1973,1975). All non-dictatorial voting methods can be manipulated.

Example 4: Committees

Divide voters into two committees of 13 for straw polls.

Entire group votes.

Plurality voting, with runoffs.

Example 4: Committees I,II

Profile Frequency Committee Joint

I II I II

A > B > C 4 4 A 4,7 4,7 A 8

B > A > C 3 3 B 3 6,6 B 9,17

C > A > B 3 3 C 6,6 3 C 9,9

C > B > A 3 0

B > C > A 0 3

Desirable Properties

• Monotonicity

• Unbiased

• Resistance to manipulation

Conclusions

• Geometry simplifies analysis and facilitates understanding.

• Problems with Condorcet explain many paradoxes.

• Borda Count is attractive.– most resistant to manipulation, minimizes paradoxes

• Runoff is usually better than plurality.• All methods have limitations, and there is no

simple way to select “best” method.