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.