Voting Geometry: A Mathematical Study of Voting Methods and Their Properties Alan T. Sherman Dept. 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 > B6 B > W > M5 W > B > M4 Total:15 What beverage should be served?

Example 1: Plurality Profile# M > W > B6 B > W > M5 W > B > M4 BWM546BWM546

Example 1: Runoff Profile# M > W > B6 B > W > M5 W > B > M4 BM96BM96

Example 1: Pairwise Comparison Profile# M > W > B6 B > W > M5 W > B > M4 W > B>M 10 : 5 9 : 6 > 9 : 6

Example 1: Borda Count Profile# M > W > B6 B > W > M5 W > B > M4 WBM WBM

Example 1: Method Determines Outcome MethodOutcome Pluralitymilk Runoffbeer Pairwisewine Borda Countwine

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 cast N (3!) 15 = 6 15 = 470,184,984,576

History Aristotle ( BC) –Politics BC Jean-Charles Borda ( ) –1770, 1984 M. Condorcet ( ) Donald Saari –1978

Modeling Voting Profiles (candidate rankings by each voter) Election Outcome Election

Profiles Frequency counts of rankings by voters P = (p 1, p 2, …, p 6 )(k = 3 candidates, P = (6,5,4,0,0,0) N = 15 voters) P = (6/15,5/15,4/15,0,0,0) normalized MB W 65 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 > Nader5 Kerry > Nader > Bush5 Nader > Bush > Kerry5 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 > N5 K > N > B5 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 B W B W M M original Condorcet reduced 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 p 1,p 2 p 1  x>y and p 2  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 FrequencyCommittee Joint IIIIII A > B > C 44 A4,74,7 A 8 B > A > C 33 B36,6 B 9,17 C > A > B 33 C6,63 C 9,9 C > B > A 30 B > C > A 03

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.