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

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)

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

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  1. Voting Geometry:A Mathematical Study of Voting Methods and Their Properties Alan T. Sherman Dept. of CSEE, UMBC March 27, 2006

  2. 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

  3. 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

  4. Question: • Does plurality always reflect the desires of the voters?

  5. Example 1: Beer, Wine, Milk Profile # Voters M > W > B 6 B > W > M 5 W > B > M 4 Total: 15 What beverage should be served?

  6. Profile # M > W > B 6 B > W > M 5 W > B > M 4 B W M 5 4 6 Example 1: Plurality

  7. Profile # M > W > B 6 B > W > M 5 W > B > M 4 B M 9 6 Example 1: Runoff

  8. Profile # M > W > B 6 B > W > M 5 W > B > M 4 W > B > M 10 : 5 9 : 6 > 9 : 6 Example 1: Pairwise Comparison

  9. 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: Borda Count

  10. Example 1: Method Determines Outcome Method Outcome Plurality milk Runoff beer Pairwise wine Borda Count wine

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

  12. 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

  13. 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 = 615 = 470,184,984,576

  14. History • Aristotle (384-322 BC) • Politics 335-323 BC • Jean-Charles Borda (1733-1799) • 1770, 1984 • M. Condorcet (1743-1794) • Donald Saari • 1978

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

  16. W 4 6 5 M B Profiles Frequency 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

  17. 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

  18. 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

  19. Pairwise Methods:Outline • Agenda • Condorcet winner • Arrow’s Theorem

  20. Example 2: Agenda • Bush > Kerry > Nader 5 • Kerry > Nader > Bush 5 • Nader > Bush > Kerry 5 • Who should win?

  21. Agenda B,K,N Example 2: Two Agendas Agenda K,N,B B 10 B 10 B 5 K 10 K 5 K 5 N 10 N 5 B > K > N 5 K > N > B 5 N > B > K 5

  22. Condorcet Winner/Loser • Condorcet Winner – wins all pairwise majority vote elections • Condorcet Loser – loses all pairwise majority vote elections

  23. Question: • Does the Condorcet winner always reflect the first choice of the voters?

  24. Problems with Condorcet Winners • Condorcet winner does not always exist • Confused voters (non-transitive preferences) • Missing intensity of comparisons election

  25. W W W 10 10 0 1 0 1 1 1 0 10 10 0 10 30 20 29 1 28 B B B Example 3: Condorcet Winner Remove confused voters! M Condorcet M M original reduced 41-40 20-28

  26. 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

  27. 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.

  28. Borda Count • “Appears to be optimal” • Unique method to represent true wishes of voters • Minimizes number and kind of paradoxes • Minimizes manipulation

  29. Additional Issues • Manipulation / Strategic voting • Apportionment

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

  31. Example 4: Committees Divide voters into two committees of 13 for straw polls. Entire group votes. Plurality voting, with runoffs.

  32. 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

  33. Desirable Properties • Monotonicity • Unbiased • Resistance to manipulation

  34. 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.

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