An Introduction to Voting Theory: History and Procedures

1. An Introduction to Voting Theory: History and Procedures Arnold B. Urken Professor of Political Science Division of Humanities and Social Science Stevens Institute of Technology aurken@stevens.edu DIMACS Workshop, May 10, 2004

2. Outline • Top Six Voting Systems • Pre-18th Century Voting Theory • 18th Century France: The Golden Age? • The Rediscovery of Voting Theory • Preference Aggregation Issues • Competence in Voting Theory

3. Top Six Voting Systems • Plurality voting • Borda voting • Condorcet scoring • Copeland scoring • Approval voting • STV (IRV)

4. Top Six Voting Systems[continued] Voting systems include rules for Vote Endowment: number of votes used to express preferences Vote Allocation: Saving or trading? Vote Aggregation: Standard for producing a collective outcome. Allocation => “fungible voting,” which allows votes to be saved and traded

5. Hypothetical Data Set • Nine voters rank • Bush • Kerry • Nader

6. Top Six Voting Procedures[continued] Plurality Voting Endowment: One vote for the most preferred choice Allocation: Trading/saving not explicitly allowed Aggregation: Choice with the most votes wins (plurality)

7. Plurality Voting Results Bush 4 votes Kerry 3 votes Nader 2 votes

8. Plurality vs. MajorityWhat’s the Difference? • Absolute vs. relative majority • Historically • Sanior et major pars • Right/healthy and greater part • Used to overturn outcomes • Sanior difficult to measure, so major used

9. Top Six Voting Procedures[continued] Borda Voting Endowment: Assign ranks to choices Allocation: Trading/saving not explicitly allowed Aggregation: Choice with the most votes wins (plurality)

10. Borda Voting Results Bush 18 points Nader 18 points Kerry 18 points Plurality aggregation not satisfied.

11. Borda and Rankings Illegal in some elections

12. Borda and Rankings[continued] Not used this way

13. Top Six Voting Procedures[continued] Condorcet Scoring Endowment: Ordinal rankings assigned to choices Allocation: Trading/saving not explicitly allowed Aggregation: Winner is the choice with the most victories in binary comparisons

14. Condorcet Scoring Results Bush 9 points Kerry 9 points Nader 9 points Plurality aggregation not satisfied.

15. Top Seven Voting Procedures[continued] Copeland Scoring Endowment: Ordinal rankings assigned to choices Allocation: Trading/saving not explicitly allowed Aggregation: Winner is the choice with greatest net score in binary comparisons

16. Copeland Scoring Results Bush 0 points Nader 0 points Kerry 0 points Plurality aggregation not satisfied.

17. Top Seven Voting Procedures[continued] Approval Voting Endowment: N votes where N = number of choices Allocation: One vote cast for each approved choice; no trading/saving Aggregation: Plurality, majority, or unanimity

18. Approval Voting Results Assuming one approval vote is cast for 1st and 2nd place choices Bush 5 points Nader 6 points Kerry 5 points Nader is the plurality winner! Based on the number of voters who approve him

19. Top Seven Voting Procedures[continued] Observations about Approval Voting • Empirical observation: Voters cast an approval vote for each choice ≥ average utility • Ties possible under plurality, majority, and unanimous aggregation rules • Definitions of base for aggregation • All allocated votes • The number of voters casting votes

20. Top Seven Voting Procedures[continued] STV (IRV--Proportional Representation) Endowment: Assign ranks to choices Allocation: One choice for each rank, trading/saving: not explicitly allowed Aggregation: Majority of first place votes, but if no choice wins, eliminate the most choice most frequently ranked last and count first place preferences again until a majority winner is produced

21. STV (IRV—Proportional) Scoring Results Bush 4 points Kerry 3 points Nader 2 points Majority aggregation not satisfied.

22. STV (IRV—Proportional) Scoring ResultsOne Round of Elimination Bush Eliminated Kerry 5 votes Nader 4 votes Kerry is the majority winner!

23. Ranking Kerry last could have eliminated him. Original data PR with Strategic Voting Strategic Voting Kerry wins

24. Summary of Results Inconsistent? Or just different?

25. Pre-18th Century Voting Theory General Observations • Theoretical insights were derived from practical problem solving • Knowledge was not cumulative • The communication of votes was an issue • “Science” was • “pre-normal” Kuhnian framework • early stage Popperian “metaphysical” research program

26. Pre-18th Century Voting Theory[continued] • Pliny the Younger • Ramon Lull • Nicolaus Cusanus • The Venetian Mehod

27. Pre-18th Century Voting Theory[continued] • Pliny the Younger • Letter to Titius Aristo, A.D. 105 • Agenda manipulation in the trial of Afranius Dexter’s slaves • Slaves accused of murdering his master • Options • Acquittal • Banishment • Death

28. Pre-18th Century Voting Theory[continued] • Execution faction leader leads switch from death to banishment • Banishment is the majority choice • Pliny’s faction favored leniency, but included less than one-half of all votes

29. Pre-18th Century Voting Theory[continued] • Pliny calls for ternary vote (with division of the whole) • Pliny knew that the opposition had the following preference orders: Death > Banishment > Acquittal Banishment > Acquittal > Death

30. Pre-18th Century Voting Theory[continued] • Why? • Neither Acquittal nor Death would get a majority in the first round of voting—in binary comparisons • In the second round of voting, the winner of the first round of voting (Acquittal or Death) would lose to Banishment • Sincere and manipulated voting produce the same outcome! • Pliny uncomfortable: inconsistent with Senate customs?

31. Pre-18th Century Voting Theory[continued] • Issues Raised • Sincere voting: honest communication of preferences • Strategic voting: changing “sincere” votes to manipulate the collective outcome • Pliny anticipates Robin Farquharson, Theory of Voting. Yale, 1969

32. Pre-18th Century Voting Theory[continued] Ramon Lull A.D. 1232-1316 • Explored methods for honest church elections • Two methods based on selections of pairs of choices from a larger set of ranked choices • Blanquera (1285) • De Arte Eleccionis (1299)

33. Pre-18th Century Voting Theory[continued] Blanquerna (1285) • Mixed method (“art”) Borda and Condorcet • Electors choose Blanquerna as bishop without following the “art” they generate an indecisive outcome and the decision must be appealed to the Pope to produce a winner • Work reflects ambivalence about preference aggregation and making the right choice.

34. Pre-18th Century Voting Theory[continued] De Arte Eleccionis (1299) • Condorcet scoring • Uses matrix notation (next used by Dodgson in the 19th century) • Method does not address collective intransitivity (later discovered by Condorcet and Arrow)

35. Pre-18th Century Voting Theory[continued] Nicolaus Cusanus (1430) • Goal: design an “honest” voting procedure to elect a Holy Roman Emperor to end a long schism in the papacy • Proposes a Borda system • Applies it to propositions with more than two choices • Criticizes manipulation of electorsand criticizes attempts to control the collective outcome by manipulating electors. • Implicitly suggests that voting by ballot is new

36. Pre-18th Century Voting Theory[continued] The Venetian Method (13th Century) • Similar to approval voting • Simplified the process of selecting 41 electors from an initial assembly of 1500 members.

37. 18th Century France: The Golden Age? • Voting in the French Academy of Sciences • Borda, Condorcet, and others • Condorcet and the French Revolution • Daunou and after • Proportional voting

38. 18th Century France: The Golden Age?[continued] Voting in the French Academy of Sciences • Scientists recommend top three candidates to the King of France • Plurality voting used since 1699, ties rare. • 1770 Borda talk about plurality voting • Borda paper not published until 1784

39. 18th Century France: The Golden Age?[continued] Voting in the French Academy of Sciences • Borda and Condorcet were political enemies • Borda fought in the American Revolution • Condorcet, a modernist, won a manipulated election as Secretary

40. 18th Century France: The Golden Age?[continued] Voting in the French Academy of Sciences • No evidence of actual voting debate • Condorcet regards Borda’s work as physicaille (petty experiments) • Condorcet’s 1785 Essai Essai sur l’application d’analyse à probabilité des décisions rendues à la pluralité des voix

41. 18th Century France: The Golden Age?[continued] Voting in the French Academy of Sciences • The 1785 Essai • Goal: analyze the probability of making a correct collective choice • Introduction: identifies collective intransitivity • Body: 13 hypothetical situations

42. 1.0 Group Voter Competence 0.5 0 0 0.5 1.0 Individual Voter Competence Condorcet “Jury Theorem” Question: How does majority rule affect the group probability of making a correct choice? • Assumptions • 50 or more voters • Binary choice • One Person, One Vote • Preferences a random variable • Individual competence statistically independent

43. 18th Century France: The Golden Age?[continued] Condorcet and the French Revolution • Creates practical voting plan for the Republican Constitution with binary agendas • Recommends jury design for the trial of the King of France • Robespierre’s hit list drives him underground • Dies in prison?

44. 18th Century France: The Golden Age?[continued] Daunou and after • FAS becomes the Institute of France • New election method needed • Napoléon interested • Borda and Daunou on commission • Daunou writes critique of Borda voting

45. 18th Century France: The Golden Age?[continued] Daunou and after (continued) • Voting theory is lost in French probability theory (Cf. Daston) • Ideas rediscovered by Dodgson (Lewis Carroll) • Nanson (Australia) refers to Condorcet’s ideas in designing elections for scientists

46. 18th Century France: The Golden Age?[continued] Daunou and after (continued) • Proportional voting developed for allocating seats in legislatures • Ideas are not integrated with voting theorists

47. The Rediscovery of Voting Theory Black • Does archival research on Condorcet • Coins “jury theorem” to explain Condorcet’s interest in competence • Develops “single-peakedness” concept to explain collective intransitivity

48. The Rediscovery of Voting Theory[continued] Arrow • Relies on Black to understand Condorcet • Invents the term “social choice” • Axiomatizes collective intransitivity problem in impossibility theorem

49. The Rediscovery of Voting Theory[continued] Arrow • Unrestricted domain or universality • Non-imposition or citizen sovereignty • Non-dictatorship • Monotonicity • Independence of irrelevant alternatives Impossible to satisfy all conditions simultaneously

50. The Rediscovery of Voting Theory[continued] Brams and Fishburn • Develop formal proposal for approval voting • Scientific societies adopt approval voting • Articulate theoretical and empirical arguments