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© 2004. Arnold B. Urken All Rights Reserved1 An Introduction to Voting Theory: History and Procedures Arnold B. Urken Professor of Political Science Division.

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Presentation on theme: "© 2004. Arnold B. Urken All Rights Reserved1 An Introduction to Voting Theory: History and Procedures Arnold B. Urken Professor of Political Science Division."— Presentation transcript:

1 © Arnold B. Urken All Rights Reserved1 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 DIMACS Workshop, May 10, 2004

2 © Arnold B. Urken All Rights Reserved2 Outline Top Six Voting Systems Pre-18 th Century Voting Theory 18 th Century France: The Golden Age? The Rediscovery of Voting Theory Preference Aggregation Issues Competence in Voting Theory

3 © Arnold B. Urken All Rights Reserved3 Top Six Voting Systems Plurality voting Borda voting Condorcet scoring Copeland scoring Approval voting STV (IRV)

4 © Arnold B. Urken All Rights Reserved4 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 © Arnold B. Urken All Rights Reserved5 Hypothetical Data Set Nine voters rank 1.Bush 2.Kerry 3.Nader

6 © Arnold B. Urken All Rights Reserved6 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 © Arnold B. Urken All Rights Reserved7 Plurality Voting Results Bush4 votes Kerry3 votes Nader2 votes

8 © Arnold B. Urken All Rights Reserved8 Plurality vs. Majority What’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 © Arnold B. Urken All Rights Reserved9 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 © Arnold B. Urken All Rights Reserved10 Borda Voting Results Bush18 points Nader18 points Kerry18 points Plurality aggregation not satisfied.

11 © Arnold B. Urken All Rights Reserved11 Borda and Rankings Illegal in some elections

12 © Arnold B. Urken All Rights Reserved12 Borda and Rankings [continued] Not used this way

13 © Arnold B. Urken All Rights Reserved13 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 © Arnold B. Urken All Rights Reserved14 Condorcet Scoring Results Bush9 points Kerry9 points Nader 9 points Plurality aggregation not satisfied.

15 © Arnold B. Urken All Rights Reserved15 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 © Arnold B. Urken All Rights Reserved16 Copeland Scoring Results Bush0 points Nader0 points Kerry0 points Plurality aggregation not satisfied.

17 © Arnold B. Urken All Rights Reserved17 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 © Arnold B. Urken All Rights Reserved18 Approval Voting Results Bush5 points Nader6 points Kerry5 points Nader is the plurality winner! Assuming one approval vote is cast for 1 st and 2 nd place choices Based on the number of voters who approve him

19 © Arnold B. Urken All Rights Reserved19 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 © Arnold B. Urken All Rights Reserved20 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 © Arnold B. Urken All Rights Reserved21 STV (IRV—Proportional) Scoring Results Bush4 points Kerry3 points Nader2 points Majority aggregation not satisfied.

22 © Arnold B. Urken All Rights Reserved22 STV (IRV—Proportional) Scoring Results One Round of Elimination BushEliminated Kerry5 votes Nader4 votes Kerry is the majority winner!

23 © Arnold B. Urken All Rights Reserved23 PR with Strategic Voting Original data Kerry wins Strategic Voting Ranking Kerry last could have eliminated him.

24 © Arnold B. Urken All Rights Reserved24 Summary of Results Inconsistent? Or just different?

25 © Arnold B. Urken All Rights Reserved25 Pre-18 th 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 © Arnold B. Urken All Rights Reserved26 Pre-18 th Century Voting Theory [continued] Pliny the Younger Ramon Lull Nicolaus Cusanus The Venetian Mehod

27 © Arnold B. Urken All Rights Reserved27 Pre-18 th 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 © Arnold B. Urken All Rights Reserved28 Pre-18 th 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 © Arnold B. Urken All Rights Reserved29 Pre-18 th 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 © Arnold B. Urken All Rights Reserved30 Pre-18 th 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 © Arnold B. Urken All Rights Reserved31 Pre-18 th 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 © Arnold B. Urken All Rights Reserved32 Pre-18 th Century Voting Theory [continued] Ramon Lull A.D 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 © Arnold B. Urken All Rights Reserved33 Pre-18 th 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 © Arnold B. Urken All Rights Reserved34 Pre-18 th Century Voting Theory [continued] De Arte Eleccionis (1299) Condorcet scoring Uses matrix notation (next used by Dodgson in the 19 th century) Method does not address collective intransitivity (later discovered by Condorcet and Arrow)

35 © Arnold B. Urken All Rights Reserved35 Pre-18 th 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 © Arnold B. Urken All Rights Reserved36 Pre-18 th Century Voting Theory [continued] The Venetian Method (13 th Century) Similar to approval voting Simplified the process of selecting 41 electors from an initial assembly of 1500 members.

37 © Arnold B. Urken All Rights Reserved37 18 th 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 © Arnold B. Urken All Rights Reserved38 18 th 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 Borda talk about plurality voting Borda paper not published until 1784

39 © Arnold B. Urken All Rights Reserved39 18 th 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 © Arnold B. Urken All Rights Reserved40 18 th 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 © Arnold B. Urken All Rights Reserved41 18 th 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 © Arnold B. Urken All Rights Reserved Individual Voter Competence Group Voter Competence Condorcet “Jury Theorem” Assumptions 50 or more voters Binary choice One Person, One Vote Preferences a random variable Individual competence statistically independent Question: How does majority rule affect the group probability of making a correct choice?

43 © Arnold B. Urken All Rights Reserved43 18 th 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 © Arnold B. Urken All Rights Reserved44 18 th 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 © Arnold B. Urken All Rights Reserved45 18 th 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 © Arnold B. Urken All Rights Reserved46 18 th 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 © Arnold B. Urken All Rights Reserved47 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 © Arnold B. Urken All Rights Reserved48 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 © Arnold B. Urken All Rights Reserved49 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 © Arnold B. Urken All Rights Reserved50 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

51 © Arnold B. Urken All Rights Reserved51 The Rediscovery of Voting Theory [continued] Saari Develops a geometric framework for comparing voting methods for three choices Does not address Ties Truncated preferences Competence

52 © Arnold B. Urken All Rights Reserved52 The Rediscovery of Voting Theory [continued] Preference Aggregation Issues Vote trading and fungible voting Manipulation: potential vs. actual Voter use of voting methods Ranking choices (STV) Identifying approved set of choices

53 © Arnold B. Urken All Rights Reserved53 The Rediscovery of Voting Theory [continued] Competence in Social Choice Young: Maximum likelihood interpretation of Condorcet’s rule Grofman (Owen, Feld) Explore models of competence Show that Condorcet solved Rousseau’s problem of reconciling “general will” and the “will of all”

54 © Arnold B. Urken All Rights Reserved54 Grofman-Shapley Theorem How to weight votes in interdependent collective decisions Don’t weight votes by using the ratio of p/1-p (ratio of competence to incompetence) Instead use ln p/1-p Experimental Confirmation The Rediscovery of Voting Theory [continued]

55 © Arnold B. Urken All Rights Reserved Individual Voter Competence Group Competence Group does better than the average individual Average individual does better than the group. Average individual competence equals group competence. Non-monotonic pattern in approval voting

56 © Arnold B. Urken All Rights Reserved Average Voter Competence Low High Voter Preferences Heterogeneous Homogeneous Optimal group competence Suboptimal performance Minimum group competence Better than minimum performance 0 Reconciling Competence and Preferences

57 © Arnold B. Urken All Rights Reserved57 Perspective History not of purely antiquarian interest Draws our attention to models and problems of integrating ideas Unresolved dualism Preference aggregation Competence


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