Resources

The literature on voting systems is extensive and can be intimidating. This page is designed to provide an easier and more organized way to access and understand the often confusing information on this subject.

The following resources are provided for those who would like to delve deeper into the study of voting systems and include not only the extensive analyses performed by election reform authority Charles T. Munger, Jr., but also others with expertise in the field. If you would like to provide any comments on, or contributions to, these resources, please use the Contact Us section of this website.

Analyses

My various papers each have individual abstracts that gives their principal results, and each has an introduction that describes the content of the separate section of the paper. The papers and sections can (mostly) be read independently of each other; think of a section as something like an entry in an encyclopedia, a book which one could if one chose read from cover to cover, but which is designed to let a reader look up and read information about one topic mostly without reading information about another.

To help a potential reader navigate, here the papers are grouped into one of three categories: The California Top-Two Primary; Condorcet Methods Versus Instant Runoff and Election Methods Without Ranked-Choice Ballots.

CONDORCET METHODS VS. INSTANT RUNOFF

Featured
Condorcet Versus Instant Runoff in Two Figures
Condorcet Versus Instant Runoff in Two Figures

Summary: Two pages, with the simplest example I know of, to show what goes wrong when three candidates face each other under plurality or Instant Runoff; why a Condorcet rule is necessary. No mathematics is needed at all, only 500 words. If you are new to Instant Runoff or Condorcet methods, start with this.

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The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-A
The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-A

Summary: The first of four linked papers A – D which analyze how best to take ranked-choice ballots and choose a single winner. The analysis includes the different incentives each method offers to candidates, causes, and voters, seeking to win, particularly to the tactics of bracketing and strategic nomination; presents examples and proofs of how the different methods perform when voters opinions are described as ranging along more than one political axis; and how the different systems rate against all the various abstract criteria used to assess the worth of election systems. In sum, any tournament method is decisively better than Instant Runoff (also known as the Alternative vote, or the Hare method); and the best of the tournament methods is Ranked Pairs; it is argued that any tournament method better than Ranked Pairs is unlikely to exist, which given its ease of implemen- tation, makes it the best choice if ranked-choice ballots are to be used. Ranked Pairs works whether voters preferences on their ballots are strict, or whether voters are permitted to rank candidates in equal groups. If Ranked Pairs is considered too different from Instant Runoff to adopt, two other runoff methods, the Direct Hybrid and the Benham method, may be used instead.

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The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-B
The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-B

Summary The first of four linked papers A – D which analyze how best to take ranked-choice ballots and choose a single winner. The analysis includes the different incentives each method offers to candidates, causes, and voters, seeking to win, particularly to the tactics of bracketing and strategic nomination; presents examples and proofs of how the different methods perform when voters opinions are described as ranging along more than one political axis; and how the different systems rate against all the various abstract criteria used to assess the worth of election systems. In sum, any tournament method is decisively better than Instant Runoff (also known as the Alternative vote, or the Hare method); and the best of the tournament methods is Ranked Pairs; it is argued that any tournament method better than Ranked Pairs is unlikely to exist, which given its ease of implemen- tation, makes it the best choice if ranked-choice ballots are to be used. Ranked Pairs works whether voters preferences on their ballots are strict, or whether voters are permitted to rank candidates in equal groups. If Ranked Pairs is considered too different from Instant Runoff to adopt, two other runoff methods, the Direct Hybrid and the Benham method, may be used instead.

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The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-C
The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-C

Summary: The first of four linked papers A – D which analyze how best to take ranked-choice ballots and choose a single winner. The analysis includes the different incentives each method offers to candidates, causes, and voters, seeking to win, particularly to the tactics of bracketing and strategic nomination; presents examples and proofs of how the different methods perform when voters opinions are described as ranging along more than one political axis; and how the different systems rate against all the various abstract criteria used to assess the worth of election systems. In sum, any tournament method is decisively better than Instant Runoff (also known as the Alternative vote, or the Hare method); and the best of the tournament methods is Ranked Pairs; it is argued that any tournament method better than Ranked Pairs is unlikely to exist, which given its ease of implemen- tation, makes it the best choice if ranked-choice ballots are to be used. Ranked Pairs works whether voters preferences on their ballots are strict, or whether voters are permitted to rank candidates in equal groups. If Ranked Pairs is considered too different from Instant Runoff to adopt, two other runoff methods, the Direct Hybrid and the Benham method, may be used instead.

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The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-D
The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs-D

Summary: The first of four linked papers A – D which analyze how best to take ranked-choice ballots and choose a single winner. The analysis includes the different incentives each method offers to candidates, causes, and voters, seeking to win, particularly to the tactics of bracketing and strategic nomination; presents examples and proofs of how the different methods perform when voters opinions are described as ranging along more than one political axis; and how the different systems rate against all the various abstract criteria used to assess the worth of election systems. In sum, any tournament method is decisively better than Instant Runoff (also known as the Alternative vote, or the Hare method); and the best of the tournament methods is Ranked Pairs; it is argued that any tournament method better than Ranked Pairs is unlikely to exist, which given its ease of implemen- tation, makes it the best choice if ranked-choice ballots are to be used. Ranked Pairs works whether voters preferences on their ballots are strict, or whether voters are permitted to rank candidates in equal groups. If Ranked Pairs is considered too different from Instant Runoff to adopt, two other runoff methods, the Direct Hybrid and the Benham method, may be used instead.

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Correction to: The best Condorcet-compatible election method: Ranked Pairs
Correction to: The best Condorcet-compatible election method: Ranked Pairs

Summary:

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The best Condorcet-compatible election method: Ranked Pairs
The best Condorcet-compatible election method: Ranked Pairs

Summary: Condorcet-compatible election methods are examined and compared. The Ranked Pairs method proves significantly better than Beatpath; that both are clone-free, and have other desirable properties, makes them much better than any alternative.

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“Spoilers” Spoil Nothing
“Spoilers” Spoil Nothing

Summary: The concept of spoilers, as defined by Fairvote, is analyzed and demonstrated to be useless for deciding whether Instant Runoff or either of the clone-free, Condorcet methods of Beatpath or Ranked Pairs is a better method to decide elections that use ranked-choice ballots. The problem of finding all patterns of ballots that generate spoilers that are not merely clones is solved for the general case of 4 candidates, and for 4 candidates restricted to a spatial model in two dimensions.

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Benham, a Condorcet method, compared to Instant Runoff, has for 3 candidates strictly fewer opportunities for tactical voting
Benham, a Condorcet method, compared to Instant Runoff, has for 3 candidates strictly fewer opportunities for tactical voting

Summary: Opportunities for tactical voting using ranked-choice ballots are examined for three-candidate elections under Instant Runoff, and under the Condorcet runoff method known as Benham. The Benham method is shown to have strictly fewer opportunities for unilateral tactical voting, as follows. If an election offers an opportunity under Benham, then it also offers an opportunity under Instant Runoff; but the converse is not true: there are elections that offer an opportunity under Instant Runoff that do not under Benham.

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Critique of the Fairvote analysis that compares Instant Runoff and Condorcet methods
Critique of the Fairvote analysis that compares Instant Runoff and Condorcet methods

Summary: A verbatim copy of the material posted by Fairvote on why Instant Runoff (which they call Ranked Choice Voting, or RCV) is inferior to a Condorcet method, interlineated however with my analysis of all the errors, which when corrected reverse the Fairvote conclusion.

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Technical Aspects of Election Methods

Featured
Tiebreaking in Beatpath, Split Cycle, and Ranked Pairs that is Free of Influence by Clones
Tiebreaking in Beatpath, Split Cycle, and Ranked Pairs that is Free of Influence by Clones

Summary:

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Dropping one candidate under Beatpath and Ranked Pairs
Dropping one candidate under Beatpath and Ranked Pairs

Summary: We report for Beatpath and Ranked Pairs an exhaustive examination of how the winning candidate changes when one candidate is dropped, for initial numbers of candidates of 4 and 5, and report sampling results of what happens for initial numbers of candidates from 6 to 18. Consistent with all the searches is the observation that if under Beatpath there is a single rank order, and if when the winning candidate drops there is also a single rank order, then in the new rank order the candidate who formerly placed second must place above the candidate who formerly placed third; and therefore we add to the proof by M. Schulze that the candidate who placed second cannot become placed last, the observation that the candidate who placed third cannot become placed first. Other than those two excluded cases, for candidates numbering from 4 to 18 we find that a candidate who placed anywhere in the original rank order could be found to be placed anywhere in the new rank order; in particular the candidate who had placed last could come to be placed first, and the candidate who had placed second could come to be placed second-to-last. These results are compared to the known properties of Ranked Pairs, and their larger political significance discussed.

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CALIFORNIA'S TOP-TWO PRIMARY

Featured
California's Top-two Primary: A Successful Reform I
California's Top-two Primary: A Successful Reform I

Abstract: This is the first of three concurrent papers on California's top-two primary, California's Top-two Primary: A Successful Reform I, II, and III.

Summary: The turnout of voters under the top-two; whether a voter when turned out actually casts a ballot in a same-party general election; whether the top-two is biased for or against either major party; whether either major party has been eliminated from a same-party general election that it could otherwise have won; the effect of the top-two on the minor political parties; the flow of money in top-two races; and the changes the top-two has wrought in the number of votes, and from which electorate, required to win and retain office: these are all analyzed, with conclusions favorable to the top-two. The effects of three other changes to the California political system| redistricting reform, which ended legislative gerrymanders of the districts; and the decision by the legislature to ban citizen initiatives from the statewide primary ballot; and changes to term limits for the Assembly and state Senate are taken into account.

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California's Top-two Primary: A Successful Reform II
California's Top-two Primary: A Successful Reform II

Abstract: This paper is the second of three concurrent papers, California's Top-two Primary: A Successful Reform I, II, and III, but it can be read independently.

Summary: Contrary to claims, it is found that no harm has been wrought by the top-two on the minor parties in California, whether judged by their voter registration; or the risk a minor party runs in not remaining ballot-qualifed; or any correlation between a minor party's voter registration and whether it has few or many candidates on the general election ballot; or any fall in the amount of money disbursed by the minor parties state committees. II. In a race in an incumbent-free Assembly district safe for one or other major party, that ends in a general election between two candidates of that party, the top-two has raised the minimum block of votes sufficient to win the seat, from 20,000 to the range of 60,000 to 80,000, and moved the required block from the primary electorate to the general; and in districts safe for their party the vulnerability of incumbents to being knocked out office in a primary election has decreased, but the number of incumbents actually losing to same-party challenges has increased, these increased losses now occurring in the general election.

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California's Top-two Primary: A Successful Reform III
California's Top-two Primary: A Successful Reform III

Abstract: This paper is the third of three, California's Top-two Primary: A Successful Reform I, II, and III, but it can be read independently.

Summary: The paper presents a table of all the candidates elected in a same-party general election in California from 2012 through 2016, and reports how often a candidate who had trailed in the primary won the general election. The paper examines whether voters who vote a general-election ballot and face, either in a statewide race or in a district race, two candidates of a party not their own, vote in that race or skip it; tallies the amount of money spent in same-party general elections, as a quantiable measure of their competitiveness and interest; examines whether the top-two primary, in creating some general election races from which one or other major party is excluded, has denied that excluded party a significant chance of electing one of their candidates; and compares the number of general election races in California, either resulting from the system of partisan primaries or from the top-two, that end with both a Democrat and a Republican on the ballot, to the number in the other states. The conclusions drawn from all three papers appear together at the end of this one.

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ELECTION METHODS WITHOUT RANKED-CHOICE BALLOTS

Featured
Approval voting is Condorcet-compatible voting under a constraint: A critique of Approval, and both Range and Star, voting
Approval voting is Condorcet-compatible voting under a constraint: A critique of Approval, and both Range and Star, voting

Summary: Two systems of electing one candidate from many are compared and proved identical: approval voting, and voting with a Condorcet-compatible method and ranked-choice ballots, but ballots on which voters must assign each candidate one of but two distinct ranks.

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