Comparison of electoral systems

From Wikipedia, the free encyclopedia
  (Redirected from Voting system criterion)
Jump to: navigation, search

Electoral systems can be compared by different means. These comparisons can focus on different aspects: on suffrage or rules for voter eligibility; on candidate eligibility and the rules governing political parties; on the way elections are scheduled, sequenced, and combined; or on the rules for determining the winner within a given election (also called the "election rule" or "voting method").

Attitudes towards systems are highly influenced by the systems' impact on groups that one supports or opposes, which can make the objective comparison of electoral systems difficult. There are several ways to address this problem. For example, criteria can be defined mathematically, such that any voting method either passes or fails. This gives perfectly objective results, but their practical relevance is still arguable. Another approach is to define ideal criteria that no voting method passes perfectly, and then see how often or how close to passing various methods are over a large sample of simulated elections. This gives results which are practically relevant, but the method of generating the sample of simulated elections can still be arguably biased. A final approach is to create imprecisely defined criteria, and then assign a neutral body to evaluate each method according to these criteria. This approach can look at aspects of voting methods which the other two approaches miss, but both the definitions of these criteria and the evaluations of the methods are still inevitably subjective.

Mathematical criteria[edit]

To compare methods fairly and independently of political ideologies, voting theorists use voting method criteria, which define potentially desirable properties of voting methods mathematically.

Using criteria to compare methods does not make the comparison completely objective. For example, it is relatively easy to devise a criterion that is met by one's preferred voting method, and by very few other methods. Doing this, one can then construct a biased argument for the criterion, instead of arguing directly for the method. There is no ultimate authority on which criteria should be considered.

The following criteria, which apply to single-winner voting methods, are considered to be desirable by many voting theorists:

Result criteria (absolute)
These are criteria that state that, if the set of ballots is a certain way, a certain candidate must or must not win.
Majority criterion (MC)
Will a candidate always win who is ranked as the unique favorite by a majority of voters? This criterion comes in two versions:
  1. Ranked majority criterion, in which an option which is merely preferred over the others by a majority must win. (Passing the ranked MC is denoted by "yes", because implies also passing the following:)
  2. Rated majority criterion, in which only an option which is uniquely given a perfect rating by a majority must win. The ranked and rated MC are synonymous for ranked voting methods, but not for rated or graded ones. The ranked MC, but not the rated MC, is incompatible with the IIA criterion explained below.
Mutual majority criterion (MMC)
Will a candidate always win who is among a group of candidates ranked above all others by a majority of voters? This also implies the majority loser criterion – if a majority of voters prefers every other candidate over a given candidate, then does that candidate not win? Therefore, of the methods listed, all pass neither or both criteria, except for Borda, which passes Majority Loser while failing Mutual Majority.
Condorcet criterion
Will a candidate always win who beats every other candidate in pairwise comparisons? (This implies the majority criterion, above.)
Condorcet loser criterion (cond. loser)
Will a candidate never win who loses to every other candidate in pairwise comparisons?
Result criteria (relative)
These are criteria that state that, if a certain candidate wins in one circumstance, the same candidate must (or must not) win in a related circumstance.
Independence of Smith-dominated alternatives (ISDA)
Does the outcome never change if a Smith-dominated candidate is added or removed (assuming votes regarding the other candidates are unchanged)? Candidate C is Smith-dominated if there is some other candidate A such that C is beaten by A and every candidate B that is not beaten by A etc. Note that although this criterion is classed here as nominee-relative, it has a strong absolute component in excluding Smith-dominated candidates from winning. In fact, it implies all of the absolute criteria above.
Independence of irrelevant alternatives (IIA)
Does the outcome never change if a non-winning candidate is added or removed (assuming votes regarding the other candidates are unchanged)?[1] For instance, plurality rule fails IIA; adding a candidate X can cause the winner to change from W to Y even though Y receives no more votes than before.
Local independence of irrelevant alternatives (LIIA)
Does the outcome never change if the alternative that would finish last is removed? (And could the alternative that finishes second fail to become the winner if the winner were removed?)
Independence of clone alternatives (cloneproof)
Does the outcome never change if non-winning candidates similar to an existing candidate are added? There are three different phenomena which could cause a method to fail this criterion:
Spoilers
Candidates which decrease the chance of any of the similar or clone candidates winning, also known as a spoiler effect.
Teams
Sets of similar candidates whose mere presence helps the chances of any of them winning.
Crowds
Additional candidates who affect the outcome of an election without either helping or harming the chances of their factional group, but instead affecting another group.
Monotonicity criterion (monotone)
If candidate W wins for one set of ballots, will W still always win if those ballots change to rank W higher? (This also implies that you cannot cause a losing candidate to win by ranking him lower.)
Consistency criterion (CC)
If candidate W wins for one set of ballots, will W still always win if those ballots change by adding another set of ballots where W also wins?
Participation criterion (PC)
Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.[2])
Reversal symmetry (reversal)
If individual preferences of each voter are inverted, does the original winner never win?
Ballot-counting criteria
These are criteria which relate to the process of counting votes and determining a winner.
Polynomial time (polytime)
Can the winner be calculated in a runtime that is polynomial in the number of candidates and linear in the number of voters?
Resolvable
Can the winner be calculated in almost all cases, without using any random processes such as flipping coins? That is, are exact ties, in which the winner could be one of two or more candidates, vanishingly rare in large elections?
Summability (summable)
Can the winner be calculated by tallying ballots at each polling station separately and simply adding up the individual tallies? The amount of information necessary for such tallies is expressed as an order function of the number of candidates N. Slower-growing functions such as O(N) or O(N2) make for easier counting, while faster-growing functions such as O(N!) might make it harder to catch fraud by election administrators.[citation needed]
Strategy criteria
These are criteria that relate to a voter's incentive to use certain forms of strategy. They could also be considered as relative result criteria; however, unlike the criteria in that section, these criteria are directly relevant to voters; the fact that a method passes these criteria can simplify the process of figuring out one's optimal strategic vote.
Later-no-harm criterion, and later-no-help criterion
Can voters be sure that adding a later preference to a ballot will not harm or help any candidate already listed?[3]
No favorite betrayal (NFB)
Can voters be sure that they do not need to rank any other candidate above their favorite in order to obtain a result they prefer?[4]
Ballot format
These are issues relating to the expressivity or information content of a valid ballot.
Ballot type
What information is the voter give on the ballot?
Equal ranks
Can a valid ballot express equal support for more than one candidate (and not just equal opposition to more than one)?
Over 2 ranks
Can a ballot express more than two levels of support/opposition for different candidates?

Note on terminology: A criterion is said to be "weaker" than another when it is passed by more voting methods. Frequently, this means that the conditions for the criterion to apply are stronger. For instance, the majority criterion (MC) is weaker than the multiple majority criterion (MMC), because it requires that a single candidate, rather than a group of any size, should win. That is, any method which passes the MMC also passes the MC, but not vice versa; while any required winner under the MC must win under the MMC, but not vice versa.

Compliance of selected single-winner methods (table)[edit]

The following table shows which of the above criteria are met by several single-winner methods. Another table can be found here.

Sort:
Criterion
Method
Maj­ority Maj.
loser
Mutual
maj.
Cond­orcet Cond.
loser
Smith/
ISDA
LIIA IIA Clone­proof Mono­tone Consis­tency Partic­ipation Rever­sal
sym­metry
Poly­time/
resolv­able
Summ­able Later-no- No
favorite
betrayal
Ballot
type
Ranks
Harm Help = >2
Appr­oval Rated[a] No No No[b][c] No No[b] Yes Yes[d] Yes[e] Yes Yes Yes Yes O(N) Yes O(N) No Yes[f] Yes Appr­ovals Yes No
Borda count No Yes No No[b] Yes No No No Teams Yes Yes Yes Yes O(N) Yes O(N) No Yes No Ran­king No Yes
Cope­land Yes Yes Yes Yes Yes Yes No No[b] Teams,
crowds
Yes No[b] No[b] Yes O(N2) No O(N2) No[b] No No[b] Ran­king Yes Yes
IRV (AV) Yes Yes Yes No[b] Yes No[b] No No Yes No No No No O(N2) Yes[g] O(N!)[h] Yes Yes No Ran­king No Yes
Kemeny–Young Yes Yes Yes Yes Yes Yes Yes No[b] Spoil­ers Yes No[b]
[i]
No[b] Yes O(N!) Yes O(N2)[j] No[b] No No[b] Ran­king Yes Yes
Majority
judgment
[k]
Rated[l] Yes[m] No[n] No[b][c] No No[b] Yes Yes[d] Yes Yes No[o] No[p] Dep­ends[q] O(N) Yes O(N)[r] No[s] Yes Yes Scores[t] Yes Yes
Mini­max Yes No No Yes[u] No No No No[b] Spoil­ers Yes No[b] No[b] No O(N2) Yes O(N2) No[b][u] No No[b] Ran­king Yes Yes
Plura­lity Yes No No No[b] No No[b] No No Spoil­ers Yes Yes Yes No O(N) Yes O(N) N/A[v] N/A[v] No Single mark N/A No
Range voting No No No No[b][c] No No[b] Yes Yes[d] Yes Yes Yes Yes Yes O(N) Yes O(N) No Yes Yes Scores Yes Yes
Ranked pairs Yes Yes Yes Yes Yes Yes Yes No[b] Yes Yes No[b] No[p][b] Yes O(N4) Yes O(N2) No[b] No No[p][b] Ran­king Yes Yes
Runoff voting Yes Yes No No[b] Yes No[b] No No Spoil­ers No No No No O(N)[w] Yes O(N)[w] Yes Yes[x] No Single mark N/A No[y]
Schulze Yes Yes Yes Yes Yes Yes No No[b] Yes Yes No[b] No[p][b] Yes O(N3) Yes O(N2) No[b] No No[p][b] Ran­king Yes Yes
Sortition, arbitrary winner[z] No No No No[b] No No[b] Yes Yes No Yes Yes Yes Yes O(1) No O(1) Yes Yes Yes None N/A N/A
Random ballot[aa] No No No No[b] No No[b] Yes Yes Yes Yes Yes Yes Yes O(N) No O(N) Yes Yes Yes Single mark N/A No

Compliance of non-majoritarian multi-winner methods (table)[edit]

The following table shows which of the above criteria are met by several multiple winner methods. This table is incomplete. If you are an electoral scientist, for the sake of electoral science, please finish it!

Criterion

Method
Propor-
tional
(½ ≤ Δ ≤ 1)
Mono-
tone
Consis-
tency
Partici-
pation
Clone-
proof
No favorite
betrayal
Semi-
honest
Universally Liked Candidates With single
winner
Ballot
type
How hard is it to understand?
Monroe's Yes Yes Yes Yes No No Approval

or range

Approvals

or

scores

5 – Moderate/somewhat hard (It's just difficult to compute, not to understand)
Elbert's Approval

or range

Approvals

or

scores

10 – EXTREMELY DIFFICULT
Psi Yes Yes Yes Yes No No No Approval

or range

Approvals

or

scores

4 – Moderate (The equation just looks scary)
Harmonic Yes Yes Yes Yes No No No Approval

or range

Approvals

or

scores

4 – Moderate (The equation just looks scary)
Proportional Approval Yes Yes No Yes No No No Approval Approvals 2 – Simple
Re-weighted Range Yes Yes No Yes No No No Range Scores 3 – Simple/Moderate
Bid voting Yes No No Yes Approval

or range

Approvals

or

scores

4 – Moderate
Single Transferable Vote Yes No No No Yes No No Yes IRV Rankings 2 – Simple
CPO-STV Yes No No No Yes No No Yes A Condorcet

Method (depends)

Rankings 6 – Somewhat hard
Schulze STV Yes Yes No No Yes No No Yes Schulze Rankings 6 – Somewhat hard (easier if you understand single winner schulze)
Single Non Transferable Vote No Yes Yes Yes No

(spoilers)

No No No Plurality Single mark 0 – Super

simple

Multiple Winner Approval No Yes Yes Yes Yes Yes Yes N/A (not proportional) Approval Approvals 0 – Super

simple

Mini-max Outcome (not to be confused with other mini-max) Approval Approvals ? (I haven't read up on it yet)
Arbitrary Winner No Yes Yes Yes Yes Yes Yes N/A (not proportional) Arbitrary

Winner

None 0 – Super

simple

Single Random Ballot No Yes Yes Yes Yes Yes Yes N/A (not proportional) Random Ballot Limited marks 0 – Super

simple

Multiple Random Ballots Approaches Yes Yes Yes Yes Yes Yes N/A (not proportional) Random Ballot Limited rankings 0 – Super

simple

Compliance of majoritarian multi-winner methods (table)[edit]

The following table shows which of the above criteria are met by several multiple winner methods. This table is incomplete. If you are an electoral scientist, for the sake of electoral science, please finish it!

Criterion

Method
Mono-
tone
Consis-
tency
Partici-
pation
Clone-
proof
No favorite
betrayal
Semi-
honest
Smith set
winners
Condorcet
winner
Condorcet
loser
With single
winner
Ballot
type
How hard is it to understand?
Multiple Winner Approval Yes Yes Yes Yes Yes Yes No No No Approval Approvals 0 – Super

simple

Multiple Winner Range Yes Yes Yes Yes Yes Yes No No No Range Scores 0 – Super

simple

Multiple Winner Schulze Yes No No Yes No No Yes Yes Yes Schulze Rankings 0 – Super

simple

At Large Yes Yes Yes No

(spoilers)

No No No No No Plurality Limited marks 0 – Super

simple

Experimental criteria[edit]

It is possible to simulate large numbers of virtual elections on a computer and see how various voting methods compare in practical terms. Since such investigations are more difficult than simply proving that a given method does or does not satisfy a given mathematical criterion, results are not available for all methods. Also, these results are sensitive to the parameters of the model used to generate virtual elections, which can be biased either deliberately or accidentally.

One desirable feature that can be explored in this way is maximum voter satisfaction, called in this context minimum Bayesian regret. Such simulations are sensitive to their assumptions, particularly with regard to voter strategy, but by varying the assumptions they can give repeatable measures that bracket the best and worst cases for a voting method.[5] To date, the only such simulation to compare a wide variety of voting methods was run by a range-voting advocate and was not published in a peer-reviewed journal.[6][7] It found that Range voting consistently scored as either the best method or among the best across the various conditions studied.[8]

Another aspect which can be compared through such Monte Carlo simulations is strategic vulnerability. According to the Gibbard–Satterthwaite theorem, no voting method can be immune to strategic manipulation in all cases, but certainly some methods will have this problem more often than others. M. Balinski and R. Laraki, the inventors of the majority judgment method, performed such an investigation using a set of simulated elections based on the results from a poll of the 2007 French presidential election which they had carried out using rated ballots. Comparing range voting, Borda count, plurality voting, approval voting with two different absolute approval thresholds, Condorcet voting, and majority judgment, they found that range voting had the highest (worst) strategic vulnerability, while their own method majority judgment had the lowest (best).[9]

Balinski and Laraki also used the same information to investigate how likely it was that each of those methods, as well as runoff voting, would elect a centrist. Opinions differ on whether this is desirable or not. Some argue that methods which favor centrists are better because they are more stable; others argue that electing ideologically purer candidates gives voters more choice and a better chance to retrospectively judge the relative merits of those ideologies; while Balinski and Laraki argue that both centrist and extremist candidates should have a chance to win, to prevent forcing candidates into taking either position. According to their model, plurality, runoff voting, and approval voting with a higher approval threshold tended to elect extremists (100%, 98%, and 94% of the time, respectively); majority judgement elected both centrists and extremists (56% extremists); and range, Borda, and approval voting with a lower approval threshold elected centrists (6%; 0.25–13% depending on the number of candidates; and 6% extremists; respectively).[10] However, their model did not take into account voters' strategic reactions to the method used, such as "lesser of two evils" voting under plurality.

Simulated elections in a two-dimensional issue space can also be graphed to visually compare election methods; this illustrates issues like nonmonotonicity, clone-independence, and tendency to elect centrists vs extremists.[11]

"Soft" criteria[edit]

In addition to the above criteria, voting methods are judged using criteria that are not mathematically precise but are still important, such as simplicity, speed of vote-counting, the potential for fraud or disputed results, the opportunity for tactical voting or strategic nomination, and, for multiple-winner methods, the degree of proportionality produced.

The New Zealand Royal Commission on the Electoral System listed ten criteria for their evaluation of possible new electoral methods for New Zealand. These included fairness between political parties, effective representation of minority[12] or special interest groups, political integration, effective voter participation and legitimacy.

Notes[edit]

  1. ^ Approval fails the majority criterion because it does not always elect a candidate preferred by over half of voters; however, it always elects the candidate approved by the most voters.
  2. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap Condorcet, Smith and Independence of Smith-dominated alternatives criteria are incompatible with Independence of irrelevant alternatives, Consistency, Participation, Later-no-harm, Later-no-help, and Favorite betrayal[clarification needed] criteria.
  3. ^ a b c In Approval, Range, and Majority Judgment, if all voters have perfect information about each other's true preferences and use rational strategy, any Majority Condorcet or Majority winner will be strategically forced – that is, win in all of one or more strong Nash equilibria. In particular if every voter knows that "A or B are the two most-likely to win" and places their "approval threshold" between the two, then the Condorcet winner, if one exists and is in the set {A,B}, will always win. These methods also satisfy the majority criterion in the weaker sense that any majority can force their candidate to win, if it so desires. Laslier, J-F (2006), "Strategic approval voting in a large electorate" (PDF), IDEP Working Papers, Marseille, France: Institut D'Economie Publique (405) 
  4. ^ a b c Approval voting, range voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power or even abstain, despite having meaningful preferences among the available alternatives. If this assumption is not made, these methods fail IIA, as they become more ranked than rated methods.
  5. ^ The original Independence of clones criterion applied only to ranked voting methods. (T. Nicolaus Tideman, "Independence of clones as a criterion for voting rules", Social Choice and Welfare Vol. 4, No. 3 (1987), pp. 185–206.) Tideman notes that "in the spirit of independence of clones", "if there were two or more candidates who were so similar that every voter would rank them as tied if given the chance to rank them [...], then the number of perfect clones present would have no effect on whether the perfect clones were in the set of winning candidates under approval voting". So, Approval Voting satisfies this mathematical criterion by definition. However, there is some disagreement about whether considerations of the voter in the process of making up his vote could be tactically influenced by clones (in a way that a voter would dispossess a candidate of his approval when a clone of him is introduced) and whether the definition of clones have to be extended to these considerations additionally to the handling of actual votes.
  6. ^ Later-No-Harm and Later-No-Help assert that adding a later preference to a strictly ordered preference ballot should not help or harm an earlier preference. An Approval ballot records approvals but does not record explicit relative (e.g. later) preferences between approvals (while preferences exist from a voter's perspective). Meanwhile, a voter approving a less preferred candidate harms the probability of any other approved candidate winning, but does not help.
  7. ^ If the number of candidates grows faster than the square root of the number of voters, this may not be the case, as ties at any point in the process, even between two non-viable candidates, could affect the final result. If the rule for resolving such ties involves no randomness, though, the method does pass the criterion.[citation needed]
  8. ^ The number of piles that can be summed from various precincts is floor ((e−1) N!) − 1.
  9. ^ Kemeny-Young does not pass the consistency criterion for winner, but the consistency criterion for full rankings, that is, if the electorate is divided in two parts and in both parts Kemeny-Young chooses the same ranking, Kemeny-Young will also choose that ranking for the combined electorate.
  10. ^ Each prospective Kemeny-Young ordering has score equal to the sum of the pairwise entries that agree with it, and so the best ordering can be found using the pairwise matrix.
  11. ^ Bucklin voting, with skipped and equal-rankings allowed, meets the same criteria as Majority Judgment; in fact, Majority Judgment may be considered a form of Bucklin voting. Without allowing equal rankings, Bucklin's criteria compliance is worse; in particular, it fails Independence of Irrelevant Alternatives, which for a ranked method like this variant is incompatible with the Majority Criterion.
  12. ^ Majority Judgment does not always elect a candidate preferred over all others by over half of voters; however, it always elects the candidate uniquely top-rated by over half of voters.
  13. ^ Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.
  14. ^ Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.
  15. ^ Balinski and Laraki, Majority Judgment's inventors, point out that it meets a weaker criterion they call "grade consistency": if two electorates give the same rating for a candidate, then so will the combined electorate. Majority Judgment explicitly requires that ratings be expressed in a "common language", that is, that each rating have an absolute meaning. They claim that this is what makes "grade consistency" significant. Balinski M, MJ; Laraki, R (2007), A theory of measuring, electing and ranking, 104 (21), USA: National Academy of Sciences, pp. 8720–25 
  16. ^ a b c d e In Majority Judgment, Ranked Pairs, and Schulze voting, there is always a regret-free semi-honest ballot for any voter, holding other ballots constant. That is, if they know enough about how others will vote (for instance, in the case of Majority Judgment, the winning candidate and their winning median score), there is always at least one way for them to participate without grading any less-preferred candidate above any more-preferred one. However, this can cease to hold if voters have insufficient information.
  17. ^ Majority judgment can actually pass or fail reversal symmetry depending on the rounding method used to find the median when there are even numbers of voters. For instance, in a two-candidate, two-voter race, if the ratings are converted to numbers and the two central ratings are averaged, then MJ meets reversal symmetry; but if the lower one is taken, it does not, because a candidate with ["fair","fair"] would beat a candidate with ["good","poor"] with or without reversal. However, for rounding methods which do not meet reversal symmetry, the odds of breaking it are comparable to the odds of an irresolvable (tied) result; that is, vanishingly small for large numbers of voters.
  18. ^ Majority Judgment is summable at order KN, where K, the number of ranking categories, is set beforehand.
  19. ^ Though Majority Judgment does not pass this or similar criteria, there are other similar median methods, such as those based on Bucklin voting, which can meet a related, weaker criterion: ranking an additional candidate below the median grade (rather than your own grade) of your favorite candidate, cannot harm your favorite. Balinski, M., and R. Laraki. “A Theory of Measuring, Electing, and Ranking.” Proceedings of the National Academy of Sciences 104, no. 21 (2007): 8720.
  20. ^ In fact, Majority Judgment ballots use ratings expressed in "common language" rather than numbers, that is, that each rating have an absolute meaning.
  21. ^ a b A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.
  22. ^ a b Since plurality does not allow marking later preferences on the ballot at all, it is impossible to either harm or help a favorite candidate by marking later preferences, and so it trivially passes both Later-No-Harm and Later-No-Help. However, because it forces truncation, it shares some problems with methods that merely encourage truncation by failing Later-No-Harm. Similarly, though to a lesser degree, because it doesn't allow voters to distinguish between all but one of the candidates, it shares some problems with methods which fail Later-No-Help, which encourage voters to make such distinctions dishonestly.
  23. ^ a b Once for each round.
  24. ^ That is, second-round votes cannot help or harm candidates already eliminated.
  25. ^ Later preferences are only possible between the two candidates who make it to the second round.
  26. ^ Sortition, uniformly randomly chosen candidate is winner. Arbitrary winner, some external entity, not a voter, chooses the winner. These methods are not, properly speaking, voting methods at all, but are included to show that even a non-voting method can still pass some of the criteria.
  27. ^ Random ballot, uniformly randomly chosen ballot determines winner. This and closely related methods are of mathematical interest because they are the only possible methods which are truly strategy-free, that is, your best vote will never depend on anything about the other voters. However, this method is not generally considered as a serious proposal for a practical method.

References[edit]

  1. ^ Vasiljev, Sergei (April 1, 2008), Cardinal Voting: The Way to Escape the Social Choice Impossibility, SSRN eLibrary, SSRN 1116545Freely accessible . Note that in practice, voters could change their votes depending on who is in the race (especially in cardinal voting methods). However, this possibility is ignored, because if it were accounted for, no deterministic method could possibly pass this criterion.
  2. ^ Consistency implies participation, but not vice versa. For example, range voting complies with participation and consistency, but median ratings satisfies participation and fails consistency.
  3. ^ Woodall, Douglas (December 1994), "Properties of Preferential Election Rules", Voting Matters (3) 
  4. ^ Small, Alex (August 22, 2010), Geometric construction of voting methods that protect voters' first choices, arXiv:1008.4331Freely accessible 
  5. ^ Poundstone, William (2008), Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), New York: Hill and Young, p. 239 
  6. ^ WDS, "Appendix", Range vote (PDF) (unpublished paper), Temple, archived from the original (PDF) on March 26, 2009 
  7. ^ Poundstone 2008, p.  257: "Range voting is still largely a samizdat enterprise on the fringes of social choice theory. The most glaring example must be Smith's pivotal 2000 paper. It has never been published in a journal."
  8. ^ Poundstone 2008, p.  240.
  9. ^ Balinski, M; Laraki, R (2011) [2007], "Election by Majority Judgement: Experimental Evidence", in Dolez, Bernard; Grofman, Bernard; Laurent, Annie, In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections, Springer 
  10. ^ Balinski & Laraki 2007.
  11. ^ These two-dimensional graphs are called Yee diagrams after their inventor, Ka-Ping Yee. His website includes some sample graphs.
  12. ^ The ballot limited designation is technically possible only if the members of the majority, dividing their votes, cannot determine, individually, the minority appointment: Buonomo, Giampiero (2001). "L''emicollegio' sceglie il rappresentante di minoranza (o di maggioranza)". Diritto&Giustizia edizione online.   – via Questia (subscription required)