Borda count

The Borda count is a family of single-winner election methods in which voters rank options or candidates in order of preference. The Borda count determines the outcome of a debate or the winner of an election by giving each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. Once all votes have been counted the option or candidate with the most points is the winner. Because it tends to elect broadly-acceptable options or candidates, rather than those preferred by a majority, the Borda count is often described as a consensus-based voting system rather than a majoritarian one.[1]

The Modified Borda Count is used for decision-making. For multi-winner elections, especially when proportional representation is important, the quota Borda system is used.

The Borda count was developed independently several times, as early as 1435 by Nicholas of Cusa,[2][3][4] but is named for the 18th-century French mathematician and political scientist Jean-Charles de Borda, who devised the system in 1770. It is currently used to elect members of the Parliament of Nauru and two ethnic minority members of the National Assembly of Slovenia,[5] in modified forms to determine which candidates are elected to the party list seats in Icelandic parliamentary elections, and for selecting presidential election candidates in Kiribati. It is also used throughout the world by various private organizations and competitions.

Voting and counting

The Borda count is a preferential, or ranked, voting system; the voter ranks the list of candidates in order of preference. So, for example, the voter gives a 1 to their most preferred candidate, a 2 to their second most preferred, and so on. In this respect, it is the same as elections under systems such as instant-runoff voting, the single transferable vote or Condorcet methods.

Points are then given to each candidate in reverse proportion to their ranking, so that higher-ranked candidates receive more points. When all votes have been counted, and the points added up, the candidate with the most points wins.

Because, from each voter, candidates receive a certain number of points, the Borda count is also classified as a positional voting system. Other positional methods include first-past-the-post voting, bloc voting, approval voting and limited voting.

The number of points assigned for each ranking varies depending on which of several variants of the Borda count is used:

Borda's system (starting at 1)

In Borda's original proposal, the number of points given to candidates for each ranking is determined by the number of candidates standing in the election. If there are five candidates in an election, candidates will receive five points each time they are ranked first, four for being ranked second, and so on, with a candidate receiving one point for being ranked last (or left unranked).[3] In other words, where there are n candidates a candidate will receive n points for a first preference, n − 1 points for a second preference, n − 2 for a third, and so on, as shown in the following example:

RankingCandidateFormulaPointsRelative points
1st Andrewn51.00
2nd Briann−140.80
3rd Catherinen−230.60
4th Davidn−320.40
5th Elizabethn−410.20

This system is used in the Slovenian parliamentary elections for two out of 90 seats.[6]

Starting at 0

Alternatively, votes can be counted by giving each candidate a number of points equal to the number of candidates ranked lower than them, so that a candidate receives n − 1 point for a first preference, n − 2 for a second, and so on, with zero points for being ranked last (or left unranked). In other words, a candidate ranked in ith place receives n−i points.[7] For example, in a five-candidate election, the number of points assigned for the preferences expressed by a voter on a single ballot paper might be:

RankingCandidateFormulaPointsRelative points
1st Andrewn−141.00
2nd Briann−230.75
3rd Catherinen−320.50
4th Davidn−410.25
5th Elizabethn−500.00

With this second weighting, in a two candidate election, a first rank vote would get 1 point, and second rank vote gets 0 points, just like plurality voting.

Dowdall system (Nauru)

The island nation of Nauru uses a variant called the Dowdall System:[8][6] the voter awards the first-ranked candidate with 1 point, while the 2nd-ranked candidate receives ½ a point, the 3rd-ranked candidate receives ⅓ of a point, etc. (A similar system of weighting lower-preference votes was used in the 1925 Oklahoma primary electoral system.) An important difference of this method from the others is that the number of points assigned to each preference does not depend on the number of candidates. Using the above example, in Nauru the point distribution among the five candidates would be this:

RankingCandidateFormulaPointsAbsolute points
1st Andrew1/11.0060
2nd Brian1/20.5030
3rd Catherine1/30.3320
4th David1/40.2515
5th Elizabeth1/50.2012

This method is far more favorable to candidates with many first preferences than the conventional Borda count; it also substantially reduces the impact of electors indicating late preferences at random because they are required to complete the full ballot.[9] It has been described as a system "somewhere between plurality and the Borda Count, but as veering more towards plurality".[6] Simulations show that 30% of Nauru elections would produce different outcomes if counted using standard Borda rules.[6] The tables below compare the official layout of the aggregated decimal vote, used in the Dowdall system, with an example of the single voting card, in raw form. Under the Nauruan system candidates 1 and 5 would be elected into parliament.

CandidateVote value
Candidate 1335.433
Candidate 2184.833
Candidate 3179.633
Candidate 4319.617
Candidate 5349.617
Candidate 6169.567
CandidateVote rankDecimal value
Candidate 11/20.5
Candidate 21/40.25
Candidate 31/50.2
Candidate 41/30.3333…
Candidate 51/11.0
Candidate 61/60.1666…

Truncated ballots

A common way in which versions of the Borda count differ is the method for dealing with truncated ballots, that is, ballots on which a voter has not expressed a full list of preferences. There are several methods:

  • Voters can simply be obliged to rank all candidates. This is the method used in Nauru.
  • The simplest method is to allow voters to rank as many or as few candidates as they wish, but simply give every unranked candidate the minimum number of points. For example, if there are 10 candidates, and a voter votes for candidate A first and candidate B second, leaving everyone else unranked, candidate A receives 9 or 10 points (depending on the formula used), candidate B receives 8 or 9 points, and all other candidates receive either zero or 1. However, this method allows strategic voting in the form of bullet voting: voting for only one candidate and leaving every other candidate unranked. This variant makes a bullet vote more effective than a fully ranked ballot.
  • Voters can be permitted to rank only a subset of the total number of candidates but obliged to rank all of those, with all unranked candidates being given zero points.
    • In Kiribati, a variant is employed which uses a traditional Borda formula, but in which voters rank only four candidates, irrespective of how many are standing.[10]
    • In Toastmasters International, speech contests are truncation scored as 3, 2, 1 for the top-three ranked candidates. Ties are broken by having a special ballot that is ignored, unless there is a tie.[11]
  • In Slovenia, legislation does not mention the truncated ballots. Consequently, in the past, election bodies dealt with them differently from district to district and from election to election. In 2004 parliamentary election, for instance, in one district unranked candidates received one point while in the other district they received zero points. In 2008, unranked candidates in both districts that use Borda count received one point.

Modified Borda count

In a modified Borda count (MBC), the number of points given for a voter's first and subsequent preferences is determined by the total number of options or candidates they have actually ranked, rather than the total number listed. This is to say, typically, on a ballot of n options/candidates, if a voter casts only m preferences (where n≥m≥1), a first preference gets m points, a second preference m–1 points, and so on. In more general terms, an 'x'th preference, if cast, gets one more point than an 'x+1'th preference (whether cast or not). The MBC involves no special weighting: the difference is always just one point.

In a BC on 5 options, voting for all 5 options gives first preference 5 points, second preference 4 points, and so on; whereas voting for only 1 option still gives first preference 5 points. In effect, conventional Borda Count encourages the voter to submit only a first preference, so it degenerates into a plurality vote.

In a 5-option MBC, by contrast, votes for only 1 option thus gives the favorite just 1 point; votes for 2 options give first preference 2 points (and second preference 1 point). To ensure the favorite gets the maximum 5 points, therefore, all 5 preferences should be cast. The MBC thus encourages voters to submit a fully marked ballot at the expense of giving candidates other than a favorite a higher rankings, increasing their chances to win as well.

Multiple winners

The system invented by Jean-Charles de Borda was intended for use in elections with a single winner, but it is also possible to conduct a Borda count with more than one winner, by recognizing the desired number of candidates with the most points as the winners. In other words, if there are two seats to be filled, then the two candidates with most points win; in a three-seat election, the three candidates with most points, and so on. In Nauru, which uses the multi-seat variant of the Borda count, parliamentary constituencies of two and four seats are used. The quota Borda system is a system of proportional representation in multi-seat constituencies that uses the Borda count.

Other systems

A number of voting systems other than the Borda count employ its system of assigning points for rankings. The Nanson and Baldwin methods are single-winner voting systems that combine elements of the Borda count and instant-runoff voting. Unlike the Borda count, Nanson and Baldwin are majoritarian and Condorcet methods.

As a consensual method

Unlike other popular voting systems, in the Borda count it is possible for a candidate who is the first preference of an absolute majority of voters to not be elected; this is because the Borda count affords greater importance to a voter's lower preferences than most other systems, including other preferential methods such as instant-runoff voting and Condorcet methods.

The Borda count tends to favor candidates supported by a broad consensus among voters, rather than the candidate who is necessarily the favorite of a majority;[1] for this reason, its supporters see the Borda count as a method that promotes unity and avoids the 'tyranny of the majority', and the resulting divisiveness and even violence that it can lead to. Advocates argue, for example, that where the majority candidate is strongly opposed by a large minority of the electorate, the Borda winner may have higher overall utility than the majority winner. On grounds such as these, the de Borda Institute of Northern Ireland advocates the use of a form of referendum based on the Borda count in divided societies such as Northern Ireland, the Balkans and Kashmir.[12][13]

Because it will not necessarily elect a candidate who is the first preference of a majority of voters, the Borda count is said by scholars to fail the majority criterion. Other voting systems that favor consensus rather than majority rule include cardinal methods such as approval voting, score voting, and their variants.[14] These are sometimes called "utilitarian voting methods" because they try to maximize the entire population's utility, rather than maximizing the majority's utility at the expense of the minority's.[15][16][17]

Example

In an election in which 100 voters express the following preferences:

No. 51 voters 5 voters 23 voters 21 voters
1st Andrew Catherine Brian David
2nd Catherine Brian Catherine Catherine
3rd Brian David David Brian
4th David Andrew Andrew Andrew

The Borda scores of the candidates are:

CandidateBase 0Base 1Nauru
Andrew15325363.25
Brian15125149.5
Catherine20530552.5
David9119143.08333...

The first totals are when the Borda system is N-1 such that last place has a point score of 0. The second totals are based on N (number of candidates) points for first choice, N-1 second, N-2 third etc. Note that using N for first place or N-1 only changes everyone's totals by the number of voters. In this case, with 100 voters, the differences are 100 for each candidate.

Under most single-winner voting systems – including 'first-past-the-post' (plurality), instant-runoff, Condorcet methods, and Nauru Borda – Andrew would have been the winning candidate; however, under the standard Borda count, Catherine has the highest Borda score and so is elected instead. Although Andrew is supported by an unambiguous absolute majority of voters, he is the last preference of 49 voters, which suggests that he may be strongly opposed by almost one half of the electorate. Catherine, though she receives only a handful of first-preference votes, is at least the second choice of all voters, implying that she is broadly acceptable to all.

Nauru Borda, relative to normal Borda, places a much stronger emphasis on first choices over second choices, which is why Catherine did so poorly under that system.

Potential for tactical manipulation

Tactical voting

Like many other voting systems, the Borda count is vulnerable to tactical voting. In particular, it is highly vulnerable to the tactics of compromising and burying. Compromising: voters can benefit by insincerely raising the position of their second choice candidate over their first choice candidate, in order to help the second choice candidate to beat a candidate they like even less. Burying: voters can help a more-preferred candidate by insincerely lowering the position of a less-preferred candidate on their ballot.

An effective tactic is to combine these two strategies. For example, if there are two candidates whom a voter considers to be the most likely to win, the voter can maximise his impact on the contest between these front runners by ranking the candidate whom he likes more in first place, and ranking the candidate whom he likes less in last place. If neither front runner is his sincere first or last choice, the voter is employing both the compromising and burying tactics at once; if many voters employ such strategies, then the result will no longer reflect the sincere preferences of the electorate.

Using the example below based on choosing the capital of Tennessee, if polls suggest a toss-up between Nashville and Chattanooga, citizens of Knoxville might change their ranking to

  1. Chattanooga (compromising their sincere first choice, Knoxville)
  2. Knoxville
  3. Memphis (burying their sincere third choice, Nashville)
  4. Nashville

If many Knoxville voters voted in this way, it would result in the election of Chattanooga. Citizens of Chattanooga could also increase the likelihood of the election of their city by voting tactically, but would require the assistance of some tactical voters from Knoxville to be successful.

The French Academy of Sciences (of which Borda was a member) experimented with Borda's system, but abandoned it, in part because "the voters found how to manipulate the Borda rule: not only by putting their most dangerous rival at the bottom of their lists, but also by truncating their lists".[18] In response to the issue of strategic manipulation in the Borda count, M. de Borda said, "My scheme is intended for only honest men".[7][18]

The academic Donald G. Saari has created a mathematical framework for evaluating positional methods which shows that, for 3-candidate races, the Borda count is more resistant to tactical voting than other positional methods such as plurality, approval, and cumulative voting.[19]

Tactical voting is common in Slovenia, where truncated ballots are allowed; a majority of voters bullet vote, with only 42% of voters ranking a second-preference candidate.[6]

Strategic nomination

The Borda count is highly vulnerable to a form of strategic nomination called teaming or cloning. This means that when more candidates run with similar ideologies, the probability of one of those candidates winning increases. Therefore, under the Borda count, it is to a faction's advantage to run as many candidates in that faction as they can. For example, even in a single-seat election, it would be to the advantage of a political party to stand as many candidates as possible in an election. In this respect, the Borda count differs from many other single-winner systems, such as the 'first past the post' plurality system, in which a political faction is disadvantaged by running too many candidates. Under systems such as plurality, 'splitting' a party's vote in this way can lead to the spoiler effect, which harms the chances of any of a faction's candidates being elected.

In 1980, William Gehrlein and Peter Fishburn compared the Borda count to other positional methods, such as plurality and approval voting. They investigated the likelihood of a positional method choosing the same candidate when the set of candidates was modified by eliminating one losing candidate from a three-candidate election, and two losing candidates from a four-candidate election. They found that the Borda count was the positional rule which maximises the probability of electing the same candidate after this modification of the choice set.

Strategic nomination is used in Nauru, according to MP Roland Kun, with factions running multiple "buffer candidates" who are not expected to win, to lower the tallies of their main competitors.[6]

Evaluation by criteria

Scholars of electoral systems often compare them using mathematically defined voting system criteria. From among these:

The variant of the Borda count that permits bullet voting satisfies the plurality criterion, but the 'modified Borda count' does not. Variants that oblige voters to rank only a certain specified number of candidates satisfy the same criteria as the conventional Borda count.

Simulations show that Borda has a high probability of choosing the Condorcet winner when one exists.[6]

Example

Tennessee and its four major cities: Memphis in the south-west; Nashville in the centre, Chattanooga in the south, and Knoxville in the east

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities and that everyone wants to live as near to the capital as possible.

The candidates for the capital are:

  • Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
  • Nashville, with 26% of the voters, near the center of the state
  • Knoxville, with 17% of the voters
  • Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)		 26% of voters
(close to Nashville)		 15% of voters
(close to Chattanooga)		 17% of voters
(close to Knoxville)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

This leads to the following point counts per 100 voters:

Voters' home cityMemphisNashvilleKnoxvilleChattanooga
Memphis42×3=12642×2=84042×1=42
Nashville026×3=7826×1=2626×2=52
Knoxville017×1=1717×3=5117×2=34
Chattanooga015×1=1515×2=3015×3=45
Total126194107173

Thus Nashville is elected.

Current uses

Political uses

The Borda count is used for certain political elections in at least three countries, Slovenia and the tiny Micronesian nations of Kiribati and Nauru.

In Slovenia, the Borda count is used to elect two of the ninety members of the National Assembly: one member represents a constituency of ethnic Italians, the other a constituency of the Hungarian minority.

Members of the Parliament of Nauru are elected based on a variant of the Borda count that involves two departures from the normal practice: (1) multi-seat constituencies, of either two or four seats, and (2) a point-allocation formula that involves increasingly small fractions of points for each ranking, rather than whole points.

In Kiribati, the president (or Beretitenti) is elected by the plurality system, but a variant of the Borda count is used to select either three or four candidates to stand in the election. The constituency consists of members of the legislature (Maneaba). Voters in the legislature rank only four candidates, with all other candidates receiving zero points. Since at least 1991, tactical voting has been an important feature of the nominating process.

The Republic of Nauru became independent from Australia in 1968. Before independence, and for three years afterwards, Nauru used instant-runoff voting, importing the system from Australia, but since 1971, a variant of the Borda count has been used.

The modified Borda count has been used by the Green Party of Ireland to elect its chairperson.[20][21]

The Borda count has been used for non-governmental purposes at certain peace conferences in Northern Ireland, where it has been used to help achieve consensus between participants including members of Sinn Féin, the Ulster Unionists, and the political wing of the UDA.

Other uses

The Borda count is used in elections by some educational institutions in the United States.

The Borda count is used in elections by some professional and technical societies.

The OpenGL Architecture Review Board uses the Borda count as one of the feature-selection methods.

The Borda count is used to determine winners for the World Champion of Public Speaking contest organized by Toastmasters International. Judges offer a ranking of their top three speakers, awarding them three points, two points, and one point, respectively. All unranked candidates receive zero points.

The modified Borda count is used to elect the President for the United States member committee of AIESEC.

The Borda count, and points-based systems similar to it, are often used to determine awards in competitions.

The Borda count is a popular method for granting sports awards in the United States. Uses include:

The Eurovision Song Contest uses a heavily modified form of the Borda count, with a different distribution of points: only the top ten entries are considered in each ballot, the favorite entry receiving 12 points, the second-placed entry receiving 10 points, and the other eight entries getting points from 8 to 1. Although designed to favor a clear winner, it has produced very close races and even a tie.

The People's Remix Competition uses a Borda variant where each voter ranks only the top three contestants.

The Borda count is used for wine trophy judging by the Australian Society of Viticulture and Oenology, and by the RoboCup autonomous robot soccer competition at the Center for Computing Technologies, in the University of Bremen in Germany.

The Finnish Associations Act lists three different modifications of the Borda count for holding a proportional election. All the modifications use fractions, as in Nauru. A Finnish association may choose to use other methods of election, as well.[23]

Sports

In sailboat fleet racing, the Borda count is used to select the winner of a regatta, with each individual race in the regatta treated as a 'vote'.

Borda-similar systems

Some voting systems described as Borda systems are variants of the Borda count. The scores candidates receive in some of those systems are significantly different from those they would receive using a strict Borda count. Fraenkel and Grofman state that "The system used on Nauru is not merely a ‘modified form of Borda’, but an important rule in its own right."[6]

An example is the case of the voting to determine the winner of the Heisman Trophy: judges vote 3,2,1 for their top three choices – the vote weights are thus (3,2,1,0,0,0, ..., 0,0,0). By contrast, the Borda vote weights in, say, a fifty-candidate election would be (49,48,47, ..., 2,1,0), which is markedly different. Heisman-style voting, when there are more than a handful of candidates, is thus more similar to plurality voting, which has weights (1,0,0,0, ..., 0,0,0), than it is to Borda.

History

A form of the Borda count was one of the voting methods employed in the Roman Senate beginning around the year 105. However, in its modern, mathematical form, the system is thought to have been discovered independently at least three times:

See also

Notes

  1. 1 2 Lippman, David. "Voting Theory" (PDF). Math in Society. Borda count is sometimes described as a consensus-based voting system, since it can sometimes choose a more broadly acceptable option over the one with majority support.
  2. ↑ Emerson, Peter (2016-01-16). From Majority Rule to Inclusive Politics. Springer. ISBN 9783319235004.
  3. 1 2 Emerson, Peter (2013-02-01). "The original Borda count and partial voting". Social Choice and Welfare. 40 (2): 353–358. doi:10.1007/s00355-011-0603-9. ISSN 0176-1714.
  4. ↑ Actually, Nicholas' system used higher numbers for more-preferred candidates
  5. ↑ "Slovenia's electoral law"
  6. 1 2 3 4 5 6 7 8 Fraenkel, Jon; Grofman, Bernard (2014-04-03). "The Borda Count and its real-world alternatives: Comparing scoring rules in Nauru and Slovenia". Australian Journal of Political Science. 49 (2): 186–205. doi:10.1080/10361146.2014.900530.
  7. 1 2 Black, Duncan (1987) [1958]. The Theory of Committees and Elections. Springer Science & Business Media. ISBN 9780898381894.
  8. ↑ Reilly, Benjamin (2002). "Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries". International Political Science Review / Revue internationale de science politique. 23 (4): 364–366. doi:10.1177/0192512102023004002.
  9. ↑ "Results of the General Election held on 19th June 2010" (PDF). Parliament of Nauru. Archived from the original (PDF) on 2012-10-29. Retrieved 16 December 2011.
  10. ↑ Reilly, Benjamin. "Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries" (PDF). Archived from the original (PDF) on 2006-08-19.
  11. ↑ SPEECH CONTEST RULEBOOK JULY 1, 2017 TO JUNE 30, 2018
  12. ↑ Emerson, Peter (2016). From Majority Rule to Inclusive Politics (1st ed.). Cham: Springer. ISBN 9783319235004. OCLC 948558369. Unfortunately, one of the worst democratic structures is the most ubiquitous: majority rule based on majority voting. It must be emphasised, furthermore, that these two practices are often the catalysts of division and bitterness, if not indeed violence and war.
  13. ↑ Emerson, Peter (2016-03-23). "Majority Rule - A Cause of War?". In Gardner, Hall; Kobtzeff, Oleg. The Ashgate Research Companion to War: Origins and Prevention. Routledge. ISBN 9781317041108.
  14. ↑ "Majority Criterion". The Center for Election Science. 2015-05-21. Retrieved 2016-12-03. Sometimes a candidate who is the Condorcet winner, or even the majority winner, isn’t the favored or “most representative” candidate of the electorate.
  15. ↑ "Utilitarian vs. Majoritarian Election Methods". The Center for Election Science. Retrieved 2018-05-13.
  16. ↑ "Vote Aggregation Methods". lorrie.cranor.org. Retrieved 2017-01-12.
  17. ↑ Hillinger, Claude (2006-05-15). "The Case for Utilitarian Voting". Rochester, NY: Social Science Research Network. SSRN 878008.
  18. 1 2 McLean, Iain; Urken, Arnold B.; Hewitt, Fiona (1995). Classics of Social Choice. University of Michigan Press. ISBN 0472104500.
  19. ↑ Saari, Donald G. (1990-01-01). "Susceptibility to manipulation". Public Choice. 64 (1): 21–41. doi:10.1007/BF00125915. ISSN 0048-5829. It is shown that the system least susceptible to micro manipulations for n = 3 candidates is the Borda Count (BC).
  20. ↑ Voting Systems
  21. ↑ Emerson, Peter (2007) Designing an All-Inclusive Democracy. Springer Verlag,Part 1, pages 15-38 "Collective Decision-making: The Modified Borda Count, MBC" ISBN 978-3-540-33163-6 (Print) 978-3-540-33164-3 (Online)
  22. ↑ Heisman.com - Heisman Trophy
  23. ↑ "Finnish Associations Act". National Board of Patents and Registration of Finland. Archived from the original on 2013-03-01. Retrieved 26 June 2011.

Further reading

  • Emerson, Peter (2007). Designing an All-Inclusive Democracy - Consensual Voting Procedures for use in Parliaments, Councils and Committees. Springer-Verlag. ISBN 978-3-540-33163-6. (Print) 978-3-540-33164-3 (online)
  • Reilly, Benjamin (2002). "Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries". International Political Science Review. 23 (4): 355–372. doi:10.1177/0192512102023004002.
  • Saari, Donald G. (2000). "Mathematical Structure of Voting Paradoxes: II. Positional Voting". Journal of Economic Theory. 15 (1). doi:10.1007/s001990050002. SSRN 195769.
  • Saari, Donald G. (2001). Chaotic Elections!. Providence, RI: American Mathematical Society. ISBN 0-8218-2847-9. Describes various voting systems using a mathematical model, and supports the use of the Borda count.
  • Toplak, Jurij (2006). "The parliamentary election in Slovenia, October 2004". Electoral Studies. 25 (4): 825–831. doi:10.1016/j.electstud.2005.12.006.
  • Adelsman, Rony M.; Whinston, Andrew B. (1977). "Sophisticated Voting with Information for Two Voting Functions". Journal of Economic Theory. 15 (1): 145–159. doi:10.1016/0022-0531(77)90073-4.
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