Condorcet methods
73-410: Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results The single transferable vote ( STV ) or proportional-ranked choice voting ( P-RCV ),
146-464: A "uniform quota" is simply set by law – any candidate receiving that set number of votes is declared elected, with surplus transferred away. Something like this system was used in New York City from 1937 to 1947, where seats were allocated to each borough based on voter turnout and then each candidate that surpassed set number of votes was declared elected, and enough others that came close to fill up
219-501: A 'cycle'. This situation emerges when, once all votes have been tallied, the preferences of voters with respect to some candidates form a circle in which every candidate is beaten by at least one other candidate ( Intransitivity ). For example, if there are three candidates, Candidate Rock, Candidate Scissors, and Candidate Paper , there will be no Condorcet winner if voters prefer Candidate Rock over Candidate Scissors and Scissors over Paper, but also Candidate Paper over Rock. Depending on
292-400: A 68% majority of 1st choices among the remaining candidates and won as the majority's 1st choice. As noted above, sometimes an election has no Condorcet winner because there is no candidate who is preferred by voters to all other candidates. When this occurs the situation is known as a 'Condorcet cycle', 'majority rule cycle', 'circular ambiguity', 'circular tie', 'Condorcet paradox', or simply
365-447: A candidate who does not need them. If seats remain open after the first count, any surplus votes are transferred. This may generate the necessary winners. As well, least popular candidates may be eliminated as way to generate winners. The specific method of transferring votes varies in different systems (see § Vote transfers and quota ). Transfer of any existing surplus votes is done before eliminations of candidates. This prevents
438-534: A contest between candidates A, B and C using the preferential-vote form of Condorcet method, a head-to-head race is conducted between each pair of candidates. A and B, B and C, and C and A. If one candidate is preferred over all others, they are the Condorcet Winner and winner of the election. Because of the possibility of the Condorcet paradox , it is possible, but unlikely, that a Condorcet winner may not exist in
511-434: A district. The key to STV's approximation of proportionality is that each voter effectively only casts a single vote in a district contest electing multiple winners, while the ranked ballots (and sufficiently large districts) allow the results to achieve a high degree of proportionality with respect to partisan affiliation within the district, as well as representation by gender and other descriptive characteristics. The use of
584-522: A large number of effective votes – 19 votes were used to elect the successful candidates. (Only the votes for Oranges at the end were not used to select a food. The Orange voters have satisfaction of seeing their second choice – Pears – selected, even if their votes were not used to select any food.) As well, there was general satisfaction with the choices selected. Nineteen voters saw either their first or second choice elected, although four of them did not actually have their vote used to achieve
657-594: A large party to take its full share of the seats, even denying a majority party a majority of seats, the Droop quota does not disadvantage larger parties. Some say the Droop quota may go too far in that regard, saying it is the most-biased possible quota that can still be considered to be proportional. Today the Droop quota is used in almost all STV elections, including those in the Republic of Ireland , Malta , Australia , Northern Ireland , and India . In some implementations,
730-452: A party from losing a candidate in the early stage who might be elected later through transfers. When surplus votes are transferred under some systems, some or all of the votes held by the winner are apportioned fractionally to the next marked preference on the ballot. In others, the transfers to the next available marked preference is done using whole votes. When seats still remain to be filled and there are no surplus votes to transfer (none of
803-455: A quota may be workable as long as rules are in place for dealing with situations where two or more tied candidates are competing for a lesser number of seats. The common quotas used in single transferable voting elections are such that no more can achieve quota than the number of seats in the district. There are two commonly-used quotas: the Hare and Droop quotas . The Hare quota is unbiased in
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#1732844648091876-448: A quota means that, for the most part, each successful candidate is elected with the same number of votes. This equality produces fairness in the particular sense that a party taking twice as many votes as another party will generally take twice the number of seats compared to that other party. Under STV, winners are elected in a multi-member constituency (district) or at-large, also in a multiple-winner contest. Every sizeable group within
949-532: A result of a kind of tie known as a majority rule cycle , described by Condorcet's paradox . The manner in which a winner is then chosen varies from one Condorcet method to another. Some Condorcet methods involve the basic procedure described below, coupled with a Condorcet completion method, which is used to find a winner when there is no Condorcet winner. Other Condorcet methods involve an entirely different system of counting, but are classified as Condorcet methods, or Condorcet consistent, because they will still elect
1022-523: A specific election. This is sometimes called a Condorcet cycle or just cycle and can be thought of as Rock beating Scissors, Scissors beating Paper, and Paper beating Rock . Various Condorcet methods differ in how they resolve such a cycle. (Most elections do not have cycles. See Condorcet paradox#Likelihood of the paradox for estimates.) If there is no cycle, all Condorcet methods elect the same candidate and are operationally equivalent. For most Condorcet methods, those counts usually suffice to determine
1095-437: A transfer if the first-preference food is chosen with a surplus of votes. The 23 guests at the party mark their ballots: some mark first, second and third preferences; some mark only two preferences. When the ballots are counted, it is found that the ballots are marked in seven distinct combinations, as shown in the table below: The table is read as columns: the left-most column shows that there were three ballots with Orange as
1168-427: A voter's choice within any given pair can be determined from the ranking. Some elections may not yield a Condorcet winner because voter preferences may be cyclic—that is, it is possible that every candidate has an opponent that defeats them in a two-candidate contest. The possibility of such cyclic preferences is known as the Condorcet paradox . However, a smallest group of candidates that beat all candidates not in
1241-564: A voter's subsequent preferences if necessary. Under STV, no one party or voting bloc can take all the seats in a district unless the number of seats in the district is very small or almost all the votes cast are cast for one party's candidates (which is seldom the case). This makes it different from other commonly used candidate-based systems. In winner-take-all or plurality systems – such as first-past-the-post (FPTP), instant-runoff voting (IRV), and block voting – one party or voting bloc can take all seats in
1314-425: Is Hamburgers, so the three votes are transferred to Hamburgers. Hamburgers is elected with 7 votes in total. Hamburgers now has a surplus vote, but this does not matter since the election is over. There are no more foods needing to be chosen – three have been chosen. Result: The winners are Pears, Cake, and Hamburgers. Orange ends up being neither elected nor eliminated. STV in this case produced
1387-452: Is a multi-winner electoral system in which each voter casts a single vote in the form of a ranked ballot . Voters have the option to rank candidates, and their vote may be transferred according to alternative preferences if their preferred candidate is eliminated or elected with surplus votes, so that their vote is used to elect someone they prefer over others in the running. STV aims to approach proportional representation based on votes cast in
1460-407: Is also a Condorcet method, even though the voters do not vote by expressing their orders of preference. There are multiple rounds of voting, and in each round the vote is between two of the alternatives. The loser (by majority rule) of a pairing is eliminated, and the winner of a pairing survives to be paired in a later round against another alternative. Eventually, only one alternative remains, and it
1533-494: Is also referred to collectively as Condorcet's method. A voting system that always elects the Condorcet winner when there is one is described by electoral scientists as a system that satisfies the Condorcet criterion. Additionally, a voting system can be considered to have Condorcet consistency, or be Condorcet consistent, if it elects any Condorcet winner. In certain circumstances, an election has no Condorcet winner. This occurs as
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#17328446480911606-411: Is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner , is formally called the Condorcet winner or Pairwise Majority Rule Winner (PMRW). The head-to-head elections need not be done separately;
1679-412: Is an election between four candidates: A, B, C, and D. The first matrix below records the preferences expressed on a single ballot paper, in which the voter's preferences are (B, C, A, D); that is, the voter ranked B first, C second, A third, and D fourth. In the matrix a '1' indicates that the runner is preferred over the 'opponent', while a '0' indicates that the runner is defeated. Using a matrix like
1752-489: Is calculated by a specified method (STV generally uses the Hare or Droop quota ), and candidates who accumulate that many votes are declared elected. In many STV systems, the quota is also used to determine surplus votes, the number of votes received by successful candidates over and above the quota. Surplus votes are transferred to candidates ranked lower in the voters' preferences, if possible, so they are not wasted by remaining with
1825-484: Is eliminated. In accordance with the next preference marked on the vote cast by the voter who voted Strawberry as first preference, that vote is transferred to Oranges. In accordance with the next preference marked on the two votes cast by the Pear–Strawberry–Cake voters (which had been transferred to Strawberry in step 2), the two votes are transferred to Cake. (The Cake preference had been "piggy-backed" along with
1898-498: Is given by the expression: It was first proposed in 1868 by the English lawyer and mathematician Henry Richmond Droop (1831–1884), who identified it as the minimum amount of support that would not possibly be achieved by too many compared to the number of seats in a district in semiproportional voting systems such as SNTV , leading him to propose it as an alternative to the Hare quota . While Hare quota makes it more difficult for
1971-401: Is holding an election on the location of its capital . The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are: The preferences of each region's voters are: To find the Condorcet winner every candidate must be matched against every other candidate in a series of imaginary one-on-one contests. In each pairing
2044-513: Is known as ambiguity resolution, cycle resolution method, or Condorcet completion method . Circular ambiguities arise as a result of the voting paradox —the result of an election can be intransitive (forming a cycle) even though all individual voters expressed a transitive preference. In a Condorcet election it is impossible for the preferences of a single voter to be cyclical, because a voter must rank all candidates in order, from top-choice to bottom-choice, and can only rank each candidate once, but
2117-440: Is no preference between candidates that were left unranked. Some Condorcet elections permit write-in candidates . The count is conducted by pitting every candidate against every other candidate in a series of hypothetical one-on-one contests. The winner of each pairing is the candidate preferred by a majority of voters. Unless they tie, there is always a majority when there are only two choices. The candidate preferred by each voter
2190-510: Is taken to be the one in the pair that the voter ranks (or rates) higher on their ballot paper. For example, if Alice is paired against Bob it is necessary to count both the number of voters who have ranked Alice higher than Bob, and the number who have ranked Bob higher than Alice. If Alice is preferred by more voters then she is the winner of that pairing. When all possible pairings of candidates have been considered, if one candidate beats every other candidate in these contests then they are declared
2263-499: Is the fault of the Hare quota or in fact the election system that was used. The Droop quota is used in most single transferable vote (STV) elections today and is occasionally used in elections held under the largest remainder method of party-list proportional representation (list PR). As well, it is identical to the Hagenbach-Bishoff quota, which is used to allocate seats by party in some list PR systems. The Droop quota
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2336-405: Is the number of votes a candidate needs to be guaranteed election. They are used in some systems where a formula other than plurality is used to allocate seats. Generally quotas are set at a level that is guaranteed to apportion only as many seats as are available in the legislature. When the electorate is divided into separate districts, the quota is commonly set by reference to valid votes cast in
2409-404: Is the winner. This is analogous to a single-winner or round-robin tournament; the total number of pairings is one less than the number of alternatives. Since a Condorcet winner will win by majority rule in each of its pairings, it will never be eliminated by Robert's Rules. But this method cannot reveal a voting paradox in which there is no Condorcet winner and a majority prefer an early loser over
2482-475: The Marquis de Condorcet , who championed such systems. However, Ramon Llull devised the earliest known Condorcet method in 1299. It was equivalent to Copeland's method in cases with no pairwise ties. Condorcet methods may use preferential ranked , rated vote ballots, or explicit votes between all pairs of candidates. Most Condorcet methods employ a single round of preferential voting, in which each voter ranks
2555-407: The simple quota or Hamilton's quota ) is the most commonly-used quota for apportionments using the largest remainder method of party-list representation . It was used by Thomas Hare in his first proposals for STV. It is given by the expression: votes seats {\displaystyle {\frac {\text{votes}}{\text{seats}}}} Specifically, the Hare quota is unbiased in
2628-441: The 23 guests. STV is chosen to make the decision, with the whole-vote method used to transfer surplus votes. The hope is that each guest will be served at least one food that they are happy with. To select the three foods, each guest is given one vote – they each mark their first preference and are also allowed to cast two back-up preferences to be used only if their first-preference food cannot be selected or to direct
2701-540: The Condorcet winner if there is one. Not all single winner, ranked voting systems are Condorcet methods. For example, instant-runoff voting and the Borda count are not Condorcet methods. In a Condorcet election the voter ranks the list of candidates in order of preference. If a ranked ballot is used, the voter gives a "1" to their first preference, a "2" to their second preference, and so on. Some Condorcet methods allow voters to rank more than one candidate equally so that
2774-460: The Condorcet winner. As noted above, if there is no Condorcet winner a further method must be used to find the winner of the election, and this mechanism varies from one Condorcet consistent method to another. In any Condorcet method that passes Independence of Smith-dominated alternatives , it can sometimes help to identify the Smith set from the head-to-head matchups, and eliminate all candidates not in
2847-546: The Copeland winner has the highest possible Copeland score. They can also be found by conducting a series of pairwise comparisons, using the procedure given in Robert's Rules of Order described above. For N candidates, this requires N − 1 pairwise hypothetical elections. For example, with 5 candidates there are 4 pairwise comparisons to be made, since after each comparison, a candidate is eliminated, and after 4 eliminations, only one of
2920-635: The Schulze method, use the information contained in the sum matrix to choose a winner. Cells marked '—' in the matrices above have a numerical value of '0', but a dash is used since candidates are never preferred to themselves. The first matrix, that represents a single ballot, is inversely symmetric: (runner, opponent) is ¬(opponent, runner). Or (runner, opponent) + (opponent, runner) = 1. The sum matrix has this property: (runner, opponent) + (opponent, runner) = N for N voters, if all runners were fully ranked by each voter. [REDACTED] Suppose that Tennessee
2993-411: The basis for defining preference and determined that Memphis voters preferred Chattanooga as a second choice rather than as a third choice, Chattanooga would be the Condorcet winner even though finishing in last place in a first-past-the-post election. An alternative way of thinking about this example if a Smith-efficient Condorcet method that passes ISDA is used to determine the winner is that 58% of
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3066-601: The candidates from most (marked as number 1) to least preferred (marked with a higher number). A voter's ranking is often called their order of preference. Votes can be tallied in many ways to find a winner. All Condorcet methods will elect the Condorcet winner if there is one. If there is no Condorcet winner different Condorcet-compliant methods may elect different winners in the case of a cycle—Condorcet methods differ on which other criteria they satisfy. The procedure given in Robert's Rules of Order for voting on motions and amendments
3139-496: The complete order of finish (i.e. who won, who came in 2nd place, etc.). They always suffice to determine whether there is a Condorcet winner. Additional information may be needed in the event of ties. Ties can be pairings that have no majority, or they can be majorities that are the same size. Such ties will be rare when there are many voters. Some Condorcet methods may have other kinds of ties. For example, with Copeland's method , it would not be rare for two or more candidates to win
3212-424: The context in which elections are held, circular ambiguities may or may not be common, but there is no known case of a governmental election with ranked-choice voting in which a circular ambiguity is evident from the record of ranked ballots. Nonetheless a cycle is always possible, and so every Condorcet method should be capable of determining a winner when this contingency occurs. A mechanism for resolving an ambiguity
3285-449: The district where it is used, so that each vote is worth about the same as another. STV is a family of proportional multi-winner electoral systems . They can be thought of as a variation on the largest remainders method that uses solid coalitions rather than party lists . Surplus votes belonging to winning candidates (those in excess of an electoral quota ) may be thought of as remainder votes – they are transferred to
3358-400: The district wins at least one seat: the more seats the district has, the smaller the size of the group needed to elect a member. In this way, STV provides approximately proportional representation overall, ensuring that substantial minority factions have some representation. There are several STV variants. Two common distinguishing characteristics are whether or not ticket voting is allowed and
3431-417: The district. The quota may be set at a number between: votes seats + 1 ≤ quota ≤ votes seats − 1 {\displaystyle {\frac {\text{votes}}{{\text{seats}}+1}}\leq {\text{quota}}\leq {\frac {\text{votes}}{{\text{seats}}-1}}} The smallest quota given above, votes/seats+1, is sometimes defended. Such
3504-474: The eventual winner (though it will always elect someone in the Smith set ). A considerable portion of the literature on social choice theory is about the properties of this method since it is widely used and is used by important organizations (legislatures, councils, committees, etc.). It is not practical for use in public elections, however, since its multiple rounds of voting would be very expensive for voters, for candidates, and for governments to administer. In
3577-416: The fewest votes and is eliminated. According to their only voter's next preference, this vote is transferred to Cake. No option has reached the quota, and there are still two to elect with five in the race, so elimination of options will continue next round. Step 4: Of the remaining options, Oranges, Strawberry and Chicken now are tied for the fewest votes. Strawberry had the fewest first preference votes so
3650-598: The first choice and Pear as second, while the right-most column shows there were three ballots with Chicken as first choice, Chocolate as second, and Hamburger as third. The election step-by-step: ELECTED (2 surplus vote) ELECTED (0 surplus votes) ELECTED (1 surplus vote) Setting the quota: The Droop quota formula is used to produce the quota (the number of votes required to be automatically declared elected) = floor(valid votes / (seats to fill + 1)) + 1 = floor(23 / (3 + 1)) + 1 = floor(5.75) + 1 = 5 + 1 = 6 Step 1: First-preference votes are counted. Pears reaches
3723-470: The following sum matrix: When the sum matrix is found, the contest between each pair of candidates is considered. The number of votes for runner over opponent (runner, opponent) is compared with the number of votes for opponent over runner (opponent, runner) to find the Condorcet winner. In the sum matrix above, A is the Condorcet winner because A beats every other candidate. When there is no Condorcet winner Condorcet completion methods, such as Ranked Pairs and
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#17328446480913796-450: The group, known as the Smith set , always exists. The Smith set is guaranteed to have the Condorcet winner in it should one exist. Many Condorcet methods elect a candidate who is in the Smith set absent a Condorcet winner, and is thus said to be "Smith-efficient". Condorcet voting methods are named for the 18th-century French mathematician and philosopher Marie Jean Antoine Nicolas Caritat,
3869-462: The last parcel of votes received by winners in accordance with the Gregory method. Systems that use the Gregory method for surplus vote transfers are strictly non-random. In a single transferable vote (STV) system, the voter ranks candidates in order of preference on their ballot. A vote is initially allocated to the voter's first preference. A quota (the minimum number of votes that guarantees election)
3942-592: The manner in which surplus votes are transferred. In Australia, lower house elections do not allow ticket voting; some but not all state upper house systems do allow ticket voting. In Ireland and Malta, surplus votes are transferred as whole votes (there may be some random-ness) and neither allows ticket voting. In Hare–Clark , used in Tasmania and the Australian Capital Territory , there is no ticket voting and surplus votes are fractionally transferred based on
4015-573: The next round. This would have left Hamburger as the last remaining candidate to fill the last open seat, even if it did not have quota. Condorcet method Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results A Condorcet method ( English: / k ɒ n d ɔːr ˈ s eɪ / ; French: [kɔ̃dɔʁsɛ] )
4088-414: The number of seats it hands out, and so is more proportional than the Droop quota (which tends to be biased towards larger parties); However, the Hare sometimes does not allocate a majority of seats to a party with a majority of the votes. [1] Droop quota guarantees that a party that wins a majority of votes in a district will win a majority of the seats in the district. The Hare quota (also known as
4161-413: The number of seats it hands out. It does suffer the disadvantage that it can allocate only a minority of seats to a party with a majority of votes. [2] In at least one proportional representation system where the largest remainder method is used, the Hare quota has been manipulated by running candidates on many small lists, allowing each list to pick up a single remainder seat. It is not clear that this
4234-425: The one above, one can find the overall results of an election. Each ballot can be transformed into this style of matrix, and then added to all other ballot matrices using matrix addition . The sum of all ballots in an election is called the sum matrix. Suppose that in the imaginary election there are two other voters. Their preferences are (D, A, C, B) and (A, C, B, D). Added to the first voter, these ballots would give
4307-419: The original 5 candidates will remain. To confirm that a Condorcet winner exists in a given election, first do the Robert's Rules of Order procedure, declare the final remaining candidate the procedure's winner, and then do at most an additional N − 2 pairwise comparisons between the procedure's winner and any candidates they have not been compared against yet (including all previously eliminated candidates). If
4380-545: The paradox of voting means that it is still possible for a circular ambiguity in voter tallies to emerge. Comparison of the Hare and Droop quotas Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results In proportional representation systems, an electoral quota
4453-426: The procedure's winner does not win all pairwise matchups, then no Condorcet winner exists in the election (and thus the Smith set has multiple candidates in it). Computing all pairwise comparisons requires ½ N ( N −1) pairwise comparisons for N candidates. For 10 candidates, this means 0.5*10*9=45 comparisons, which can make elections with many candidates hard to count the votes for. The family of Condorcet methods
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#17328446480914526-408: The quota with 8 votes and is therefore elected on the first count, with 2 surplus votes. Step 2: All of the voters who gave first preference to Pears preferred Strawberry next, so the surplus votes are awarded to Strawberry. No other option has reached the quota, and there are still two to elect with six options in the race, so elimination of lower-scoring options starts. Step 3: Chocolate has
4599-411: The quota) or until there are only as many remaining candidates as there are unfilled seats, at which point the remaining candidates are declared elected. Suppose an election is conducted to determine what three foods to serve at a party. There are seven choices: Oranges, Pears, Strawberries, Cake (of the strawberry/chocolate variety), Chocolate, Hamburgers and Chicken. Only three of these may be served to
4672-498: The remaining candidates' votes have surplus votes needing to be transferred), the least popular candidate is eliminated. The eliminated candidate's votes are transferred to the next-preferred candidate rather than being discarded; if the next-preferred choice has already been eliminated or elected, the procedure is iterated to lower-ranked candidates. Counting, eliminations, and vote transfers continue until enough candidates are declared elected (all seats are filled by candidates reaching
4745-486: The result. Four saw their third choice elected. Fifteen voters saw their first preference chosen; eight of these 15 saw their first and third choices selected. Four others saw their second preference chosen, with one of them having their second and third choice selected. Note that if Hamburger had received only one vote when Chicken was eliminated, it still would have won because the only other remaining candidate, Oranges, had fewer votes so would have been declared defeated in
4818-469: The same number of pairings, when there is no Condorcet winner. A Condorcet method is a voting system that will always elect the Condorcet winner (if there is one); this is the candidate whom voters prefer to each other candidate, when compared to them one at a time. This candidate can be found (if they exist; see next paragraph) by checking if there is a candidate who beats all other candidates; this can be done by using Copeland's method and then checking if
4891-426: The set before doing the procedure for that Condorcet method. Condorcet methods use pairwise counting. For each possible pair of candidates, one pairwise count indicates how many voters prefer one of the paired candidates over the other candidate, and another pairwise count indicates how many voters have the opposite preference. The counts for all possible pairs of candidates summarize all the pairwise preferences of all
4964-459: The transfer to Strawberry.) Cake reaches the quota and is elected. Cake has no surplus votes, no other option has reached the quota, and there is still one choice to select with three in the race, so the vote count proceeds, with the elimination of the least popular candidate. Step 5: Chicken has the fewest votes and is eliminated. The Chicken voters' next preference is Chocolate but Chocolate has already been eliminated. The next usable preference
5037-482: The voter might express two first preferences rather than just one. If a scored ballot is used, voters rate or score the candidates on a scale, for example as is used in Score voting , with a higher rating indicating a greater preference. When a voter does not give a full list of preferences, it is typically assumed that they prefer the candidates that they have ranked over all the candidates that were not ranked, and that there
5110-420: The voters, a mutual majority , ranked Memphis last (making Memphis the majority loser ) and Nashville, Chattanooga, and Knoxville above Memphis, ruling Memphis out. At that point, the voters who preferred Memphis as their 1st choice could only help to choose a winner among Nashville, Chattanooga, and Knoxville, and because they all preferred Nashville as their 1st choice among those three, Nashville would have had
5183-448: The voters. Pairwise counts are often displayed in a pairwise comparison matrix , or outranking matrix , such as those below. In these matrices , each row represents each candidate as a 'runner', while each column represents each candidate as an 'opponent'. The cells at the intersection of rows and columns each show the result of a particular pairwise comparison. Cells comparing a candidate to themselves are left blank. Imagine there
5256-529: The winner is the candidate preferred by a majority of voters. When results for every possible pairing have been found they are as follows: The results can also be shown in the form of a matrix: ↓ 2 Wins ↓ 1 Win As can be seen from both of the tables above, Nashville beats every other candidate. This means that Nashville is the Condorcet winner. Nashville will thus win an election held under any possible Condorcet method. While any Condorcet method will elect Nashville as
5329-523: The winner, if instead an election based on the same votes were held using first-past-the-post or instant-runoff voting , these systems would select Memphis and Knoxville respectively. This would occur despite the fact that most people would have preferred Nashville to either of those "winners". Condorcet methods make these preferences obvious rather than ignoring or discarding them. On the other hand, in this example Chattanooga also defeats Knoxville and Memphis when paired against those cities. If we changed
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