The City of Cambridge, Massachusetts has used the single transferable vote to elect its city council and school committee since 1941. The statute providing the counting rules is Chapter 54A of Massachusetts General Laws, the relevant portions of which are included below. Note that § 16(b) allows Cambridge to use any method for transfering surplus votes that was in use in 1938, and Cambridge has chosen to use the Cincinnati method.
The City of Cambridge describes the Cincinnati method as follows: The ballots of the candidate who has a surplus are numbered sequentially in the order in which they have been counted (that is, in the sequence dictated by the random draw of precincts) and then every nth ballot is drawn and transferred to a continuing candidate until the original candidate is credited with ballots equaling no more than quota. n is nearest whole number computed by the formula
n = Candidate's Total Ballots
Surplus Ballots.
A ballot selected by this method that does not show a preference for a continuing candidate is skipped and remains with the original candidate. If not enough ballots are removed when ballots n, 2n, 3n, .... have been transferred, the sequence starts again with n+1, 2n+1, 3n+1, ....
For more information, see http://www.cambridgema.gov/election/Proportional_Representation.cfm.
Since candidates with fewer than 50 votes are eliminated, this method should not be used with a small number of ballots. OpenSTV's implementation of Cambridge STV has been validated against official Cambridge results from 1999 to present.
OpenSTV provides the option of saving the winning candidates' ballots to separate files. The Cambridge rules use these ballots to elect a replacement candidate in the event of a vacancy.
Massachusetts General Laws, Chapter 54A
ELECTION OF CERTAIN CITY AND TOWN OFFICERS BY PROPORTIONAL REPRESENTATION OR PREFERENTIAL VOTING
§ 9. Rules for counting ballots, and determining results. Ballots cast under proportional representation shall be counted and the results determined under the supervision of the director of the count appointed pursuant to section six, according to the following rules:-
(a) The ballots in each ballot receptacle shall be examined for validity and those which are found to be blank or otherwise invalid shall be separated from the valid ballots. The number of valid ballots from each precinct and the total number of valid ballots shall be recorded. If a ballot does not clearly show which candidate the voter prefers to all others, or if it contains any word, mark or other sign apparently intended to identify the voter, it shall be set aside as invalid. Every ballot not thus invalid shall be counted according to the intent of the voter, so far as that can be clearly ascertained, whether marked according to the directions printed on it or not. No ballot shall be held invalid because the names of candidates thereon for whom the voter did not mark a choice have been stricken out, unless such striking out constitutes an identifying mark. A single cross on a ballot on which no figure 1 appears shall be considered equivalent to the figure 1. If a ballot contains both figures and crosses, the order of the choice shown by the figures shall be taken as the voter's intention in so far as the order is clearly indicated. If the consecutive numerical order of the figures on a ballot is broken by the omission of one or more figures, the smallest number marked shall be taken to indicate the voter's first choice, the next smallest his second, and so on, without regard to the figure or figures omitted.
(b) Each candidate shall be credited with one vote for every valid ballot that is sorted to him as first choice, or otherwise credited to him as hereinafter provided, and no ballot shall ever be credited to more than one candidate at the same time.
(c) A "quota" is the smallest number of votes which any candidate must receive in order to be assured of election without more candidates being elected than there are offices to be filled. It shall be determined by dividing the total number of valid ballots by one more than the total number of candidates to be elected and adding one to the result, disregarding fractions. Whenever at any stage of the counting the number of ballots credited to a candidate becomes equal to the quota, he shall be declared elected, and no ballots in excess of the quota shall be credited to him except as provided in rule (f) or (1) of this section.
(d) The ballots shall be sorted according to the first choices marked on them, the ballots from each polling place being handled together, and those from different polling places being handled in the order of polling places determined under the provisions of section eight.
(e) If a candidate is elected while the ballots are being sorted according to first choices, any subsequent ballots which show him as first choice shall each be credited to the second choice marked on it, or, if the second choice also has been elected, to the next choice marked on it for a candidate not yet elected.
(f) If during the first sorting of ballots, ballots are found which are marked for a candidate already elected as first choice, but show no clear choice for any unelected candidate, such ballots shall at the end of the sorting be given to the candidate of their first choice, and in their place an equal number, as nearly as possible, of the last ballots sorted to that candidate which show a clear choice for unelected candidates, all as determined by the director of the count, shall be taken and re-sorted to unelected candidates as if they were then being sorted for the first time.
(g) When all the ballots have been thus sorted and credited to the first available choices marked on them, every candidate who is credited with fewer ballots than the number of signatures required for his nomination shall be declared defeated.
(h) All the ballots of the candidates thus defeated shall be transferred, each to the candidate indicated on it as next choice among the continuing candidates. A "continuing candidate" is a candidate not as yet either elected or defeated. Any ballot taken for transfer which does not clearly indicate any candidate as next choice among the continuing candidates shall be set aside as "exhausted".
(i) When all the ballots of the candidates thus defeated have been transferred, the one candidate who is then lowest on the poll shall be declared defeated and all his ballots transferred in the same way.
(j) Thereupon the candidate who is then lowest shall be declared defeated and all his ballots similarly transferred; and in like manner candidates shall be declared defeated one at a time and all their ballots transferred.
(k) If, when a candidate is to be declared defeated, two or more candidates are tied at the bottom of the poll, that one of the tied candidates shall be declared defeated who was credited with fewest ballots immediately prior to the last transfer of ballots. If two or more of the tied candidates were tied at that stage of the count, also, the second tie shall be decided by referring similarly to the standing of candidates immediately prior to the last transfer of ballots before that. This principle shall be applied successively as many times as may be necessary, a tie shown at any stage of the count being decided by referring to the standing of the tied candidates immediately prior to the last preceding transfer of ballots. Any tie not otherwise provided for shall be decided by lot.
In interpreting this and other rules contained in this section the transfer of all ballots from candidates defeated together under rule (g) of this section, and the transfer of all ballots from each candidate defeated thereafter shall each constitute a single separate transfer.
(l) Whenever candidates to the number to be elected have received the quota, any transfer of ballots in progress when the last quota was reached shall be completed, but immediately thereafter all continuing candidates shall be declared defeated and the election shall be at an end. Whenever all ballots of all defeated candidates have been transferred, and it is impossible to defeat another candidate without reducing the continuing candidates below the number still to be elected, all the continuing candidates shall be declared elected and the election shall be at an end.
(m) A record of the count shall be kept in such form as to show, after each sorting or transfer of ballots, the number thereby credited to each candidate, the number thereby set aside as exhausted, the total for each candidate, the total set aside as exhausted, and the total number of valid ballots found by adding the totals of all candidates and the total set aside as exhausted.
(n) Every ballot that is transferred from one candidate to another shall be stamped or marked so that its entire course from candidate to candidate can be conveniently traced.
(o) If at any time after the first sorting of the ballots a ballot is found to have been credited to the wrong candidate, it may be transferred, as part of the transfer that is in progress, to the continuing candidate, if any, to whom it should have been credited at the time the error was made, or, if it should previously have become exhausted, may be set aside as exhausted as part of the transfer that is in progress; provided, that if the number of misplaced ballots found is sufficient to make it possible that any candidate has been wrongly defeated, so much of the sorting and transferring as may be required to correct the error shall be done over again before the count proceeds.
If in correcting an error any ballots are re-sorted or re-transferred, every ballot shall be made to take the same course that it took in the original count unless the correction of an error requires its taking a different course. The principles of the rules of this section shall apply also to any recount which may be made after the original count has been completed.
(p) The director of the count and his assistants shall proceed with reasonable expedition in the counting of the ballots, but may take recesses at the discretion of the director. The city or town clerk shall make proper provision for the safekeeping of the ballots while the counting is not in progress.
(q) The candidates, their witnesses, alternate witnesses and representatives accredited under section seven, representatives of the press, and, as far as may be consistent with good order and with convenience in the counting and transferring of the ballots, the public shall be afforded every facility for being present and witnessing the counting and transferring of the ballots.
(r) Each of the candidates entitled to appoint witnesses of the central count as provided in section seven shall be entitled to appoint a member of a board of review of the central count. Such appointment shall be made within the time and in the manner prescribed for the appointment of such witnesses of the central count. In the central counting place a board of review so constituted shall be given facilities for examining all the ballots in the quota of each elected candidate in order to make sure that all the ballots of such quota are rightfully credited to the candidate toward whose election they have been counted, that the number of ballots therein is actually equal to the quota prescribed in this section, and that"exhausted" ballots have been properly so designated. Any errors discovered by such a board of review shall be reported to the director of the count.
(s) When the election with respect to any particular body or office is at an end the director of the count shall publicly announce the result of the vote for such body or office. The provisions of section one hundred and seven of chapter fifty-four relative to presiding officers and other election officers at polling places shall, so far as apt, apply to the director of the count and his assistants with respect to all ballots, records, copies of records, envelopes and ballot boxes, transmitted to the central counting place under section eight and to all other papers, records and apparatus used in counting the votes at the central counting place, except that ballots cast for a particular body or office, as well as those spoiled and returned and those not given out, shall be enclosed, and the envelopes sealed and delivered or transmitted to the city or town clerk as soon as may be after the public announcement of the result of the vote for such body or office.
(t) No canvass or count of the vote shall be made on the Lord's day.
§ 10. Ballots; preservation; examination. The ballots cast at each election by proportional representation or preferential voting shall be preserved by the city or town clerk until the term of office of the members of the body or of the officer elected thereby has expired, and shall be available for examination continuously throughout the business day, under supervision of the city or town clerk, on written application signed by not less than one hundred voters of the city or town and the payment of a fee of twenty-five dollars for each day on which such inspection is held. Such application shall name not more than three representatives of the applicants to make such examination.
§ 11. Publication of statements regarding ballots cast. Within thirty days after an election to elect members of a body by proportional representation or an officer by preferential voting, the city or town clerk shall cause the ballots cast for such body or office to be examined and shall publish a statement showing-
(a) The number of first-choice ballots cast for each candidate at each polling place.
(b) The number of ballots from each polling place finally counted for each of the elected candidates.
(c) The number of the exhausted ballots from each polling place which showed one or more choices for elected candidates and the number which did not show any such choice.
(d) The number of blank ballots cast for each body or office at each polling place.
(e) The number of ballots otherwise invalid cast for each body or office at each polling place.
(f) The number of first choices, second choices, third choices, and so on, used in the election of each of the elected candidates.
(g) Such other information in regard to the ballots as the city or town clerk may deem of interest.
A copy of such statement shall be kept on file in the office of the city or town clerk open to public inspection.
§ 12. Recount of ballots. Partial or complete recounts of the ballots cast for any body or office in an election by proportional representation or by preferential voting shall take place in the manner provided in sections one hundred and thirty-four to one hundred and thirty-seven, inclusive, of said chapter fifty-four, except that any petition shall be submitted on or before five o'clock in the afternoon of the third day following the public announcement by the director of the count of the result of the vote for such body or office and shall be on a form approved and furnished by the city or town clerk and be signed in a town by ten or more voters of such town, in a city, except Boston, by fifty or more voters of such city and in Boston by two hundred and fifty or more voters of said Boston and except that any such recount in any city or in any town divided into precincts shall be conducted for the entire city or town instead of for specified precincts. If a partial or complete recount of the ballots cast in such an election shall in fact take place, it shall be conducted according to the rules prescribed for the original count as nearly as is practicable.
§ 13. Vacancies in bodies elected by proportional representation; filling. When a vacancy occurs in an elective body whose members were elected by proportional representation, such vacancy shall be filled for the remainder of the unexpired term by a public recount of the ballots credited at the end of the original count to the candidate elected thereby whose place has become vacant. Except for the following special rules, the provisions governing the original count shall be in effect:
(a) All choices marked for candidates theretofore elected or who have become ineligible or have withdrawn shall be disregarded:
(b) The ballots shall be sorted each to the earliest choice marked on it for any of the eligible candidates.
(c) If any candidate has to his credit more than half of the ballots which show any preference among the eligible candidates he shall be declared elected to the vacant place.
(d) If no candidate receives more than half of such ballots, the candidates lowest on the poll shall be declared defeated one after another and after each candidate is defeated his ballots shall be transferred among the continuing candidates.
(e) The process hereinbefore provided shall be continued until one candidate is credited with more ballots than all the other undefeated candidates together, when he shall be declared elected to the vacant place.
If a vacancy in an elective body occurs for which no regularly nominated candidate remains it shall be filled for the unexpired term by a majority vote of the remaining members; and if but a single member remains or if a majority vote of the remaining members is not obtained within thirty days after the vacancy occurs, it shall be filled by a special election, in the case of a single vacancy, by preferential voting or, in case two or more vacancies exist at the same time, by proportional representation.
§ 14. Ballots; rules for counting where election by preferential voting. Ballots cast under preferential voting shall be counted in the central counting place under the supervision of the director of the count, in accordance with the following rules:-
(a) The ballots shall first be sorted according to the first choices marked on them, and the total number of valid ballots thus sorted to each candidate shall be ascertained. The validity of ballots shall be determined according to the principles laid down for the count of ballots in an election by proportional representation in rule (a) of section nine.
(b) If any candidate is found to have been marked as first choice on more than half of the valid ballots he shall be declared elected.
(c) If no candidate is so elected after the count of first choices, every candidate who is credited with fewer ballots than the number of signatures required for his nomination shall be declared defeated.
(d) All the ballots of the candidates so defeated shall be transferred, each to the candidate indicated on it as next choice among the undefeated candidates. Any ballot taken for transfer which does not clearly indicate any candidate as next choice among the undefeated candidates shall be set aside as "exhausted".
(e) If, after this or any subsequent transfer of ballots, one candidate is credited with more than half of the valid ballots which have not become exhausted, he shall be declared elected.
(f) If no candidate is so elected after the transfer of the ballots of candidates defeated under rule (c), the one candidate who is then lowest on the poll shall be declared defeated and all his ballots transferred in the same way.
(g) Thereupon, if no candidate is yet elected, the candidate who is then lowest shall be declared defeated and all his ballots similarly transferred. Thus candidates shall be deemed defeated one at a time, and all their ballots transferred until some candidate has received the necessary majority of the ballots which have not become exhausted and is accordingly declared elected.
(h) Ties shall be decided, a record of the count kept, errors corrected, recesses taken, and candidates and others permitted to be present according to the principles prescribed for elections by proportional representation in rules (k), (m), (o), (p) and (q) of section nine.
§ 15. Vacancies in single elective offices; filling. All provisions of law from time to time applicable in the case of a vacancy in an elective office shall continue to apply after the filling of such office by preferential voting, except that any election to fill such vacancy shall also be by preferential voting.
§ 16. Mechanical or other voting devices; methods of counting first choices.
(a) In conducting any election by proportional representation or preferential voting, mechanical or other devices may be used, subject, however, to the provisions of sections thirty-two to thirty-nine, inclusive, of chapter fifty-four, if the city council or the town passes a vote providing expressly that such devices shall be used in such election; and said sections, so far as apt, shall be applicable in all respects in case of such devices so used. In case such devices are to be used in any city or town, the city or town clerk may modify the form of ballot, the rotation of names thereon, the directions to voters and other details in respect to the election process; provided, that no change shall be made which will alter or impair the principles of voting or counting the ballots governing elections by proportional representation or preferential voting, as the case may be, but the voter may be limited to not less than fifteen choices for any particular body or office.
(b) In any city or town where elections by proportional representation are to be held, any method of counting the voters' first choices and treating any such choices in excess of the quota, provided for under any system of proportional representation which on January first, nineteen hundred and thirty-eight was in effect for the purpose of municipal elections in any city of the United States, may be substituted for the method of counting such choices set forth in this chapter, if the registrars of voters determine that such substitution is advisable; provided, that they issue regulations embodying the method so substituted and provided, further, that such regulations shall not be effective with respect to any election unless at least thirty days prior thereto copies of such regulations are available for delivery to such of the voters as may request them.