# preference voting for three choices **URL:** \ **Category:** Factual Questions\ **Created:** [October 6, 2010, 7:53pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288 "2010-10-06T19:53:57Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [October 6, 2010, 7:53pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/1 "2010-10-06T19:53:57Z")
What is the “best” ballot design for a choice in our small community of three designs for an interior design matter? And how do you tabulate it? I’m assuming that the design choices the consultant will be providing might be fairly similar so that there will be no majority winner, but we don’t want to do multiple ballots. Thanks. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 6, 2010, 8:03pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/2 "2010-10-06T20:03:02Z")
It’s been proven ([Condorcet’s paradox](http://en.wikipedia.org/wiki/Voting_paradox)) that there can be no perfect system, even with three voters and three choices, but the best practical system without multiple ballots is: (1) Each voter numbers their choices in order of preference 1, 2 and 3. (2) You tally the no. 1 votes. (3a) If one choice has a majority (more than 50%), it wins. (3b) If no choice has a majority, then take the ballot papers for the least popular choice, count the no. 2 votes on those ballot papers, and add them to the no. 1 votes for those candidates. Then the choice with a majority wins. You also need a rule for tie-breaking, and generally in real-life a tie is broken by a drawing lots. ---
**Author:** ![Munch](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/munch/32/5281_2.png) [@Munch](https://boards.straightdope.com/u/Munch)\ **Post date:** [October 6, 2010, 8:10pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/3 "2010-10-06T20:10:22Z")
> [@Giles](#): > > (3b) If no choice has a majority, then take the ballot papers for the least popular choice, count the no. 2 votes on those ballot papers, and add them to the no. 1 votes for those candidates. Then the choice with a majority wins. This portion doesn’t read as clear as I think it could. How about “take the ballots whose #1 choice was the 3rd most popular, and reassign them according to their #2 choice”? ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 6, 2010, 8:26pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/4 "2010-10-06T20:26:41Z")
> [@Munch](#): > > > [@Giles](#): > > > > (3b) If no choice has a majority, then take the ballot papers for the least popular choice, count the no. 2 votes on those ballot papers, and add them to the no. 1 votes for those candidates. Then the choice with a majority wins. > > This portion doesn’t read as clear as I think it could. How about “take the ballots whose #1 choice was the 3rd most popular, and reassign them according to their #2 choice”? Yes, that’s another way of putting it. ---
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [October 6, 2010, 8:29pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/5 "2010-10-06T20:29:15Z")
Thanks! ---
**Author:** ![Saltire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saltire/32/553_2.png) [@Saltire](https://boards.straightdope.com/u/Saltire)\ **Post date:** [October 6, 2010, 8:30pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/6 "2010-10-06T20:30:20Z")
Giles described Instant Runoff Voting, which is a common means of determining a winner in a preferential vote. It’s easy to explain, but isn’t the best method, in the opinion of a lot of experts. I prefer the [Condorcet Method](http://en.wikipedia.org/wiki/Condorcet_method), which is tallied by treating every ballot as if it were voting on several two-party elections. Compare each choice against each other choice, and see who would have won if only those two were running. Note each win in these sub-races for each candidate. The one that gets the most sub-wins gets the overall win. It leads to fewer ways to end up with everyone’s less-preferred choice as the winner. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 6, 2010, 8:42pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/7 "2010-10-06T20:42:26Z")
Yes, but: (1) When you have a lot of candidates (say 10 or 12), you have to do a lot of counting in the Condorcet method. (2) It can give rise to situations where the Condorcet paradox happens, and you need to resolve those situations as well as tied votes. I’ve seen a lot of real-life elections in Australia where instant runoff voting was used, including elections where three or four candidates were in serious contention for a single-member seat, and I don’t think I’ve seen a situation “with everyone’s less-preferred choice as the winner”. If it were to happen in real life, then I think it’s likely that the paradox would become real as well. ---
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [October 6, 2010, 9:16pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/8 "2010-10-06T21:16:06Z")
> [@Saltire](#): > > Giles described Instant Runoff Voting, which is a common means of determining a winner in a preferential vote. It’s easy to explain, but isn’t the best method, in the opinion of a lot of experts. > > I prefer the [Condorcet Method](http://en.wikipedia.org/wiki/Condorcet_method), which is tallied by treating every ballot as if it were voting on several two-party elections. Compare each choice against each other choice, and see who would have won if only those two were running. Note each win in these sub-races for each candidate. The one that gets the most sub-wins gets the overall win. > > It leads to fewer ways to end up with everyone’s less-preferred choice as the winner. But the rules for casting the ballot are the same? I mean the same design? About counting. So if somebody votes: choice a =1st choice b =2nd choice c =3rd You count this as what? You tally that as two votes for choice a (a\>b, a\>c) and one vote for choice b (b\>c)? So with 25 voters, who all complete the full ballot, you get 75 votes to distribute? Thanks again. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 6, 2010, 9:22pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/9 "2010-10-06T21:22:01Z")
> [@drhess](#): > > So if somebody votes: > > choice a =1st > choice b =2nd > choice c =3rd > > You count this as what? You tally that as two votes for choice a (a\>b, a\>c) and one vote for choice b (b\>c)? > > So with 25 voters, who all complete the full ballot, you get 75 votes to distribute? > > Thanks again. (1) In instant runoff voting, that’s 1 vote for choice a. If choice a is the least popular, and neither of the other choices have a majority of the first preferences, then on the second count it’s a vote for choice b. (2) In the Condorcet method, you have three counts. (i) For choice a versus choice b, it’s 1 vote for choice a (ii) For choice a versus choice c, it’s 1 vote for choice a (iii) For choice b versus choice c, it’s 1 vote for choice b In both cases, on every count, you only count 25 votes, and 13+ votes is a majority. ---
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [October 6, 2010, 9:57pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/10 "2010-10-06T21:57:36Z")
> [@Giles](#): > > > [@drhess](#): > > > > So if somebody votes: > > > > choice a =1st > > choice b =2nd > > choice c =3rd > > > > You count this as what? You tally that as two votes for choice a (a\>b, a\>c) and one vote for choice b (b\>c)? > > > > So with 25 voters, who all complete the full ballot, you get 75 votes to distribute? > > > > Thanks again. > > (1) In instant runoff voting, that’s 1 vote for choice a. If choice a is the least popular, and neither of the other choices have a majority of the first preferences, then on the second count it’s a vote for choice b. > > (2) In the Condorcet method, you have three counts. > (i) For choice a versus choice b, it’s 1 vote for choice a > (ii) For choice a versus choice c, it’s 1 vote for choice a > (iii) For choice b versus choice c, it’s 1 vote for choice b > > In both cases, on every count, you only count 25 votes, and 13+ votes is a majority. That’s not clear to me. Under Condorcet, you got three three votes there, so what is that ballot counted as? If there are 25 ballots, what do you do with those three votes on one ballot? ---
**Author:** ![Saltire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saltire/32/553_2.png) [@Saltire](https://boards.straightdope.com/u/Saltire)\ **Post date:** [October 6, 2010, 10:02pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/11 "2010-10-06T22:02:27Z")
[Here](http://www.wordiq.com/definition/Condorcet_method#Condorcet_compared_to_Instant_Runoff_and_First-past-the-post) is a comparison of IRV, Condorcet, and plurality voting. It shows an example where the person that was least-preferred by more people ends up winning in IRV, but loses in Condorcet. Of course, for almost all votes, both IRV and Condorcet will end up with the same winner. Plurality will often differ, however. ---
**Author:** ![RealityChuck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/realitychuck/32/195_2.png) [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)\ **Post date:** [October 6, 2010, 11:21pm UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/12 "2010-10-06T23:21:36Z")
> [@Giles](#): > > It’s been proven ([Condorcet’s paradox](http://en.wikipedia.org/wiki/Voting_paradox)) that there can be no perfect system, even with three voters and three choices, but the best practical system without multiple ballots is: > (1) Each voter numbers their choices in order of preference 1, 2 and 3. > (2) You tally the no. 1 votes. > (3a) If one choice has a majority (more than 50%), it wins. > (3b) If no choice has a majority, then take the ballot papers for the least popular choice, count the no. 2 votes on those ballot papers, and add them to the no. 1 votes for those candidates. Then the choice with a majority wins. > > You also need a rule for tie-breaking, and generally in real-life a tie is broken by a drawing lots. In science fiction fandom, this is called an Australian ballot. Hugo Awards (and up until last year, Nebulas) were chosen this way. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 7, 2010, 1:07am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/13 "2010-10-07T01:07:47Z")
> [@Saltire](#): > > Giles described Instant Runoff Voting, which is a common means of determining a winner in a preferential vote. It’s easy to explain, but isn’t the best method, in the opinion of a lot of experts. That’s not quite accurate. In general, the experts regard IRV as a terrible method. No method is perfect (assuming that we have the right method of perfect), but Condorcet voting and [approval voting](http://en.wikipedia.org/wiki/Approval_voting) are generally regarded as the least bad. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 7, 2010, 1:24am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/14 "2010-10-07T01:24:42Z")
> [@RealityChuck](#): > > In science fiction fandom, this is called an Australian ballot. Hugo Awards (and up until last year, Nebulas) were chosen this way. That’s an odd use of the term “Australian ballot”: it usually means a secret ballot using ballot papers prepared by the state, and containing the names of all the candidates. According to the [Wiki article](http://en.wikipedia.org/wiki/Secret_ballot), that was first used in Tasmania, Victoria and South Australia in 1856. ---
**Author:** ![RealityChuck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/realitychuck/32/195_2.png) [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)\ **Post date:** [October 7, 2010, 1:37am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/15 "2010-10-07T01:37:30Z")
> [@Giles](#): > > > [@RealityChuck](#): > > > > In science fiction fandom, this is called an Australian ballot. Hugo Awards (and up until last year, Nebulas) were chosen this way. > > That’s an odd use of the term “Australian ballot”: it usually means a secret ballot using ballot papers prepared by the state, and containing the names of all the candidates. According to the [Wiki article](http://en.wikipedia.org/wiki/Secret_ballot), that was first used in Tasmania, Victoria and South Australia in 1856. I know. No one knows where the term came from and everyone in SF knows it has nothing to do with the general use of the term, but it’s been used in Hugo voting for over 60 years, and Nebula voting until last year (I’m still not sure why that was changed – a single plurality means that you could have 80% of the voters not liking the winner (with six nominated works)). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 7, 2010, 2:09am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/16 "2010-10-07T02:09:20Z")
> [@](#): > > That’s not quite accurate. In general, the experts regard IRV as a terrible method. Which is perhaps an overstatement, given that the experts also agree that it’s far better than, say, what most of the US currently uses. And approval voting, while it has advantages, also has a whole set of oddities peculiar to it, stemming from the fact that “approval” or “disapproval” isn’t really a binary choice. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 7, 2010, 2:10am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/17 "2010-10-07T02:10:07Z")
> [@ultrafilter](#): > > > [@Saltire](#): > > > > Giles described Instant Runoff Voting, which is a common means of determining a winner in a preferential vote. It’s easy to explain, but isn’t the best method, in the opinion of a lot of experts. > > That’s not quite accurate. In general, the experts regard IRV as a terrible method. No method is perfect (assuming that we have the right method of perfect), but Condorcet voting and [approval voting](http://en.wikipedia.org/wiki/Approval_voting) are generally regarded as the least bad. Well, I’d write off approval voting for a start. Any system which can elect the Condorcet loser has to be pretty bad. Instant runoff voting cannot elect the Condorcet loser, and almost always in real life elects the Condorcet winner. When IRV doesn’t, it’s because there are three main candidates, with the least popular one in the middle, so that the least popular one (on first preferences) would be the Condorcet winner. For example: A (on the left) – 45% B (on the right) – 45% C (in the middle) – 10% With IRV, the result of the election depends on the second preferences of the votes for B, but with Condorcet Candidate B wins 55-45 against each of A and B. With approval voting, any of the candidates could win. Both IRV and approval voting are subject to strategic voting, in which some voters don’t vote for the preferred candidate in order to stop another candidate winning. I’ll give a real life example where (in my opinion) a significant number of voters didn’t vote for their preferred candidate in order to try to get another candidate elected. However, it didn’t work – though in fact the winning candidate was probably the Condorcet winner. It’s the [2007 New South Wales state election for the seat of Newcastle](http://www.elections.nsw.gov.au/state_government_elections/electoral_districts/all_districts_/newcastle/results_2007/perferential_count). There were 9 candidates in that election, but 4 can be ignored because they got very few votes. After distributing their preferences (on count 5 on the page I just cited), the votes were: BABAKHAN 4,298 GAUDRY 8,974 McKAY 13,274 OSBORNE 4,817 TATE 10,308 From left to right on the political spectrum: Osborne was the Green Party candidate. Gaudry was the sitting member, and had been on the left wing of the Labor Party. However, the right-wing state branch had decided to make McKay the Labor candidate, and Gaudry was standing as an independent. McKay was right-wing Labor. Tate was a centrist independent, and the Lord Mayor of Newcastle. Babakhan was the Liberal Party candidate – i.e., he belonged to the main opposition party in New South Wales. Note that the candidate from the main opposition party came 5th in the ballot, below even the Greens. Why? Because his party didn’t run a strong candidate, and because some people who would normally vote Liberal were voting for a strong centrist independent in order to keep Labor out. They did the same thing back in 1988, when an independent defeated the Labor candidate in a nominally safe Labor seat, and they wanted it to happen again. However, the strategic voting didn’t quite work. Most of Babakhan’s preferences went to Tate (as you’d expect); most of Osborne’s preferences went to Gaudry (as you’d expect); and Gaudry’s preferences split 60-40 between Tate and McKay (the 40% returning to the Labor fold, but the 60% voting for the centrist local guy against the candidate imposed by Labor Party head office). It was close, but McKay won as the probable Condorcet winner (though she might have not won in a 2-way fight against Gaudry) – and if Tate had won, he would have been the Condorcet winner. It’s closely fought elections with multiple serious candidates like this one that give me faith in IRV. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [October 7, 2010, 5:31am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/18 "2010-10-07T05:31:57Z")
Actually, it’s the Arrow impossibility theorem that says that it’s impossible to create a voting system that can fairly choose a winner among three or more candidates at all times: > **[Arrow's impossibility theorem](https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem)** > > Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting the specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alterna... ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [October 7, 2010, 8:01am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/19 "2010-10-07T08:01:33Z")
> [@Giles](#): > > It’s been proven ([Condorcet’s paradox](http://en.wikipedia.org/wiki/Voting_paradox)) that there can be no perfect system, even with three voters and three choices, but the best practical system without multiple ballots is: > (1) Each voter numbers their choices in order of preference 1, 2 and 3. > (2) You tally the no. 1 votes. > (3a) If one choice has a majority (more than 50%), it wins. > (3b) If no choice has a majority, then take the ballot papers for the least popular choice, count the no. 2 votes on those ballot papers, and add them to the no. 1 votes for those candidates. Then the choice with a majority wins. > > You also need a rule for tie-breaking, and generally in real-life a tie is broken by a drawing lots. Suppose 49% of ballots are 1) A 2) B 3) C 49% of ballots are 1) C 2) A 3) B 2% of ballots are 1) B 2) C 3) A B is least popular, so step 3b increases C’s vote to 51% and he wins. Despite that A’s 1st-or-2nd votes outnumber C’s 98 to 51. Perhaps some quantitative compromise among the run-off methods would be better. Another electoral-related matter with no good mathematical answer is [apportioning Congressional Representatives by population](http://en.wikipedia.org/wiki/Apportionment_paradox) where, for example, > [@](#): > > The Alabama paradox was discovered in 1880, when it was found that increasing the total number of seats would decrease Alabama’s share from 8 to 7. > … in 1910, Virginia lost a seat to Maine although its population had grown faster than Maine’s. > When Oklahoma became a state in 1907, a recomputation of apportionment showed that the number of seats due to other states would be affected even though Oklahoma would be given a fair share of seats and the total number of seats increased by that number. ---
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [October 7, 2010, 9:00am UTC](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288/20 "2010-10-07T09:00:36Z")
Using this calculator for Condorcet: [http://www.wiley.com/college/mat/gilbert139343/java/java01.html](http://www.wiley.com/college/mat/gilbert139343/java/java01.html) I entered ino the calculator this election with 11 ballots cast: 2 ballots for A\>B\>C 2 ballots for B\>C\>A 3 ballots for B\>A\>C 4 ballots for C\>A\>B I then hit calcuate for condorcet and it said there was no condorcet winner. Does that seem right? It seems B isthe winner for all the other methods they offer. [Next page](https://boards.straightdope.com/t/preference-voting-for-three-choices/556288.md?page=2)