Yes, that’s right. A beats B by 6 to 5; B beats C by 7 to 4; and C beats A by 6 to 5. So no candidate wins against both other candidates in two-way contests.
Then you should be interested to know that approval voting is actually the method MOST LIKELY to elect the Condorcet winner when tactical voting is taken into account; more often than any proper Condorcet method, and a HECK of a lot more often than instant runoff; instant runoff will quite often eliminate the Condorcet winner INSTANTLY, leaving the RUNOFF between the lesser candidates.
http://rangevoting.org/StratHonMix.html
(You have to scroll down pretty far to find IRV, past not only approval, but a whole bunch of obscure Condorcet methods. But hey, it does beat plurality, I guess that’s something…)
Also, Arrow’s theorem only applies to voting methods that operate on voter’s ranked-order preferences; so it can’t be applied to methods like approval or score voting (AKA range voting).
(Although, those methods aren’t perfect either; it’s just that Arrow’s construction isn’t sufficient to prove that. They are noticeably CLOSER to perfect though, so clearly there’s room for a more-robust theorem (might win someone a Nobel, too!))
Most of the answers so far are to the OP’s 2nd question, how to tabulate the ballot.
I’m going to respond to the first question, on ballot design.
- make 3 different versions of the ballot, rotating the order of the choices between them.
- don’t change anything else except the order.
- interleave blank ballots so that each voter gets a different version than the previous voter (that is, do NOT use version A for the firs hour of voting, version B for the 2nd hour, etc.)
- design the ballots so that each option has the same emphasis as the others: same size typefont used, same size picture, same colors/black & white for each one, etc.
In general, you want the 3 designs to look equally desirable on the ballot, only varying in order.
So I take it that the method used on most informal questionnaires, that of assigning point values, doesn’t work? I would assign the winner 2 votes, the second place 1 vote, and last place would get 0.
Also, what about range voting? Wouldn’t that give you more information? That’s another thing questionnaires do. Heck, even the psychological ones, which are specifically designed to be accurate tests.
If everyone voted honestly, then range voting would be ideal (in fact, I think that “ideal” is probably defined as what you’d get if everyone voted honestly in a range voting system). The problem is that the incentives to vote dishonestly are even stronger in a range voting system than in most others.
I’m going to take a guess at the problem, and what comes out is less dishonesty than failure of any system to accurately gauge people’s opinions on candidates.
Candidates A and B are very popular, and C is likely to not win despite you liking him the best. While you’re voting full marks for C and giving minimum rating to A since you think he’s a detestable slime bag, you’re not sure what to vote for B who you could definitely live with despite his not being nearly as awesome as C. You come to the conclusion that since you really don’t want A to win, you have to vote full marks on B. There’s going to be people who feel similar but opposite to you about A and B who vote full marks on A and none for B, so if you vote less than full for B they will get more of their share of opinion expressed. So despite liking C quite a bit more than B, you effectively are forced to dishonestly rank them the same. Anyone who wants to minimize the chance of a certain candidate winning is going to think along the same lines.
But is this really dishonesty? If you hate A so much that you’re willing to make sure B gets in, isn’t it the case that you effectively like B and C equally? The problem is that the voting system forces the same range on every voter and assumes that everyone’s full marks and no marks are equivalent and that there is a linear relationship between preference and the percentage of vote you give each candidate. That’s actually where the system breaks down. There’s no way you can accurately describe just how much more you like C compared to B because they are both so far above A. So while it’s nice to be able to provide a little more information about your preferences, the necessary simplifying assumptions make the system utterly unworkable.
Interestingly enough, this reminds me of a topic the professor brought up in an economics class I had in college: TV ratings. Shows tend(ed) to get canceled/renewed based on the number of people watching the shows, not based on how much those people actually like the show or would pay for it if they had to. You could watch a show you only sorta like every once in a while, but if you like the show most out of things at its time slot (assuming you don’t use a DVR for everything) and are going to watch TV at that particular time, you might as well watch the show you like the best. Thus while you might like Show C a ton and would do anything to see it around, think Show B is the best available at its time but it only OK, and would never ever watch Show A, in self-interest you’re going to show equal preference for shows B and C.
Neither of these claims is true.
If everyone voted honestly, even under range voting, there would still be times where the best candidate didn’t win. HOWEVER, this occurs much less frequently under range voting than under any other commonly-suggested voting system. And when it does occur, range more-often picks a better candidate than other systems (say, the 2nd-best, instead of the 4th-best.)
Meanwhile, the “damage” caused by strategic voting under range is significantly less than under any other commonly-suggested alternative.
Both of these points are obvious by examining the graphic I already shared:
http://rangevoting.org/BayRegsFig.html
You’ll also note that strategic approval voting is equally-good as strategic range. Approval is a also the simplest/cheapest/easier voting system to implement, other than plurality. If you’re looking for bang-for-the-buck, approval is a no-brainer. If it’s bang-no-matter-the-cost, go with range. All ranked voting methods are simply not as good.
The method you describe is called the Borda count, and for a three-candidate election with honest voters, it IS the best ranked-ballot-based method.
But the Borda count loses a lot from strategy (Borda himself said “My system is meant for honest men.”) Furthermore, it doesn’t perform as well as range or even approval voting (which are not ranked-ballot-based methods.) Finally, Borda has an issue called “teaming”, where (in contrast to our familiar plurality-voting environment) each candidate is helped by having more candidates on the ballot who are similar to them; so you might find it difficult to keep the ballot down to just three candidates, as every one tries to recruit larger and larger teams.
Yup; range is best.
Oops… seems I hadn’t previously shared that image. Sorry about that; can’t berate people for not reading something I hadn’t shown them yet!
Is range voting used anywhere? I’d think it sounds like an interesting idea for allowing a small number of voters to pile on votes for a candidate in a multi-seat district to get at least one seat to represent them. If the top three candidates are selected, a minority group could pack their votes for one candidate and shoot for at least one seat. Perhaps has they feel more candidates are addressing their community they would feel less of a need to do that.