Elections (and Choices) with more than two options
©2007 John D. Stackpole, CPP, PRP
Abstract
There is rather surprising (mathematical) proof that it is simply not possible to run a completely reasonable or "perfect" election system (or make a selection between options) when there are more than two candidates/choices to select from. This is true even when the system, whatever it may be, is run completely honestly. This essay will describe what "reasonable" means and then go into various familiar elections systems to indicate, with examples, how they are indeed unreasonable in one way or another, and why. Various systems do show different degrees of unreasonableness and this essay will conclude by discussing a particular voting system, the Borda Count, that appears to have the "least bad" characteristics.
Many, if not most, organizations have a long tradition of decision making by majority vote, in conformance with the democratic process that has evolved over some three thousand years or so, following the rules described in Robert’s Rules of Order Newly Revised (RONR). In the particular case of elections to office, a majority election is all but automatically assured when there are only two candidates in the running, tie votes being the one relatively rare exception. But if there are more than two candidates for any one office, a problem immediately crops up: it is perfectly possible, and not at all unlikely, that no candidate will get a majority of the votes cast. Under these circumstances (as with a tie vote in a two-way election), RONR requires the voting to be repeated with all candidates remaining on the ballot (unless an individual chooses to withdraw), in the (sometimes wan) hope that a majority winner will emerge.
Unfortunately, there is no assurance that a second round of voting (or third, or fourth…) will produce a majority winner. Indeed, if positions are hardened, for example, into two camps, neither of which can (quite) achieve a majority for their favorite, there could be many repeated ballots. RONR suggests that this can lead to the eventual selection of a “dark horse” who might be the third (or a newly nominated fourth) candidate who is, in effect, a “second choice” of a majority of the voters. Unfortunately, there is again no assurance that this will happen. Voting can go on for a long, long time.
For a small organization this can be inconvenient, annoying, frustrating, and time consuming. For a large organization, this can also be quite expensive. In an extreme case, a large annual convention, say, could have to adjourn (because the facilities were no longer available) without completing an election for one or more offices, leaving both the association and the candidates in a parliamentary limbo.
If the decision is not an election but a choice between multiple (more than two) courses of action, RONR includes methods of making such choices that will allow the assembly to reach an apparent majority decision in a reasonably prompt manner. If there are three choices, the standard method of making a motion (containing choice “A”), then an amendment (choice “B”), and a secondary amendment (choice “C”), assures that the assembly, by a series of three majority votes, will arrive at a final selection, guaranteed. If there are more than three choices, the method of “filling the blank” can accomplish the same task with up to as many rounds of voting as there are choices, plus one more to adopt (or defeat) the main motion after the blank has been filled. The “blank-filling” system is, however, not guaranteed to work as it is possible that none of the choices will get the majority vote needed to fill the blank. Then, much as in an election, repeated voting may be necessary, with the same attendant difficulties.
Furthermore, in the case of the motion-amendment-secondary amendment system, it is perfectly possible, under not unusual circumstances, that the first choice offered, the main motion choice, will always be the final choice, no matter which of the three choices, “A”, “B” or “C” is offered first as the main motion. And what is even more insidious, there will be no indication during the moving, amending, and voting that this is a real possibility. (This was discussed in an earlier paper, Stackpole (1), and will be touched on in what follows, although multiple candidate elections will be the main focus of this essay.)
It would be certainly helpful to be able to suggest a “good” election process to many organizations that avoided the multiple ballot difficulty, and, as a bonus, automatically defaulted to a regular two person election when there were only two candidates in the running.
There are any number of ways to assure that repeated balloting will not be necessary, some better than others, some with possibly uncomfortable consequences, and none of them (sad to say) ideal. The options can be placed in two broad categories: 1) The nomination process and 2) the voting and vote tabulating system.
Many organizations and their nominating committees follow the tradition of putting two candidates forward for each office, usually with an “if possible” contingency to avoid pointless nominations. Others instruct the committee to nominate just one candidate for each office. Both varieties of organization, wisely, do not forbid nominations from the floor and some set up petition systems for additional nominations well ahead of the election date. The nomination of two candidates assures that an election will take place (which many organizations feel is a “good thing”) but it also means that if there are candidates put forth by floor nomination or petition there will be a three (or more) way election. Furthermore, per RONR, write-in votes are in order, which can further dilute the vote away from a majority decision on the first ballot.
An obvious possibility is to mandate, via the bylaws, that the nominating committee present only one candidate per office – its “best” choice. This does not completely eliminate the possibility of multiple candidates – floor nominations, petitions, and write-ins remain possibilities – but it certainly reduces the probability of its happening. It also puts considerable (indirect) power in the hands of the nominating committee as its single candidate for a given office could be perceived as the “official” candidate. Floor nominations and petitions (and write-ins) could also be ruled out, via the bylaws, but this would not seem to be particularly wise. The possibility of floor nominations and petitions serves as a sort of goad, reminding the nominating committee to make every effort to come up with good candidates.
In conclusion, the “only one candidate” proposal appears to be the only potential way to at least minimize the possibility of multiple candidate elections prior to the election itself. And it is none too satisfactory.
So that leaves us with contemplating various voting and vote tabulating systems to see if any can be expected to reduce or eliminate repeated balloting in a completely reasonable way. They can’t, but at least some are better than others.
Before discussing some (by no means all) of the ways that the choice, or preferences, of a collection of individuals can be combined or aggregated into what the psephologists and sociologists call a "rational collective choice", i.e., an election, a few general comments are appropriate. For one, there is a huge literature dealing with voting schemes, their details, their foibles, and their quirks, when multiple choices are involved. Much of the literature, including some portions that are heavily mathematical, is given over to describing each author's favorite scheme, how good it is, and how awful the other peoples’ schemes are. This essay is not going to recapitulate all that. Just some of it, via a few examples.
The basic reason for all that literature is that none of the systems appear to “work” entirely satisfactorily. No matter what ingenious method of voting is put forth, it is, it seems, always possible to construct examples where the outcome of the system, whatever it is, is contrary to what one thinks it “ought” to be. Either that or all sorts of paradoxical, inconsistent, and just plain strange results show up. The distressing fact, however, is that this (historical) failure to think up a “perfect” voting system is not due to lack of intelligence or not trying hard enough. It just isn’t even possible, but it wasn’t until relatively recently that this was fully understood.
In 1951 K. J. Arrow proved that whenever there are more than two candidates in the running there simply is no "perfect" system for voters to use (and how votes are to be tallied). It is worth a moment to look briefly at what Arrow considered to be a “perfect” voting system and how it failed to be realized. He proposed four entirely reasonable conditions on how the voters and the system should behave, based on how election systems have been constructed over the years:
1) Voters are “rational” and free. A “rational voter”, in this context, is one with consistent preferences. That is, if he prefers candidate (or option) A over B, and B over C, then, if asked, he will prefer A over C. (To save some typing, “prefers A over B”, &c., will be written A>B. So if a rational voter has preferences A>B and B>C (or A>B>C), he will also have a preference A>C.) Now we all know people who have a preference of A>B>C but still prefer C>A. I may prefer vanilla ice cream over chocolate, and chocolate over strawberry, but if I am confronted with only a vanilla/strawberry choice, I may well go for strawberry, at least some of the time. This is not a problem at the dairy bar, but if I start making these kinds of choices all the time and base my life on them, I’ll probably end up behind (institutional) bars, for my own protection, if nothing else. By keeping irrational voters out of the mix, Arrow was able to simplify his analysis – he was concerned with how the system worked with rational voters (that, it turned out, was bad enough), not what could go wrong in addition if the voters were nuts (too).
“Free” simply means that there are no constraints on how the voters rank the candidates, as long as they rank all of them. Any ordering is fine. Sometimes these rational voters are called “sincere” – they stick to their (ranked) choices no matter what.
This first condition describes the voters themselves. The remaining three conditions state how the “perfect” voting system should work as it produces the final results, what sociologists call the “societal outcome”.
2) The final results of the system will also be “rational” as above, with one possible exception. One or more candidates may be tied in the results even though the voters all rank their choices completely.
3) If all the voters prefer A, say, as their first choice, the voting system should produce A as the winner – it had better! Or if all the voters prefer A>B>C, the election system should do so too in the results. In the books this goes by the name “Pareto”, but it could also be called the “Everybody Loves Raymond” requirement.
4) “Independence of Irrelevant Alternatives” (IIA). That’s a mouthful, sorry, but it is a label that concisely describes a very reasonable requirement to place on the voting/election system. What it amounts to is the assertion that the outcome of the election, the collective ranking of A vs. B, say, should depend only on how the voters individually rank A vs. B, when they are all aggregated together. In this process, an individual voter’s ranking of A>C>B is equivalent to A>B>C or C>A>B as far as his ranking of A and B is concerned. The voter prefers A over B and what he thinks of C relative to A or B should make no difference in the final result. That is, how the voters may rank C against A or B should have no impact on the outcome of the A vs. B race. In still more words, the outcome of the election of A vs. B should be “independent” of all the B vs. C or A vs. C rankings by the voters. B vs. C and A vs. C are “irrelevant alternatives” to the decision of how A and B rank in the outcome. If you claim otherwise, folks may soon be doubting your powers of logic, or your sanity.
People have looked long and hard at these four “reasonableness” criteria and have not come up with anything that they consider to be “more reasonable”. It looks very much like these are the requirements for a “perfect” voting/election system. There is only one small problem: it isn’t possible to construct a system that follows all these criteria. The criteria are incompatible in one way or another; you simply cannot put them all together to build a “perfect” voting system. That is what Arrow proved in 1951.
Well, actually, what Arrow showed was that in order to have a system that did have all four of these desirable criteria you need one more requirement, an additional special requirement on the voters, not on the election system:
5) A “dictatorship”, by which is meant that of all the voters one person has one vote (or holds most all the proxies, or owns most of the stock) and his single vote, his preferences and his alone, is the only one that counts. That one voter is “The Decider”. Most folks in our society would agree that this is not a very desirable criterion for democratic elections.
Well. If we can’t have perfection, we will have to settle for something less. Let us see what is available and what we think we can live with.
What Arrow’s result doesn’t tell us, unfortunately, is exactly how his proof that “You can’t do that!” manifests itself in difficulties when you try to do it anyway, i.e., what goes wrong when you adopt some voting system, or systems, that, on analysis, is or are apparently in violation of one or more of these criteria. In some cases the “violation” of one of Arrow’s criteria may be subtle, and the troublesome problems may seem unrelated to the criteria. In other cases, it will be obvious. But that doesn’t really matter, You can expect trouble for sure and investigation of how a particular voting system works will reveal the problems, if not the precise relationship to Arrow’s theorem.
One thing that Arrow’s investigation did do, perhaps subtly, was change the way people went about looking at voting systems. Instead of striving to come up with the “best” system, and arguing between themselves as to what was “best”, the emphasis shifted to looking for the “least bad” system. And this has turned out to be a rather fruitful approach. It appears to be easier to decide what is “more awful” about a voting system than to agree on what is “more better”.
A critical, and necessary, part of analyzing voting/election systems is that we, the analysts, get to be omniscient – we assume we know all the preferences, the A>B>C, C>B>A, &c. rankings of all the (rational) voters and then we work with that information to see how the election system under consideration responds. In some cases, as with a plurality/majority election where the voters just vote-for-one, their top choice, the election system does not use the full range of the available information – the full preference rankings – to find out who wins. But we will use that information to show the problems and defects in the system that, it turns out, are ultimately caused by the failure to use the full range of information.
An important thing to keep in mind is that the individual voter only knows what his or her individual preferences are – we are omniscient, but not the voters. In some cases this can make all the difference, in explaining the generation of inconsistent, strange, and paradoxical surprises in the outcomes.
O.K. enough with the introductions, already.
Returning to the problem that initiated this discussion, “endless balloting”, the reason the balloting can go on for a long time is, of course, the “must win by majority” requirement. So let us do away with that requirement and settle for a plurality election system, where the candidate who simply gets the most votes, where each voter votes for his or her favorite, wins.
It turns out that this is one of the worst ways to go about finding the true “will of the people”, whatever it may be. Consider: suppose there is an approximately divided electorate, with a large minority favoring candidate Alice, and a majority favoring a male candidate, any male candidate. But there are two male candidates, Bob and Charlie, in the running. Result: Bob and Charlie split (more or less) the “male” vote and Alice wins by plurality. The result is that the “winning” candidate is actually the least favored by a majority of voters. (There is pretty good evidence that this is how Jesse Ventura won as governor in Minnesota some years ago.) Had the voters used an “inverted” system where they vote against their least favored candidate, Alice would have been immediately eliminated, by a majority vote. Two closely related systems (“vote for your favorite” or “vote against your least favored”) have come up with completely different outcomes. This is certainly an uncomfortable result. It isn’t clear who should be the winner at this point, perhaps, but it is apparent that the voting and tabulating systems have somehow skewed the outcome.
Here is a numerical example, first dreamed up by Borda in the late 18th century – these arguments have been going on for a very long time – for those readers who may want to see it for themselves. There are a total of 12 voters and their full preferences (remember, we are omniscient) line up like this:
5 voters: Alice>Charlie>Bob (A>C>B)
4 voters: Bob>Charlie>Alice (B>C>A)
3 voters: Charlie>Bob>Alice (C>B>A)
Since the voters cast ballots only for their first choices in a plurality (or majority) system, the (ranked) results will be Alice(5)>Bob(4)>Charlie(3) and Alice is the plurality winner. Had, as suggested above, the voters cast a ballot against their bottom ranked candidate, the ranked outcome would be (with least favored last): Charlie(0)>Bob(5)>Alice(7). Alice is not just the plurality but the majority looser. Not only that but the overall ranking is completely reversed from the vote-for-first-choice system. If you vote for your favorite the final results are exactly reversed from what you get when you vote against your least favorite. If that doesn’t make you feel uncomfortable, I don’t know what might.
To increase your discomfort, consider a more recent example of a plurality election: Bush/Gore/Nader in Florida, 2000. For our purposes here we will ignore all the difficulties with chads, counting, and the Supreme Court, and assume that the numerical results are essentially correct. There is fairly good evidence that Nader was a “spoiler”: had his partisans decided that they didn’t like him all that much after all and voted instead for Gore, Bush would have lost. So? That’s the way it goes, right? Well, what about those “Irrelevant Alternatives”, Arrow’s fourth criterion for a “reasonable” election system? What the Florida results suggest is that many of the voters ranked Nader over Gore and therefore Bush won over Gore. WHAT! Why should the Gore vs. Nader rankings make any difference (rationally) in the Bush vs. Gore contest? Allowing this to happen is exactly what Arrow, and lots of other people, agreed was an unreasonable voting system, and Arrow tried to rule it out by the IIA criterion. But there it is in the plurality election system. We are so used to it that it doesn’t seem strange. But it sure is.
A very common “fix” to a plurality election is to eliminate the lowest vote-count candidate, after the first round of voting, thus assuring, if there were three candidates to start, a two person race, with a majority winner. If there is a first round majority, then that is the end of it. The majority winner is the winner, no question. This wouldn’t entirely eliminate the problem of multiple ballots, but at least it will assure that there are no more than two rounds of voting.
A close relative to runoff elections is Instant Runoff Voting (IRV). In this arrangement, the voter specifies his “second choice” in addition to his first choice preference. Then, if there is no majority of first choice selections, the ballots originally voted for the third place candidate get assigned to whichever candidates were indicated as the second choice on those particular ballots. Thus one of the original “top two” will get a majority of the votes. This does assure that there will be only one round of voting, although the ballots may have to be counted twice.
Philosophically, this isn’t (quite) a true majority win in the end because the final winner’s majority collection of votes is, in part, made up of the second choices of the people who voted for the eliminated candidate first time around – it is a compromise that is forced on the voters by the system. And, interestingly, the “second choice” votes that get tabulated into the result are only those of the third (eliminated) candidate’s voters. The second choices of the other voters, whatever they may be, are simply ignored. Why should the loser’s second choices be the only ones that count? Doesn’t seem equitable.
Runoff voting doesn’t eliminate the problem of independent alternatives effecting the final outcome, it just makes it explicit. Had IRV been in effect in Florida, say, it is likely that Gore would have been the eventual winner, presuming that the Naderites indicated Gore as their second choice (but there is no way to be sure of that as there is no record of the voter’s second choices).
Another startling problem: If there are four, or more, candidates running, the question comes up of how many should be eliminated after the first round. All but the top two, or just the lowest vote-recipient? Eliminating all but the top two assures no more than two rounds of voting. Eliminating the lowest candidates one at a time increases the opportunities for choices but adds additional rounds of voting, depending on how many candidates are in the initial field. The kicker is that under not unusual situations, the final result will be different depending on which system is used. If Alice and Bob are the “top two” after the first round of voting, and all the others are eliminated at once, Alice, say, might be the final majority winner. But if the elimination is done one candidate at a time, it is entirely possible that Alice will not show up in the “top two” after one of those intermediate votes, and thus be eliminated. The final pairing could be Bob vs. Charlie (or Dave). It all depends on how those second (and possibly third) choices fall in the voter’s minds. Change the system, slightly, and the outcome changes. So, what do the voters really want?
A more serious difficulty with the runoff system is what it does to a possible dark horse compromise candidate. Basically, it completely eliminates any opportunity for him to be selected. Suppose there is, again, a nearly equally divided electorate in which Alice's supporters can't stand the sight of Charlie, and vice versa. Just under half favor Alice, and (not quite) the other half favor Charlie. A few favor Bob, just (barely) enough so that neither Alice nor Charlie can get a majority on the first round of voting. Both major camps rank each other’s candidate as a dead last choice but view Bob as bearable; he would be their second choice, if they even thought about that possibility. (They would be asked to think about it in an IRV system). But what happens in a runoff? Bob is eliminated, and the few voters who preferred Bob vote for Alice or Charlie (their second choices) and one of them wins. The split in the voting body remains and is, possibly, intensified. The inability to ever come up with a compromise winner is forced on the voters by the system.
Further, there are not unreasonable situations in which the actual winner of the runoff is not the (first) "winner" in the three-way plurality election - a situation that can lead to some pretty strong arguments about "honesty" or the competence of the tellers, etc. Or at least wonderment.
IRV introduces a really remarkable new possibility. Because the voters have no opportunity to change their “second choice” between tabulation rounds, the following can happen: suppose (with reliable pre-election polling) it is clear that Alice is going to win by IRV, and a number of voters decide to jump on the band wagon and go out and vote for Alice instead of Bob (they shift Alice ahead of Bob as their first choices). It is quite possible that, because of second place choices and the elimination and counting method, Alice then loses even though her first place vote count actually increased! Is that what the voters want?
This possibility is so paradoxical that it is worth an example. There are 93 voters in all, with these rankings before the polling results are revealed:
42 voters: Alice>Bob>Charlie
27 voters: Bob>Charlie>Alice
24 voters: Charlie>Alice>Bob
Alice has a plurality, but not a majority of first place votes, so the lowest (Charlie) would be dropped. His second place votes go to Alice and she would win over Bob, 66 to 27. But then, after the polling is announced (or Alice makes a real stem-winder of a speech) 4 of the 27 voters who preferred Bob decide they really like Alice best. So going into election day the rankings now line up like this:
46 voters: Alice>Bob>Charlie
23 voters: Bob>Charlie>Alice
24 voters: Charlie>Alice>Bob
Alice is still (just) short of a majority, but now Bob is dropped, and the runoff is between Alice and Charlie. Bob’s second place votes go to Charlie who then (just barely) beats Alice 47 to 46! So Alice increased her (first place) lead but lost, where she would have won before the poll results were revealed (or her super speech). Sure a funny way to run an election.
This is a bit like IRV from the voter’s point of view except instead of ranking his first and second choice, the voter can simply vote for as many of the candidates as he “approves” of. Whoever gets the most votes wins – no majority decision required. This will cause the dark horse (almost everybody’s second choice, if there is one) to be the winner, but it is a system with other strange quirks, detailed in the literature. One of the strangest is that it can cause a “clear” victory of one candidate (with an overwhelming number of first place votes) to turn into a tie with another candidate.
In preferential voting the voter ranks all the candidates, not just the top two choices. Of course, in a three-person election that is the same as just ranking the top two as the third is last by default. Then a complicated (and rather time consuming) shuffling of ballots and votes results in one winner eventually. There is no need for repeated balloting . Unfortunately, this system has the same defects as the runoff system – the lowest "first place" candidate is eliminated and the second place votes on his ballots get distributed to the other candidates. Thus a possible "everybody's second choice" candidate is eliminated right from the start, and no account is made of the second choices of the initially top ranked candidates. Also, in the preferential system it is possible (not probable, but possible) that particular configurations of votes will result in the necessity for a completely arbitrary choice of who to eliminate. And which choice is made can, in turn, change the final outcome. Any system in which an arbitrary choice in the tabulating system can change the outcome is, to say the least, suspect. A detailed example of this is found in Stackpole(2)
In three remarkable books (and associated research papers) Dr. Donald Saari (a mathematician now at UC Irvine) has looked at these and other election/voting problems in great detail and arrived at a quite remarkable conclusion: virtually all, if not all, of the strange results sketched above, and many others, can be attributed to the voting system’s failure to take proper account of “balanced voters” (my term, not Saari’s more mathematical one).
So what are “balanced voters”? Very simply they are sets of voters who, when you look closely at their preferences together, should not make any difference in the outcome of an election, or decision. A familiar example is found in the newspaper reports of congressional votes on “important” issues. Often, when congressmen have to be away from congress for those votes, they will be reported as “paired for” and “paired against” indicating how they would have voted had they been there. This informs the home folks of their positions but it is clear that had they, and their paired congressmen, both shown up to vote there would be absolutely no change in the outcome. The vote counts would be increased, to be sure, but who won would not be changed. It doesn’t matter one bit (for that particular issue, of course) whether these “balanced voters” were present and voting or on the other side of the country. And the reason this works, in a mathematical sense, is because those balanced voters are voting in a two-choice decision, “for”, or “against”.
In a three way choice, a pair of balanced voters would have preferences that matched up like this, for example:
1 voter: A>B>C
1 voter: C>B>A
In a plurality/majority election system, remembering that these two (or 20, or 200) voters will vote for their first choice only, it is obvious that A will get a vote, and C will get a vote – they balance off. Also if you were to match these voters up into pairwise votes (a different system - more on this later) one voter would prefer A>B, B>C, and A>C and the other voter would select B>A, C>B, and C>A, exactly the reverse of the first voter. These voters are as balanced as can be. So what’s the problem?
It just doesn’t work right.
Here’s an example. Suppose we have a set of voters with a clear cut, unambiguous, collective decision, thus:
6 voters: A>B>C
2 voters: B>A>C
1 voter : A>C>B
Clearly A, with 7 first place votes, is not only the majority winner but is a “landslide” winner as well with a 77.8% supermajority. There is absolutely no question that A is the “people’s choice”. But now suppose, before the polls close, a bunch of balanced pairs of voters, 6 to be exact, show up to vote, with these preferences:
6 voters: B>A>C
6 voters: C>A>B
Put them in the mix and the new set of voters line up like this:
6 voters: A>B>C
8 voters: B>A>C
6 voters: C>A>B
1 voter : A>C>B
And the outcome? B is now the plurality winner! A’s supermajority has not only vanished but he isn’t even in first place!. Well, B did not get a majority, so we will do a runoff between the top two, A and B. And A wins 13 to 8, a pretty substantial margin. Whew!. The people’s choice wins after all. Maybe.
Suppose, instead of 6 balanced pairs as above, there were 12. Then the voting body looks like this:
6 voters : A>B>C
14 voters: B>A>C
12 voters: C>A>B
1 voter : A>C>B
B continues to be a plurality winner (no matter how many balanced pairs are added, B cannot, mathematically, become a majority winner), so another runoff: But now the runoff is between B and C and B wins! The people’s choice, A, has simply been swamped.
What is going on here? This is not a “contrived” example; the discovery that “enough” balanced pairs can cause the outcome of a three-way (and more than three-way) plurality/majority election (with or without a runoff) to shift is one of Dr. Saari’s major results. The mathematics proving this is a bit much, but the example captures the essence of it. Adding sets of demonstrably balanced pairs of voters changes the outcome.
The basic problem is that a plurality/majority election system simply does not take proper account of obviously balanced pairs of voters – and there is no way to “fix” this using the system.. This is a fundamental result. And it certainly brings into question the use of any vote-for-your-first-choice system.
Not only that but Dr. Saari demonstrated that any ranking system, save one, has this same defect: the outcome can change from what it “should be” if enough balanced pairs of voters are in the mix. If this applied to the U.S. Congress (which it doesn’t, of course), the absence or presence of those “paired voters” could change the outcome of votes. It doesn’t effect Congress but it certainly can effect any plurality election in the country, which is almost all the elections that take place.
Just to be clear, a plurality/majority election system is a ranked, and weighted, voting system: when you make your first choice selection, you have ranked all the candidates, but then given a weight of One to your top choice and Zero to all the others. The IRV system gives a weight of One to the top two choices, and then ignores the #2 choice if there is a majority winner. Approval voting gives a weight of One to as many candidates as the voter wishes, then adds them all up. None of these systems will properly account for balanced pairs of voters.
Should this be a real concern in real life? In any election involving a large electorate, whether it is a state election primary or a officer election at a large convention, there are bound to be balanced pairs of voters. Whether there are “enough” of them to make a difference in the outcome is impossible to tell, but the possibility is always lurking there. That is not a confidence building thought.
Just to pound the point home, look back at the Borda example from some pages back The voters had these preferences:
5 voters: Alice>Charlie>Bob (A>C>B)
4 voters: Bob>Charlie>Alice (B>C>A)
3 voters: Charlie>Bob>Alice (C>B>A)
Notice that 8 of the voters form a balanced set
4 voters: Alice>Charlie>Bob (A>C>B)
4 voters: Bob>Charlie>Alice (B>C>A)
Since we agree that these voters are balanced and should not make a difference in the outcome (if you disagree, I would invite you to tell me how they should change things), let us remove them from the mix, giving:
1 voter : Alice>Charlie>Bob (A>C>B)
3 voters: Charlie>Bob>Alice (C>B>A)
as the “core” voters, the ones that determine how the election should turn out. Obviously Charlie beats Alice 3 to 1, and Bob gets no votes at all. This is a bit more reasonable that before, when Alice was the plurality winner, with Bob the runoff winner, and closer to the result when the voters are asked to vote against someone. The presence of the balanced voters caused the “peoples choice” to be hidden from view, to lose, simply because the voting system did not do what any sensible system should do: ignore or cancel out the balanced voters.
In describing how these balanced voters really were balanced among themselves, I used the example of pairwise choices, indicating how each pairwise vote (A against B, B against C, &c.) all balanced out. For every A>B choice there was a matching B>A choice for no net gain for either choice. (This is the same as the Congressional paired votes.) This suggests that there might be another voting system that is immune to the balanced pairs of voters described above. And indeed there is, the Condorcet Choice system. (But don’t get your hopes up; we are not out of the woods yet.)
This system, which does not involve (explicit) ranking, is one with a long and respectable history, dating from the 18th Century. It caries the name of its first proponent, the Marquis de Condorcet. In this arrangement the voter is asked to select a (majority) winner by voting for the various candidates in pairs: A vs. B; A vs. C; B vs. C; each vs. D, etc. for as many pairs as the set of candidates allow. Then the Condorcet Winner is the candidate who “beats all comers”, i.e., wins the pairwise elections against all the others. (Obviously, each voter’s unstated ranking of the candidates comes into play when he is asked to decide between two candidates neither of which is his “first” choice. Presumably he then opts for his second, or third, etc., choice, if he is rational.) In those cases where there is a split electorate and a potential “everybody’s second choice” dark horse exists, this system will produce the dark horse as the Condorcet Winner. And, obviously, if there are only two candidates, the Condorcet system defaults to the familiar one-on-one election system. This is an attractive system with a clear basis in majority choice and, indeed, is the basis for all the voting systems, except elections, found in RONR. It can involve a complex ballot because of the requirement to vote in each of the pairwise choices; this can be alleviated (somewhat) by requiring the voters to produce a full ranking of the candidates and then extracting the pairwise winners from that information. The “extraction” can be an intricate computation and the ranking process takes away the appearance of pairwise voting from the individual voters. However, these are only technical problems that machine voting could overcome.
Unfortunately (as could be anticipated by Arrow’s theorem) it is not a “perfect” system; it doesn’t work “right” all of the time. It doesn’t work right when, just like the plurality/majority system, it is confronted with a collection of balanced voters, although the balanced voters don’t come in pairs, they come in bunches of three, triplets. Here is what a set of “Condorcet Balanced Triplets” look like:
1 voter: A>C>B
1 voter: C>B>A
1 voter: B>A>C
Clearly each candidate receives one vote for each of the three positions, first second and third, and there is no question that they are balanced. If you were to tabulate all the pairwise votes you would get these numerical results:
A>C by 2 to 1
C>B by 2 to 1
B>A by 2 to 1
Clearly there is no “Condorcet Winner” who beats all the others.
Also the collective ranking of the candidates is
A>C, C>B, B>A
which is a little strange when you look at it; the collective choice goes in a circle; you might even call it “irrational”. It is a bit unsettling that the Condorcet voting system should take three perfectly rational voters with clear choices and generate a collective result that is clearly not rational. If these three voters were the only ones, this is not much of a problem since no clear-cut outcome could be expected. But, still, a system that turns rational voters into a collectively irrational result seems a little suspect. And that is just the tip of the iceberg.
It is quite possible, in real situations, that triplets of voters, if there are enough of them (and “enough” can be a remarkably small number), can cause the outcome to produce no winner, and indeed produce the rather strange cyclic pattern. Or what is worse, these “Condorcet triplets” which, upon inspection, clearly should have no influence on an election (analogous to absent “paired voters” in U.S. Congressional votes) can actually change the outcome from what would be the case were they not present. It is disconcerting to discover that a collection of voters, clearly all tied (or “tripled”) when considered as an independent unit, if added to a set of voters with a clear (Condorcet) preference, can change the winner or even cause there to be no winner at all.
As an example, we can go back to the unambiguous set of voters introduced previously:
6 voters: A>B>C
2 voters: B>A>C
1 voter : A>C>B
If you tabulate the pairwise contest results of these voters, you will get:
A>B by 7 to 2
A>C by 9 to 0
B>C by 8 to 1
A is the Condorcet winner as she beats all the other candidates in a pairwise races. And if you were to use this information to rank the results, you would get
A>B>C
which is the same as the plurality/majority system gave earlier with this set of “core” voters.
Just as we did before, let us now introduce a set of 18 “triplet” voters who form a balanced set
6 voters: A>C>B
6 voters: C>B>A
6 voters: B>A>C
and add them to the original, giving:
6 voters: A>B>C
7 voters: A>C>B
6 voters: C>B>A
8 voters: B>A>C
This is getting to be a bit of work, but the pairwise tabulations of this set of voters comes out to:
B>A by 14 to 13
B>C by 14 to 13
A>C by 21 to 6
which points out that the introduction of the balanced triplet of voters has changed the Condorcet outcome from A to B. Just the sort of problem we encountered before. Doesn’t look good.
And just to make matters worse, if we were to add in another 18 triple-balanced voters, the outcome would be:
A>C by 33 to 12
C>B by 25 to 20
B>A by 26 to 19
or a completely irrational cycle with no “winner” of any kind at all. The clear preference of the “core” voters for A has been completely obliterated by the accumulation of the balanced triplet sets of voters.
Just as the plurality/majority system (and most other ranked systems) fails to take proper account of balanced pairs of voters, the Condorcet system fails to take proper account of balanced triplets of voters. And there is no way out – that is what Dr. Saari demonstrated for each system.
Another example of how the presence of balanced triplets can corrupt, in a very subtle manner, the outcome of pairwise voting is described in Stackpole (1). The example brings into question much of the decision-making process in RONR.
It is interesting, although of no particular value in the real world, to note that the plurality/majority system is impervious to the balanced triplets of voters, and the Condorcet system is equally impervious to balanced pairs of voters. The election outcomes do not change when balanced voters are added (or removed) in this “crossed over” arrangement. It is left as an exercise for the reader to demonstrate this for himself by an example or two.
To emphasize, both the plurality system and the Condorcet system suffer from this serious problem: the addition of clearly tied voters (in balanced pairs or triplets) to an original set of voters with a clear and unambiguous choice of “who wins” can alter the result to cause someone else to be the winner. Is that what the voters want?
Furthermore, if a group of voters get together before the election and discover that they form plurality “balanced pairs” (or Condorcet balanced triplets) they may well think that they needn’t bother to stand in long lines to vote because they are obviously tied – their votes just couldn’t or shouldn’t make a difference. So they don’t bother to vote. And yet it is quite possible that the absence of these neutral voters does make a difference in the election outcome. But that is because of the voting system itself, not because those voters are “really” not all tied, tripled, or balanced. They really are tied, but the voting system doesn’t recognize that fact and causes them, or their absence, to make a difference in the outcome.
What is needed is a system of voting that does take proper account of both balanced pairs and balanced triplets of voters, i.e. assures that their presence or absence in any particular collection of voters makes no difference in the outcome of any three (or more) way decision. It won’t be a “perfect” system – Arrow makes that clear – but at least the imperfections won’t be so blatant as to allow balanced voters to change the results.
Is there one? Yes. And it, just like the Condorcet system, has been around since the 18th century, first dreamed up and then vigorously promoted by Jean-Charles Borda, and, more recently, by Dr. Saari.
The Borda count system is a ranked system, requiring the voters to rank all the candidates, but manages to properly account for the balanced pairs of voters that the other ranked systems fail to do. It also perfectly accommodates the balanced triplets of voters that the Condorcet system fails to handle properly. It does so by the judicious choice of the weights applied to the ranked choices of each voter, top to bottom. Commonly, if there are N candidates, the top ranked candidate gets (N-1) points, the next one down gets (N-2) points, etc., and the last gets (N-N) (zero) points. The actual numbers don’t matter as long as there is a linear progression on down. In a three-person election, these weights are 2, 1, 0, and in a two person “normal” election the weights are, obviously, 1 and 0. All the points for each candidate are summed and the winner is the one with the most points. One can also get the “societal ranking” of how the candidates are ranked overall, or collectively, by the voters by looking at the summed point scores for all the candidates.
The key thing about the Borda system is that it takes complete and proper care of those tied, tripled, or, in general, balanced sets of voters – they can come and go as they please (in balanced groups, of course) and the Borda count ranking will not change. They are truly treated as “tied” in that their weighted votes are equal for all candidates, and thus they have no impact on the outcome of the election. So it doesn’t matter whether they vote or not in any one election. The societal ranking remains constant, certainly suggesting that Borda measures “what the voters really want”, or surely comes close to it.
And just to clarify with an example, let us compute the Borda Count for our troublesome balanced voters: the balanced pairs, and the balanced triplets.
Here’s the balanced pair from before:
1 voter: A>B>C
1 voter: C>B>A
The Borda points from the first voter are A(2), B(1), C(0), and from the second voter,
A(0), B(1), C(2). Add the points for each candidate up and, clearly, each gets 2 points – a clear tie.
Here’s the balanced triplet:
1 voter: A>C>B
1 voter: C>B>A
1 voter: B>A>C
Clearly, each candidate picks up 2 points for his first place vote, 1 for the second, and none from the third, for a total of three points for each candidate – also a tie.
It works! And it works right!
Are we done? Well, not quite.
As noted at the start, nothing is perfect. Arrow proved that. The major departure from “standard”, i.e., what we think is the “right way” to win, is that Borda is not a majority system. It is perfectly possible that a candidate who is not a plurality winner, or even is bottom ranked by plurality count (counting only first choice votes), will still be the Borda selection. This comes about if the candidate has lots of “second place” votes, i.e., is “(almost) everybody’s second choice” as discussed above.
Repeating Borda’s original example:
5 voters: (A>C>B)
4 voters: (B>C>A)
3 voters: (C>B>A)
As seen previously, the plurality vote outcome is A(5)>B(4)>C(3). The Borda count outcome is C(15)>B(11)>A(10), a complete reversal of the results, but in agreement with the “core” voters that were left when the four balanced pairs (4 voters: B>C>A, 4 voters: A>C>B) were removed. The Borda system has “removed” (or, better, properly accounted for) the balanced pairs of voters and given what is the “true” will of the voters.
What is perhaps more unsettling is that it is also perfectly possible that someone can be the majority winner and not be the Borda winner. Again it is a matter of the second place votes. Here is an example constructed by simply adding three A>C>B voters to Borda’s set:
8 voters: (A>C>B)
4 voters: (B>C>A)
3 voters: (C>B>A)
A is now the (bare) majority winner of the 15 votes cast, leading the plurality outcome ranking of A(8)>B(4)>C(3). But, in effect because C got lots of second place and no last place votes, the Borda count ranking is C(18)>A(16)>B(11) – C is here an example of “everybody’s second choice” actually being the winner.
The careful reader might note that there are four balanced pair voters (4 voters: A>C>B, 4 voters: B>C>A) in the above set of 15 – removing them might change things? Nope. Remember, the Borda ranking will not change when balanced pairs (or balanced triplets) are added or removed from a set of “core” voters. If those eight balanced voters “leave”, the remaining electorate is:
4 voters: A>C>B
3 voters: C>B>A
A is still the majority winner (4 to 3) and the Borda ranking is C(10)>A(8)>B(3).
Allowing this possibility – selecting the “second choice” compromise candidate even in the face of a (first choice) majority selection – is a substantial philosophical leap from common practice. The basis of the change is the notion that the full range of the voters’ choices should be considered in deciding “what the voters want”, not just their first choices. And this consideration should extend to all the voters, not just the ones who voted for the eliminated candidate as in the runoff and IRV systems. The system detects the candidate with the broadest overall appeal, not just “first place” appeal. Whether one views this semi-automatic selection of the compromise candidate as a good thing or not depends on one’s outlook toward life, presumably.
There is one saving grace to all this, however. If the winner has a substantial supermajority of first choice votes, he will be selected by the Borda system (and most all other systems). It is only in the relatively close races that the problems described above come up. See Stackpole (3). This is simply a manifestation of Arrow’s third and fifth conditions: if (almost) all the voters like the same candidate, they are (approximately) satisfying the “Everybody Loves Raymond” (or “Pareto”) condition, or alternatively, the large block of voters all voting the same form a “dictatorship” and the system works. All the paradoxes vanish!.
This is a relatively small point, but the Borda count system does not allow for any write-in votes – the full set of candidates must be available to all the voters so that the ranking weights can be consistently applied. And all the candidates have to be ranked – this can or could be a difficulty in crowded primary races for example. It is a trade off between the manifest problems of the plurality system, vs. the lesser ones of the Borda system. But in the usual association with rarely more than three or four candidates running for a single office, ranking them all does not seem to be much of a problem.
From time to time, in discussions about voting systems, the question of “manipulation” comes up. By this is meant that an individual voter chooses not to vote his first choice (or changes his ranking) in an effort, he hopes, to cause someone more to his liking (but not his first choice) to be the winner. He is an “insincere” or “strategic” voter. Back to Florida in 2000: had more Nader voters voted “insincerely”, i.e., for Gore, say, their least favored candidate Bush (I’m guessing) would not have won. They could have manipulated the system by voting strategically. (And, of course, a more insidious manipulation could take the form of the Bush people supporting Nader’s candidacy to assure his being on the ballot in order to draw voters away from Gore.)
There is another theorem (it goes by the name of “Gibbard-Satterthwaite”) that asserts that any system of voting and tabulating can be "manipulated" to some extent to cause a different result to be the outcome, by knowledgeable voters casting their votes in a way that would appear to be contrary to their own preferences. It applies to Borda count systems, too, of course, although the examples are more involved than in the plurality (Florida) example. The important thing to note is the word “knowledgeable” – to vote successfully in a strategic manner an individual has to know an awful lot about how the “other guys” are planning to vote and adjust his vote accordingly. And at the same time the “other guys” are (maybe) doing the same to him so it could all balance out. It seems unlikely that “manipulation” of the Borda system need be of much concern in the real world (as opposed to academic voting system studies). It can certainly effect plurality elections, as noted above.
It appears that there are good reasons (lots of them!) to abandon the vote-for-first-choice system with a plurality winner (even with a following runoff) in favor of a ranked Borda count system. However, allowing for the possibility that a majority of the voters’ first choices might not be the winner may well go too far for comfort. After all, some three thousand years of decision making by majority rule is not something to be lightly cast aside (although perhaps it should be).
Here is a compromise proposal: The voters would do a complete ranking of the candidates on their ballots, but there would be a first pass through the ballots to see if there was a true majority winner, counting the first place votes only. If so, that would be the end of it, he’s elected. And if not, of course, the full Borda count would be performed to find out who the candidate with the widest overall appeal, the Borda winner, would be (“everybody’s second choice”, if there was one). This would assure that there was only one round of voting, with the added cost of requiring the candidate rankings by the individual voters as a “just in case” contingency. The rather extensive calculations of the weighed vote totals could be programmed into electronic voting machines and not present undue delays or problems (as long as there are back-up paper ballots!). And since the Borda system defaults to a “regular” election when there are only two candidates, the same tabulating programs would work in all situations.
But, as must be re-emphasized, no system is “perfect”: if there is a “true” majority winner imbedded at the core of the full set of voters, he can be hidden by those balanced pairs of voters that are misrepresented by the plurality/majority system of vote-for-one. One of the other candidates then could be selected as a plurality (never majority) winner, and the Borda count finishes the election. And again, there is no assurance that the Borda count system will find or select that “hidden” majority winner. Second choices may well overwhelm him.
There is no way out of that sort of dilemma with multiple candidates. There simply is no perfect multi-candidate voting system, singly or in combination. Borda may come close, but it can’t get all the way there. Nothing can.
None of the above is original with the author of this essay, although I have emphasized things a bit differently than in the sources. All of the arguments and statements can be found, backed up with varying degrees of mathematical proof and specific examples, in three remarkable books by Dr. Donald Saari, the “Distinguished Professor: Mathematics and Economics, and Professor: Logic and Philosophy of Science”, University of California, Irvine.. Dr. Saari is also a Fellow of the National Academy of Sciences. (He was elected a Fellow, but it is not clear what voting system was used.) See << www.math.uci.edu/~dsaari/ >>
The three books are:
Basic Geometry of Voting, Springer-Verlag, New York, 1995.
Heavy (very heavy) going in places, this contains many of the full proofs of various theorems that are the basis for the results described above.
Chaotic Elections, A Mathematician Looks at Voting, American Mathematical Society, 2001
Decisions and Elections, Explaining the Unexpected, Cambridge University Press, 2001
These are two (semi-)popular accounts of voting problems and paradoxes with an emphasis on the worthiness the Borda count; they are the primary sources for this report. The first emphasizes various voting systems and the difficulties inherent in them; the second is a somewhat more theoretical and general study of why there are no “perfect” decision-making systems, based on Arrow’s theorem and some extensions. It touches upon some of the surprising consequences of the imperfections (even well outside the field of “voting” as such), and what can be done about them, at least in part..
The “combination compromise” described above was first proposed by Duncan Black (another early big name in the business), in a slightly different form, in The Theory of Committees and Elections, Cambridge University Press, New York, 1958. It is described and commented upon in the Saari books.
All of the Saari books have substantial numbers of references to original peer-reviewed publications in scholarly journals and one book even has exercises for the reader. Tough exercises.
Other References:
Stackpole (1): Why the RONR System of Main-Motion/Primary-Amendment/Secondary-Amendment is a Really Lousy Way to Make a Three-Way Decision; Parliamentary Journal, October, 2001, pp. 129-144
Stackpole (2): A Bad Hare Day; Parliamentary Journal, July, 1997, pp. 105-109
Stackpole (3): Why Two-Thirds? Or Why not 63%, or 74%, or Whatever; Parliamentary Journal, April, 2000, pp. 35-42