Bullet Voting

A common criticism of Score Voting and/or Approval Voting is that they will degrade into sincere Plurality Voting, because voters won’t want to hurt their favorite candidates by voting for anyone else. An example:

…because ‘approving’ a second choice may help defeat the voter’s first choice, most experts agree that it [Approval Voting] is likely to devolve to typical vote-for-one pluarlity [sic] voting.
Terrill Bouricius [Source]

We first look at a simple counter-argument, and then analyze some typical supporting arguments associated with this claim.

Further we show the irony that this argument typically comes from proponents of Instant Runoff Voting, which actually does suffer from tactical degradation to Plurality.


Imagine a voter who prefers a weak candidate but tactically casts an insincere vote for one of the frontrunners under Plurality Voting. A common example is a voter who preferred Green Party candidate Ralph Nader in the 2000 USA election, but voted for Democrat Al Gore. If suddenly given the option to vote for an unlimited number of candidates, do you believe that most voters under such circumstances would most likely:

  1. Cast additional votes for Nader (and any other candidates preferred to Gore)
  2. Switch from Gore to Nader, still casting only a single vote

If you chose the first option, then we think you agree with us that bullet voting is not a problem.

The second option would imply that the freedom to vote for more than one candidate turns tactical Plurality voters into sincere Plurality voters, which is not supported by any theoretical or empirical evidence, and seems implausible on its face.

We offer a more thorough look at Score/Approval threshold strategy here.

“Sanity check”

In case the above argument wasn’t quite convincing, consider this alternate way of looking at it. The bullet voting argument is tantamount to claiming that ordinary Plurality Voting is virtually free from tactics, since no voter will want to harm his favorite candidate by voting for a lesser-liked candidate. For instance, IRV advocates claim that a voter whose favorite candidate was Ralph Nader (in the 2000 U.S. Presidential election) would not vote for Al Gore, because doing so could cause Gore to defeat Nader. Yet clearly they know better, since they argue the complete opposite in the context of touting IRV’s tactical superiority to Plurality Voting. For instance:

…many minor candidates genuinely seek to raise important issues. Their supporters must make a tough decision: to vote for their favorite candidate, knowing that the candidate won’t win and might even throw the race to the supporters’ least preferred candidate, or to settle on a less preferred candidate who has a chance to win. In other words, voters must accurately judge not only which candidate they prefer, but whether that candidate has a chance of winning.
Rob Richie, Caleb Kleppner, and Terrill Bouricius [Source]

Misleading rationale

Justification for the bullet voting argument is rare, is almost invariably based on experience with the multiple-winner form of Plurality Voting, in which voters get some number of votes, and can vote for that many candidate. This system is significantly different than Approval Voting from a strategic and mathematical perspective, despite superficial appearances. When this argument is made, it is generally just asserted as fact, with no factual basis whatsoever. For example:

…because indicating support for a lesser choice counts directly against your favorite choice (violating the later-no-harm criteria, as referenced earlier), these systems also lead to immediate incentives to vote insincerely, unlike instant runoff voting where theoretical scenarios are too convoluted to affect voter behavior.
Rob Richie [Source]

Our page on the Later-No-Harm Criterion explains why Rob is mistaken to cite it as support for this bullet voting argument. Ironically, this Plurality-like behavior is actually a problem for Rob’s preferred method, Instant Runoff Voting.

Real world data

I. In the French approval voting study (thousands of voters, 16 candidates, presidential election of 2002; probably the largest approval voting study ever), the plurality vote totalled 100% and the approval votes totalled 315%, and the percentage of “bullet style” (approves exactly one) ballots was 11.1%. Let us compare that head to head with the (similar parameters) San Francisco Mayoral election of 2007 (12 candidates, 143359 voters). The group FairVote touted SF’s adoption of IRV as a “great success” and Hertzberg himself listed SF as an example IRV city in his very New Yorker blog post we quoted from above. This is, as far as I know, the largest IRV election ever carried out on US soil during the 50 years prior, if not all time. Checking the full ballot dataset we see that over 76063 (53%) of San Francisco’s IRV ballots were “bullet” style. The total number of candidates ranked on those ballots amounted to below 187% of the total number of ballots.

There were 67590 “bullet” votes for Newsom, 3825 for Hoogasian, 2539 for Pang, 590 for Sumchai, and 349 for Rinaldi out of 143359 total accepted IRV ballots.

Thus, this head-to-head comparison suggests that “bullet” voting is more common with the IRV system brought to SF by FairVote, than it is with approval voting. (Indeed, in this case, hugely more common.)

(A subsequent similar study was conducted in Germany.)

II. Now let us perform a second head-to-head comparison of similar-parameter elections. The Burlington Mayoral Election of 2009, another IRV election for which I have the full ballot set (6 candidates), featured 1481 “bullet style” votes out of 8980 valid ballots (16.5%) and 21.5% of ballots ranked exactly two candidates.

The UN secretary general election of 2006 (approval voting, 6 candidates) featured 39 approvals, 35 disapprovals, and 16 “no opinion” votes from 15 voters, an approval fraction of 260%. Since the ballots were secret I do not actually know the percentage of approve-1-disapprove-rest “bullet style” ballots, but it is possible to tell from the data they did publish, that at most 3 of the 15 voters cast a bullet-style ballot. I.e, the percentage of bullet-voters was at most 20%. This is only an upper bound. The lower bound is 0. There overall were more approvals than disapprovals, the exact opposite of what would have happened if there had been a lot of bullet voting. Also, if there really were 3 bullet-ballots (meeting the upper bound) then the remaining 12 ballots would each have had to have approved exactly 3 of the 6 candidates – or somebody must have approved at least 4 of the 6. The uniqueness of this (1,1,1,3,3,3,3,3,3,3,3,3,3,3,3) configuration and the fact it contains a “gap” at 2 both make it seem unlikely; and it also seems unlikely (especially to believers in the prevalence of “bullet voting”) that any voter approved 4. Therefore it is likelythat the 20% upper bound can be decreased to 13.3%. Hence the truth probably is either 0, 1/15=6.7%, or 2/15=13.3%.

III. For our third head-to-head comparison, we can contrast the first four USA presidential elections (having similarities to approval voting) with the three Irish IRV presidential elections.

The early USA conducted its first 4 presidential elections with approval voting, except it was forbidden to approve 3 or more candidates; and the 2nd-place finisher became vice president as an almost worthless “consolation prize.”

All 69 electors each approved the maximum allowed number (2) among the 12 candidates.
All 132 electors each approved the maximum allowed number (2) among the 5 candidates.
All 138 electors each approved the maximum allowed number (2) among the 12 candidates. There were thus zero bullet-style votes. It has been argued, though, that 9 second-approvals were “effectively not there” since they were not for Thomas Pinckney whom they “should have been” for, but rather for no-hopers, as “another means of casting an effectively-bullet” vote. I do not agree such should qualify as “bullet votes,” but if you want to count them that way they would be 9/138=6.5%.
There were 138 electors. Of these 137 approved the maximum allowed number (2) among the 5 candidates, while one – Anthony Lispenard from New York – bullet-voted for Aaron Burr. But this bullet-vote was disallowed because the US Constitution forbade an elector’s first approval from being for anybody from his own state, and Burr was from New York. Lispenard had written “Burr” for both of his allowed approvals (which also was technically illegal since it would have given Burr 2, as opposed to 1; he should have just written “Burr” and “Nobody”). Lispenard demanded a secret ballot in which case his disobeyals of the rules would not have been detectable and he would have gotten away with it and made Burr the victor! However, his demand was rejected because New York State law forbade ballot secrecy in NY electoral votes. The Electoral College after consulting with New York’s delegation, decided therefore to change Lispenard’s vote (against his will?) to “approve Burr & Jefferson.” This led to a Burr-Jefferson tie. Lispenard could have caused Burr to win outright by simply not casting his first approval (or voting it for some no-hoper), although he did not know that when he voted.

The percentage of bullet-style ballots in all 4 of the US presidential elections carried out with (restricted) approval voting, then, was either 0 or0.2% depending on how we view Lispenard (or 2.1% even if you count both Lispenard and all 9 alleged Pinckney-denials). It is plausible that there would have been 3-approving ballots if the rules had allowed it. This contrasts with, e.g. the entire history of Irish presidential elections, all of which were carried out with IRV. Counting only the 3 elections (1945, 1990, 1997) with at least 3 candidates running so that “full ranking” actually could meaningfully differ from “bullet voting,” it appears that somewhere between 9.5% and 31.8% of the ballots were bullet style (based on the percentages of “nontransferable” votes among those ballots that “tried” to transfer).

In 1990 after Currie was eliminated, his 267902 votes “tried” to transfer, but 25548 failed to do so because they were (either intentionally or accidentally) bullet-style ballots, a rate of 25548/267902=9.5%. In 1945, also a 3-candidate election, after McCartie was eliminated, his 212834 votes “tried” to transfer, but 67748 failed to do so, a rate of 67748/212834=31.8%.

Actually, these estimates are probably all underestimates since they were based on voters for “underdogs” and who hence would have had high incentive to rank further choices. The voters for “overdogs” would have had less such incentive, i.e. would have been more likely to “bullet vote.” So probably 9.5% and 31.8% are merely lower bounds.

IV. Dartmouth College’s alumni association used Approval Voting during 1990-2007 to fill vacancies as they arose on its 18-member Board of Trustees. Each election involved 3 “nominated” candidates plus perhaps additional “petition” candidates (usually 3 or 4 in all). The final Approval Voting election, held in 2007, had 4 candidates. It was won by S.F. Smith with 9984 approvals on 18186 ballots (54.9% approval). There were 32941 approvals in all, i.e. 181%. This implies that at most 59.5% of the ballots were bullet-style, and the only way it would be possible to meet this upper bound would be if every ballot approved either 1 or 3 candidates (never 2). If instead every ballot approved either 1 or 2 then the fraction of approve-1 ballots would have had to be 19%. So the bullet fraction, we estimate, was between 19% and 59.5%. Robert Z. Norman, a Dartmouth math professor, explains:

“The claims about bullet voting in the Dartmouth Alumni election [by Rob Richie and other IRV proponents] remind me that with a per voter average of voting for 1.8 candidates, the proportion of bullet votes has to be fairly small. The alternative..is that nearly everyone voted for one or three candidates but not two. Unlikely as that might be, it would suggest that most of those who voted followed a strategy of either voting for the petition candidate or voting for all [3 opposing] nominated candidates, in which case Richie’s claim that the opposition was disorganized falls apart, as does the claim by some of the Alumni Council people that in a 1 on 1 situation the petition candidate would been defeated.”

Meanwhile Dartmouth’s students used instant runoff voting to elect their Student President. You can see their 2006 election results here.

“Frankly, this 2006 election seems like an absurd disaster for IRV because there were 176 candidates. Only three of these 176 candidates were on-ballot (Chick, Patinkin, and Zubricki); the other 173 were “write-ins,” including the eventual winner, Timothy A. Andreadis. Most of the write-ins got zero votes, which was strange. (Couldn’t you vote for yourself? Or was their computer system defective?) Obviously, it was not going to be attractive for voters to provide a full rank ordering of all 176, and indeed I doubt that any voter provided such an ordering nor that any voter even knew who most of the 176 even were. Nevertheless FairVote applauded Dartmouth for adopting IRV and featured this exact election on their web page.”

There were 2435 voters. In the 10th and final round of IRVing Andreadis’s 1127 votes defeated David S. Zubricki’s 913. Really, though, this was only a 3-man race between Andreadis, Zubricki, and Adam Patinkin. The other 173 could have been eliminated immediately if Dartmouth had used better software. That’s because A, Z, and P got 1025, 577, and 554 top-rank votes immediately while the remaining 173 candidates allcombined into an imaginary “supercandidate” (call it “S”) only got 279 (11.5%). Restrict attention, then, to the 4 candidates A, Z, P, and S.

  • The 279 S-voters also ranked somebody in {A,Z,P} 148 times, so there were 131 bullet-type S-votes (47.0%) – not all of which necessarilyreally were “bullet” votes because remember that S is an imaginary supercandidate. (But considering the great unpopularity of S, it seems likely that most of them really were.)
  • The 554 P-voters also ranked somebody in {A,Z} at most 341 times, so there were ≥213 votes (≥38.4%) each of which either was a bullet-vote for P, or ranked P and S only.
  • The 577 Z-voters also ranked A at most 142 times, so there were ≥435 votes (≥75.4%) which either were a bullet-vote for Z, or ranked Z and {P and/or S} only.

In view of the above, it seems reasonable to estimate that about 40% of the IRV ballots were bullet-style.


There is not a great deal of evidence available due to the relative paucity of historically important IRV and approval elections for which we have ballot data (and in some cases we were forced to work from incomplete ballot data, and thus could get only approximate results). But it appears that every one of the first three approval election sets above, involved smaller percentages of “bullet style” ballots than every one of the first three IRV election sets.

“We can’t tell for IV because the data is too imprecise. Actually IV is not a good comparison anyhow because its elections were quite dissimilar; its only similarity is the electorates – Dartmouth alumni & students. (Also it has no historical importance.) For those worried that IV may be an exception to the pattern that IRV elections have more bullet voting than approval elections, note that (as one would a priori have expected) there seems to be a trend for more bullet voting in approval elections with fewer candidates. I, II, and III all compare elections with the same (or close) numbers of candidates. IV is different: Since it compares a 176-candidate IRV election(!) with a 4-candidate approval election, this is no ‘exception’ at all.”

“Bullet-voting” percentages.
Comparison Approval Voting Instant Runoff (IRV)
I France 2002 study: 11.1% (315% approved in total) San Francisco 2007: 53% (<187% ranked in total)
II UN secy genl election of 2006: 260% approved in total, between 0 and 20% (almost certainly at most 13.3%) bullet voting rate Burlington 2009 mayoral: 16.5% bullet voting rate
III Early USA presidential: 0-2% bullet voting rate* Irish presidential: 9.5 to 31.8% bullet rate, or more (average≈21%?)
IV Dartmouth 2007 alumni assoc: 19-59.5% (181% approved in total) Dartmouth 2006 student presidential: ≈40%

*Note that this early presidential election is not really approval voting but just has similarities. Multiple positions are elected, and voters are limited in their number of candidates they can choose. In traditional approval voting, one candidate is elected and voters are not limited in the number of approvals they can make.

We also have a large set of range-style and approval-style polls and in them, again, bullet-voting obviously occurred only at small levels and was not a problem.

Criticisms (ranging from silly to not-so-silly)

  1. Lest IRV advocates want to charge us with some sort of bias in cherrypicking “atypically bad” IRV elections, we point out that advocates lauded the Burlington, San Francisco, and Ireland 1990 elections as “great successes” and has often touted them as examples to show how “wonderful” IRV is and how well it performs; IRV advocates were indeed instrumental in making the first two enact IRV. Anyhow, if should be obvious there was no such bias; the elections were chosen to try to (a) make the members of each pair similar in some important way, e.g. #candidates, (b) to try to make them historically-important, and (c) to try to chose elections where the data was available.
  2. Also, in case IRV advocates want to charge us with choosing approval elections with few voters (UN & USA) – a correct charge – we point out that the strategic incentive for approval voters to bullet-vote is greater with fewer voters, i.e. this “charge” actually only makes our argument more valid.
  3. Lest IRV advocates want to claim III’s early USA elections were not “really” approval voting, we point out that the paper identifying it as such was written by Prof. Jack H. Nagel, a member of FairVote’s (then the ‘Center for Voting & Democracy’) own “advisory committee.” And this paper was quoted from by Rob Richie, head of FairVote, who at the time tried to use it to argue that the Early-USA elections did not work out so well and hence approval voting was a poor voting system.
  4. The main differences between the early USA rules and genuine approval voting were (a) the 2-approval limit (which only helps us here by making our argument more valid) and (b) the vice-president “consolation prize.” I think (b) is a valid criticism (although a isn’t). But: Obviously, in any approval election there always is a “consolation prize” available – the honor and publicity of finishing second! This is important since it can enable that finisher’s entire future political career. Quite possibly, that importance is comparable to the importance of the vice presidency in relation to the president (“Not worth a bucket of warm piss” – USA vice president Jake Garner). So, you can judge for yourself how greatly to be impressed by (b). One should be impressed by it, but the question is how much.

The criticism which seems to me to be the most valid, is simply the fact we’ve only got four pairs, i.e. not a lot of data. Our statistical confidence, therefore, that IRV elections really have more bullet voting than approval elections, is only about 90% (as opposed to, say, 99.999%, which is the sort of number you generally get when you have lots of data). Even so, though, this is enough to cast a cloud of great skepticism over this entire “horrible bugaboo” of “bullet voting” raised by Score/Approval opponents and IRV supporters all without presenting any evidence of their own, e.g. Hertzberg & Bouricius in their essays.

So in short, had Hertzberg, Bouricius, and FairVote consulted the available evidence they would have found that their “bullet voting bugaboo” argument contradicted it. Their “argument” for IRV and against approval voting, in view of this evidence really is an argument for approval and against IRV.