### I. France-San Francisco Study

In the French approval voting study [article] (thousands of voters, 16 candidates, presidential election of 2002; probably the largest approval voting study ever), the plurality vote totaled 100% and the approval votes totaled 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). One advocacy group 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 can be determined) the largest US IRV election 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 67,590 “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 than it is with approval voting. (Indeed, in this case, hugely more common.)

(A subsequent similar study was conducted in Germany.)

### II. UN Secretary General-Burlington Comparison

Here’s a second head-to-head comparison of similar-parameter elections. The Burlington IRV Mayoral Election of 2009 (ballot set) had 6 candidates; featured 1,481 “bullet style” votes out of 8,980 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 it’s hard to 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. US-Irish Presidential Election Comparison

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.”

- 1788-9: All 69 electors each approved the maximum allowed number (2) among the 12 candidates.
- 1792: All 132 electors each approved the maximum allowed number (2) among the 5 candidates.
- 1796: 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. Even if you decided to qualify those as “bullet votes,” they would still only add up to be 9/138=6.5%.
- 1800: 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 a methods loosely resembling approval voting, then, was either 0 or 0.2% 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 Alumni-Student Comparison

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 18,186 ballots (54.9% approval). There were 32,941 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 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 the 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. Dr. Norman again:

“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 doubtful that any voter provided such an ordering nor that any voter even knew who most of the 176 even were.”

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 all combined 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 necessarily really 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.