Everyone knows vote splitting exists. Nobody seems to have measured how often it changes the answer in an ordinary group decision — a team lunch, a feature list, a meeting time. So we simulated it.
With three options, single-choice voting is mostly fine — the two methods disagreed 12.7% of the time. From four options upward it degrades quickly, and by five the winner almost never holds a real majority.
The sharpest finding is arithmetic rather than statistical: if a group who prefer similar things splits between two of them, they lose under single-choice voting unless they hold more than two-thirds of the vote. Below that threshold, the win goes to an option most of the group did not want.
We ran 20,000 simulated votes at each option count with 25 voters and independently placed options, then compared the plurality winner against the instant-runoff winner.
The second chart is the more troubling one. At five options, the leading choice held less than half the first preferences in 84.3% of votes. At eight options, 98.3%. In other words, when you put eight things to a single-choice vote, the winner almost certainly represents a minority of the room.
| Options | Different winner | No majority | IRV winner beats plurality head-to-head | Avg winner share | Avg rounds |
|---|---|---|---|---|---|
| 3 | 12.7% | 37.6% | 12.7% | 55.6% | 1.38 |
| 4 | 21.8% | 67.8% | 21.5% | 46.5% | 2.25 |
| 5 | 28.7% | 84.3% | 27.9% | 40.4% | 3.34 |
| 6 | 35.2% | 93.1% | 33.8% | 35.8% | 4.51 |
| 7 | 40.0% | 96.6% | 38.2% | 32.5% | 5.63 |
| 8 | 43.2% | 98.3% | 40.8% | 29.9% | 6.71 |
The fourth column is the strongest test. It is not merely that the methods disagreed — it is that in those cases, the ranked-choice winner would have beaten the single-choice winner in a direct head-to-head vote. At eight options that happened in 40.8% of all votes.
The classic vote-splitting worry is that two similar options divide their supporters and hand victory to a third. We isolated that scenario: a bloc of voters who prefer either of two similar options, against one distinct alternative.
| Bloc share | Single-choice picks the minority option | Ranked choice picks the bloc's option |
|---|---|---|
| 52% | 100.0% | 100.0% |
| 55% | 100.0% | 100.0% |
| 60% | 100.0% | 100.0% |
| 65% | 40.9% | 100.0% |
| 70% | 0.0% | 100.0% |
At 52%, 55%, and 60% the single-choice vote handed the win to the minority-preferred option every time. At 65% it happened in 41% of runs. At 70% it never happened.
The threshold is exact and can be derived without simulation. A bloc holding share s, splitting evenly between two options, gives each s/2. The distinct alternative holds 1−s. The bloc loses when s/2 < 1−s, which resolves to s < 2/3.
If your team has two similar options on the list — two Italian restaurants, two variants of the same feature, two adjacent time slots — and the people who prefer that general direction are fewer than two-thirds of the group, a single-choice vote will systematically pick something else.
We expected small groups to be less affected, since fewer voters means less fragmentation. They were not.
| Voters | Different winner | No majority |
|---|---|---|
| 6 | 28.4% | 78.4% |
| 10 | 28.0% | 82.4% |
| 15 | 27.8% | 78.4% |
| 25 | 28.8% | 84.7% |
| 50 | 29.5% | 90.4% |
| 100 | 30.6% | 91.6% |
Across groups from 6 to 100 with five options, disagreement stayed between 27.8% and 30.6%. A six-person team choosing between five options faces essentially the same risk as a hundred-person department. Group size is not the variable that matters — the number of options is.
A Condorcet winner is an option that beats every other option in a direct one-against-one comparison. Where one exists, it is a reasonable definition of the "right" answer. We measured how often each method found it.
| Options | Condorcet winner exists | Single-choice found it | Ranked choice found it |
|---|---|---|---|
| 3 | 99.3% | 85.1% | 96.3% |
| 4 | 98.4% | 73.4% | 90.6% |
| 5 | 97.3% | 64.1% | 84.5% |
| 6 | 96.4% | 55.4% | 77.9% |
| 7 | 95.1% | 49.2% | 72.8% |
| 8 | 94.3% | 44.3% | 67.7% |
Ranked choice was closer in every condition. With five options it found the head-to-head winner 84.5% of the time against single-choice's 64.1%. With eight, the gap widened to 67.7% against 44.3% — meaning a single-choice vote with eight options found the head-to-head winner less than half the time.
Neither method is perfect, and no method can be. Arrow's impossibility theorem rules out a ranking system satisfying every fairness criterion simultaneously. The question is not which is flawless but which is closer, and in these conditions ranked choice was closer throughout.
Voters and options were placed in a two-dimensional latent preference space drawn from a standard normal distribution. Each voter ranked options by Euclidean distance from their own position, with Gaussian noise of σ = 0.10 added per voter-option pair to represent individual idiosyncrasy.
This is the spatial model standard in social choice research. It produces correlated preferences — voters who favour one option tend to favour nearby ones — which uniform random rankings do not. Using random rankings would have inflated the disagreement figures considerably.
Plurality: the option with the most first preferences. Ties broken at random.
Instant-runoff: count first preferences among continuing options; if one exceeds 50% of continuing ballots it wins; otherwise eliminate the lowest and redistribute to each ballot's next continuing preference. Repeat. Ties for elimination broken at random.
Condorcet: the option beating every other in pairwise comparison, computed from a full preference matrix. Reported as absent when no such option exists.
Before generating any results, the instant-runoff implementation was tested against a worked example with a known outcome — a four-option vote with first preferences of 35 / 28 / 22 / 15 — and reproduced the published elimination sequence and winner exactly.
18 primary conditions × 20,000 trials = 360,000 simulated votes, plus 15 vote-splitting conditions × 20,000 = 300,000. Total 660,000. Seeded at 20260731 for reproducibility.
Two or three genuinely distinct options: single-choice is fine. Disagreement was 12.7% at three options and the extra effort of ranking is not worth it.
Four or more options: disagreement roughly doubles at four and the majority rate collapses. This is where ranking starts to earn its cost.
Any two options that are similar to each other: the two-thirds rule applies regardless of how many options there are in total. If the people who want that general direction are fewer than two-thirds of the group, single-choice will systematically choose against them.
Any decision where the outcome needs legitimacy: a winner holding 30% of first preferences is difficult to defend as the group's choice. If people have to live with the result, the margin matters as much as the answer.
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