By Warren D. Smith, Oct. 2016

IRV (instant runoff) proponents unfortunately have often claimed that switching from plain plurality voting to IRV will "eliminate" the "spoiler problem."

That claim is false. (Here is a useful counterexample election for refuting many common false claims about IRV.)

Not only can spoilers still occur with IRV, in some ways they can actually be worse. For example, with IRV, an "obvious" spoiler candidate can occur, meaning one that can be proven to be a spoiler by simply examining the ballots. In contrast with plain plurality voting, spoilers always are "non-obvious," meaning you need to know information beyond that available on the ballots alone, in order to be sure he was a spoiler. For example, in a distorted and oversimplified version of USA presidential Florida 2000 race

#voters their preferences 23 Nader>Gore>Bush 28 Gore>Nader>Bush 49 Bush>Gore>Nader

both Nader and Gore are spoilers using plurality voting, but this is due to the 2nd preferences of the voters, which simply are not written on plurality ballots. Meanwhile, in an IRV-spoiler election like

#voters their preferences 23 Nader>Bush>Gore 28 Gore>Nader>Bush 49 Bush>Gore>Nader

Gore would be a spoiler (if he dropped out, Nader would win, but by running, he causes Bush, the most-hated candidate in the view of Gore voters, to win) and this is knowable from the ballots alone.

But we shall focus on a lesser claim/hope. Is it the case that with
IRV, spoiled elections become *less frequent*?

We are going to consider both IRV and plurality elections in each of the 3 different probability models defined here (all involving a number of voters V tending to infinity with C held fixed) where there are C candidates; and we shall consider both the case where C=3 and the limit C→∞.

A "spoiler" candidate shall mean a nonwinner who by dropping out of the race, would alter the winner. A "spoiled election" means an election containing at least one spoiler candidate.

In the 3-candidate case we shall show with computer aid that in all 3 models, the chance of a spoiled election is smaller with IRV than for plain plurality voting. In fact it is approximately

- 2.0000 times smaller in Quas' "1D political spectrum" model (suggesting this ratio is exactly 2)
- 2.54 times smaller in the Random Elections Model
- 4.538 times smaller in the Dirichlet model (suggesting this ratio might be exactly 59/13≈4.53846).

In the C-candidate case in the limit when C becomes large we shall outline a proof that in all 3 models, the chance of a spoiled election goes to 100% with plain plurality voting. We then shall outline an argument which is not fully rigorous (at least not at present) but nevertheless I find it convincing, that the same is true in all 3 models with instant runoff voting.

So in this limit instant runoff has no advantage (as far as reducing the chance your election has spoilers is concerned) over plain plurality – or more precisely its advantage tends to zero.

One might then enquire whether that tiny advantage, tending to zero, actually always has the "correct sign," i.e. favors IRV over plain plurality for every C=3,4,5,... in all three probability models. I do not know the answer to that question, and do not even advance a conjecture.

Consider the general 3-candidate ranked ballot election

#voters their preferences t A>B>C u A>C>B v B>A>C w B>C>A y C>A>B z C>B>A

where t,u,v,w,y,z all are nonnegative not all zero. An election is said to be "spoiled" if it contains at least one spoiler. A "spoiler" is a nonwinner who, by dropping out, changes the winner.

Warning: with IRV we can have "nonmonotonicity" where some voters can cause X to win by voting dishonestly against X. This perhaps could be viewed as an additional kind of "spoiler" (X "spoils himself"!) but for simplicity we will not take that view here. Also with IRV X can by losing some but not all votes, sometimes change the winner, e.g. to himself(!), but we for simplicity will not count those as spoilers either.

Wlog the IRV winner is A, and C is eliminated first, meaning y+z<v+w, y+z<t+u, y+t+u>v+w+z. Then an IRV spoiler (who necessarily is B) occurs if and only if w+y+z>t+u+v. Meanwhile in this election, the plain plurality winner must be either A or B. We have that

- A wins and C=spoiler iff: t+u>v+w and z+v+w>t+u+y
- A wins and B=spoiler iff: t+u>v+w and w+y+z>t+u+v
- B wins and A=spoiler iff: t+u<v+w and u+y+z>t+v+w
- B wins and C=spoiler iff: t+u<v+w and y+t+u>v+w+z.

So... trying it on a computer (the three probability models described here ) four independent runs each model

Election Prob. Model | #elections tried | #spoiled IRV | #spoiled Plur | ratio |
---|---|---|---|---|

Dirichlet | 10^{10} | 902778540 | 4097213715 | 4.53845 |

Dirichlet | 10^{9} | 90282798 | 409733353 | 4.53833 |

Dirichlet | 10^{9} | 90266216 | 409694306 | 4.53873 |

Dirichlet | 10^{9} | 90276472 | 409739388 | 4.53872 |

R.E.M. | 10^{8} | 12155926 | 30918546 | 2.54350 |

R.E.M. | 10^{7} | 1217051 | 3092980 | 2.54137 |

R.E.M. | 10^{7} | 1217752 | 3093316 | 2.54019 |

R.E.M. | 10^{7} | 1217786 | 3092975 | 2.53983 |

Quas1D | 10^{10} | 1944426007 | 3888899123 | 2.000024 |

Quas1D | 10^{9} | 194454013 | 388905437 | 1.999987 |

Quas1D | 10^{9} | 194436146 | 388893244 | 2.000108 |

Quas1D | 10^{9} | 194453186 | 388909920 | 2.000018 |

**Conclusion:**
*Yes*, IRV does reduce "spoiled" 3-candidate elections versus plain
plurality, making spoiled elections 2.00 to 4.54 times rarer
depending on which of these three probability models we use.

If I had included nonmonotone "self spoilers" and partial spoilers too, then 3-candidate IRV would look a little (perhaps 10%) worse, not enough to change the conclusion that 3-candidate IRV does improve over plain plurality about spoilers. (The reason this has little effect is in the large majority of instances where these problems happen, garden-variety spoilage also happens, so that election was already counted.)

Warning: we shall use "extreme value statistics" without much explanation or care. However, I claim the amount of care & precision I use, is sufficient to see the results, at least for readers with enough mathematical and statistical maturity so that I do not need to explain a lot of basic stuff. The 3 probability models are described here.

The results of the present analysis are: With plurality voting, in Random Elections Model, Dirichlet, and Quas1D models (all three) the chance a spoiler exists goes to 100%.

Now for the analyses.

In the random elections model, the total number of votes for each candidate is (or rather can equivalently be regarded as, after appropriate rescale and translate) i.i.d. standard normal deviates:

Cumulative(x) = [1+erf(x/√2)]/2 ∼ 1 - exp(-x

Hence the candidate with the most votes among C candidates is almost surely going to get about √(2lnC) votes, and the runner-up is almost surely going to get about √(2ln(C-1)) with the difference being about 1/(C√(2lnC)). Meanwhile, the second-preference vote-totals among the fraction≈1/C voters for any particular candidate S (who is not either the winner or runner-up) are going to be normally distributed with variance=1/C not 1. The difference between the two frontrunners' vote-totals among this fraction is thus normally distributed with variance=2/C. The chance S is a spoiler is the chance a normal random variable with variance=2/C i.e. StdDeviation=√(2/C) is going to exceed 1/(C√(2lnC)). That is the same as the chance a standard normal (with unit variance) is going to exceed 1/(2√(ClnC)). But this chance in the limit C→+∞ is 1/2. So each non-frontrunner candidate is a spoiler with chance→50% in the limit, so the likelihood a spoiler exists should go to 100%.

This same reasoning also works in the alternative "Dirichlet model" of random elections (since central limit theorem can be used to see normalities for appropriate vote-total quantities).

Now in the Quas1D model, the number of votes a candidate gets is the area of his 1D Voronoi region on the real interval [0,1]. These areas have mean about 1/C but are distributed (in the limit) like the sum of two i.i.d. exponential deviates. (This is a consequence of limit Poisson statistics.) Without loss of generality (after scaling vote-counts by C) we may regard these as the sum of two standard exponential deviates, i.e. having

Cumulative(x) = 1 - exp(-x) (1+x)

The winning candidate will almost surely have about ln(C) votes, and runner-up ln(C-1), the difference being about 1/C. If a candidate neighboring the runner-up is eliminated, then the runner-up's Voronoi region will expand, i.e. he will get more votes, in fact typically (on this scale) order 1 more votes. Hence the chance→100% that each neighbor of the runner-up will be a spoiler.

Q.E.D.

You may ask how quickly the chance of a spoiled plurality election
goes to 100% as C→∞.
That more precise answer seems to be roughly 1-2^{2-C}
for the R.E.M. and Dirichlet models, and a lower bound of about 1-k/C for some
positive constant k, for the Quas1D model (although it should be possible to
improve that lower bound a lot by also considering the 3rd-placer, the 4th-placer, etc
as potential new-winners, not just the runner-up).

In this section I will attempt to examine the frequency of spoilers in IRV elections in the limit which the number C of candidates becomes large (and the number of voters is assumed to tend to infinity much faster). I will use a "mass deletion" trick which looks valid but I have not proven that, so you might want to regard this as an imperfectly-rigorous, but still pretty convincing, proof. The 3 probability models are described here.

The results of the present analysis are: With instant runoff voting (IRV), in Random Elections Model, Dirichlet, and Quas1D models (all three) the chance a spoiler exists goes to 100% in the limit where the number C of candidates becomes large.

Now for the analyses.

The "mass deletion trick" is this. Let there be C candidates
with C large. Delete all but √C of them at random.
[You may prefer to replace √C by, e.g, C^{0.49}
if that makes it easier to rigorize the argument.]
Then claim that the winner of the resulting much-smaller IRV election
is essentially random among those √C candidates.
More precisely, claim that if those √C included the IRV-winner of
the original
(all C candidates) IRV election, then the chance he still wins, goes to 0
as C→∞.

This claim is supposed to be pretty obvious in all three models – so I'm
not going to prove it here, I'm just going to claim/hope it is pretty
obvious to those
familiar with behaviors... e.g. consider
the Quas distribution,
the model-independent "100% ignoring" theorem for IRV,
and the
fact
that asymptotically 100% of elections contain no Condorcet winner and
the IRV winner differs from the Condorcet winner (even when the latter exists)
asymptotically 100% of the time
...
and then *assume* it as the starting point of our
main argument.

Now this claim does not directly prove the existence of a single "spoiler"
candidate; it merely proves there exists a *large set* of candidates
whose collective
deletion changes the winner, and indeed claims in our limit that almost every
candidate-subset of cardinality C-√C (which does not
include the IRV-winner of the original election) is such a "collective spoiler."

But what this tells us is: as we delete those candidates one by one, there will come a time (maybe many times) when the winner changes. In that particular reduced election, that candidate was a (genuine) spoiler. The total number of (reduced and partly reduced) elections we are talking about is

which sum is dominated by the terms with M≈C/2.
The other terms, even combined, are relatively
negligible in the limit C→∞.
And the total number of genuine "spoiler events", meaning a time when
a single candidate is deleted from one of those elections to change its winner,
in aggregate over all those elections, is at least 1-o(1) per "deletion path."
But note the total number of "deletion *paths*" in "election space"
is far greater than the number of elections themselves, indeed
in each election with K candidates we have K-1 choices about who to delete next
so the total number of paths is C!/(√C)!.
Indeed, this number of paths also in our limit,
is far greater than the *square* (or any fixed power)
of the number of elections themselves
(the number of elections is less than 2^{C}).
And each path is short (only C-√C long).

But now comes the trick: all these reduced and intermediate
elections actually are statistically
speaking the *same* as the original election
(maybe with different but always-large C, but always sampled from the exactly correct
distribution for that C; this works in all 3 probability models)!
Therefore the chance an IRV spoiler-event is available in an
election randomly chosen from our set of elections must go to 100%.

This reasoning actually works even *better* if, say, the spoilers
tended to occur nonuniformly mainly at the high-#candidates end (or
prefer to occur mainly near the low-#candidates end) of the paths;
then we'd conclude the elections near one end of our set had to
include spoilers with probability→1, which would suffice for
our purposes.

Therefore, the chance should go to 100% in all three models that a C-candidate IRV election contains an IRV-spoiler.