## How often is Instant Runoff (IRV) worse than plain plurality voting?

By Warren D. Smith, August 2015

In general, one intuitively would expect the IRV winner to be better than the plain plurality winner (when they differ) more often than the reverse. This is since all rounds of the IRV process after the first, use additional information from the voters, beyond that perceived or perceivable by the plurality process; and use it in a way that seems likely to "improve" things.

However, as we shall show by example, the plain plurality winner can be superior to the IRV winner, and indeed sometimes in a very clear way. We then enquire how often those phenomena happen.

We will not fully answer that question. But we shall be able to make it pretty clear that the IRV and plurality winners become equally likely to be the better in the limit of a large number C of candidates. In other words, IRV's advantage diminishes as C increases, ultimately to arbitrarily small values. This "no advantage in limit" conclusion holds in a very wide class of settings. Meanwhile the 3-candidate situation is pretty much fully understood and can simply be looked up in our tables. The C=3 and C→∞ results then are

#CandidatesProbability Model IRV winner betterPlur winner better Winners same
C=3REM22.1884%2.2777%75.5339%
C=3Dirichlet10.7637%1.5625%87.6737%
C=3Quas 1D25.0000%0.0000%75.0000%
C→∞Any of above50.0000%50.0000%  0.0000%

We also argue that our 5-candidate example structure occurs often enough to show that the Plurality winner is not merely better, but actually clearly visibly better, than the IRV winner, at least some fraction F of the time, where in "1-dimensional politics" with 5 candidates F is somewhere between about 0.38% and 1.25%.

### The 5-candidate example

The following election based on one by Brian W. Goldman, illustrates how it is possible for IRV to produce clearly worse election results than plain plurality voting, in an election with the same ballots for both systems, in a hopefully-realistic scenario arising from "one dimensional politics." The example has 5 candidates named C,L,R,F,G (Centrist, Leftist, Rightist, FarLeft, FarRight respectively along a "one dimensional spectrum"), and 14 voters:

#votersTheir Vote
4C > L > R > F > G
3L > C > R > F > G
3R > C > L > G > F
2F > L > C > R > G
2G > R > C > L > F

With plain plurality, the results are

C=4, L=3, R=3, F=2, G=2,

meaning C wins.

C is also the "beats-all (Condorcet) winner" because: C>L on 4+3+2=9 versus 3+2=5 ballots, C similarly beats R by 9:5, and C beats F by the even huger margin 12:2, and finally C beats G similarly by 12:2.

C also would be the score voting winner with any scores S1≥S2≥S3≥S4≥S5, (with S1,S2,S3 not all equal), being given by each voter to her 1st, 2nd, ..., 5th choice. Indeed, the final finish order with score voting under any such circumstance would always be "C>L>R>F>G," where the letters here denote that candidate's summed-score. You can see all that after first realizing that the summed scores for each candidate would be given by

 C = 4S1+6S2+4S3 L = 3S1+6S2+3S3+2S4 R = 3S1+2S2+7S3+2S4 F = 2S1                 +7S4+5S5 G = 2S1                 +3S4+9S5

So we contend it is pretty clear that in this election C is the "best" winner, in addition to being the winner under plain plurality voting.

However, in this election C is not the instant runoff voting (IRV) winner, but rather finishes third! The IRV election proceeds thusly:

Round #1:   C(4) - F(2) - G(2) - L(3) - R(3);   eliminate G
Round #2:   C(4) - F(2) - G(X) - L(3) - R(5);   eliminate F
Round #3:   C(4) - F(X) - G(X) - L(5) - R(5);   eliminate C
Round #4:   C(X) - F(X) - G(X) - L(9) - R(5);   eliminate R
Winner:   L

If IRV breaks the tie between F and G the other way in round 1 (i.e. eliminating F first) that does not matter; the top three according to instant runoff voting still are L,R,C in that order.

So plainly in our example election the IRV winner is worse than the winner under plain plurality voting, with both voting systems seeing the same ballot set.

### How often that happens

The event that the IRV winner is "worse" than the plurality winner is somewhat undefined, in general. We have designed the above example to make it very clear its IRV winner is worse, but there are also many election situations in which the IRV (or plurality) winner is worse, but that is not clear to observers.

We now point out that in our example election if we were to change the top-rank vote counts C=4, L=3, R=3, F=2, G=2 in the first column of the defining table at the start to any other positive numbers C,L,R,F,G obeying:

1. The numbers are distinct and ordered (from most to least) according to one of these 6 orderings: CLRFG, CRLFG, CLRGF, CRLGF, CLFRG, CRGLF,
2. F+L>C,
3. G+R>C.

then the same thing will happen: the plain-plurality winner will be C, who also will defeat every rival pairwise, but the IRV winner will not be C (he instead will either be L or R) and indeed C necessarily will finish in third place or worse under IRV. (Also, if we only allowed the first 4 of those 6 orderings, then in addition C will be the winner under score voting with any scores S1>S2>S3>S4>S5, being given by each voter to her 1st, 2nd, ..., 5th choice. Call this strengthened condition "i.b" to distinguish it from the weaker condition "i.")

What are the chances that our two necessary conditions i and ii will both be satisfied?

1. If the 5 numbers C,L,R,F,G are chosen independently from any probability density then condition i will be satisfied with chance 6/120=1/20, but the stronger condition i.b only with chance 4/120=1/30. (Note 5!=120.)
2. If F and L were selected independently uniformly from the real interval [0,C] and similarly for G and R, then the chance condition ii would be satisfied would be exactly 1/2, and similarly 1/2 for condition iii (and 1/4 for both).
3. If instead C,L,R,F,G were identically exponentially distributed (e.g. we chose 3 independent samples from the exponential density e-x on x>0, then denoted the maximum by C and the other two by L, F) then the chance condition ii would be satisfied would be (e-1)-2≈0.3386969 and similarly for condition iii, with the chance of both ii and iii being satisfied being (e-1)-4≈0.1147156.

### Conclusion (1D politics, clear "IRV worse" scenarios with 5 candidates)

This all somewhat vaguely and approximately suggests that the frequency of 5-candidate elections, in "1-dimensional politics" situations, in which the IRV winner is worse than the plurality winner (same ballot set for both voting systems), and worse in a very clearly observable way, ought to be lower bounded by some number somewhere between about (e-1)-4(1/30)≈0.38% and (1/4)(1/20)=1/80=1.25%.

### I. 3-Candidate scenarios

Meanwhile, in 3-candidate elections, the IRV winner is never "clearly worse" than the plain plurality winner (if the same ballot set is used for both voting processes). These two winners differ, according to these tables, with chances

24.4661%,   12.3263%,   25.0000%

in the three different probability models defined there; and whenever they differ the IRV winner usually arguably is "better" than the plain plurality winner. Since the IRV winner defeats the plurality winner pairwise in the final IRV round, the only way to try to argue the plurality winner is better is in situations with a Condorcet cycle. In those same three probability models, the chances that Condorcet cycles exist (given that the IRV & plurality winners differ) are respectively

18.6193%,   25.3526%,   0.0000%.

So if we assume the IRV and plurality winners are equally likely to be better in 3-candidate situations where (a) they differ and (b) there is a condorcet cycle, we get the results summarized at the start of this page.

### II. C-candidate Scenarios in the limit C→∞

See puzzle 34 here, to realize that in any limit where #candidates=C→∞ and #voters=V→∞ with V0.51≤C≤V:

1. IRV never examines a more than a fraction of the number CV of "cell entries" on the V ballots, that goes to zero;
2. IRV fails to elect Condorcet winners (even when they exist) in a fraction of elections that goes to 100%.

Those arguments also work (more simply too) for plurality winners – the plain plurality voting system never examines more than V of the CV cell entries, i.e. a fraction 1/C which goes to zero (since it only examines each voter's top choice, ignoring all others); and due to this huge-ignoring, plurality also will virtually always guess wrong about who is the Condorcet winner.

We will now argue that the IRV winner is better than the plurality winner (when they differ) slightly more than half the time (albeit not necessarily "clearly visibly better," just "better") which in our limit-scenario goes to exactly half.

To explain my (somewhat nonrigorous given the lack of a definition, but I still like it) reasoning behind that:

IRV ignores fewer ballot data than plurality but both ignore asymptotically 100% of the data on the ballots as was shown in puzzle 34A. Hence the reason the plurality winner is better than the IRV winner (in cases when it is), or vice versa, generally arises from the parts of the ballots that both IRV and plurality ignore. Since these ignored parts are asymptotically 100%, we can expect the question "who is better -- IRV winner or plurality winner?" to have answer that simply cannot be determined from the few ballot cells that the IRV and plain plurality processes examine, and which simply cannot even be guessed with correctness probability above K, if K>50%. The answer to the question of which winner is "better" is effectively dominated by random noise in a random election (i.e. regarding the parts of the ballots IRV and plurality ignore as random and unknown, while the parts that are not ignored are solid and known). Therefore answer frequency should be near 50-50, with a bias in favor of IRV which should diminish toward 0 in the limit.

This claim should hold in any setting in which the question of "who is better, X or Y?" cannot be guessed by examining only a vanishingly small fraction of the VC data cells on the V ballots – or at least, in which any such guess can achieve correctness fraction tending at best to 50% in our limit.

The reason IRV and plurality both become nearly useless and approximately equivalent in quality in the large #candidates limit, is the fact they both always ignore a fraction →100% of the input vote (full ranking) data. Meanwhile, voting methods such as Condorcet and Score voting ignore nothing.