## Example of severe IRV (Instant Runoff Voting) misbehavior

By Stephen Unger (emeritus professor, Columbia University, NY)

Previous dramatic IRV pathologies have included the following:

1. The IRV winner is top-ranked by exactly 2 voters (out of 2k+1 for any k>1; there are k+1 candidates running). For every other candidate, with one exception, a large supermajority of voters prefers that candidate over the winner. The IRV winner is preferred over exactly one other candidate, and this by a 1-vote margin.
E.g. see http://www.electology.org/irv-worst-case-scenario.
2. The IRV winner would still be the winner if every voter's expressed preferences were reversed!
E.g. see http://rangevoting.org/IrvRevFail.html.
3. And here's a nasty combination of many other IRV pathologies all in a single simple election example:
http://rangevoting.org/CompleteIdioticIRV.html.

The contribution of this page is to show how both pathologies of types (1) and (2) can happen simultaneously in a single election example with k+1 candidates and 2k+1 voters. The simplest example, with k=2, is shown below, where A wins with either the IRV votes shown, or if all votes are reversed. There are 5=22+1 voters and 3=2+1 candidates (named "A," "B," and "C"). Actually, we scaled up the number of voters to 5000 just to make the point that the problems here have nothing to do (as one nationally prominent IRV-proponent claimed!) with the "unrealistically small number of voters," since that can be arbitrarily scaled. (Similarly one can perturb the numbers, e.g. changing 2000 to 2137, and the pathologies will still happen.) Nevertheless this is not a very interesting example.

#voters their vote
2000 C > B > A
1000 B > A > C
2000 A > C > B

Two examples are shown below where the numbers are large enough to be interesting, and to demonstrate how to extend them to arbitrarily higher numbers. In all cases, A is the winner.

For k=6   (7 candidates, 65 voters. The X's represent "don't care" candidates, i.e. all those not shown explicitly in the row. These can be in any order – it does not matter which, since the IRV procedure never examines them.)

#voters their vote
32 G>F>E>D>C>B>A
16 F>E>D>C>B>A>G
8 E>D>C>B>A>G>F
4 D>C>B>A>G>F>E
2 C>B>A>X>X>F>D
1 B>A>X>X>F>D>C
2 A>X>X>X>X>F>B

What happens: Starting with IRV round #1, candidates B, C, D, E, F, and G are eliminated in that order, leaving A the winner over G by 33-32 in the final round. But in any two-candidate A-versus-somebody IRV election with the preferences shown above, voters would prefer anybody except G over the winner A – indeed always by large majorities, for example D is preferred over A by 60-to-5. (A beats G by a single vote.)

For the reverse election (whose purpose is to determine the worst candidate), reverse all the ">" signs. C is eliminated in IRV round #1, then B, D, and E (as the votes for F accumulate). Then G is eliminated, making A the "winner" by 48-17 over F. (According to IRV, the "best" candidate is the same as the "worst" candidate, refuting claims IRV "elects majority winners" etc.) In all cases, elimination starts with the next-to-bottom line, then proceeds to the bottom line, and then up to the 3rd line from the bottom and up from there a line at a time until we skip over the 3rd line from the top to eliminate the second row and terminate. This should make it clear how larger examples can be constructed.

Now here is the k=7 case, with 129 voters:

#voters their vote
64 H>G>F>E>D>C>B>A
32 G>F>E>D>C>B>A>H
16 F>E>D>C>B>A>H>G
8 E>D>C>B>A>H>G>F
4 D>C>B>A>X>X>G>E
2 C>B>A>X>X>X>G>D
1 B>A>X>X>X>G>D>C
2 A>X>X>X>X>X>G>B

Every candidate except H would have defeated IRV winner A by a large margin in a 2-candidate election. For example B would have defeated A by 127-2. A's only pairwise victory would be a 65-64 win over H.

For determining the biggest loser, the candidates eliminated are, in order, C, B, D, E, F, H before A "defeats" G by 96-33.

Pathologies such as these (many of which, simulations show, can occur with probabilities far above microscopic) should be enough to persuade any reasonable person that IRV is off the wall and so should be off the table. And this is apart from the need for central tabulation, which in my opinion by itself should make IRV unacceptable because it destroys election transparency, thus helping fraudsters.