Instant Runoff Voting: Worst Case Scenario

The following IRV election has 1,048,577 voters. The candidates are labeled using letters of the alphabet.
The winner is Z. However…
  • Z is ranked first by only 2 voters, which is one ten-thousandth of one percent of the voters.
  • Every other voter aside from those two prefers A over Z.
  • 524288 voters, or just under 50%, rank Z in last place.
  • A majority of voters prefers every other candidate (except T) over Z.
  • S is ranked first by almost 25% of the voters, and ranked second by almost 50% of the voters, and preferred to Z by nearly 75% of the voters.
 # of voters  their ranking
1  A>Z
2  Z
2  B>A>Z
4  C>B>A>Z
8  D>C>B>A>Z
16 E>D>C>B>A>Z
32 F>E>D>C>B>A>Z
64 G>F>E>D>C>B>A>Z
128 H>G>F>E>D>C>B>A>Z
256 I>H>G>F>E>D>C>B>A>Z
512 J>I>H>G>F>E>D>C>B>A>Z
1024 K>J>I>H>G>F>E>D>C>B>A>Z
2048 L>K>J>I>H>G>F>E>D>C>B>A>Z
4096 M>L>K>J>I>H>G>F>E>D>C>B>A>Z
8192 N>M>L>K>J>I>H>G>F>E>D>C>B>A>Z
16384 O>N>M>L>K>J>I>H>G>F>E>D>C>B>A>Z
32768 P>O>N>M>L>K>J>I>H>G>F>E>D>C>B>A>Z
65536 Q>P>O>N>M>L>K>J>I>H>G>F>E>D>C>B>A>Z
131072 R>Q>P>O>N>M>L>K>J>I>H>G>F>E>D>C>B>A>Z
262144 S>R>Q>P>O>N>M>L>K>J>I>H>G>F>E>D>C>B>A>Z
524288 T>S>R>Q>P>O>N>M>L>K>J>I>H>G>F>E>D>C>B>A>Z

The hypocrisy

Proponents of Instant Runoff Voting commonly criticize Score Voting, including its simplified form called Approval Voting, for potentially allowing a candidate who is the favorite of a majority of voters to lose. But when presented with the above example of paradoxical results with IRV, they typically dismiss it as “theoretically possible, but very unlikely”.

We note the irony here. So too is it unlikely that a candidate who is the favorite of a majority of voters would lose in a Score Voting or Approval Voting election.

The moral here is that worst case scenarios are not a good indicator of overall performance. Instead, we have to look at typical or “average” performance. The only truly comprehensive and objective way to do that is via Bayesian regret.