Can Instant Runoff (IRV) be worse than plain plurality voting?

By Brian W. Goldman (PhD in computer science from Michigan State University, East Lansing)

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. Here is an example election with 5 candidates, 24 voters, and voting compatible with a "1-dimensional political spectrum":

#votersTheir Vote
7Centrist > Left > Right > FarLeft > FarRight
6Left > Centrist > Right > FarLeft > FarRight
6Right > Centrist > Left > FarRight > FarLeft
2FarLeft > Left > Centrist > Right > FarRight
3FarRight > Right > Centrist > Left > FarLeft

With plain plurality, the results are

Centrist=7, Left=6, Right=6, Far Left=2, Far Right=3,

meaning Centrist wins.

Centrist is also the "beats-all (Condorcet) winner" as Centrist>Left on 7+6+2=15 versus 6+3=9 ballots, Centrist>Right on 7+6+2=15 versus 6+3=9 ballots, and the "Far" candidates are only better than Centrist on 2 and 3 ballots.

However, in this election Centrist is not the IRV winner. FarLeft has the fewest first-round top-rankings and hence is eliminated, giving us:

#votersTheir Vote
7Centrist > Left > Right > FarRight
8Left > Centrist > Right > FarRight
6Right > Centrist > Left > FarRight
3FarRight > Right > Centrist > Left

for round 2; then FarRight has the fewest and is eliminated to get

#votersTheir Vote
7Centrist > Left > Right
8Left > Centrist > Right
9Right > Centrist > Left

for round 3; now Centrist is eliminated to yield in the 4th and final round

#votersTheir Vote
15 Left > Right
9Right > Left

so IRV elects Left even though the voters preferred Centrist>Left by 15:9, and Centrist also won with plain plurality.

I also have constructed such examples with only 4 candidates (here's one; see also this), but they did not seem as realistic as this 5-candidate example.

How often does this happen?

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