Monotonicity Criterion - Instant-runoff Voting and The Two-round System Are Not Monotonic

Instant-runoff Voting and The Two-round System Are Not Monotonic

Using an example that applies to instant-runoff voting (IRV) and to the Two-round system, it is shown, that these voting systems violate the mono-raise criterion. Suppose a president were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51.

Suppose the votes are cast as follows:

Number of votes	1st preference	2nd preference
28	Right	Center
5	Right	Left
30	Left	Center
5	Left	Right
16	Center	Left
16	Center	Right

According to the 1st preferences, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes, thus all candidates lack an absolute majority of first preferences. In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens (in this example) under IRV, Center gets eliminated, and Left wins against Right with 51 to 49 votes.

But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote 1st Left, 2nd Right, then Left would be defeated by these votes in favor of Left. Let's assume that two voters change their preferences in that way, which changes two rows of the table:

Number of votes	1st preference	2nd preference
3	Right	Left
7	Left	Right

Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and there is again no candidate with an absolute majority of first preferences. But now Right gets eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). And Center beats its opponent Left with a remarkable majority of 60 to 40 votes.