This method fails the monotonicity criterion. This disadvantage is somewhat technical in nature, and it's hard to prove that nonmonotonic behavior occurred if you have two rounds rather than an ordinal ballot, but it is a weird and arguably undesirable property.
Intuitively, this means that rating a candidate higher can cause that candidate to lose.
A voting method satisfies the monotonicity if the following property holds. In order to define it, I need a few other definitions first. Note that we can define this property without talking about agents/rational actors at all; it can be defined using just collections of ballots.
In order to define this property, I'll introduce the nonstandard notion of X-superiority.
Let A and B be ballots. Let X be a candidate. A is X-superior to B if and only if, for all pairs of distinct candidates (Y, Z), A and B have the same ranking of Y, Z when neither Y nor Z is X, and for any pairwise comparison involving X, A ranks X higher than B does or A and B have the same ranking.
Following Wikipedia's example, X > Z > Y is X-superior to Z > X > Y, but X > Y > Z is NOT X-superior to Z > X > Y because the order of Y and Z is reversed.
Let A and B be I-indexed sets of ballots. A is X-superior to B if and only if, for all i in I, it holds that A[i] is X-superior to B[i].
A voting method fails the monotonicity criterion if and only if, there exist I-indexed sets of ballots C and D such that the candidate X wins in C, the candidate X does not win in D and D is X-superior to C.
Here's the nonmonotonicity example from Wikipedia with an explanation.
C > B > A 28
C > A > B 5
A > B > C 30
A > C > B 5
B > A > C 16
B > C > A 16
In this election, B
is eliminated first and A wins in the next round.
Next consider the election below
C > B > A 28
C > A > B 3
A > B > C 30
A > C > B 7
B > A > C 16
B > C > A 16
Now, C
is eliminated first and B
subsequently wins.
Two C > A > B
were shifted to A > C > B
ballots in this example, increasing candidate A's votes, and that caused candidate A to lose.