Compromise Criterion for Rank-Order systems
I will define the compromise criterion for rank-order systems as follows:
If the ballot set consists only of
which are perfect reversals of each other, then a system satisfies the compromise criterion if and only if it elects either C((N+1)/2) if N is odd, or one of C(N/2) or C(N/2+1) if N is even.
For example, IRV does not satisfy the compromise criterion: if the ballots are
then the compromise candidate y is immediately eliminated.
Bucklin voting does satisfy the compromise criterion. Borda count does not, but concave score systems do. Plurality fails. Generally Condorcet methods are agnostic on the compromise criterion, and whether they satisfy it or not depends on how situations without Condorcet winners are handled. For example, Black's system (Condorcet//Borda) fails. Approval voting “conditionally" passes or fails depending on “approval thresholds” (technically the criterion does not apply, since it is defined for rank order systems).
For three candidates we can generalize and strengthen the compromise criterion:
For every ballot set of that is a multiple of
for any P in [0,1], the winner is y. In this case, a weak Condorcet method that prioritizes candidates with positive margins of victory over ties and that satisfies the weak compromise criterion will also satisfy this stronger compromise criterion.