Consensual Condorcet by Positional Domination

cfrank

I wasn't sure what else to call this, I already described this method as an aside in Condorcet with Borda Runoff, but unlike that method this one seems to be resistant to tactical nomination.

As described, one defines a sequence of "Nth-order Condorcet winners" (assuming they exist in all relevant cases and deferring to another reasonable method otherwise) as being the Condorcet winner when all lower-order Condorcet winners are removed from the election. Then, the winner is the lowest-order Condorcet winner who has a successor (the Condorcet winner of next highest order) and who positionally dominates that successor.

Just to be clear, what I mean by a candidate A "positionally dominating" a candidate B in this case is that for every rank, there are at least as many voters who placed A at or above that rank as voters who did the same for B. I think that term is used elsewhere to possibly mean something different, correct me if I'm wrong.

The problem with this method is that there may not be an Nth-order Condorcet winner (or substitute) who positionally dominates their successor. In that case, I am suggesting to iteratively ignore the extremal ranks when considering positional dominance until a winner is found.

If this method is indeterminate, some reasonable alternative method will be needed to choose the winner.

Here are a few examples I worked out:
First, if the ballots are

C>B>D>A [38%]
A>B>C>D [30%]
A>C>D>B [22%]
B>D>A>C [10%]

Then the primary, secondary, tertiary and quaternary Condorcet winners are A, C, B and D in that order. B is the lowest-order Condorcet winner who positionally dominates their successor, so is elected. One informal interpretation is that candidates A and C were too divisive, and B was identified as a good compromise for a broad supermajority of the electorate (notice that 78% have B ranked in the top two positions).

As a second example, let the ballots be as follows:

A>A'>A''>B>C [40%]
A>A''>A'>B>C [11%]
C>B>A''>A'>A [30%]
B>C>A>A'>A'' [10%]
C>B>A'>A>A'' [9%]

The candidates labeled with A are meant to be candidates strategically nominated to crowd out competition against a common platform. Reasonably, the Nth-order Condorcet winners/substitutes in sequence would be A, A', A'', B, C. None of these candidates positionally dominate their successor over all ranks. However, once the most extremal ranks are ignored, B satisfies the criterion and is elected. The strategic nomination tactic failed. An informal interpretation is that this method was in a sense able to navigate the political spectrum and find a relative middle ground candidate. If B were removed from the ballots with all else kept equal, candidate A'' would have been elected, which reflects the majority platform.

As a third example, we can have a less divisive election, such as

C>A>B>D [21%]
A>B>C>D [39%]
A>C>D>B [30%]
B>D>A>C [10%]

Clearly the Condorcet winner is A, and the secondary Condorcet winner is C. Candidate A also positionally dominates C, so is elected. An informal interpretation is that the Condorcet winner was not too divisive and won the election with broad consensual support.