[EM] Improvement to Ranked Robin method

[Kristofer Munsterhjelm]

On 05.04.2022 02:23, Richard, the VoteFair guy wrote:
> On 4/4/2022 3:49 AM, Kristofer Munsterhjelm wrote:
>> Could you give an example without truncation or equal rank, where your
>> count elects a different candidate than the Borda count restricted to
>> the same candidates?
> 
> The big weakness of the Borda count is that many voters choose to mark
> their ballot using "equal ranks" (multiple candidates ranked at the same
> level) and truncated ballots.
> 
> If, somehow, all voters can be forced to rank each candidate at a
> different rank, and assuming the number of ranks equals the number of
> candidates, then the Borda count produces very nice results.

I wouldn't say that it does. Consider this simple example with complete
ballots:

66: A>B
34: B>A

A wins by the majority criterion and has got a 66% supermajority. Now
clone B into B1 and B2, and by convention say the positional score
values are n-1 for first place and 0 for last:

66: A>B1>B2
34: B1>B2>A

A obtains 2*66 = 132 points. B1 obtains 1*66 + 2*34 = 134 points and
wins. An almost two thirds supermajority is wiped out by cloning, which
I wouldn't really call a nice outcome.

For a more general result, some algebra then gives that k clones can
overturn a supermajority ever so slightly smaller than k/(k+1), e.g.

74: A>B1>B2>B3
26: B1>B2>B3>A

B1 obtains 226 points to A's 222 points and wins.

This also suggests that if equal rank existed, it wouldn't be used by
tactical voters because there's more power to be had by ranking the
viable opposition last (e.g. the A-voters responding by voting
A>B3>B2>B1 above).

> In other words, your constraints rule out the very reason I'm suggesting
> the switch from Borda count to pairwise support counts (for the Ranked
> Robin method).
> 
> If you want evidence that pairwise support/opposition counts are better
> than the Borda count (without marking constraints), then here's a link
> to the scatter plot that shows Instant Pairwise Elimination (IPE) and
> Borda count having significantly different failure rates for CI (clone
> independence) and IIA (independence of irrelevant alternatives):
> 
> https://www.rankedchoiceoregon.org/img/clone_iia_success_rates.jpg
> 
> In a general way IPE is an upside-down version of the improved version
> of Ranked Robin that I'm suggesting.

I don't think IPE is really comparable to Borda, though. According to
its Electowiki page, IPE proceeds by eliminating the Condorcet loser
whenever one exists. So in all of the clone examples above (which have
no Condorcet cycles anywhere in the rank order), the majority candidate
would be the winner in IPE but not in Borda.

Every (majoritarian deterministic) ranked voting method can be made to
fail IIA by starting with an election where there's a Condorcet cycle.
But Condorcet methods never fail when there's no such Condorcet cycle.
So it would seem reasonable that a method that passes Condorcet loser
and majority (like IPE does) would fail IIA less than a method that
passes neither, like Borda, all else equal.

So in an apples-to-apples comparison, the right comparison wouldn't be
Borda vs IPE, but a Borda variant that uses the pairwise count vs one
that uses your count. Or Ranked Robin with one tiebreaker vs with the other.

-km