[EM] Unanimity test works with many candidates. How it works.

MIKE OSSIPOFF nkklrp at hotmail.com
Tue Jan 6 00:59:03 PST 2004

The unanimity test, for detecthing order-reversal works at least as well 
with many candidates, as well as with 3 candidates. It merely requires a 
strictly-observed 1-dimensional political spectrum.

Though we don't have _exactly_ that, it isn't an unreasonable approximation.

Say it's A, B, & C.

As usual, B is the middle Condorcet winner.

The A voters order-reverse against B, making a strategic circular tie.

First of all, if B is majority-supported against A, then the only way that A 
can win is if, when his voters have voted C>B, they create a cycle in which 
everyone is majority beaten more than A is.

So the unanimitly test, like the other voter-option order-reversal tests, is 
triggered when there's an all-majority-defeats.

We're assuming that voters and candidates are on a 1-dimensional political 
spectrum, and that everyone perceives the candidates in that way.

The political spectrum order is A B C.

All the C voters prefer B to A. I always assume that C has the support of 
the C voters. If he doesn't, then all bets are off. Since all the C voters 
prefer B to A, they all rank B over A.

That tells the method that B is between C and A.

Since B is between A and C, some B voters will likely vote BAC and some will 
likely vote BCA . That is a further indication that B is between A and C.

So the unanimity of the C voters and the nonunanimity of the B voters both 
agree in indicating that B is between A and C.

But some of the A voters order-reverse against B. Say all the A voters 
order-reverse against B.

They're saying that C is between A and B. But the C voters are saying that B 
is between A and C.
They disagree. The B voters, however, voting as if they're between A and C, 
confirm the evidence of the C voters. The method concludes correctly that 
the A voters are order-reversing.

If only some of the A voters order-reverse, then they're suggesting that A 
is between B and C.

Again, that's contradicted by both the C voters and the B voters. The method 
concludes again that the A voters are order-reversing.

This works just as well with more candidates. Then there are even more 
voters to the sides, agreeing about the order of those candidates in that 

So I suggest that the unanimity test is a good way of detecting 
order-revesal, to the extent that the votrers and candidates are on a 
1-dimensional political spectrum which is strictly observed. I suggest that 
that's a good enough approxmation to make the unanimity test useful.

It would be an option that the voter could choose by marking a box on his 
ballot. Doing so tells the method that that voter wants to delete from his 
ranking any candidate whose voters are judged by the unanimity test to be 

When the voter chooses the "spectrum order estimate option", he indicates 
his estimate of the spectrum order of the candidates, indicating that he 
wants the method to drop from his ranking any candidate whose voters seem to 
be order-reversing based on that spectrum order.

A voter could choose both of those options.

When marking a tie-drop line on his ballot, he's saying that if there's an 
all-majority-defeats circular tie with some members above the line and some 
below, he wants to drop those who are below the line.

The voter could mark a tie-drop line in addition to using one or both of the 
automatic order-reversal test and drop options.

In fact the voter could mark several tie-drop lines on his ballot.  Don't 
mark a tie-drop line above your important tie-drop line. But marking another 
one below it is harmless. If you mark a 2nd most important line, below your 
important one, don't mark a 3rd most important one above the 2nd most 
important one. Each less imporant one should be below the more important 
ones. In that way it doesn't interfere with the operation of the more 
important tie-drop line and the protecion of important candidates.

For instance, I could mark a line, the most important one, below Nader & 
Sharpton, and another one below Kucinich (ranked below Sharpton), and 
another below Dean, and another between the Democrats and the Republicans.  
I've mentioned those lines in the order of their importance.

These anti-order-reversal enhancements are luxuries that would be nice to 
have, enhancements that would further reduce Condorcet wv's already 
practically nonexistent defensive strategy need.

The candidatre withdrawal option is also a very effective way of getting rid 
of defensive strategy need. A candidate can withdraw after a count, have his 
name deleted from the ballots, so that if he was 1st on a ballot, now the 
2nd choice becomes first, and call for a recount with his name deleted.

So there are many good ways of getting rid of whatever defensive strategy 
need wv has.

As I was saying, every nonprobabilistic 1-balloting method shares the same 
offensive and defensive strategies of order-reversal and truncation, and the 
possible defensive strategy of equal 1st choice ranking. That is, every 
method that doesn't have worse strategy problems.

So all the newer 1-balloting methods that have been defined here could 
benefit from these anti-order-reversal enhancements if people considered 
order-reversal a probem and wanted to deal with it. All these best methods 
are truncation resistant. Their only defensive strategy need occurs if 
offensive order-reversal is attempted.

Mike Ossipoff

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