# Condorcet Intended "Margin of Defeat" in Tiebreak

Steve Eppley seppley at alumni.caltech.edu
Thu Feb 20 19:07:43 PST 1997

```Hugh T wrote:
-snip-
> I have now examined the original; Condorcet wrote that one
> discards successively from the impossible opinion "les propositions
> qui ont une moindre pluralité."  In the immediate context, given
> the usage of "pluralité" in surrounding text (for example,
> contrasted with "minorité"), the most natural reading is that it
> refers to the total number of votes, not the difference.  However,
> in the larger context of Condorcet's Essai, I now think it very
> likely that he intended what Demorep said -- the margin of victory.
-snip-

Given M. de Condorcet's assumptions that voters will express strict
(i.e., no pairwise indifference) sincere, complete preference
orders, the pairwise margin and pairwise opposition variations both
produce the same result.  So is there some evidence to suggest that
he evaluated both variations, and chose one over the other on
merits that only come into play when his assumptions are violated?

I think we in EM have endorsed the best variation.  Condorcet's
assumption that voters won't truncate their preference orders is
probably incorrect, so truncation resistance is relevant.  And the
"Consensus-like" property from counting "pairwise opposition" is
attractive, and easy to explain: Elect the candidate which
will minimize the number of voters who prefer a different winner.

We've already discussed the relative merits of pairwise margin and
pairwise opposition.  And we could discuss variations which look at
"pairwise support"--if we have already, I've forgotten--which would
be more "intuitive" for a lay audience:

Example:

The preferences:

,------------------ A
|    ,------------- B
|    |    ,-------- C
|    |    |

1    2    2   ....46 voters
2    1    2   ....20 voters
3    2    1   ....34 voters

The tally:
pairwise support:
A    B    C
A vs B       46   54W
A vs C       46W       34
B vs C            20   34W
---- ---- ----
smallest support:  46   20   34
smallest win:      46   54   34
largest support:   46   54   34
largest win:       46   54   34

B's smallest win is largest, so elect B?

Variations involving pairwise support would probably be far easier
for people to understand than counting "pairwise opposition", since
the matrix elements are more intuitive.  But they would have
undesirable results in plausible scenarios, not apparent from the
acceptable B result in this example.

Here's a bad example of "largest smallest pairwin."  (I don't like
the bulky format of the preference orders in the example above, so
I'll revert to our traditional compact preference order notation.)

46: A
10: BAC
10: BCA
34: CBA

pairwise support:
A    B    C
A vs B       46   54W
A vs C       56W       44
B vs C            20   34W
---- ---- ----
smallest support:  46   20   34
smallest win:      56   54   34
largest support:   56   54   44
largest win:       56   54   34

Truncation elects A.

I prefer a variation which minimizes incentives to misrepresent
preferences, especially the ranking of a "lesser evil" equal to or
ahead of one's favorite(s), and "smallest largest loss" appears to
do so.  Ending the spoiler dilemma, the flip side of the lesser evil
dilemma, is key to busting the "only two viable candidates" system
(a.k.a. the two-party system) that prevents potential candidates from
raising issues which the "big two" aren't addressing, and thereby
prevents voters from expressing their preferences on those issues.

But perhaps Hugh would like to conduct a poll asking Condorcet
advocates to rank some Condorcet variations.  If we do that poll,
I'd prefer it wait until after the upcoming poll in which we rank
standards and criteria.

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)

```