[EM] Antidote to Baldwin Induced Burial

[Forest Simmons]

Let's look at the simplest scenario of three geometrically positioned
factions tightly distributed around their respective candidates A,B,and C:

Without loss, assume A and B are the furthest apart, and that B and C are
farther apart than A and C.

Then assuming preferences are consistent with distances, we arrive at the
following ballot profile:

a ACB
b BCA
c CAB,

where the lower case letters give the respective sizes of the three
factions.

To keep things interesting, we also assume that none of the factions has a
majority: that is max(a, b, c)<(a+b+c)/2

Then the pairwise defeats are
C beats A ... b+c to a
C beats B ... a+c to b
A beats B ...  a+c to b.

So C is the sincere Condorcet Winner.

Could either the A or B faction improve its outcome by burying the CW?

If the B faction buries C the profile becomes

a ACB
b BAC
c CAB

Then A becomes the (insincere) ballot CW beating B ... a+c to B, and
beating C ... a+b to c.

This makes A the new winner under any method that meets the Condorcet
Criterion of always electing the ballot CW when one exists.

So we see that the B faction's outcome is not helped by this unilateral
order reversal.

How about B switching places with A or C on the B faction ballot?

It's easy to check that the first would benefit only A, while the second
would retain C as the winner.

On the other hand, burial of C by the A faction is much more interesting,
because the resulting ballot set no longer has a ballot CW .... which means
that the Condorcet Criterion is no longer adequate to decide the winner:
Different Condorcet methods will elect different winners, depending on the
faction sizes.

The pairwise defeats are now ...
A beats B ... a+c to b
B beats C ...  a+b to c
C beats A  ... c+b to a

Classical Condorcet (MinMax, Ranked Pairs, and CSSD) elect the member of
the cycle suffering the weakest defeat ... so A is elected when a is
largest ... the largest faction gets its way.

So in this context, under classical Condorcet, the A faction is rewarded
for their burial of the sincere CW when the A faction is largest.

How about under Black? Black elects the Borda winner when there is no
ballot CW.

The respective Borda scores are (up to a constant shift) a-b, b-c, and c-a.

So A will win  iff a-b is greater than max(b-c, c-a) ... which will be true
iff
a-b>b-c and a-b>c-a ... that is iff ...
a>2b-c and a>(b+c)/2 ... i.e.
a>max(2b-c, (b+c)/2).I

How about Baldwin?

Baldwin eliminates the Borda loser, and elects the pairwise winner of the
other two candidates.

So under Baldwin, A will win when C is the Borda loser .... i.e. iff
c-a<min(a-b, b-c) ... i.e. iff
(b+c)/2<a and 2c-b<a ... i.e. iff
max((b+c)/2, 2c-b)<a

How about Benham?

Under Benham, when the smallest faction candidate is eliminated, the
pairwise winner of the other two is elected.

A will win (i.e. be rewarded for burial) when the C faction is smallest.

How about our max defeat strength winner?

Two cases:

I. Defeat strength is Winning Votes. Then A wins when a+c>max(b+a, c+b) ....
i.e. c>b and a>b ....
 i.e. ehen the B faction is smallest.

II. Defeat strength is ...
WinningVotes minus LosingMaxPairwiseSupport.

A wins iff ...
a+c-(a+b)=c-b>max(a-c,b-a) ... i.e.
c>max((a+b)/2 ,2b-a)

and B wins when (and only when) ...
a>max((b+c)/2, 2c-b) ...

... precisely when A wins under Baldwin!

Our WV-LosingMaxPS method backfires on the burying faction precisely when
Baldwin rewards the burying faction!

Why do academic Condorcet advocates keep proposing Baldwin?

Because they are not acquainted with the basic facts of burial (not to
mention the basic facts of Clone Dependence).

Methods like Baldwin, Nanson, and Black unwittingly sully the Condorcet
name.

-Forest
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