[Election-Methods] Burial comparison: WV and C//A

[Kevin Venzke]

Hello,

In the past I've suggested that Condorcet//Approval (including ICA or
Schwartz//Approval, etc.), with ranked candidates as implicitly approved,
is a good anti-burial method that doesn't sacrifice minimal defense. In
this post I will compare it to WV with the typical scenario.

Start with the "sincere" scenario:

x A
y B>A
z C>B

x, y, and z are distinct values that sum to 100%, and each is smaller than
50%.

For this scenario to make sense, B needs to win this election. WV elects B
always: B is the only candidate who could be CW; if he is not CW, then the
cycle can only be C>B>A>C; the only defeat here which is less than a
majority is C>B, so that B must win.

CA (Condorcet//Approval) elects B unless x>z>y. In that case B loses
pairwise to C, and A wins on approval. x>z>y is one of six possible
situations of course.

Now the A voters will attempt to bury B, using C, in order to steal the
election:

x A>C
y B>A
z C>B

With no CW possible, WV elects A if 
(y+z < x+y and y+z < x+z)
= (z < x and y < x)
or in other words, the A faction is the largest. This seems quite possible
if the A faction is contemplating burial. If y is the largest, the burial
attempt has no effect and B still wins. Only if z is the largest does the
burial attempt backfire and elect C: This is troubling because surely the A
voters would not attempt this strategy if they thought that the C faction
was the largest.

Under CA, again no CW is possible, and A is elected if
(x+y > y+z and x+y > x+z)
= (x > z and y > z)
or in other words, the burial only succeeds if C faction is the *smallest*.
While this may also be predicted by the A faction, to my mind I question
how likely it is that the B faction would reliably rank A, when C is in
fact hardly a contender.

The burial attempt backfires and elects C under CA if y is the smallest.
And as I just suggested, the likeliest scenario for B voters to be ranking
A is if B is expected to be a weak faction. In other words, the likeliest
scenario for A voters to receive support from B voters is the scenario
where burial strategy backfires.

Note that if x>z>y, the way this backfires under CA is not to simply move
the win from B to C, but to move it from A to C, so that the A faction has
defeated their own candidate.


Now let's look at what happens if the B voters do not rank A, in
self-defense. Original scenario:

x A
y B
z C>B

Under both WV and CA, B wins unless C is the largest faction, which would
not be the scenario we expect if the A voters are contemplating a burial
strategy.

Now the A voters employ their strategy, guessing incorrectly that they have
support from the B voters:

x A>C
y B
z C>B

Note for simplicity that A and B cannot be the CW here, and the direction
of a cycle must be B>A>C>B. The A>C is the only win we can't count on.

WV elects A if:
(y+z < x and y+z < z) and (x > z) <-- C must beat A
= (false and false) and (x > z)
So the strategy at least doesn't work.

WV elects B if:
(z < y+z and z < x) and (x > z)
= false (x and z need to be larger and smaller than the other)
This doesn't work out either. So this strategy always backfires here under
WV.

Under CA:
x A>C
y B
z C>B

A can't win; he can be neither the CW nor the approval winner.

B can't win as CW, but will win if
(y+z > x and y+z > x+z) and (x > z)
= (true and y > x) and (x > z)
So if y>x>z; i.e., the A voters are using burial strategy against a larger
B faction, the strategy is ineffective and B still wins.

When does the strategy backfire? C already wins if z is the largest
faction, so that must be kept in mind.
(z < x or z < y) and
C can be the CW:
((z+x > y and z > x) or
or else C can have the most approval:
(x+z > x and x+z > y+z))

simplifying a bit
(z < x or z < y) and
((true and z > x) or
(true and x > y))

So if x>y>z, x>z>y, or y>z>x, the strategy backfires. So it's the same as
WV except for the y>x>z scenario, where the strategy is just ineffective.

My feeling from all this is at least that Condorcet//Approval or ICA are
more likely than WV to punish offensive strategy, in the situation that
other voters are not doing anything to discourage it. When the "abused"
faction does defensively truncate, WV is a little more likely to punish the
strategy, but is just the same (i.e. perfect) at preventing the strategy
from working.

Any thoughts?

Kevin Venzke


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