# Hybrid Theory

```Focusing on just A.6 again, in this draft:

(*) Weakest defeats can now be eliminated: before a defeat of the default
option is eliminated, all options which fail to meet their supermajority
requirements are deleted.
(*) When artificial supermajority defeats are eliminated the corresponding
option is also deleted.
(*) Terms expressed using mathematical shorthand have been named.
(*) Other minor fixes

______________________________________________________________________

A.6 Vote Counting

1. Each voter's ballot ranks the options being voted on.  Not all
options need be ranked.  Ranked options are considered preferred
to all unranked options.  Voters may rank options equally.
Options left unranked by the voter are considered to be ranked
equally with one another and below any ranked options.  The other
details of how ballots may be filled out will be included in

2. We construct the Schwartz set based on undropped options and
defeats:

a. An option A is in the Schwartz set if A has not been dropped
and if for all options B, either A transitively defeats B,
or B does not transitively defeat A.

b. An option A transitively defeats an option C if A defeats C
or if there is some other option B, where A defeats B AND
B transitively defeats C.

c. An option A defeats an option B, if the strength
N(A,B)*V(A,B) is larger than N(B,A)*V(B,A), and if the
(A,B) defeat has not been dropped.

d. Given two options A and B, the votes V(A,B) is the number
of voters who prefer option A over option B.

e. If a majority of n:1 is required for A, and if B is the
default option, majority requirement N(B,A) is n.  In all
other cases, N(B,A) is 1.

3. If any options involved in the weakest defeats between options
in the Schwartz set are options in superdefeats, we drop the

a. A defeat (A,X) is weaker than a defeat (B,Y) if V(A,X)
is less than V(B,Y).  Also, (A,X) is weaker than (B,Y) if
V(A,X) is equal to V(B,Y) AND V(X,A) is larger than V(Y,B).

b. A weakest defeat is a defeat that has no other defeat weaker
than it. There may be more than one such defeat.

c. An option B is superdefeated by an option A if A defeats B,
and if V(A,B) is not larger than V(B,A).

c. Given a defeat (A,B) the options A and B are involved in
this defeat.

d. An option A is dropped by dropping all defeats which involve
A and also stipulating that option A is not a member of
the Schwartz set.

4. If there are defeats between members of the Schwartz set, we

5. If there are no defeats within the Schwartz set, then the winner
is chosen from the undropped options in the Schwartz set where
at least Q voters ranked that option above default option, where
Q is the quorum requirement for the ballot. If there is only one
such option, it is the winner. If there are multiple options, the
elector with a casting vote chooses which of those options wins.

"RATIONALE": Options which voters rank above the default option are
options they find acceptable.  Options ranked below the default
option are unacceptable options.  Supermajority options require
some approximation of unanimity before they can be accepted.

______________________________________________________________________

This changes a few of the outcomes for tests posted in
http://lists.debian.org/debian-vote/2002/debian-vote-200211/msg00321.html

Here's the test cases which have changed:

A requires 2:1 majority; D is the default option
3 ABD
1 BDA
1 DAB

A defeats B 4:1
B defeats D 4:1
D superdefeats A 4:3

eliminate A

B defeats D 4:1

B wins
in previous draft, this was a tie between B and D
the requirement was that D did not win

Z is the default option
40 A
25 B
35 ZBA

B defeats A 60:40
A defeats Z 40:35
Z defeats B 35:25

eliminate 35:25

B defeats A 60:40
A defeats Z 40:35

B wins
in previous draft, Z won
the requirement was that A not win

Quorum of n, no supermajorities, D is the default option:
25 DAB
30 BDA
35 ABD

B defeats D 65:25
A defeats B 60:30
D defeats A 55:35

eliminate 55:35

B defeats D 65:25
A defeats B 60:30

A wins unless quorum is not met in which case D wins.
In the previous draft B won unless quorum was not met in which case D wins.

Additionally, here's the result of the test proposed by Clinton Mead in
http://lists.debian.org/debian-vote/2002/debian-vote-200212/msg00020.html

A requires 3:1 majority; D is the default option
4 ABD
1 BDA
1 DAB

A defeats B 6:1
D superdefeats A 6:5
B defeats D 5:2

eliminate A

B defeats D 5:2

B wins
in the previous draft, D won
the requirement is that B wins

______________________________________________________________________

If there's any ambiguities or problems with this, please let me know.
[And please bear with me if I seem to be asking inane questions in the
course of understanding a possible problem.]

Thanks,

--
Raul

```