[EM] Another Approval-Condorcet compromise method

[Martin Harper]

I've been much taken with Dyadic Approval, especially as it offers a middle
path between Approval and Condorcet. I thought I'd suggest another method
which similarly offers a middle path, using similar (but marginally different)
ballot types.

Contents - just read what you find helps you most effectively judge voting
methods, and skip what you don't::
1) Explanation
2) Criteria Compliance
3) Examples

Have fun,
Martin

--
EXPLANATION
The ballots work as follows:

a ranking is decided, say A>B>C>D
The voter adds to that the order determing which preferences are most
important. For example:

A>B>>>C>>D

states that the B>C comparison is most important, followed by the C>D
comparison, followed by the A>B comparison. We can use this ballot to
determine an approval vote for any subset of candidates. For example , the
vote in {C,D} is C - while the vote in {A,B,C} is A,B - because the B>C
comparison is more important than the C>D comparison. The vote in {A,B,C,D} is
also A,B.

Condorcet can be thought of as follows:
for every set containing two candidates, hold an appoval election between
them. The winner is the person who wins all elections that they are part of.

Approval can be thought of as follows:
for every set containing all candidates {there is only one such set}, hold an
approval election between them. The winner is the person who wins all
elections that they are part of.

Universal Approval is done as follows:
for every set of candidates, containing any number of candidates {there are
2^n such sets}, hold an approval election between them. The winner is the
person who wins all elections that they are part of.

Now, cycle resolution is clearly the tricky part, but the same options as are
available in Condorcet will often work. For example, stealing MinMax - the
margin of defeat for X in an election which elects Y is votes(Y) - votes(X).
Find the candidate with the minimum maximum margin of defeat. You might also
use Condorcet(EM), if you prefer size of opposition to margins.

I prefer margins in this case - I'll calculate both in the examples later on -
but I'll be talking about margins in the criteria section.
--
CRITERIA COMPLIANCE

I may talk about the Universal Smith Set(USS) here. For all elections which
include a candidate in the USS, a candidate in the USS wins the election. The
USS is the smallest such set. Universal Approval with USS//Minmax as a
completion method corresponds to the Smith//Minmax Condorcet method. From here
on in, Universal Approval refers to the USS/Minmax variation.

Similarly a Universal Condorcet Winner (UCW) is the sole member of the USS if
it is empty.

USS criterion: the winner must be a member of the USS.Obviously PASS
UCW criterion: if there is a UCW, then it must be elected: Obviously PASS
ULIIA criterion: as for LLIAC, but we're talking about the USS. Obviously PASS

Note that both Condorcet and Approval pass both these criteria.

Monotonicity criterion. PASS

Without loss of generality, suppose we have some vote which includes
......X~>~Y~>>>~Z..... somewhere in it, where ~>~ is actually >>>... with m
'>'s and ~>>>~ is actually >>>... with n '>'s {n>m}.

Then, a change in that vote to X~>>>~Y~>~Z represensts a decrease in support
for Y, all else remaining equal. Consider the elections this change might
affect:

- includes none or only one of X,Y,Z, or X and Z but not Y.
- - NO effect.

- the largest comparison not between one of X,Y,Z is smaller than m or larger
than n.
- - NO effect

- X and Y are in the set, but Z is not. The largest other comparison is
between m and n in size.
- - the vote will change to one which includes X but not Y. This is either an
increase in support for X, or a decrease in support for Y. {LOSS}

- includes Y and Z, but not X. The largest other comparison is between m and n
in size.
- - the vote will change to one which either includes Y and Z, or includes
neither Y nor Z. This is either an increase in support for Z, or a decrease in
support for Y. {LOSS}

- includes all of X, Y, Z, and the largest other comparision is smaller than
n.
- - the vote will change to one which does not include Y. {LOSS}

Hence, Universal Approval is monotonic.

Independance from Clones. PASS

In Universal Approval, clones are a set of candidates which are always
adjacent, and seperated by the smallest possible gap. I'll show independance
from a clone pair - it generalises in the obvious way.

Hence, we have X1 and X2, and in all cases X1 and X2 are adjacent, and
seperated by only one '>'. What happens if we replace them both by X. There
are the following cases:

- no candidates except X1 and/or X2
- - merges to the {X} election, which X always wins anyway. {it's hard to lose
when you're unapposed}.

- neither X1 or X2
- - NO effect

- both X1 and X2
- - NO effect. the > between X1 and X2 will be ignored in favour of a larger
comparator.

- just one or the other
- - discarded, with NO effect. The set with X1 in, and the set with X2 in in
the same position will both give identical results. They'll also give
identical results to the same set with X in the same position. Since we're
looking at worse margins, it doesn't matter if there are two examples of a
worst margin, or one.

Hence, Universal Approval is independant from clones.

I'll give some more criterion checks later, but this is enough for tonight. If
anyone has any criteria they would particularly like me to look at, do say.
--
EXAMPLES

Sample election with a Condorcet paradox, where B has least support, but the
support it has is unwilling to compromise. A and C have mores supporters, but
they are more likely to compromise.

sincere votes:
12 A>B>>C
8 B>>C>A
10 C>A>>B

elections:
{A,B} 22A, 8B
{A,C} 12A, 18C
{B,C} 20B, 10C
{A,B,C} 22A, 20B, 10C

MinMax elects A, Approval elects A, IRV elects C, Bucklin elects A. Borda
elects A. Universal Approval with MinMax elects A. When every other sensible
method agrees on a winner, so does Universal Approval.

If you assume that A>B>>C means sincere CR ratings of 1.0A, 0.33B, 0.0C, and
similar for the others, then you 'should' elect A - A gets 18.67, B gets16.0,
C gets 12.67 - of course, if you did this then you'd have wild strategy
problems...

Same as previous example, but the C faction  is up two, and the A faction is
down two.

sincere votes:
10 A>B>>C
8 B>>C>A
12 C>A>>B

elections:
{A,B} 22A, 8B
{A,C} 10A, 20C
{B,C} 18B, 12C
{A,B,C} 22A, 18B, 12C

MinMax elects C, Approval elects A, IRV elects C, Bucklin elects A, Borda ties
A and C.

Universal Approval with MinMax ties A and C:
A's worst loss is in {A,C} = 20-10 = 10
B's worst loss is in {A,B} = 22-8 = 14
C's worst loss is in {A,B,C} = 22-12 = 10

Universal Approval with Condorcet(EM) elects A:
A's worst loss is in {A,C} = 20
B's worst loss is in {A,B} and {A,B,C} = 22
C's worst loss is in {A,B,C} = 22

The guess at ratings from last time would give A=18.0, B=14.67, C= 14.67.

Now, the standard low utility Condorcet example, much adored by Approval
advocates... A and C are the extremists, B is the low utility compromise. In
this particular example, the B faction are more of a BC faction, but only
just.

sincere votes:
10 A>>B>C
8 C>>B>A
3 B>C>>A

elections:
{A,B} 10A, 11B
{A,C} 10A, 11C
{B,C} 13B, 8C
{A,B,C} 10A, 3B, 11C

Minmax elects B, Approval elects C, IRV elects C, Bucklin elects B, Borda
elects B.

Universal Approval with MinMax elects A.
A's worst defeat: {A,B} and {A,C} = 11-10=1
B's worst defeat: {A,B,C} = 11-3 = 8
C's worst defeat: {B,C} = 13-8 = 5

Universal Approval with Condorcet(EM) elects B:
A's worst loss is in {A,C} = 11
B's worst loss is in {A,B,C} = 10
C's worst loss is in {B,C} = 13

The ratings estimate used last time would give A=10, B=9, C=10.