[EM] attempt of a grand compromise

[Jobst Heitzig]

Hello folks!

Below is an attempt to formulate a compromise proposal which combines
what I think are the most useful ingredients to a good Condorcet method.

The GOALS are: (a) Use simple ballots which allow to express strong and
weak individual preferences. (b) Use all preferences to find defeats,
but only strong preferences to measure defeat strength. (c) Use a method
to resolve cycles which guarantees that the winner is immune and which
has good consistency properties. (d) When resolving a cycle, rather
ignore a defeat whose defeated candidate has also other defeats against
it, than one which is the only defeat against its defeated candidate.
(e) Use a method which allows for an easy graphical explanation and
justification of the result.

Jobst

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Here's the proposed method:

1. BALLOTS
   • Ranked ballot
   • Ties allowed
   • Optional approval-cutoff,
     default = approve of all ranked candidates

2. DEFEATS
   • Determine all defeats as usual.
   • Strength of defeat X>Y
     = number of voters which approve of X but not of Y
     (= number of voters which "strongly prefer X to Y")

3. WINNER
   • Process defeats by descending strength.
   • Affirm the defeat if that does not introduce
     a branching or cycle.
   • One option remains undefeated and wins.

After having found the winner, the fourth part of the method deals with
explaining and justifying the result:

4. DIAGRAM
   • Draw the tree-like diagram of all affirmed defeats.
   • Label each arrow with the defeats strength.
   • Process all unaffirmed defeats again by descending strength.
   • Add the defeat as a dashed arrow to the diagram
     if that does not introduce a cycle.
   • From all beatpaths contained in the diagram, determine
     the strongest ones from the winner to each other candidate.
   • Remove all dashed arrows which are not on these beatpaths.
   • This gives a diagram of
     n-1 solid arrows and at most n-1 dashed arrows that contains
     optimal beatpaths which are consistent with the affirmed defeats.

(see below for more details)

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PROPERTIES:

Since step 3 uses the River method's technique to break cycles, we have
immunity (implies Pareto efficiency, Majority criterion, Condorcet
efficiency, Smith criterion, and Steve Eppley's strategy criteria),
monotonicity, summability (except for the tiebreaking rank order),
resoluteness, composition consistency (implies clone-proofness),
independence of strongly dominated alternatives (ISDA, implies IPDA),
and k-set-consistency (see below).

Because of step 2, the method is a variant of James' Cardinal Weighted
Pairwise and is thus robust against compromising and burying.

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COMPARISON WITH OTHER METHODS:

Although it has the same appealing robustness to strategic voting, it
uses much simpler ballots than Cardinal Weighted Pairwise, hence it is
much easier to tally manually. Since it uses the notion of "approval"
instead of "rating" or "utility", it is also easier to tell whether a
ballot is filled out sincerely.

Other than Beatpath and Ranked Pairs which also meet goal (c) and share
most criteria with the above River-based proposal, the latter meets
goals (d) and (e) and fulfils ISDA in addition.

The more voters only tie some candidates for top and leave the others
unranked, the more the method resembles Approval Voting insider the
Smith set.

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DETAILS:

Unranked candidates are understood as tied for bottom.

Ranks must be integers but can skip some numbers.

If the voter uses marks instead of numbers, they are understood as 1s.

The approval cutoff is specified as the number of the last approved rank.

The order in which defeats are processed is, more precisely:
i)   by decreasing number of voters which approve of X but not of Y
ii)  if equal,
     by increasing number of voters which approve of Y but not of X
iii) if still equal,
     by decreasing number of voters which prefer X to Y
iv)  if still equal,
     by increasing number of voters which prefer Y to X
v)   if still equal,
     by increasing tiebreaking rank of X
vi)  if still equal,
     by decreasing tiebreaking rank of Y

The tiebreaking rank order is a random voter hierarchy determined from
the strong preferences. In other words, we process randomly chosen
ballots and put X>Y whenever the ballot approves of X but not of Y until
every pair is distinguished. Remaining ties are resolved randomly.

A "branching" is a pair of defeats defeating the same candidate.

Although it would make the ballots still more simple, I did not include
the recent ABCDF-proposal since those labels can be confused too easily
with absolute judgements instead of rankings which indicate a relative
order only.

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k-SET-CONSISTENCY:

This is a new criterion developed by Forest and me in the last days
which is a natural generalization of the Condorcet and Smith criteria. A
method is k-set-consistent, where k is some integer larger than 1, if
the following holds:

   Suppose there is a set A of candidates such that
      whenever the candidates are restricted to any set of
          k candidates which contain at least one candidate from A,
      the winner of that set belongs to A.
   Then the overall winner must also belong to A.

In particular, when some candidate wins against every set of k-1 other
candidates, it must be elected.

It is easy to see that a method which fulfils the majority criterion is
2-set-consistent if and only if it is Smith consistent (with the
smallest A being the Smith set).

It can be proven that Beatpath, Ranked Pairs, and River (and thus also
Cardinal Weighted Pairwise) fulfil k-set-consistency for every k>1. I
will post the proofs some later time.