Here's an election method I thought of a few weeks ago, called
Proportionality of Solid Coalitions by Condorcet Loser Elimination
(PSC-CLE). It has the advantage of simplicity, but the disadvantage of
having a Borda-like "teaming" effect.
DEFINITION AND EXAMPLE
Consider an election for 2 seats with the ballots
33 A>D>B>C
33 B>D>A>C
32 C>D>A>B
2 D>A>B>C
**Step 1: Choose a Condorcet ranking method and use it to rank the
candidates. This ranking will be the inverse of the elimination order.**
For the example, I will use Ranked Pairs, which gives the order D>A>B>C,
so the elimination order will be C, B, A, D.
**Step 2: Calculate the quota.**
quota = votes/(seats+1)+epsilon=33.333...+epsilon
**Step 3: Count the votes for each solid coalition, as in DSC.**
First preferences:
{A} has 33 votes
{B} has 33 votes
{C} has 32 votes
{D} has 2 votes
First 2 preferences:
{A,D} has 34 votes
{B,D} has 33 votes
{C,D} has 32 votes
First 3 preferences:
{A,B,D} has 68 votes
{A,C,D} has 32 votes
First 4 preferences:
{A,B,C,D} has 100 votes
**Step 4: Divide all the vote totals by the quota and discard the
remainder**
{A} has 0.98999... quotas -> 0
{B} has 0.98999... quotas -> 0
{C} has 0.95999... quotas -> 0
{D} has 0.05999... quotas -> 0
{A,D} has 1.01999... quotas -> 1
{B,D} has 0.98999... quotas -> 0
{C,D} has 0.95999... quotas -> 0
{A,B,D} has 2.03999... quotas -> 2
{A,C,D} has 0.95999... quotas -> 0
{A,B,C,D} has 2.999... quotas -> 2
**Step 5: List the "proportionality rules". Each set of C candidates
with Q quotas is entitled to min(C, Q) seats.**
Elect at least 1 candidate from {A, D}.
Elect at least 2 candidates from {A, B, D}.
Elect at least 2 candidates from {A, B, C, D}.
**Step 6: If there are any proportionality rules of the form "Elect at
least N candidates from this set of N candidates," then declare those N
candidates elected, and remove them from the elimination order. If the
required number of candidates has been elected, then stop.**
So far, there aren't any.
**Step 7: Eliminate the next candidate in the elimination order.**
C is eliminated. That leaves us with the candidates {A, B, D} and the
proportionality rules:
Elect at least 1 candidate from {A, D}.
Elect at least 2 candidates from {A, B, D}.
**Step 8: Go back to Step 6.**
Nothing happens then, but at step 7, candidate B is eliminated. Then we
go back to Step 6 and declare A and D elected, and are done.
CRITERION COMPLIANCE
* (Single-winner) Condorcet Criterion: PASS
Suppose there is a Condorcet winner.
If there are no proportionality rules, then the CW wins by virtue of
being last in the elimination order.
Suppose for contradiction that there is a proportionality rule requiring
the election of a candidate other than the CW. That is, there is a set
of candidates S, not including the CW, that are ranked above the
complement S' (including the CW) on at least one quota (i.e., a simple
majority) of the ballots. But this means that the candidates in S
pairwise beat the CW, which is impossible by definition. Therefore, if
there is a proportionality rule, it must include the CW, who will win
because the other members of the coalition would be eliminated first.
* Proportionality of Solid Coalitions: PASS.
This follows from the definition of the method.
* Independence of Clones: FAIL.
Consider the example election (in which A and D won) with D replaced by
D1>D2>D3:
33 A>D1>D2>D3>B>C
33 B>D1>D2>D3>A>C
32 C>D1>D2>D3>A>B
2 D1>D2>D3>A>B>C
The elimination order is C, B, A, D3, D2, D1. The coalition {A, D1, D2,
D3} is entitled to 1 seat, and {A, B, D1, D2, D3} is entitled to 2
seats. The winners are D1 and D2, which gives the D clone set an extra
seat compared to the original election.
* (Polynomial) Summability Criterion: FAIL.
AFAICT, this method requires a summation array of size 2^n+n(n-1) = O(2^n).
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