[EM] Approval/Condorcet Compromise

[Forest Simmons]

Kevin's Approval Runoff in which low approval candidates are eliminated 
until there is a Condorcet Winner, can also be described as follows:

Pick the lowest approval score candidate that beats all of the candidates 
with greater approval scores.

Proof of equivalence:

Kevin's winner KW has to beat all candidates with greater approval because 
they do not get eliminated before the winner, which beats all remaining 
candidates. And if any lower candidate were to beat all candidates above 
it (including KW), then it would become the CW before KW.

Note that this new characterization of Kevin's winner does not require 
talking about elimination or Condorcet Winners.

The voter just needs to understand "approval score" and "beats."

Here's a more procedural definition of the method:

List the candidates in order of approval scores, from lowest to highest.

Go up the list until you first come to a candidate that is not beaten 
(head to head) by any of the candidates above it.

That candidate is the winner.

One could dispense with approval if ballots are to be strictly ordinal 
rankings:

List the candidates in order of average rank, from lowest to highest.

Go up the list until you first come to a candidate that is not beaten 
pairwise by any candidate further up the list.

This can be done with any list:

List the candidates in order of number of first place preferences, etc.

Jobst would use random approval ballot order instead of some deterministic 
order.

Here's my randomization, which can be done with either a deterministic 
order or one of Jobst's random orders (for good measure).

Let P be the set of candidates none of which is defeated pairwise by 
anybody further up than she in the order.  Pick from P by random ballot.

If both the initial order and the picking are done by random ballot, the 
method is hard to second guess.

If approval order or random ballot order is used, then the method is 
monotone, clone free, and independent from pareto dominated alternatives, 
if I am not mistaken.  [I have been wrong before.]

This method requires no concept any more difficult than approval order, 
random ballot, or pairwise defeat.

Forest