[EM] The "Fresh Egg" winner - beyond Condorcet's pairs

[Tom Ruen]

Hi Dave, Adam and others,

Perhaps I wrote too much to be convincing of anything. Let me try again and focus on my primary interest.

My primary interest is in recognizing the existing of "degrees" of strength of a winner. Condorcet does this too, but not quite as completely. That is, Condorcet also is incomplete (when pairwise cycles exist), and requires refined measures to determine a winner, and those refinements can require further refinements! My interest is to show that perhaps Condorcet's Winner could use some refining itself.

I think it is useful to attempt to label the strength of a candidate in an election, even if it isn't unconditionally used to pick the winner.

When I first started learning about EMs over two years ago Craig Carey demanded from me to define what I meant by a majority among 3 or more candidates (He claimed it didn't exist). Whatever Craig's funny ideas are, it is a worthy question.

At the time, I though it was obvious - A majority is more than half the votes, but of course that idea is of limited value by itself among 3 or more candidates since there may be no candidate that passes the test. Still this is a reasonable test since not all methods satisfy it:

1. A Sovereign Majority (SM) - If a candidate exists that is never below first place among any subset of competitors. (Also more simply defined If a candidate exists that has more plurality votes than all others combined.)

This rule is certainly a good test of an election method, and two well known methods can break it - Approval and Borda.

Examples: 
    Approval: [XY] means a vote for both X and Y, but prefer X.
        [AC]=51%, [BC]=49% - 51% prefer A, but C wins.
    Analysis: Foolish approval "overvoting" can hide a sovereign-majority preference.

    Borda:
        Election 1: AC=65%, CA=35%
              A=2*65+1*35=165
              C=1*65+2*35=135
        Election 2: ACB=65%, CBA=35% (Same preference between A and C)
              C=2*65+3*35=235
              A=3*65+1*35=230
              B=1*65+2*35=135
    Analysis: Introducing a losing candidate hides a top-majority preference!

This tells me that if I'm interested in supporting a "Sovereign-Majority" (SM) criterion, I don't want to support Approval or Borda.

Then we have another lower criterion that can be applied if there is no sovereign-majority - Condorcet's winner.

3. A Condorcet Majority (CM) - If a candidate exists that is always above last place among all pairwise subset elections with competitors. ("Above-last" is equivalent to saying "first" among pairwise comparisons.)

I call this "Concercet Majority" because it defines a unique winner, in parallel how the SM works. Two candidates can never simultaneously satisfy this criterion, although in some elections (with cyclic preferences), there may again be no candidate that satisfies it.

Whether I want to use this rule for picking a winner, it is a real measure of a candidate. AND we all know that IRV can break this rule and pick a different candidate sometimes.

So, if I'm interesting in ALWAYS picking the CM winner, then IRV is not a method I want to support.

I misnumbered the CM as (3) above because there is an intermediate criterion which I chose to call the "Fresh Egg Majority".

2. A Fresh Egg Majority (FEM) - If a candidate exists that is always above last place along all subset elections with competitors.

Looking at FEM is similar to CM, except it is a more strict requirement. Sometimes a CM winner won't exist, but more often a FEM winner won't exist.

On the surface this would appear to be back-stepping, being less decisive than CM, less useful in practice, however I choose to see it as being more careful, more demanding, and choosing to recognize the difference between a candidate that can merely do well in pairwise contests, and one that also does well in the plurality set and other subsets.

This criterion suggests to me that the Condorcet Majority definition, however nice, is not the end-all-be-all rule for picking a winner, and that it is reasonable to consider election methods that don't always pick the Condorcet winner.

As I said before an interesting result of the FEM definition is that when it exists, both Condorcet AND IRV must agree on this winner.

I don't think I should have to define FEM as a concept that is meaningful for recognizing candidate's strength, although I admit many may not consider it useful for defining a better method.

For me it is interesting to me to define these. I call them "majorities" but only the first (SM) REALLY qualifies as a true (50%+1) majority. The FEM and CM criteria succeed by defining a unique winner sometimes, and uncertainty elsewhere. These are perhaps best called pseudo-majorities for being a pass/fail test that can have at most one winner. Condorcet supporters might talk of "strongest majority", although I'd call that phrase as overstepping their case since a measure of what is strongest can be subjective.

Obviously other such criteria may be named, for example: 

1+1/2. A Noble Majority (NM) - If a candidate exists that is always above the average vote along all subset elections with competitors. (Average = total_votes/subset_candidates)

Avoiding last place (FEM) is a good sign of a winner, but ALWAYS being above the middle is even stronger. This criterion is between SM and FEM. It occurs more often than SM but less often than FEM.

I'm note sure if this one has any use, but it is an attractive criterion to satisfy.

I think it is useful for recognizing some winners will be stronger than others, AND it may be useful to grade winners in this way. Condorcet is the method that first demanded this complexity of giving degrees of winners, and there is still disagreement in the hardest cases. I am merely continuing his story and seeing what Condorcet chose to ignore.

If I have a point, it is to suggest that Condorcet may be more negligent than supporters wish to admit, and even if everyone agrees that Condorcet picks a best winner, I'd like recognition when a pairwise winner is picked without core support of plurality counts.

Thanks for listening.

Sincerely,
Tom Ruen
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