[EM] How many candidates are treated fairly?

[Tom Ruen]

Greetings,

I've not written to EM list much, mostly because the debate is overwhelming and so often seems far from practical level methods that have a chance to be implemented in political elections.

However I've been developing a useful line of thought and I hope it might be worthy of discussion. My personal agenda is towards Condorcet, but considering practical steps away from plurality that are almost as good as Condorcet.

Given any election method, I can ask the question, "How many candidates can be guaranteed to be treated fairly?"

I decided to classify different methods on limitations on "fairness" in the Condorcet sense of direct competition without distraction.

ELIMINATION LIMITED FAIRNESS - eliminated candidates fail to get equal treatment.
1. Plurality - Trivially treats 1 candidate fairly (top-one!)
2. Top-two runoff - Treats at least 2 candidates fairly  (top-two!)
3. Bottom-up runoff - Treats at least 2 candidates fairly (Possibly a different two from the top-two runoff)

PAIRWISE LIMITED FAIRNESS - Fair treatment, but candidates can win without being anyone's favorite.
1. Condorcet (Ignoring cycles for now) - Treats ALL candidates fairly

APPROVAL LIMITED FAIRNESS - "Equal treatment", but not "Fair treatment" since lower preference votes can hurt higher ones (comparing levels of preference truncation).
1. Approval - No rankings
2. Borda
3. Bucklin

I'm sure there's many other interesting methods and variations, but I just wanted to give some basic differences. I admit others might group them differently. Plurality and Approval might not fit any single classification with the others.

I included "approval methods", but find myself unable to deal well with analysis here. They are "fair" in the sense of no elimination, but unfair since they DEMAND strategic truncation by most voters to guarantee a majority winner be identified. Voters who have better polling data have more influence for knowing better how to truncate their choices to affect the winner. Approval elections are a game of chicken, always depending other voters to truncate less. (I know I'll get slammed for my opinions here, and they aren't worth any further debate for me.)

Primarily I looked at the idea of fairness in consideration of runoff methods which eliminate candidates.

I think it is valuable to recognize that all runoffs, instant or not, (with a 50% majority requirement) are limited to guaranteed fair treatment of two candidates - the final two survivors. If the Condorcet candidate is in that final two-set, he'll be guaranteed to win, and other candidates don't get that same opportunity.

This is a significant failing, but at least it is clear that it is better than plurality which offers no minimum support needed to win, besides the trivial (non)limit: (votes/candidates+1) - approaches zero as candidates gets larger.

I also think it is interesting to notice that on these simple grounds a "top-two" runoff is "just as fair" as a "bottom-up" runoff - both only promise that two candidates will have undistracted competition. The only way to go beyond TWO candidate fairness is to go to Condorcet-style competition where there are no distractions.

Now I don't want at all to be critical of Condorcet's approach, but I can see Plurality and Condorcet as opposites with different valuable properties. Plurality counts reward candidates that can stand out as special among full competition. Condorcet rewards candidates that stand out only under pairwise competition. I can imagine that without limits to competition, there is no incentive in pure Condorcet for voters and candidates to unify before the election. Condorcet offers no incentive for parties to organize or rally behind one choice.

I consider perhaps it is more conservative to offer compromise solutions - methods that have one "Plurality" count to recognize the strongest candidates under full competition, and a second Condorcet counting round among the survivors of the first round. A single plurality-based elimination round allows serious candidates to be separated from the less serious ones, and encourages voters make a hard choice which candidate to support.

Interestingly to me a "top-two runoff" is the simplest (trivial) form of this two round approach.

I like the idea of supporting IRV reform, except limiting it to a two-round (top-two) instant runoff because it has the purity of a single elimination round of plurality-strong candidates and equal treatment of survivors.

This approach is most defendable to people who see virtue in the plurality placings and fear IRV rewards weak candidates. 

Someday if too many elections like France 2002 occur, we can consider election rules that allow more than two candidates to survive elimination.
http://en.wikipedia.org/wiki/French_presidential_election%2C_2002

In that election everyone will agree there were three strong candidates most worthy of consideration. A pure bottom-up IRV election would be horrible due to so many rounds of elimination that would demand recounts when eliminations were close among weak candidates. A pure 16 candidate Condorcet election would probably be safe, but with 3% spoiled ballots, pretty much meaningless to think voters could offer much wisdom in all pairwise contests. It is just a little scary to perform all pairwise counts, even if it is extremely rare for a plurality weak candidate to win Condorcet.

What is most interesting to me to consider that the next (simplistic) reform beyond a "top-two runoff" may NOT be "top-three runoff" as recursion implies (or worse a recursive "bottom-up-runoff") but instead a "top-three-Condorcet" as a plurality round followed by 3 pairwise rounds.

A two-round Plurality-Condorcet approach allows different methods to be defined that can treat any desired number of candidates equally. Or more sensibly it can have elimination rules that allow a variable number of plurality survivors based on their judged strength in the plurality count.

For instance here's three sensible rules that can work together:
If there's no outright majority winner, let candidates survive by rules:
    1. At least two candidates must survive. (top-two)
    2. At least the top 50% of votes for candidates must survive (top-median)
    3. All candidates with at least 20% of the vote must survive. (individual threshold)

Rule #2 is most used when there's many candidates and none very strong. In the French 2002 election, rule #2 would keep the top-3 candidates in the plurality round, whom combined have 53% of the vote. Rule#3 applies more when there's more than 2  strong candidates, allowing up two 4 strong candidates to survive.
 
Rules like these define thresholds that candidates must reach to be guaranteed fair treatment of a Condorcet round. Nonmontonicity can exist when a possible Condorcet winner fails the thresholds, but clear threshold requirements at least help voters decide their strategy in how to get at least one favored candidate into the "fairly treated" set.

Pre-election polling data measures "plurality strength" so a direct prediction can be made which candidates will make the first round cut. (unlike bottom-up IRV elections which offers no guarantees to the strongest plurality candidates unless they hold an outright majority)

Anyway, I really like this approach of measuring methods (by how many candidates are treated fairly) and limited one-round elimination by plurality strength. Everyone should recognize that runoffs, however much put-down as offering poor results in strong 3-way contests, are a major step forward in treating two candidates equally at least.

My reform steps for single winner elections would be:
1. Plurality (beats dictatorship, fair treatment for ONE top candidate)
2. Top-two runoff (majority requirement, fair treatment for at least TWO top candidates)
3. Plurality-Condorcet (fair treatment for a SET of top candidates)
4. Pure Condorcet (fair treatment for ALL candidates)

I offer these methods as either one election (Voting with one set of ranked ballots) or two rounds of voting. When there are many candidates and less informed voters, it can be distracting to have to do everything in one round. If voters are willing I'd prefer allowing two rounds of actual voting to make sure all preferences offered are well considered.

Thanks for listening. I apologize for being too long-winded. :)

Sincerely,
Tom Ruen
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