Why the 'half' cutoff in rule d?

How the precise wording of this rule evolved

Originally we suggested the criterion that to be elected, a candidate would need to have at least 50% numeric scores. That is: you cannot be elected if more than 50% of the voters regard you as "unknown" and hence vote "intentional blank" for you.

But that causes a problem: If mutually-detesting candidates A, B, and C have 26%, 26%, and 48% support respectively, then the A- and B-supporters can conspire to all write "X" (intentional blank) as C's score, preventing C from getting his deserved victory and as their reward getting a 50-50 chance of electing either A or B. (They could also just agree to maximally support the other, but that would not work in practice because each side would worry the other one would renege on the deal. In contrast, with the X-based deal we are suggesting here, neither side is motivated to renege.)

That collusion problem is solved by changing "50%" to "33%" in rule d. However, even then, a 3-way conspiracy among A-, B-, and C-supporters could similarly deny a deserved victory to D. That problem in turn is solved by further decreasing it to "25%."

A 4-way conspiracy could still work, but then this is becoming pretty ridiculously unlikely so we are not going to worry about it. On the other hand, if we decrease the threshold too much, it then becomes too worrying to some that a "stealth candidate" unknown to most of society, could be elected. We regard it, however, as ridiculously unlikely that anybody could be well known to and heavily supported by 25% of the voters, but at the same time the remaining 75% of the voters would know nothing about him and all would want to write intentional blank scores for him (with few or none wanting to give him a low score). So "25%" seems like a good balance between these two bad possibilities which keeps them both ridiculously unlikely.

So then our criterion was that

a candidate, to be elected, must get at least 25% of the number of numeric scores, as the candidate with the greatest number of numeric scores.

But that phrasing has a flaw.

Flaw and rephrasing to avoid flaw

But then Abd ul-Rahman Lomax pointed out that

26% of the voters could conspire to elect their candidate as a write-in candidate not on the ballot. They write him in and give him the maximum score. Most voters would have given him the minimum score but it did not occur to them to do so since he was not on the ballot. This conspiracy candidate could beat even a well known on-ballot candidate whose average rating was 90% of maximum.
Oops – that was a flaw in the 25% criterion. Actually, we consider this flaw to be pretty unlikely (you could not organize such a huge conspiracy without alerting the opposing teams so they could take countermeasures) but it is worthy of note.

Our revised quorum rule, designed both to blunt that attack and also to be a simpler rule, is just

To win, a candidate must get at least half the score-sum as the candidate with the greatest score-sum.
This has an effect similar to the old 25% cutoff, but is less vulnerable to the write-in-conspiracy attack.

Why the "sliding scale"?

By making the threshold depend on the candidate with the greatest score-sum (or greatest number of numeric votes) we create a sliding scale so that even in races where no candidate is well known enough to get a lot of numeric scores, our criterion still does something sensible. I.e. at least one candidate will always be electable. (Just saying "25%" would have failed miserably in a race in which every candidate got below 25% numeric scores – then nobody could be elected!)

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