[EM] A CMJ-like numeric tiebreaker equivalent to MJ

[Jameson Quinn]

CMJ (continuous majority judgment) is an MJ-like system using the numeric
tiebreaker*:

Median¹ + ((Votes_above_median - Votes_below_median) / (2 *
Votes_at_median))

The CMJ formula is actually a simplified version of the trimmed-mean-like
formula:

Median + ((Votes_above_median - Votes_below_median) / (Votes_at_median +
|Votes_above_median - Votes_below_median|))

This full version is not equivalent algebraically, and it gives slightly
different scores; but it is fully equivalent in all cases in terms of the
order of candidates and specifically the winner.

A similar formula can be given for MJ itself:

Median + ((Votes_above_median - Votes_below_median) / |Votes_above_median -
Votes_below_median|) * ((Votes_at_median - |Votes_above_median -
Votes_below_median| + 2) / (4 * ((Votes_at_median - |Votes_above_median -
Votes_below_median| + 1))

This is equivalent to a subset of the MJ tiebreaker. That is, if it breaks
a tie it always does so in accordance with MJ, and it breaks most of the
ties that MJ does; but if the MJ tiebreaker has to cross more than one
rating boundary to break the tie, then this formula will not break the tie.

I suspect that the above formula could be simplified further.

Jameson Quinn

¹ "Median" is the median grade, converted to an integer
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