Outcome error

Outcome error is the difference between what voters want and what they actually get. It can apply to individuals (what one person wanted vs. what they got) as well as being summed all participants in an election. The latter gives an idea of how well the voting system serves as a way of determining how well any given voting system serves the votership in any given circumstance.

Note that "want" is much more than just "I want Option X to win". Most voters have some idea of not only what outcome they'd most prefer, but also a range of other outcomes which would be acceptable, others which would be tolerable, and still others which would be bad or even absolutely unacceptable. In the event that Option X does win the election, the degree of individual outcome error goes up as we move from "preferred" to "absolutely unacceptable".

Range voting alone allows an unambiguous expression of this complexity for each of the possible options, allowing compromise so that (e.g.) an option that most people agree is tolerable could prevail, despite violent disagreement about which option would be best.

Terminology

• individual outcome error (for one voter): "individual outcome error", or E_i
• collective outcome error across all voters: "summed outcome error", or E_all

Summing

There are at least a couple of different reasonable ways of calculating E_all. My default is to use RMS:

SUM((Ei^2)^0.5) (summed across all votes for a given issue)

In any system where voters can express a position on more than one candidate (this applies both to range voting and ranked voting), we also need to sum up those positions to determine how much they disagree with the winning choice. This calculation depends on the meaning of those expressions, or (more precisely) what meaning the voter is trying to express through them.

Range Voting

In the case of range voting with a range from -X to +X, the meaning is pretty obvious: minus X indicates the strongest disapproval, and plus X represents the strongest approval.

Another question which arises is whether we count the voter's opinions on choices which didn't win.

Example, where we use a range from -10 to +10:

option →	A	B	C
voter 1 ratings:	+10	0	-10
voter 2 ratings:	+5	+10	-5

Let's play out the scenario where A wins:

• Voter 1 would seem to be completely satisfied. They're not going to object that B didn't win, because they were lukewarm about it, and they're delighted that C didn't win either. It seems fair to say that they're 100% satisfied with this outcome.
• Voter 2, however, is less sanguine. One of the options they liked (A) did win, and the one option they disliked (C) did not -- but there was another option they liked better (B) which didn't win. It seems fair to say that they have a positive feeling about the outcome, but less than 100%.

For the sake of working out an algorithm to calculate this:

• Let's call the voter's ranking for each option the "option's voter-rating", O_v.
• Let's call the value assigned to each option to express how it did in the election the "option's outcome-value" (O_o)

An algorithm which seems to result in this unitary evaluation of each voter's satisfaction would be:

• Each losing option gets an outcome value (O_o) of -10.
• Any winning option gets a O_o of +10.
• We then take the difference between O_o (the outcome for each option) and O_v (how the voter felt about that option), and sum them across all options - and since we want the magnitude of the disagreements to sum, we'll use RMS again:

  SQRT(SUM((Ov)2-(Oo)2) (summed across all votes for a given issue) - dissatisfaction rating (0 = not at all dissatisfied; positive values = more dissatisfied)

Let's add some more voters to illustrate additional forms of dissatisfaction:

• voter 3: really disliked the option which won (A), but wasn't super-crazy about the others
• voter 4: was dead-set on B, hated A, neutral on C
• voter 5: was dead-set on B, hated both A and C

Adding those voters and this algorithm's results to the table, we get:

option →	A	B	C	SUM(δ^2)	SQRT()
Oo (outcome value):	+10	-10	-10
voter 1 Ov:	+10	0	-10
Oo-Ov:	0	-10	0	(0+100+0)=100	10
voter 2 Ov:	+5	+10	-5
Oo-Ov:	+5	-20	-5	(25+400+25)=450	21.2
voter 3 Ov:	-10	0	0
Oo-Ov:	+20	-10	-10	(400+100+100)=600	24.5
voter 4 Ov:	-10	+10	0
Oo-Ov:	+20	-20	-10	(400+400+100)=900	30
voter 5 Ov:	-10	+10	-10
Oo-Ov:	+20	-20	0	(400+400+0)=800	28.3

This does correctly reflect the fact that voter 1 is happier with the outcome than voter 2, that voter 3 is less happy than voter 2, that voter 4 is even less happy, and that voter 5 is slightly happier than voter 4 because they were really worried about an option which didn't win (whereas 4 didn't care much) but still less content than voter 3 (narrative unclear on this one; will have to think about it).

NTS: should probably check these figures over again.