Instant Runoff Normalized Ratings: Difference between revisions

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Based on a ratings ballot, IRNR seeks to give every voter equal power and encourage honest ratings.

The first step is normalizing, which can happen in two ways:

Divide each rating by the sum of the absolute values of the ratings. The sum of absolute values of the ratings will then be 1.
- This shall be called IRNR[1] since the normalization factor is the L1 norm.

Divide each rating by the square root of the sum of the squared ratings. The vote will then be a vector with magnitude 1.
- This shall be called IRNR[2] since the normalization factor is the L2 norm.

One could more generally consider IRNR[p], based on the Lp norm, for any fixed real p with $1\leq p\leq \infty$ . (To avoid difficulties with dividing by 0, we agree to ignore votes that rank all candidates 0.)

Formula for IRNR[n] normalization:

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}  {C_{new}} =\frac{C_{old}}{\sqrt[n]{\sum \left(\bigl| C_{i}\bigr|^{n}\right)}}  \end{equation}}

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}{C_{old}}\end{equation}}
 = rating of candidate C in the vote, before the normalization.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}{C_{new}}\end{equation}}
 = rating of C, after the normalization.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}{C_{i}}\end{equation}}
 = ratings of each candidate in the vote, before the normalization.

Sum up the normalized ratings for each candidate. If there are two choices, the highest rated is the winner. If there are more than two choices, disqualify the lowest rated choice.

The process repeats with a normalization step that ignores disqualified choices. A voter's voting power is thus redistributed among the remaining choices.

If it were not for the "runoff," then generally the best strategy in IRNR[p] is simply to (strategically) plurality-vote, i.e. giving all candidates except one a rating of zero. This is true whenever there are two "frontrunner" candidates judged to be far more likely to win than the others and p is finite (then vote for the best among these two), and its truth is unaffected by the runoff by induction on rounds.

If p is infinite, IRNR without the runoff would just become equivalent to range voting in the range [-1, 1] with an extra rule demanding that the best- or worst-rated candidate must have a rating with absolute value 1. The best strategy is then the same as for approval voting and again this statement's validity is unaffected by adding the runoff.

Notes

It is possible to normalize by first observing the highest score the voter gave to any candidate, and pretending that is the maximum allowed score when interacting with that voter's ballot. In other words, a voter who gave their favorite a 3 out of 5 could have their ballot normalized such that the highest score they give to any candidate in any round of IRNR would be a max of 3 out of 5.

Related Systems

Distributed Voting (specific variant, based on L1 norm)

External link

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 * One could more generally consider '''IRNR[p]''', based on the Lp norm, for any fixed real p with <math>1 \le p \le \infty</math>. (To avoid difficulties with dividing by 0, we agree to ignore votes that rank all candidates 0.)
+Formula for '''IRNR[n]''' normalization:
+ <math>\begin{equation}
+ {C_{new}} =\frac{C_{old}}{\sqrt[n]{\sum \left(\bigl| C_{i}\bigr|^{n}\right)}}
+ \end{equation}</math>
+ <math>\begin{equation}{C_{old}}\end{equation}</math> = rating of candidate C in the vote, before the normalization.
+ <math>\begin{equation}{C_{new}}\end{equation}</math> = rating of C, after the normalization.
+ <math>\begin{equation}{C_{i}}\end{equation}</math> = ratings of each candidate in the vote, before the normalization.
 Sum up the normalized ratings for each candidate. If there are two choices, the highest rated is the winner. If there are more than two choices, disqualify the lowest rated choice.