# Instant Runoff Normalized Ratings

Instant Runoff Normalized Ratings, or IRNR is a method devised by Brian Olson.[1] Based on a ratings ballot, the method seeks to give every voter equal power and encourage honest ratings.

The first step is normalizing, which acts to equalize the effect of a ballot. Some ways of doing this are:

• 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 is 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 is 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 ${\displaystyle 1\leq p\leq \infty }$. (To avoid difficulties with dividing by 0, we agree to ignore votes that rank all candidates 0.)

Formula for IRNR[n] normalization:

${\displaystyle {C_{new}}={\frac {C_{old}}{\sqrt[{n}]{\sum \left({\bigl |}C_{i}{\bigr |}^{n}\right)}}}}$
${\displaystyle {C_{old}}}$ = rating of candidate C in the vote, before the normalization.
${\displaystyle {C_{new}}}$ = rating of C, after the normalization.
${\displaystyle {C_{i}}}$ = ratings of each candidate in the vote, before the normalization.


The second step consists of summing 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, the lowest rated choice is disqualified.

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

## 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.