# 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 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.

- This shall be called

- 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.

- This shall be called

- One could more generally consider
**IRNR[p]**, based on the Lp norm, for any fixed real p with . (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 (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_{old}}\end{equation}}= rating of candidate C in the vote, before the normalization.Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}{C_{new}}\end{equation}}= rating of C, after the normalization.Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}{C_{i}}\end{equation}}= 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.

## Related systems

- Distributed Voting (specific variant, based on L1 norm)

## External links

## References

- ↑ "Election Methods Defined".
*bolson.org*. Retrieved 2021-12-18.