# Difference between revisions of "Instant Runoff Normalized Ratings"

m (State author in the introduction, and fix some grammar/tense errors.) |
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− | Based on a [[ratings ballot]], |
+ | '''Instant Runoff Normalized Ratings''', or '''IRNR''' is a method devised by Brian Olson.<ref>{{cite web | title=Election Methods Defined | website=bolson.org | url=https://bolson.org/voting/methods.html#IRNR | ref={{sfnref | bolson.org}} | access-date=2021-12-18}}</ref> 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 |
+ | 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. |
* 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. |
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− | This shall be called IRNR[1] since the normalization factor is the L1 norm. |
+ | ** 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. |
* Divide each rating by the square root of the sum of the squared ratings. The vote will then be a vector with magnitude 1. |
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− | This shall be called IRNR[2] since the normalization factor is the L2 norm. |
+ | ** This shall be called '''IRNR[2]''' since the normalization factor is the L2 norm. |

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− | <math>1 \le p \le \infty</math> |
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− | TeX bug here |
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− | 1<=p<=infinity. |
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− | (To avoid difficulties with dividing by 0, we agree to ignore votes that rank all candidates 0.) |
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+ | Formula for '''IRNR[n]''' normalization: |
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+ | <math>\begin{equation} |
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+ | {C_{new}} =\frac{C_{old}}{\sqrt[n]{\sum \left(\bigl| C_{i}\bigr|^{n}\right)}} |
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+ | \end{equation}</math> |
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+ | <math>\begin{equation}{C_{old}}\end{equation}</math> = rating of candidate C in the vote, before the normalization. |
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+ | <math>\begin{equation}{C_{new}}\end{equation}</math> = rating of C, after the normalization. |
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+ | <math>\begin{equation}{C_{i}}\end{equation}</math> = ratings of each candidate in the vote, before the normalization. |
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− | If it were not for the "runoff," then |
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− | generally the best strategy in IRNR[p] is simply to (strategically) plurality-vote, i.e. |
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− | giving all candidates except one a rating of zero. |
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− | This is true whenever there are two "frontrunner" candidates judged to be far more likely |
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− | to win than the others and p is finite |
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− | (then vote for the best among these two), |
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− | and its truth is unaffected by the runoff by induction |
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− | on rounds. |
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− | If p is infinite, IRNR without the runoff would just become equivalent to [[range voting]] |
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− | in the range [-1, 1] with an extra rule demanding that the best- or worst-rated |
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− | candidate must have a rating with absolute value 1. |
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− | The best strategy is then the same as for [[approval voting]] and again this statement's |
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− | validity is unaffected by adding the runoff. |
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− | == |
+ | == 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. |
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+ | ==Related systems== |
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+ | * [[Distributed Voting]] (specific variant, based on L1 norm) |
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+ | |||

+ | == External links == |
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* [http://bolson.org/voting/vote_util/org/bolson/vote/IRNR.java Java code that implements IRNR] |
* [http://bolson.org/voting/vote_util/org/bolson/vote/IRNR.java Java code that implements IRNR] |
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+ | *[http://bolson.org/voting/IRNR_explaination.pdf Instant Runoff Normalized Ratings: an Election Method by Brian Olson] |
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+ | |||

+ | ==References== |
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+ | <references/> |
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+ | |||

− | [[Category:Single-winner voting |
+ | [[Category:Single-winner voting methods]] |

+ | [[Category:Cardinal voting methods]] |
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+ | [[Category:Sequential loser-elimination methods]] |

## Latest revision as of 15:04, 18 December 2021

**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:

= rating of candidate C in the vote, before the normalization. = rating of C, after the normalization. = 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[edit | edit source]

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[edit | edit source]

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

## External links[edit | edit source]

## References[edit | edit source]

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