# Difference between revisions of "Instant Runoff Normalized Ratings"

Aldo Tragni (talk | contribs) (Added normalization formula) |
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− | Based on a [[ratings ballot]], IRNR seeks to give every voter equal power and encourage honest ratings. |
+ | 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 |
+ | 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 |
+ | * 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 '''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 equal to 1. |
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+ | * 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. |
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+ | ** This shall be called '''IRNR[2]''' since the normalization factor is the L2 norm. |
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− | * One could more generally consider IRNR[p] for any fixed real p with 1<=p<=infinity. |
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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.) |
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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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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. |
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. |
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The process repeats with a normalization step that ignores disqualified choices. A voter's voting power is thus redistributed among the remaining choices. |
The process repeats with a normalization step that ignores disqualified choices. A voter's voting power is thus redistributed among the remaining choices. |
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+ | 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. |
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− | Unfortunately usually 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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+ | 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. |
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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. (Then vote for the best among these two.) |
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+ | == Notes == |
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− | If p is infinite IRNR just becomes equivalent to [[range voting]] |
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+ | 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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− | 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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+ | ==Related Systems == |
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+ | * [[Distributed Voting]] (specific variant, based on L1 norm) |
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== External link == |
== External link == |
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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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− | [[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 12:00, 29 August 2020

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.

- 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.

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[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)