FAIR-V

First-Approval Instant-Runoff (FAIR) is a Single-Winner Cardinal voting systems proposed by Aldo Tragni.

The objectives of this voting system is the balance between simplicity, resistance to strategies, and elect the utilitarian winner.

Procedure
Ballot use range [0,3].

Eliminate the candidate considered worst by the most voters.

The above elimination is repeated until only one candidate is left, who is the winner.

Note: worst candidates for a voter are those (among the remaining) with the lowest rating in his ballot.

The original idea was FAIR-Max, which was then simplified to FAIR-V.

Normalization
Norm: set -1 the minimum value of the ballot to normalize, and the others all to 0.

The actual FAIR-V algorithm uses this norm and eliminates the candidate with the lowest point sum.

This norm is applicable also to ranges with more than 4 ratings.

Name derivation
First-Approval Instant-Runoff Voting:


 * "First-Approval": the vote is initially treated as a multiple-choice. However, if all the worst candidates are eliminated in a vote, then the initial multiple-choice is reduced and can become a single-choice, during the count. It's like a single-choice (refer to FPTP) masked at the beginning by multiple-choice (refer to Approval voting).
 * "Instant-Runoff": refers to the fact that, by eliminating one candidate at a time, only two will remain at the end, obtaining the "Instant-Runoff" (comparison of the top two candidates).

FAIR-nV: the FAIR-V norm works with ranges of different sizes and n indicates the number of ratings used in the range, minus 1.


 * FAIR-1V: it's equivalent to AV, with ratings in [0,1].
 * FAIR-V: is the default definition, with ratings in [0,3].
 * FAIR-5V: uses ratings in [0,5].
 * FAIR-9V: uses ratings in [0,9].

FAIR-Max
It's FAIR-V, with range [0,3], in which:


 * the elimination ends when 2 candidates remain (finalists).
 * in each vote, if 1 of the 2 finalists has obtained rating 0, then his opponent receives rating 3.
 * the finalist with the highest sum of ratings is the winner.

FAIR-Max resists maximization strategies (like FAIR-V), elect the utilitarian winner (much more than FAIR-V), but it's more complicated to explain (than FAIR-V).

FAIR-S: by removing step 2 from the list, the method resists a little less to maximization strategy, but becomes simpler and remains utilitarian.

mdM norm (min-do-Max norm)
Given a range vote, with 2 candidates:


 * if one of the two candidates has the minimum rating of the range, then his opponent receives the maximum rating of the range (no changes are made if both or none have a minimum rating).

Mdm norm (Max-do-min norm)
Given a range vote, with 2 candidates:


 * if one of the two candidates has the maximum rating of the range, then his opponent receives the minimum rating of the range (no changes are made if both or none have a maximum rating).

mM-do-Mm norm
Apply both mdM and Mdm norms.

Min-maxing
Properties of FAIR-V:
 * increase rating of the candidate X in one vote doesn't change the chance of victory for candidates rated below the old rating of X.
 * decreasing rating of the candidate X in one vote doesn't change the chance of victory for candidates rated below the new rating of X.

These properties mean that in FAIR-V a voter cannot favor a candidate more than the worst ones, by increasing his rating (avoid the strategy of maximization but not the one of minimization).

Example, given this honest vote: A[0] B[1] C[2] D[3]  -->   Normalized: A[-1] B[0] C[0] D[0] if the voter only wanted to increase the chance of victory of B,C,D with respect to A, then vote like this is useless: A[0] B[3] C[3] D[3]  -->   Normalized: A[-1] B[0] C[0] D[0]

Voting lesser of two evils
Consider 2 frontrunners F1 and F2, among which the voter considers F1 > F2.

The properties indicated in the previous section ensure that the only interest of the voter is to decrease the rating of F2, leaving the rating of F1 unchanged.

If only F1 and F2 remain at the end of the count, it's sufficient only that those have two different ratings to ensure that the weight of the vote is maximum in favor of F1. This specifically ensures that F1 receives rating 0 if it's the worst candidate among all, or receives 1 if there are candidates much worse than F1 but not frontrunners (minorities); in both cases, the vote would remain very honest.

Monotonicity failure
Using the Yee diagram it was possible to observe that FAIR-V procedure is extremely resistant to the failure of monotonicity, so the Push-over strategy can be considered practically absent.

IRV
IRV assigns 1 point to the candidate with the highest rating, while FAIR-V assigns -1 point to the candidate with the lowest rating; both eliminate at each step the candidate with the lowest sum of points.

A big difference between the 2 types of counting is that in FAIR-V the failure of monotonicity is practically absent, while IRV is one of the systems in which it's most present.

Coombs
Use ranked votes instead of range [0,3] and win the candidate with the absolute majority, if there is after an elimination.