# Cardinal-weighted pairwise comparison

Cardinal-weighted pairwise comparison (also known as "Cardinal pairwise" or "CWP") is a method that uses rated ballots, but also evaluates the implicit ranking information of the candidates on each ballot.

## History

This method was first proposed by James Green-Armytage in June of 2004.[1] Green-Armytage then published a paper in Voting Matters in November 2004.[2]

## Definition

Cardinal pairwise differs from traditional pairwise count methods (Condorcet methods) in that it uses cardinal (rating) information in addition to ordinal (ranking) information.

CWP uses the ordinal information to determine the direction of pairwise defeats, exactly as most Condorcet methods do. However, it uses the cardinal information to determine the strength of the pairwise defeats.

Thus, in essence, CWP can be thought of as a defeat strength definition. If A pairwise defeats B, CWP finds the strength of the defeat as follows:

For each voter who ranks A over B, and only for these voters, subtract B's rating from A's rating, to get the rating differential. Sum these rating differentials to get the defeat strength.

This is akin to a cardinal version of winning votes, since only the preferences of voters who prefer the winner of the matchup are counted.

The name "cardinal pairwise" also implies that a Smith-efficient, defeat-dropping base method will be used, for example Schulze, ranked pairs, or river.

### Ballot types

1. One way to ballot for CWP is to have a separate ordinal and cardinal ballot, and to require that if a voter gives candidate R a higher rating than candidate S, then that voter must also give candidate R a higher ranking than candidate S.

2. A simpler way to ballot for CWP is to use only a cardinal ballot, and to derive the ordinal information from the cardinal information. The only disadvantage of this is that it creates an additional compromising-compression incentive not found in the first version. However, this additional incentive should be extremely minor if the scale is sufficiently fine.

For example, assume that the scale consists of integers from 0 to 100. If my sincere preferences are J>K>L, and I want to make the J>K defeat as weak as possible while making the K>L defeat as strong as possible, I can vote J:100, K:99, L:0. There is only a very small temptation to vote J: 100, K:100, L:0. This temptation can be reduced even further by allowing decimal ratings, e.g. J:100, K:99.99, L:0. It can be mostly eliminated by allowing voters to use an approval threshold to indicate their cardinal support.

## Approval-weighted pairwise

"Approval weighted pairwise", "AWP", or "approval pairwise" is the special case of cardinal pairwise in which the only available ratings are 0 and 1. AWP can use a ranked ballot with an approval cutoff.

## Example

10 A:10 B:2

9 B:10 A:0

A Pairwise beats B 10 voters to 9, and has a rating differential of 80 points (equivalent to 8 votes on a scale of 0 to 10) against B.

## References

1. Author: James Green-Armytage (jarmyta at antioch-college.edu), Thread: "approval vs. IRV: false claim about MMC examples?", Publisher: election-methods list, Date: Sun Jun 6 18:57:01 PDT 2004
2. Green-Armytage, James, "Cardinal-weighted pairwise comparison", Date: November 2004, Journal-issue: Voting Matters: Volume 19, Publisher: Voting Matters, Current link: https://www.votingmatters.org.uk/ISSUE19/I19P2.PDF , Old link: http://www.mcdougall.org.uk/VM/ISSUE19/I19P2.PDF