"Cardinal pairwise" and "CWP" are shorter names for "cardinal-weighted pairwise comparison", a method first proposed by James Green-Armytage in June of 2004.
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.
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.
"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.