Given two elements \(i\) and \(j\), go through each coalition \(S \in 2^{N \setminus \lbrace i, j \rbrace}\).
\(D_{ij}(\succsim)\) then contains all coalitions \(S\) where
\(S \cup \lbrace i \rbrace \succsim S \cup \lbrace j \rbrace\) and \(D_{ji}(\succsim)\) contains all coalitions where
\(S \cup \lbrace j \rbrace \succsim S \cup \lbrace i \rbrace\).
The cardinalities
\(d_{ij}(\succsim) = |D_{ij}|\) and
\(d_{ji}(\succsim) = |D_{ji}|\) represent the score of the two elements, where
\(i \succ j\) if \(d_{ij}(\succsim) > d_{ji}(\succsim)\) and
\(i \sim j\) if \(d_{ij}(\succsim) == d_{ji}(\succsim)\).
cpMajorityComparison() tries to retain all that information. The list returned contains the following information.
Note that in this context the two elements \(i\) and \(j\) refer to element 1 and element 2 respectively.
$e1: list of information about element 1
$e1$name: name of element 1
$e1$score: score \(d_{ij}(\succsim)\). \(d_{ij}(\succ)\) if strictly == TRUE
$e1$winningCoalitions: list of coalition vectors \(S \in D_{ij}(\succsim)\). \(S \in D_{ij}(\succ)\) if strictly == TRUE
$e2: list of information about element 2
$e2$name: name of element 2
$e1$score: score \(d_{ji}(\succsim)\). \(d_{ji}(\succ)\) if strictly == TRUE
$e1$winningCoalitions: list of coalition vectors \(S \in D_{ji}(\succsim)\). \(S \in D_{ji}(\succ)\) if strictly == TRUE
$winner: name of higher scoring element. NULL if they are indifferent.
$loser: name of lower scoring element. NULL if they are indifferent.
$tuples: a list of coalitions \(S \in 2^{N \setminus \lbrace i, j \rbrace }\) with:
$tuples[[x]]$coalition: vector, the coalition \(S\)
$tuples[[x]]$included: logical, TRUE if \(S \cup \lbrace i \rbrace\) and \(S \cup \lbrace j \rbrace\) are in the power relation
$tuples[[x]]$winner: name of the winning element \(i\) where \(S \cup \lbrace i \rbrace \succ S \cup \lbrace j \rbrace\). It is NULL if \(S \cup \lbrace i \rbrace \sim S \cup \lbrace j \rbrace\)
$tuples[[x]]$e1: index \(x_1\) at which \(S \cup \lbrace i \rbrace \in \sum_{x_1}\)
$tuples[[x]]$e2: index \(x_2\) at which \(S \cup \lbrace j \rbrace \in \sum_{x_2}\)
The much more efficient cpMajorityComparisonScore() only calculates $e1$score.
Unlike Lexcel, Ordinal Banzhaf, etc., this power relation can introduce cycles. For this reason the function
cpMajorityComparison() and cpMajorityComparisonScore() only offers direct comparisons between two elements
and not a ranking of all players. See the other CP-majority based functions that offer a way to rank all players.