appendMissingCoalitions {socialranking}R Documentation

Append missing coalitions

Description

Append an equivalence class to a power relation with all coalitions of elements that do not appear in the power relation.

Usage

appendMissingCoalitions(powerRelation, includeEmptySet = TRUE)

Arguments

powerRelation

A PowerRelation object created by PowerRelation() or as.PowerRelation()

includeEmptySet

If TRUE, include the empty set in the last equivalence class if it is missing from the power relation.

Details

For a given set of elements N={1,...,n}N = \lbrace 1, ..., n \rbrace, a PowerRelation object describes a total preorder of its subsets, or coalitions, P2N\mathcal{P} \subseteq 2^N, where 2N2^N is the superset of elements.

If P2N\mathcal{P} \neq 2^N, that means that there are some coalitions S2N,SPS \in 2^N, S \notin \mathcal{P}, such that we cannot compare STS \succsim T or TST \succsim S for every TPT \in \mathcal{P}.

This may be caused by 2N2^N having too many coalitions to consider. In certain cases, it may be more interesting to only consider the top ranking coalitions and "shoving" all remaining coalitions into the back.

For this use-case, appendMissingCoalitions() takes the set 2NP2^N \setminus \mathcal{P} and attaches it in form of an equivalence class to the back of the power relation.

I.e., take as an example 1213(12)12 \succ 13 \succ (1 \sim 2). Here, we have

2N={123,12,13,23,1,2,3,}P={12,13,1,2}2NP={123,23,3,}. \begin{aligned} 2^N &= \lbrace 123, 12, 13, 23, 1, 2, 3, \emptyset \rbrace\\ \mathcal{P} &= \lbrace 12, 13, 1, 2 \rbrace\\ 2^N \setminus \mathcal{P} &= \lbrace 123, 23, 3, \emptyset \rbrace . \end{aligned}

Adding the missing coalitions to the power relation then gives us 1213(12)(123233)12 \succ 13 \succ (1 \sim 2) \succ (123 \sim 23 \sim 3 \sim \emptyset).

Value

PowerRelation object containing the following values:

See Also

Other helper functions for transforming power relations: makePowerRelationMonotonic()

Examples

pr <- as.PowerRelation(list(c(1,2), 3))
# 12 > 3

appendMissingCoalitions(pr)
# 12 > 3 > (123 ~ 13 ~ 23 ~ 1 ~ 2 ~ {})

appendMissingCoalitions(pr, includeEmptySet = FALSE)
# 12 > 3 > (123 ~ 13 ~ 23 ~ 1 ~ 2)


[Package socialranking version 1.2.0 Index]