Calculate the Kramer-Simpson-like scores. Lower scores are better.

kramerSimpsonRanking returns the corresponding ranking.

## Usage

kramerSimpsonScores(powerRelation, elements = NULL, compIvsI = FALSE)

kramerSimpsonRanking(powerRelation, compIvsI = FALSE)

## Arguments

powerRelation

A PowerRelation object created by newPowerRelation()

elements

vector of elements of which to calculate their scores. If elements == NULL, create vectors for all elements in pr$elements compIvsI If TRUE, include CP-Majority comparison $$d_{ii}(\succeq)$$, or, the CP-Majority comparison score of an element against itself, which is always 0. ## Value Score function returns a list of type KramerSimpsonScores and length of powerRelation$elements

(unless parameter elements is specified). Lower scoring elements are ranked higher.

Ranking function returns corresponding SocialRankingSolution object.

## Details

Inspired by the Kramer-Simpson method of social choice theory (Simpson 1969) (Kramer 1975) , the Kramer-Simpson-like method compares each element against all other elements using the CP-Majority rule.

For a given element $$i$$ calculate the cpMajorityComparisonScore against all elements $$j$$, $$d_{ji}(\succeq)$$ (notice that $$i$$ and $$j$$ are in reverse order). $$\max_{j \in N \setminus \lbrace i \rbrace}(d_{ji}(\succeq))$$ then determines the final score, where lower scoring elements are ranked higher.

## Note

By default this function does not compare $$d_{ii}(\succeq)$$. In other terms, the score of every element is the maximum CP-Majority comparison score against all other elements.

This is slightly different from definitions found in (Allouche et al. 2020) . Since by definition $$d_{ii}(\succeq) = 0$$ always holds, the Kramer-Simpson scores in those cases will never be negative, possibly discarding valuable information.

For this reason kramerSimpsonScores and kramerSimpsonRanking includes a compIvsI parameter that can be set to TRUE if one wishes for $$d_{ii}(\succeq) = 0$$ to be included in the comparisons. Put into mathematical terms, if:

 compIvsI Score definition FALSE $$\max_{j \in N \setminus \lbrace i \rbrace}(d_{ji}(\succeq))$$ TRUE $$\max_{j \in N}(d_{ji}(\succeq))$$

## References

Allouche T, Escoffier B, Moretti S, Öztürk M (2020). “Social Ranking Manipulability for the CP-Majority, Banzhaf and Lexicographic Excellence Solutions.” In Bessiere C (ed.), Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, 17--23. doi:10.24963/ijcai.2020/3 , Main track.

Simpson PB (1969). “On defining areas of voter choice: Professor Tullock on stable voting.” The Quarterly Journal of Economics, 83(3), 478--490.

Kramer GH (1975). “A dynamical model of political equilibrium.” Journal of Economic Theory, 16(2), 310--334.

Other CP-majority based functions: copelandScores(), cpMajorityComparison()

Other score vector functions: copelandScores(), cumulativeScores(), lexcelScores(), ordinalBanzhafScores()

## Examples

# 2 > (1 ~ 3) > 12 > (13 ~ 23) > {} > 123
pr <- newPowerRelation(
2,
">", 1,
"~", 3,
">", c(1,2),
">", c(1,3),
"~", c(2,3),
">", c(),
">", c(1,2,3)
)

# get scores for all elements
# cpMajorityComparisonScore(pr, 2, 1) = 1
# cpMajorityComparisonScore(pr, 3, 1) = -1
# therefore the Kramer-Simpson-Score for element
# 1 = 1
#
# Score analogous for the other elements
# 2 = -1
# 3 = 2
kramerSimpsonScores(pr)
#> $1 #> [1] 1 #> #>$2
#> [1] -1
#>
#> $3 #> [1] 2 #> #> attr(,"class") #> [1] "KramerSimpsonScores" # get scores for two elements # 1 = 1 # 3 = 2 kramerSimpsonScores(pr, c(1,3)) #>$1
#> [1] 1
#>
#> $3 #> [1] 2 #> #> attr(,"class") #> [1] "KramerSimpsonScores" # or single element # result is still a list kramerSimpsonScores(pr, 2) #>$2
#> [1] -1
#>
#> attr(,"class")
#> [1] "KramerSimpsonScores"

# note how the previous result of element 2 is negative.
# If we compare element 2 against itself, its max score will be 0
kramerSimpsonScores(pr, 2, compIvsI = TRUE)
#> \$2
#> [1] 0
#>
#> attr(,"class")
#> [1] "KramerSimpsonScores"

# 2 > 1 > 3
kramerSimpsonRanking(pr)
#> 2 > 1 > 3