nucleoluspcvalue {TUGLab}R Documentation

Per capita nucleolus

Description

Given a game, this function computes its per capita nucleolus.

Usage

nucleoluspcvalue(v, binary = FALSE, tol = 100 * .Machine$double.eps)

Arguments

v

A characteristic function, as a vector.

binary

A logical value. By default, binary=FALSE. Should be set to TRUE if v is introduced in binary order instead of lexicographic order.

tol

A tolerance parameter, as a non-negative number.
By default, tol=100*.Machine$double.eps.

Details

Given a game v\in G^N and an allocation x \in I(v), the per capita excess of each coalition S \in 2^{N} with respect to x is defined as

e^{p}(v,x,S) = \frac{v(S)-\sum_{i \in S}x_{i}}{|S|}.

The per capita excesses of all non-empty coalitions, sorted in non-increasing order, are stored in the per capita excesses vector, \theta^{p}(x). For any game v\in G^N with a non-empty set of imputations, the per capita nucleolus is defined as the only imputation pcn(v) \in I(v) that satisfies \theta^{p}(pcn(v))_{i} \leqslant \theta^{p}(y)_{i} for each i \in \{1,\dots,2^{N}-1\} and for all y \in I(v). This function is programmed following the algorithm of Potters, J.A., et al. (1996).

Value

The per capita nucleolus of the game, as a vector.

References

Grotte, J. (1970). Computation of and Observations on the Nucleolus, the Normalized Nucleolus and the Central Games. Master’s thesis), Cornell University, Ithaca.

Potters, J. A., Reijnierse, J. H., & Ansing, M. (1996). Computing the nucleolus by solving a prolonged simplex algorithm. Mathematics of Operations Research, 21(3), 757-768.

See Also

excesses, leastcore, nucleolusvalue, prenucleolusvalue

Examples

nucleoluspcvalue(c(1,5,10,6,11,15,16))
nucleoluspcvalue(c(0,0,0,30,30,80,100))

# Computing the per capita nucleolus of a random essential game:
n <- 10 # number of players in the game
v <- c(rep(0,n),runif(2^n-(n+1),min=10,max=20)) # random essential game
nucleoluspcvalue(v)

# What if the game is a cost game?
cost.v <- airfieldgame(c(1,5,10,15)) # cost game
-nucleoluspcvalue(-cost.v) # per capita nucleolus of the cost game

[Package TUGLab version 0.0.1 Index]