leastcore {TUGLab} | R Documentation |
Least core
Description
Given a game, this function computes its least core.
Usage
leastcore(v, binary = FALSE, tol = 100 * .Machine$double.eps)
Arguments
v |
A characteristic function, as a vector. |
binary |
A logical value. By default, |
tol |
A tolerance parameter, as a non-negative number. |
Details
Given a game v\in G^N
and a number \varepsilon \in \mathbb{R}
, the \varepsilon
-core of v
is defined as
C_{\varepsilon}(v)= \{ x\in \mathbb{R}^n : x(N)=v(N) \text{ and } x(S) \ge v(S)-\varepsilon \ \forall S \in 2^N \setminus \{\emptyset,N\} \},
where x(S)=\sum_{i\in S} x_i
.
The least core of v
is defined as the intersection of all non-empty \varepsilon
-cores of v
:
LC(v) = \{ \bigcap_{\varepsilon \in \mathbb{R} \ : \ C_{\varepsilon}(v) \neq \emptyset} C_{\varepsilon}(v) \}.
The implementation of this function is based on the algorithm presented in Derks and Kuipers (1997) and on the MATLAB package WCGT2005 by J. Derks.
Value
This function returns four outputs:
t |
The excess value that defines the least core. |
sat |
The positions (binary order positions if |
x |
A least core allocation, as a vector. |
vt |
The game whose core is the least core of |
References
Derks, J. & Kuipers, J. (1997). Implementing the simplex method for computing the prenucleolus of transferable utility games.
Software by J. Derks (Copyright 2005 Universiteit Maastricht, dept. of Mathematics), available in package MatTuGames,
https://www.shorturl.at/i6aTF.
See Also
excesses, nucleoluspcvalue, nucleolusvalue, prenucleolusvalue
Examples
v <- c(0,0,0,0,10,40,30,60,10,20,90,90,90,130,160)
( vt <- leastcore(v)$vt )
# Plotting the core and the least core of v:
plotcoresets(games = rbind(v,vt), imputations = FALSE)
# What if the game is a cost game?
cost.v <- c(2,2,2,3,4,4,5) # characteristic function of the cost game
-leastcore(-cost.v)$t # the excess value that defines the least core of cost.v
leastcore(-cost.v)$sat # the saturated coalitions
-leastcore(-cost.v)$x # a least core allocation
-leastcore(-cost.v)$vt # the cost game whose core is the least core of cost.v