TUGLab (version 0.0.1)

marginalgame: Marginal game

Description

Given a game and a coalition, this function returns the characteristic function of the corresponding marginal game.

Usage

marginalgame(v, S, binary = FALSE)

Value

The characteristic function of the S-marginal game, as a vector in binary order if binary=TRUE and in lexicographic order otherwise.

Arguments

v

Characteristic function, as a vector.

S

The position of a coalition, as an integer.

binary

A logical value. By default, binary=FALSE. Should be set to TRUE if v and S are introduced according to binary order instead of lexicographic order.

Details

Given a game \(v\in G^N\) and a coalition \(S\in 2^N\), the S-marginal game, \(v^S\in G^N\), is defined by $$v^S(R)=v(R\cup (N\backslash S))-v(N\backslash S)+v(R\cap (N\backslash S))\text{ for all }R\in 2^N.$$

References

Sánchez Rodríguez, E., Mirás Calvo, M.A., Quinteiro Sandomingo, C., & Núñez Lugilde, I. (2024). Coalition-weighted Shapley values. International Journal of Game Theory 53, 547-577.

Examples

v <- c(0, 0, 0, 2, 3, 10, 20)
marginalgame(v, 5, binary = TRUE) # coalition {1,3}
n <- 3
for (i in 1:(2^n - 1)) {
  cat("[", i, "]", paste(marginalgame(lex2bin(v),codebin2lex(n,i),binary=TRUE)),"\n")
}
for (i in 1:(2^n - 1)) {
  cat("[", i, "]", paste(marginalgame(v,i)),"\n")
}