TUGLab (version 0.0.1)

normalizedgame: Normalized game

Description

Given a game, this function returns the characteristic function of its 0-1-normalization, its 0-(-1) normalization or its 0-0 normalization, as appropriate.

Usage

normalizedgame(v, binary = FALSE)

Value

The characteristic function of the 0-1-normalized game, the 0-(-1) normalized game or the 0-0 normalized game; as a vector in binary order if binary=TRUE and in lexicographic order otherwise.

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.

Details

A game $v\in G^N$ is: 0-1 normalized if $v(i)=0$ for all $i\in N$ and $v(N)=1$; 0-0 normalized if $v(i)=0$ for all $i\in N$ and $v(N)=0$; and 0-(-1) normalized if $v(i)=0$ for all $i\in N$ and $v(N)=-1$.

If $v(N)>\sum _{i\in N}v(i)$, the 0-1 normalized game of $v$, $v_{0,1}\in G^N$, is defined by $$v_{0,1}(S)=\frac{v(S)-\sum _{i\in S}v(i)}{v(N)-\sum _{i\in N}v(i)}$$ for all $S\in 2^N$.

If $v(N)<\sum _{i\in N}v(i)$, the 0-(-1) normalized game of $v$, $v_{0,-1}\in G^N$, is defined by $$v_{0,-1}(S)=-\frac{v(S)-\sum _{i\in S}v(i)}{v(N)-\sum _{i\in N}v(i)}$$ for all $S\in 2^N$.

If $v(N)=\sum _{i\in N}v(i)$, the 0-0 normalized game of $v$, $v_{0,0}\in G^N$, is defined by $$v_{0,0}(S)=v(S)-\sum _{i\in S}v(i)$$ for all $S\in 2^N$.

See Also

strategicallyequivalentcheck, zeronormalizedcheck, zeronormalizedgame

Examples

v <- c(1, 5, 11, 6, 11, 15, 16)
normalizedgame(v, binary = TRUE)
w <- c(4, 3, 8, 16, 17, 18, 15)
normalizedgame(w)
z <- c(2,3,5,10,12,14,5)
normalizedgame(z)