normalizedgame {TUGLab} | R Documentation |
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)
Arguments
v |
A characteristic function, as a vector. |
binary |
A logical value. By default, |
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
.
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.
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)