convexcheck {TUGLab} | R Documentation |
Convex check
Description
This function checks if the given game is convex.
Usage
convexcheck(v, binary = FALSE, instance = FALSE)
Arguments
v |
A characteristic function, as a vector. |
binary |
A logical value. By default, |
instance |
A logical value. By default, |
Details
A game v\in G^N
is convex if v(S \cap T) + v(S \cup T) \ge v(S)+v(T)
for all
S,T \in 2^N
. Zumsteg, S. (1995) shows that v
is convex if v(S \cup {i}\cup {j}) + v(S) \ge v(S\cup {i})+v(S\cup {j})
for all
S\in 2^N
and i,j\in N\backslash S
such that i\not=j
.
A game v\in G^N
is concave if -v
is convex.
Value
TRUE
if the game is convex, FALSE
otherwise. If instance=TRUE
and the game is not convex, the function also returns the positions (binary order positions if binary=TRUE
; lexicographic order positions otherwise) of a pair of coalitions violating the Zumsteg convexity characterization.
References
Zumsteg, S. (1995). Non-cooperative aspects of cooperative game theory and related computational problems. PhD thesis, ETH Zurich.
See Also
strategicallyequivalentcheck, superadditivecheck
Examples
v1 <- c(5, 2, 2, 1, 8, 8, 6, 4, 3, 3, 12, 10, 10, 6, 14)
convexcheck(v1)
v2 <- c(0, 0, 0, 2, 2, 1, 3)
convexcheck(v2, binary = FALSE, instance = TRUE)
# How to check if a game is concave:
v.conc <- c(4, 3, 3, 2, 6, 6, 5, 5, 4, 4, 7, 6, 6, 6, 7) # concave game
convexcheck(-v.conc)