TUGLab (version 0.0.1)

corecentervalue: Core-center

Description

Given a game, this function computes its core center.

Usage

corecentervalue(v, binary = FALSE, tol = 1e-12)

Value

The core-center, as a vector.

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.

tol

A tolerance parameter, as a non-negative number. By default, tol=1e-12.

Details

The core of a game \(v\in G^N\) is the set of all its stable imputations: $$C(v)=\{x\in\mathbb{R}^n : x(N)=v(N), x(S)\ge v(S)\ \forall S \in 2^N\},$$ where \(x(S)=\sum_{i\in S} x_i\). A game is said to be balanced if its core is not empty.

The core-center of a balanced game \(v\), \(CC(v)\), is defined as the expectation of the uniform distribution over \(C(v)\), and thus can be interpreted as the centroid or center of gravity of \(C(v)\). Let \(\mu\) be the \((n-1)\)-dimensional Lebesgue measure and let \(V(C)=\mu(C(v))\) be the volume (measure) of the core. If \(V(C)>0\), then, for each \(i\in N\), $$CC_i(v)=\frac{1}{V(C)}\int_{C(v)}x_i d\mu$$.

References

Gonzalez-Díaz, J. & Sánchez-Rodríguez, E. (2007). A natural selection from the core of a TU game: the core-center. International Journal of Game Theory, 36(1), 27-46.

See Also

balancedcheck, corecenterhitrun, coredimension, corevertices, corevertices234

Examples

v1 <- claimsgame(E=8,d=c(3,5,6))
corecentervalue(v1)
plotcoreset(v1,solutions="corecenter")

v2 <- c(0,0,0,0,0,0,0,0,1,4,1,3,6,8,10)
corecentervalue(v2)
plotcoreset(v2,solutions="corecenter")

# What if the game is not full-dimensional because of a dummy player?
v3 <- c(440,0,0,0,440,440,440,15,14,7,455,454,447,60,500)
dummynull(v3) # player 1 is a dummy in v3, so the core is degenerate
# For coredimension to detect that, tolerance has to be appropriate:
coredimension(v=v3,tol=100*.Machine$double.eps) # tolerance too small
coredimension(v=v3) # default tolerance, 1e-12, big enough

# Now how to compute the corecenter?
# When given a degenerate game, corecentervalue computes an approximation:
( cc.approx <- corecentervalue(v=v3) ) # approximate core-center
# However, knowing that player 1 is a dummy and that the core-center assigns
# dummies their individual worth...
v3.without1 <- subgame(v=v3,S=14) # subgame without player 1
( cc.exact <- c(v3[1],corecentervalue(v3.without1)) ) # "exact" core-center

# Plotting both results:
plotcoreset(v3,allocations=rbind(cc.approx,cc.exact),projected=TRUE)