Nucleolus and prenucleolus of a transferable-utility game

The imputation (or, with pre = TRUE, the efficient preimputation) that lexicographically minimises the non-increasingly ordered vector of coalition excesses \(v(S) - x(S)\) (Schmeidler 1969), computed by the sequence of linear programs of Maschler, Peleg and Shapley (1979): each stage minimises the largest free excess, then fixes the coalitions whose excess cannot fall below that level, until the fixed coalitions determine the allocation. The first level of the prenucleolus is the least-core value.

Usage

Nucleolus(v, pre = FALSE, tol = 1e-09)

Arguments

v

Worths in binary order, length \(2^n - 1\): v[S] is the worth of the coalition with bitmask S.

pre

Prenucleolus (no individual-rationality bounds).

tol

Tolerance for tight and fixed excesses.

Value

List with x (the allocation) and levels (the successive maximal excesses).

References

Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics 17, 1163-1170.

Maschler, M., Peleg, B. and Shapley, L. S. (1979). Geometric properties of the kernel, nucleolus, and related solution concepts. Mathematics of Operations Research 4, 303-338.

Examples

Nucleolus(c(0, 0, 60, 0, 60, 60, 72))
#> $x
#> [1] 24 24 24
#> 
#> $levels
#> [1] 12
#>