An LP approach to compute the pre-kernel for cooperative games

Abstract

We present an algorithm to compute the (pre)-kernel of a TU-game 〈 N , ν 〉 with a system of n 2 + 1 linear programming problems. In contrast to the algorithms using convergence methods to compute a point of the (pre)-kernel the emphasis of the chosen method lies not on efficiency and guessing good starting points but on computing large parts or in good cases the whole (pre)-kernel of a game. The chosen algorithm computes on a first step by relying on linear programming the n 2 largest bi-symmetrical amounts δ ij ε which can be transferred from player i to j while remaining in the strong ε -core. The associated payoff vector is a midpoint of the ε -core segment in i – j direction and is therefore a candidate that satisfies the bisection property. From these results we can determine in a sophisticated pattern-matching procedure the constraints which are needed to construct the final linear programming problem for computing at least a (pre)-kernel point of the game. From the derived final linear program large parts or the whole (pre)-kernel can be easily calculated. Finally, the program checks if the computed (pre)-kernel candidate belongs to the (pre)-kernel. In cases where the candidate does not pass the (pre)-kernel check, the function is called a further time with additional informations about the game. A further call could be necessary if the intersection set of the n 2 solution sets is empty and no correction of the final LP is applied for, in this case, for at least one distinct pair of players the largest bi-symmetrical transfer is of no importance to compute a (pre)-kernel point, that is, no candidate of the final linear problem satisfies the bisection property. This implies that at least one largest bi-symmetrical transfer δ ij ε is greater than the maximal transfer in i – j direction that is possible at a (pre)-kernel point y ⇀ while remaining in the core, that is, δ ij ε > δ ij ε ( y ⇀ ) , with y ⇀ ∈ K * ( Γ ) . Hence, if the solution intersection set is non-empty, then all payoff vectors in the intersection set possess the bisection property and are therefore (pre)-kernel elements. The (pre)-kernel of a TU-game with an empty core can be computed, for instance, by providing the epsilon value for the least-core as an additional information.