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 XN, v \ with a system of In2M + 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 In2M largest amounts ¯∆i j which can be transferred from player i to j while remaining in the core or in the strong epsilon core. The associated payoff vector is a midpoint of the 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 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 doesn’t pass the (pre)-kernel check, the function is called a further time with additional informations about the game. According to our tests a further call is necessary if the intersection set of the In2M solution sets is empty, in this case no candidate of the final linear problem satisfies the bisection property. This implies that at least one largest transfer ¯∆i j is greater than the maximal transfer in i- j direction that is possible at a (pre)-kernel point Óx while remaining in the core, that is, ¯∆i j> ∆i j(Óx), with Óx Î K*(G). Hence, if the solution intersection set is non-empty, then all payoff vectors in the intersection set possess the bisection property and are therefore kernel elements. The kernel of a TU-game with an empty core can be computed by providing the epsilon value for the least-core as an additional information.