On a Cooperative Game in the Knapsack Problem

Abstract

The knapsack problem with indivisible items as agents is considered. Each agent has certain weight and utility and wants to be in a knapsack. Such situation is treated as a cooperative game with transferable utility. A characteristic function of this game generalizes the characteristic function associated with the bankruptcy problem but, in contrast to the latter case, it is not convex. Nevertheless, it turns out that the core of this game is non-empty. At the end of the paper some special cases of the knapsack problem are studied. For these cases, the Shapley value, the τ-value and also the nucleolus are found in the explicit form.