Some Excess-Based Solutions For Cooperative Games With Transferable Utility

For a finite player cooperative cost game, we consider two solutions that are based on excesses of coalitions. We define per-capita excess-sum of a player as sum of normalized excesses of coalitions involving this player and view it as a measure of player's dissatisfaction. So, per-capita excess-sum allocation is that imputation that minimizes the maximum per-capita excess-sums of players. We provide a closed form expression for an allocation, which is the per-capita excess-sum allocation if it is also individually rational. We propose a finite step algorithm to compute per-capita excess-sum allocation for a general game. We show that per-capita excess-sum allocation is coalitionally monotonic. Next, we consider excess-sum solution wherein a player views entire coalition's excess as a measure of dissatisfaction. This excess-sum solution also has above properties. In addition, we consider a super set of core and show that excess-sum allocation can be viewed as an imputation that is a certain center of this polyhedron. We introduce a class of cooperative games that can model cost sharing among divisions of a firm when they buy items at volume discounts. We characterize when excess-based allocations coincide with Shapley value, nucleolus, etc. in such games.