Stable Allocations for k -Additive Cost Games with Critical Players Ana Meca • Greys Soˇ si´ c † Center for Operation Research, University Miguel Hern´ andez, 03202 Elche, Spain Marshall School of Business, University of Southern California, Los Angeles, California 90089 ana.meca@umh.es • sosic@marshall.usc.edu Tel. *34-96-665 8717 • *1-213-821 3632 June 9, 2011 Abstract In this paper, we study some special classes of cost games that satisfy k-additivity and possess players who contribute to cost reduction of all members of the alliance that they belong to (which we call critical players). This type of games can be found in knowledge-sharing games, inventory games, holding cost games, etc. While papers analyzing these games study specific examples of k-additive cost functions and analyze how players can reduce their cost through cooperation, our goal is to provide an analysis of stability for more general classes of k-additive games with critical players. For arbitrary k ≥ 1, we identify core allocations and a class of population-monotonic allocations. When k = 1, we pay special attention to the situations in which critical players may not see any savings on their own if the core allocations are implemented. We identify alternative stable outcomes in which the total cost remains unchanged, but critical players receive a positive reward for their contribution, which incentivizes their participation. † Corresponding author