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We use cooperative game theory to investigate multiplayer allocation problems under the almost diminishing marginal contributions (ADMC) property. This property indicates that a player’s marginal contribution to a non-empty coalition decreases as the size of the coalition increases. We develop ADMC games for such problems and derive a necessary and sufficient condition for the non-emptiness of the core. When the core is non-empty, at least one extreme point exists, and the maximum number of extreme points is the total number of players. The Shapley value may not be in the core, which depends on the gap of each coalition. A player can receive a higher allocation based on the Shapley value in the core than based on the nucleolus, if the gap of the player is no greater than the gap of the complementary coalition. We also investigate the least core value for ADMC games with an empty core. To illustrate the applications of our results, we analyze a code-sharing game, a group buying game, and a scheduling profit game.
Bibliographical noteThe authors are grateful to the Department Editor (Professor Chung Piaw Teo), the Associate Editor, and two anonymous reviewers for their insightful comments that helped improve this paper.
The first author (Mingming Leng) was supported by the Research Committee of Lingnan University under Direct Grant No. DR20A1, and was also supported by Hong Kong Institute of Business Studies of the Faculty of Business at Lingnan University under Research Program Grant No. HCRG/190/005.
The second author (Chunlin Luo) was supported by the National Natural Science Foundation of China under Research Project Grant Nos. 71862014 and 71461009.
The third author (Liping Liang) was partially supported by the Research Committee of Lingnan University under Faculty Research Grant No. DB17B2.
- coalitional games
- diminishing marginal contributions
- the core
- the nucleolus
- the Shapley value