在合作对策中引入归一化权重系统，给出加权Solidarity值的递归定义，在收益分配过程中照顾到联盟中的弱势参与者，并运用权重参数度量参与者之间的差异、调整对弱势参与者的保护程度。通过定义解的期望变异加权对称性，从代数角度给出公理化加权Solidarity值的新方法。设计加权Solidarity值的递归实现算法，并通过案例将加权Solidarity值与其他经典解进行比较，分析加权Solidarity值的合理性。
A new recursive definition of the weighted Solidarity value is provided based on the normalization weight system. The weighted Solidarity value not only supports the vulnerable players in payoff allocation problems, but also evaluates the difference of players and adjusts the degree of protection for the vulnerable participants flexibly applying the weight coefficient. Through defining the weighted symmetry for the expected variation, a new axiomatization for the weighted Solidarity value is proposed in the point of the view of algebra. Ultimately, we design the recursive algorithm to implement the weighted Solidarity value. The rationality of the weighted Solidarity is analyzed through comparing with other classical solutions in an actual case.
[1] Shapley L S. A value for n-person games[J]. Contributions to the theory of Games, 1953, 2(28):307-317.
[2] Sprumont Y. Population monotonic allocation schemes for cooperative games with transferable utility[J]. Games and Economic Behavior, 1990, 2(4):378-394.
[3] Nowak A S, Radzik T. A solidarity value for n-person transferable utility games[J]. International Journal of Game Theory, 1994, 23(1):43-48.
[4] Béal S, Casajus A, Huettner F, et al. Characterizations of weighted and equal division values[J]. Theory and Decision, 2016, 80(4):649-667.
[5] Kalai E, Samet D. On weighted Shapley values[J]. International Journal of Game Theory, 1987, 16(3):205-222.
[6] Shapley L S. Additive and non-additive set functions[D]. Princeton:Princeton University, 1953.
[7] Calvo E, Gutiérrez-López E. Axiomatic characterizations of the weighted solidarity values[J]. Mathematical Social Sciences, 2014, 71:6-11.
[8] 胡勋锋, 李登峰, 刘家财, 等. 基于分量差的线性及匿名合作对策值的简化算法[J]. 管理科学学报, 2017, 20(6):32-41.
[9] Calvo E. Random marginal and random removal values[J]. International Journal of Game Theory, 2008, 37(4):533-564.
[10] Calvo E, GutiWrrez E. The Shapley-Solidarity value for games with a coalition structure[J]. International Game Theory Review, 2013, 15(1):1350002.
[11] van den Brink R, Funaki Y. Axiomatizations of a class of equal surplus sharing solutions for TU-games[J]. Theory and Decision, 2009, 67(3):303-340.
[12] Xu G, Dai H, Hou D, et al. A-potential function and a non-cooperative foundation for the Solidarity value[J]. Operations Research Letters, 2016, 44(1):86-91.
