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运筹学学报 >

2015 , Vol. 19 >Issue 4: 114 - 120

DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2015.04.011

运筹学

Shapley值与Winter值的解析关系

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  • 1. 福州大学经济与管理学院, 福州 350108

收稿日期: 2014-09-01

  网络出版日期: 2015-12-15

基金资助

1.国家自然科学基金重点项目(No. 71231003),2.国家自然科学基金(No. 71171055),

3.高等学校博士学科点专项科研基金(No.20113514110009), 4.国家教育部新世纪优秀人才支持计划(No. NCET-10-0020)

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Analytic relationship between Shapley and Winter values

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  • 1.School of Economics and Management, Fuzhou University, Fuzhou 350108, China

Received date: 2014-09-01

  Online published: 2015-12-15

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摘要

鉴于 Shapley 值和 Winter 值都是局中人边际贡献的平均值,探究了它们之 间的解析关系.证明了 Shapley 值是 Winter 值在层次结构集上对称概率分布下的期望均值. 作为这一结论的一个推论, 证明了 Shapley 值是 Winter 值在层次结构集的任意相似类中的平均值. 最后,还指出了这一结 论与推论的等价性.研究结果不仅扩展了 Shapley 值和 Owen 值与此对应的解析关系, 还大大简化了这些关系的已有证明.

关键词： 合作对策; Shapley值; Owen值; Winter值; 联盟结构; 层次结构

本文引用格式

胡勋锋, 李登峰 . Shapley值与Winter值的解析关系[J]. 运筹学学报, 2015 , 19(4) : 114 -120 . DOI: 10.15960/j.cnki.issn.1007-6093.2015.04.011

Abstract

As both the Shapley and Winter values are averages of players' marginal contributions, this paper explores their analytic relationship. Specifically, the result that Shapley value is Winter value's expectation with respect to symmetric probability distributions on level structure set is proved. As a corollary, the argument that
Shapley value is Winter value's average with respect to any similar class in level structure set is also attested. Finally, the equivalence of this result and corollary is presented. The research results not only expand corresponding relationship between Shapley and Owen values, but also simplify the proofs of these correspondingrelationship enormously.

Key words： cooperative game; Shapley value; Owen value; Winter value; coalition structure; level structure

参考文献

Shapley L S. A value for n-person games [C] Contributions to the Theory of Games II, Princeton: Princeton University Press,
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Owen G. Values of games with a priori unions [C] Mathematical Economics and Game Theory, Berlin: Springer-Verlag, 1977, 76-88.

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\'Alvarez-Mozos M, Tejada O.  Parallel characterizations of a generalized Shapley value and a generalized Banzhaf value for cooperative games with level structure of cooperation [J]. Decision Support Systems, 2011,  52(1): 21-27.

\'Alvarez-Mozos M, Tejada O. Share functions for cooperative games with levels structure of cooperation [J].  European Journal of Operational Research, 2013,  224(1): 167-179.

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L\'opez S, Martha S. On the relationship between Shapley and Owen  values [J].  Central European Journal of Operations Reasearch, 2009, 17(4): 415-423.

 
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