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合作博弈在供水工程建设成本分摊中的应用研究
李娟 1，李鹤 2
1河海大学商学院，南京（210098）
2深圳市同洲电子股份有限公司，南京（210000）
摘 要：供水工程在居民生活中起着重要的作用，由于其需要资金之多，因此此类工程建设
往往由相邻几个城市合资建设，但是，各方具体投入多少成了需要研究的重要问题。本文比
较分析了合作博弈的三种分摊方法——核心分摊法、夏普利值（Shapley）分摊法、核仁分摊
法，并建立核仁分摊模型将各方在供水工程的建设投资比例估算出来，以此解决投资过程中
的矛盾，实现供水工程的效益最大化。
关键词：跨流域调水工程，成本分摊，合作博弈
1. 引言
供水系统是解决地区用水不足的问题的，但是建造一个供水工程的成本是很大的，因此，
如何能找到行之有效的降低成本的方法，显得尤为必要。
既然各市都需要建设成本如此昂贵的供水系统，那么相邻的各市是否可以合资建设，以
此来降低成本，达到收益最大呢？举个例子说明，假设地域毗邻的 A 市和 B 市和 C 市都需
要建设市政供水系统，A 市单独建设需要 100 万元，B 市单独建设需要 60 万元，C 市单独
建设需要 70 万元，如果他们合作建设一个能够满足两市需要的大供水系统则需要 190 万元。
容易看出，在这种情况下，合作建设将是有利可图的(可以节省 100+60+70-190=40 万元)。 它
们将选择合作建设。但是合作的各方只要有一方对合作不满意，合作就无法达成，因此，如
何分摊成本能保证各方的满意度最大是需要研究的重要问题。
2. 供水工程的建设成本分摊
A、B、C 城市的合作博弈
博弈论是近几年新兴起的一种方法，它具有很好的经济背景，通常可以用合作博弈解的
概念（核心、夏普利值、核仁）求解。这个问题通常有两大难点：第一是建模困难，即许多
实际问题难以转化为博弈论中严格要求的模型结构；第二是求解过程复杂[1]。
将供水工程项目的建设资金投入视为合作博弈中的建设成本分摊问题，将 A、B、C 城
市看作合作博弈中的三个局中人，成本分摊的一个突出特点是，不同的分摊方法对应着不同
的分摊结果，因此，选择一种恰当的分摊方法将使得局中人得到一个相对满意的分摊结果。
通过此得出的几个城市的投资比例是最合理的，各合作人能获得的收益是相对最大的，在这
种情况下，合作最有效。
合作博弈的分摊方法介绍：
核心分摊法
对于一个n人合作（n 较大）的博弈对策 ( )VN , ，分配集 ( )VE 中被任何分配向量所优
超的分配，其全体称为核心，记为 ( )VNC , 或 ( )VC 。
分配向量在核心 ( )VC 中的充分必要条件是：
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①
( )NVxn
i
i =∑
=1
②
( )SVx
i
i>∑
<1 ，对所有 NS ⊂
这样，若博弈问题中存在核心，就可以将联盟总收益 ( )NV 按此方式分配给各个局中人，
使之满足分配向量具有的个体合理性和整体合理性条件。即任何联盟在这种分配方式下的所
得都不应小于它独干时的所得。
核心分摊法在具有如此优越性的同时，它的致命缺陷是：核心的解集经常为空或者是一
个区域（集合）[2]。
夏普利值（Shapley）分摊法
当建设成本的分摊问题用核心分摊法是一个空集或者集合时，对合作问题的分配不能做
出结论或唯一结论，因此引入夏普利值分摊法，可以克服核心分摊法的弊端。这样一种分配
方案得到的结果直接与局中人的编号有关，由于用夏普利值的方法进行成本分摊适用于合作
博弈中有多个局中人参与的情况，故本文对此方法的具体应用不加探讨。
核仁分摊法
对于n个局中人的合作博弈的成本分摊的核仁分摊法的理解，首先了解核仁的概念，
假设： ( ) ( ) ( )SxSVSe −= ， S ——在n个局中人中随意挑选人组成的联盟 ( )nS< ；
( )SV ——联盟 S人合作下的各成本之和； ( )Sx ——联盟中的 S个人在全体合作下的各成本
之和； ( )Se ——联盟 S关于 x的超出值。 ( )Se 越大，越令人满意。
核仁具有以下四个性质：①满足个体合理性和整体合理性；②每个博弈问题有一个而且
只有一个核仁；③假如存在核心，则核仁是核心的一部分；④处于对称地位的局中人所得支
付相同。核仁分摊法特别适用于在局中人数目不大时，用作图法求解核仁[3]。
供水水工程的建设成本分摊
由以上分析可以看出，本文中求 A、B、C 城市的投资问题可以用核仁分摊法求解。为
了确保各投资方的相对满意度最大（这直接影响着今后供水工程中各方的合作积极性问题），
因此，它们选择的三方合作（而不是两方合作）是最优的选择将是建设成本投资比例分摊合
理的一个重要前提。
为了更好的用合作博弈中的核仁分摊法解决建设成本中的三市的投入比例问题，在这
里，作以下假设：
将参与工程建设成本投入的 A 城市，B 城市，C 城市设为合作博弈中的三个局中人。选
择三方合作达成的条件是：
CBACBA xxxCCC ++>++ ； （1）
BABA xxVV +>+ 00 ， CACA xxVV +>+ 01 ， CBCB xxVV +>+ 11 ； （2）
假设 aCA = ， bCB = ， cCC = ， 0mCxxx ABCCBA ==++ ， 100 mVV BA =+ ，
2
01 mVV CA =+ ， 311 mVV CB =+ ， 3210 ,,,,,, mmmmcba 均为已知数值。上面一组式子变为：
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0 mxxx CBA =++ ， cxbxax CBA <<< ,, ； （3）
BA xxm +>1 ， CA xxm +>2 ， CB xxm +>3 ； （4）
( )CBAiCi ,,= 表示各局中人单独建设的资金； ABCC 为三市合作下的建设资金，
( )CBAixi ,,= 表示三方合作下的各局中人的建设资金； ( )CBAiVi ,,= 表示两方合作下的
各局中人的建设资金；如
0
AV ，
0
BV ，表示 A、B 合作情况下得各自出的资金；
1
AV ，
0
CV 表示
A、C 合作情况下得各自出的资金
1
BV ，
1
CV 表示 B、C 合作情况下得各自出的资金。
为了方便求出这个分摊问题的图示，将其作一个变量变换，定义各局中人的合作收益为
( )CBAixCy iii ,,=−= ，由此式将上式变换得出：
0 mcbayyy CBA −++=++ ； （5）
1 mbayy BA −+>+ ， 2 mcayy CA −+>+ ， 3 mcbyy CB −+>+ ； （6）
令 00 nmcba =−++ ， 11 nmba =−+ ， 22 nmca =−+ ， 33 nmcb =−+ ，得到：
0 nyyy CBA =++ ； （7）
321 , , nyynyynyy CBCABA >+>+>+ ； （8）
下图 1 描述了这个分摊问题的核仁解，三个顶点 CBA ,, 分别代表将全部收益分别只分
摊 给 一 市 的 情 况 ， 三 顶 点 为 ( ) ( ) ( )000 ,0,0,0,,0,0,0, nCnBnA 。 三 个 约 束 条 件
321 , , nyynyynyy CBCABA >+>+>+ ，构成的黑三角区部分就是这个分摊问题的核仁，也
即是三方局中人分摊的比例解。核仁中的点就是使三方相对满意的分摊比例，核仁的中心就
是最终的唯一解。
图 1 大型供水工程建设成本分摊问题的核仁
A
B C
1nyy BA =+
2nyy CA =+
3nyy CB =+ C A
B
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由上图可以看出，三角形 ABC 部分就是成本分摊的核仁部分，在这个三角区中的点就
是使局中人相对满意的解。核仁的中心就是最优解，经过计算得出，合作分摊结果为
( )**** ,, cbay = ，建设成本出资的比例是 ( )**** ,, ccbbaax −−−= 。
3. 结论
在大型供水工程建设成本分摊的合作博弈过程中，介于参与合作博弈的局中人很少，采
取核仁分摊法比较有效，它能使得外来资本在资金投入建设的过程中，得到相对满意的预期，
从而能够保证其在投入建设过程中的积极性，并刺激资本市场的发展。
但是本章研究的关于大型供水工程的建设成本分摊问题，是在假定了参与合作博弈的局
中人只有三个的情况下得出的核仁解，在现行的大型供水工程的建设资本筹资过程中，一些
工程涉及的合作博弈局中人往往不止三个，这就需要我们采取更复杂的分摊方法。
参考文献
[1]施锡铨.博弈论.上海：上海财经大学出版社[M]，1999（2）
[2]荆朝霞，段献忠，问福拴，等.输电系统固定成本分摊问题.电力系统自动化[M]，2003
[3]鲁栗，基于合作博弈核仁解的输电固定费用分摊[J]，2005
The Applied Research about the Cooperative Games in the
Cost Allocation for the Water Supply Engineering
Li Juan1，Li He2
1 Hohai University，Nanjing（210098）
2 Shenzhen Coship Electronics CO., LTD，Nianjing (210000)
Abstract
Water supply engineering plays an important role in people’s everyday life, as the huge money is needed
for this, so the engineering always be built through several cities’ corporation. However, the money that
every city should spend becomes a key problem for this. This essay is aimed at solving such problem
though the comparation among three Cooperative games, and build a new modern to compute the fee of
every city, through that approach, hope to resolve the contradiction in the process of investment and
realize the utmost benefit.
Keywords: the water transfer project, cost allocation, Cooperative games
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/NOR <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>
/SVE <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>
>>
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [ ]
>> setpagedevice
