Kumamoto Gakuen University Repository / 熊本学園大学　機関リポジトリ

Lwin, Maung Maung

6498

Importance of Decentralized Development Planning and the Role of Government under Market Oriented Economy : The Case of Myanmar (マング・マング・ルウィン教授 退職記念号)

斎藤, 恒

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Japanese

What I thought in the survey under the Minamata Disease Special Measures Law

熊本学園大学博士(経済学)2018年度doctoral thesi

福間, 比呂志

English

多国籍企業の所有権構造と参入戦略

山西, 裕美

6389

周, 典芳

6390

Issues on the children's parental rights and joint custodyⅡ : Comparing Japan with Taiwan from the results of research in both countries

新村, 太郎

丸本, 幸治

6401

野田, 和俊

6155

Traditional Chinese

Evolution of Drones and an Attempt to Volcano Monitoring

金, 栄緑

1214

熊本学園大学　機関リポジトリ
原口行雄教授の退職記念号の発刊に寄せて
著者 金 栄緑
雑誌名  熊本学園大学論集『総合科学』 
巻 25
号 1
発行年 2019-12-25
URL http://id.nii.ac.jp/1113/00003249/
原口行雄教授の退職記念号の発刊に寄せて
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্͛͠·͢ɻ
経済学部長　金　栄　緑


Greek

原口行雄教授の退職記念号の発刊に寄せて

渡辺, 功

6404

町田, 直美

6405

Apparent transformation accompanies entrained motion

山下, 壽文

6464

Bookkeeping and Accounting : Research on Japanese Written Works in the early 20th Century

松本, 勝明

6461

Application of EU Competition Law to Social Insurance Insurers

宇野木, 広樹

6506

熊本学園大学　機関リポジトリ
α-サブモジュラー性を満たす利得関数の一考察 (
マング・マング・ルウィン教授 退職記念号)
著者 宇野木 広樹
雑誌名 熊本学園大学経済論集
巻 25
号 1-4
ページ 109-132
発行年 2019-03-31
URL http://id.nii.ac.jp/1113/00003338/
α－サブモジュラー性を満たす利得関数の一考察
1 
 
 
 
 
 
 
要  旨 
 
Calvó-Armengol and lk l ç(2009)
-
(2007)
 
 
はじめに 
1 2
                                                  
* JSPS 17K03744 2
 
**   unoki@nkjc.ac.jp 
1 Calvo-Armengol and Jackson(2004)  
2 Wang and Watts(2005), Kranton and Minehart(2001)  
宇野木　広　樹
2 
 
3  
Jackson and Wolinsky(1996) Bala 
and Goyal(2000)
Johonson and Gilles(2000) Jackson and Wolinsky(1996)
0 1
McBride(2008) Bala and Goyal(2000)
Galeotti, 
Goyal and Kamphorst (2006) Bala and Goyal(2000)
Galeotti, Goyal and Kamphorst (2006)
Galeotti, Goyal and Kamphorst (2006)
 
Calvó-Armengol and lk l ç(2009)
-
(2007)
 
                                                  
3 Furusawa and Konishi(2005, 2007), Goyal and Joshi(2006)  
α－サブモジュラー性を満たす利得関数の一考察
3 
 
1. 連結モデル 
Jackson and Wolinsky(1996)
(pairwise stability)
Jackson and Wolinsky (1996)
 
 
1.1.リンクとネットワーク 
2
4
5
 
 
 
1.2. 利得関数 
                                                  
4  
5 (complete network)
 
宇野木　広　樹
4 
 
6  
 
(1) 0
2
2 7
1
 
0 0
 
 
 
 
                                                  
6 (2007) Jackson and Wolinsky(1996)
 
7 (path) (simple path)  
α－サブモジュラー性を満たす利得関数の一考察
5 
 
1  
 
1 5
1
(1)  
 
 
  
              
                   
 
2. ネットワークの均衡概念 
2
 
 
2.1. ネットワーク形成戦略 
宇野木　広　樹
6 
 
ij
8  
 
2.2. ペア安定性とペアナッシュ均衡 
 
 
 
  
 
 
1
1
2
1
 
                                                  
8 ji,
 
α－サブモジュラー性を満たす利得関数の一考察
7 
 
9
 
 
 
  
 
 
30
536870912
30
30
29
30  
 
 
 
                                                  
9  Dutta and Mutuswami(1997) Jackson and van den 
Nouweland(2000) (strong stability)  
宇野木　広　樹
8 
 
3. α-サブモジュラー性 
Calvó-Armengol and lk l ç(2009)
Calvó-Armengol and lk l ç(2009)
-
 
 
3.1. -サブモジュラー性 
Calvó-Armengol and lk l ç(2009) Calvó-Armengol and lk l ç(2009)
 
 
 
Calvó-Armengol and lk l ç(2009) 1  
-
 
 
-
-
 
 
-  
-  
 
 
α－サブモジュラー性を満たす利得関数の一考察
9 
 
(2)
(2)
1
(2)
 (2) 0
 
-
-
-
10  
 
3.2. サブモジュラー性を満たす利得関数 
-
Calvó-Armengol and lk l ç(2009)
Clame1 Jackson and Wolinsky(1996)
 
1 Calvó-Armengol and lk l ç(2009)
2 Jackson and Wolinsky(1996) (1) 1
Calvó-Armengol and lk l ç(2009) Cla e1
1  
 
                                                  
10 30 10
(2) 0
10 (
)
 
宇野木　広　樹
10 
 
 
1 ( ) ( )
 
( )
(3)  
                           (3)  
(  
        (4)  
 
( )  
         (5)  
 
 
(4)
(4)
11 3
(5)
(5)
                                                  
11 (4)  
 
 
 
α－サブモジュラー性を満たす利得関数の一考察
11 
 
12  
(1) ( ) ( 3) (1)
(6)  
 
(6)
 
13  
 
(6)
(6)
 
(6) ( 2)
(6) ( )  
(6)
                                                  
12 (5)  
 
 
 
13 1 1 4
1  
 
 
4 5  
2
 
 
宇野木　広　樹
12 
 
 
 
 
 
 
(7) 1
(7) 1  
 
(7)
(7) (8)  
 
 
(8) ,  
(6)
 
 
α－サブモジュラー性を満たす利得関数の一考察
13 
 
 
 
 
(9) 1
(9) 1  
 
(9)
(9) (10)
 
 
 
 
 
 
(10) , 
 
1 2 (8) (10) RHS1 RHS3 RHS2 RHS4
宇野木　広　樹
14 
 
1 2 3 4 2
 
 
1 RHS1 RHS3  
 
2 RHS2 RHS4  
 
1 2 (1) 3  
3 (1) ( )
 
 
(1) 3  
4 (1) ( 3)
 
 
3 4 2  
 
2 (1)  
 
おわりに 
(2007)
 
(2007)
-
 
 
 
α－サブモジュラー性を満たす利得関数の一考察
15 
 
参考文献 
[1] Bala V. and S. Goyal (2000), “A Non-cooperative Model of Network Formation”, 
Econometrica, 68, 1181-1229. 
[2] Calvó-Armengol, A. and lk l ç (2009). “Pairwise-stability and Nash equilibria in 
network formation”, International Journal of Game Theory, 38, pp.51–79.  
[3] Calvó-Armengol, A. and Jackson, M.O. (2004) “Social Networks in Determining 
Employment: Patterns, Dynamics, and Inequality”, American Economic Review, 94, 
pp.426-454. 
[4] Dutta, B., and Mutuswami, S. (1997) “Stable Networks”, Journal of Economic Theory, 76, 
pp.322-344. 
[5] Furusawa, T. and H. Konishi (2005) “Free Trade Networks with Transfers”, Japanese 
Economic Review, 56, pp.144-164. 
[6] Furusawa, T. and H. Konishi (2007) “Free Trade Networks,” Journal of International 
Economics, 72, pp.310-335. 
[7] Galeotti, A., Goyal, S., Kamphorst, J., (2006), “Network Formation with Heterogeneous 
Players”, Games and Economic Behavior, 54, pp. 353-372. 
[8] Goyal, S. and Joshi, S. (2006) “Bilateralism and Free Trade,” International Economic 
Review, 47, pp.749-778. 
[9] Kranton, R. and D. Minehart (2001) “A Theory of Buyer-Seller Networks”, American 
Economic Review, 91, pp.485-508. 
[10] Jackson, M. O. and A. Wolinsky (1996), “A Strategic Model of Social and Economic 
Networks”, Journal of Economic Theory, 71, pp.44-74. 
[11] Jackson, M.O. and van den Nouweland, A. (2005) “Strongly Stable Networks,” Games 
and Economic Behavior, 51, pp.420-440. 
[12]Johnson, C. and Gilles, R.P. (2000) “Spatial Social Networks,” Review of Economic Design, 
5, pp.273-300. 
[13] Wang, P. and A. Watts (2005) “Formation of Buyer-Seller Trade Networks in a 
Quality-Differentiated Product Market,” Canadian journal of Economics, 39, pp.971-1004. 
[14]  (2007)
, 
, 1 , pp.73-88. 
宇野木　広　樹
16 
 
数学付録 
1  
 ( 2) , 
 
 
 
 
 
 
 
 
 
 
 
(A1) (A5)
 
 
 
 
 
 
(A6) ( 3)
 
α－サブモジュラー性を満たす利得関数の一考察
17 
 
 
 
 
 
(A6) (A7)  
  
=  
 
 
(A8) (2)
 
Q.E.D. 
 
1  
RHS1 RHS3 RHS1
RHS3
 
 
 
 
宇野木　広　樹
18 
 
(A9) RHS1 RHS3  
 
 
(A10)
 
Q.E.D. 
 
2  
RHS2 RHS4 RHS2
RHS4
1
4
4 1
RHS2 RHS4  
( )
1
14  
 
                                                  
14 1 1 4 2
 
1 4 2 1 
2 4 3 1 
1 12 1 4
1  
 
α－サブモジュラー性を満たす利得関数の一考察
19 
 
 
2
1
( 1)
 
 
( 1)
(A11)
( 1)
 
 
(A11) (A12) ( 1)
 
 2  
3
( 2)
 
 
宇野木　広　樹
20 
 
( 2)
 
 
(A13) (A14) ( 2)
 
 
4
( 3)  
 
i
( 3)
 
 
α－サブモジュラー性を満たす利得関数の一考察
21 
 
i
i
( )
 
 
 (A15) (A16) (A17) ( 3)
 
4  
2
5
1
(
4)  
 
宇野木　広　樹
22 
 
( 4)
 
 
 (A18) (A19) ( 4)
 
1 2 (
1) ( 4) ( 1) ( 4)
RHS2 RHS4  
Q.E.D. 
 
2 3 (1)
( 2)  
 
 
 
Q.E.D. 
 
4  
(1)
 
 
 
 
 
α－サブモジュラー性を満たす利得関数の一考察
23 
 
 
( 3)  
Q.E.D. 
 
2  
(1) ( 1) 3 4 ( 2)
( 3) (1)  
Q.E.D. 
 
 
  
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宇野木　広　樹
24 
 
Summary 
 
A Study of Payoff Function that Satisfies Submodularity 
 
Hiroki Unoki 
 
 “Network formation game” is one of economics research field, which 
focuses on behavior and incentives of each node, where each node would 
strategically choose whether or not to form a link and network is formed as 
a result of such choices. In network formation game study, there exists 
several concept of network equilibrium. 
In this paper, we compare pros and cons of “pairwise stability” and  
“pairwise Nash equilibrium”. In Calvó-Armengol and lk l ç (2009), 
-submodularityis introduced as an important factor to make pairwise 
stability and pairwise Nash equilibrium coincide. In this paper, we show that 
the sufficient condition to achieve submodularity and payoff function in 
Sakagami, Kato, Unoki (2007) satisfies this sufficient condition. 


A Study of Payoff Function that Satisfies Submodularity

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Importance of Decentralized Development Planning and the Role of Government under Market Oriented Economy : The Case of Myanmar (マング・マング・ルウィン教授退職記念号)