法律家が数学者に助けを求める理由

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

法律家が数学者に助けを求める理由

寺本, 振透

九州大学大学院法学研究院 : 教授

http://hdl.handle.net/2324/2545036

出版情報：2019-12-11 バージョン：

権利関係：

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Lawyers Seek the Help of Mathematicians

法律家が数学者に助けを求める理由

December 11, 2019

IMI Colloquium, Institute of Mathematics for Industry, Kyushu University 九州大学マス・フォア・インダストリ研究所 IMIコロキウム

Shinto TERAMOTO（寺本振透）

Professor, Faculty of Law, Kyushu University（九州大学法学研究院教授）

[email protected], [email protected]

1

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•

Our society can be represented by a network, which is also denoted by a matrix. The role of rules, including laws and contracts, is to

intervene in society by means of connecting or disconnecting specific pairs of nodes belonging to such a network. Naturally, designing rules requires help and justification by utilizing the concept of a network or matrix.

•

Besides, rules often include measures to incentivize or disincentivize a party or a citizen to engage in or refrain from specific actions or

behaviors. Lawyers drafting such rules often envision that non-linearly increasing or decreasing values are suitable for the purpose of such measures. However, such black-letter rules employ a stepwise change of values, and lawyers are often frustrated with the disparity between their vision and the resulting provisions, which could be mitigated with the help of mathematicians.

•

Thus, we, lawyers, seek the help of mathematicians.

3

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•

社会はネットワークあるいは行列で表現できます。法律や契約といったルールの役割は，ネットワークから特定の頂点を選び，頂点同士をつないだり，つながりを切断することで，社会に介入することです。

そういうわけで，ルールの設計には，ネットワークや行列の概念を利用した支援と正当化が必要になります。

•

また，ルールは，当事者や市民が特定の行動をするように，あるいは，避けるように，インセンティブやディスインセンティブを定めることがあります。ルールを起案する法律家は，しばしば，非線形に増加または減少する値を思い浮かべます。しかし，文書で表現されたルールでは，インセンティブやディスインセンティブが階段状に変化するという不満足な結果に終わります。このギャップも，数学の支援により，緩和することができそうです。

•

こういう次第で，法律家たちは，数学者の助けを必要としています。

4

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•

法律家の，数学に対する需要は，このセミナーのために考え出したことでは，ありません。

•

そのことを示すため，法学府国際コースの授業や，学会報告で用いた資料を，なるべくそのままお示しするようにします。

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社会は，ネットワークあるいは行列で表現できる。

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

3 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

4 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0

6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

7 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

8 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0

9 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0

11 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1

12 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0

13 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

15 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

16 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0

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Anna is good at singing and writing songs

• Anna always writes funny

songs, and sings these songs to Bob.

• Bob really enjoys Anna singing funny songs.

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8

Indicate a person by a dot or a small circle

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9

Indicate a dependency of a person on another person to gain access to information with a line with direction (an arrowed line)

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Cindy goes on a date with Bob

•

Cindy has no direct contact with Anna.

•

Bob is always singing his favorite songs in the car when he goes on a date with

Cindy.

•

Almost all of Bob’s favorite songs are originally created and sung by Anna.

•

Cindy also becomes attached to Anna’s songs that she has learned from Bob,

and she sometimes sings the songs when she goes on a date with Bob.

10

(11)

11

Cindy

Cindy goes on a date with Bob

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Bob and Dann

• Dann is a colleague of Bob at their office.

• Bob and Dann often go for a drink after work.

• Bob always sings Anna’s

songs when he is in a good mood after drinking.

• So, the melodies of Anna’s songs stick to Dann’s

memory.

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13

Cindy

Bob and Dann

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Dann and Elena

• Dann is humming the

melodies of Anna’s songs, but he modifies or mutilates them funnily, when doing yard work in his backyard.

• Elena, Dann’s partner, hears

Dann’s humming Anna’s songs in the backyard.

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15

Cindy

Dann and Elena

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Anna, Elena and Fujiko

•

Anna and Elena are members of an amateur band, where Anna is the songwriter and vocalist, while Elena plays an electronic piano, and Fujiko plays an electronic guitar.

•

They perform the songs of Anna.

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Anna

Elena Fujiko

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17

Cindy

Anna, Elena and Fujiko

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• A graph gives us an intuitive understanding of the relationship between actors.

• However, we have to utilize numerics in order to analyze the relationship between actors using a computer.

18

Cindy

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We can use a matrix (a “sociomatrix”) to represent the relationship between actors by numerics.

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A B C D E F A

B C D

E F

Cindy

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Each column corresponds to one actor.

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A B C D E F A

B C D

E F

Cindy

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Each row also corresponds to one actor.

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A B C D E F A

B C D

E

F

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The order of actors aligned in the rows must correspond to the order in the columns.

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A B C D E F A

B C D

E

F

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If actor

i

sends an arc to actor

j

, put “ 1 ” in the cell (i, j),

which means the cell on which the row of actor

i

meets the column of actor

j

.

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i j

i 1

i j j

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Be careful about the direction of arcs.

If actor

j

does not send an arc to actor

i

, put “0” in the cell (j, i).

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i j

i 1

j 0

i j

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Complete the sociomatrix.

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A B C D E F A

B C D

E F

Cindy

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Complete the sociomatrix.

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A B C D E F A 0 0 0 0 1 1 B 1 0 1 0 0 0 C 0 1 0 0 0 0 D 0 1 0 0 0 0 E 1 0 0 1 0 1 F 1 0 0 0 1 0

Adjacency Matrix

Cindy

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Assuming that we neglect any self-returning arcs, every cell on the diagonal line from upper left to lower right is “0.”

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A B C D E F A 0 0 0 0 1 1 B 1 0 1 0 0 0 C 0 1 0 0 0 0 D 0 1 0 0 0 0 E 1 0 0 1 0 1 F 1 0 0 0 1 0

Cindy

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頂点同士の distance を縮めることが，多くのビジネスの核心

28

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•

The “length” of a path is the number of lines (arcs or edges) contained in the path.

•

“Distance (from actor B to actor A)” means the length of the

shortest path which enables actor B to directly or indirectly reach actor A.

A B

A 1 1 B

1 2

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Measure the distance from each actor to actor

1

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(1) Prepare the adjacency matrix representing the network.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

3 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

4 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0

6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

7 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

8 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0

9 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0

11 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1

12 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0

13 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

15 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

16 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0

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(2) Define the adjacency matrix.

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(3) Load the “sna” package (“sna” is a package containing a range of tools for social network analysis).

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(4) Execute the “geodist” command (“geodesics” means the shortest path(s) between a pair of actors).

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> geodist(sparse)

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(5) The “geodist” command outputs the distances between each pair of vertices.

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The distances from actor1,2,3,…16 to actor1.

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Actor

2

may start a service to reduce the distances

from other actors to actor

1

.

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actor1

actor1 0

actor2 1

actor3 6

actor4 7

actor5 4

actor6 Infinity

actor7 4

actor8 5

actor9 5

actor10 2

actor11 2

actor12 1

actor13 3

actor14 4

actor15 Infinity

actor16 3

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Add the arc sent by every actor to actor

2

.

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(1) Put “1” in every cell corresponding to the arc potentially sent from each actor to actor2.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

3 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0

4 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

5 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0

6 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0

7 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0

8 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0

9 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0

10 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0

11 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1

12 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0

13 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0

14 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1

15 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0

16 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0

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(2) Define the new sociomatrix.

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> onehub<-matrix(c(

+ 0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0, + 1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0, + 0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0, + 0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0, + 0,1,0,0,0,0,0,1,1,0,0,0,1,0,0,0, + 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0, + 0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0, + 0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0, + 0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0, + 0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0, + 0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1, + 1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0, + 0,1,0,0,1,0,1,0,0,1,0,0,0,0,0,0, + 0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1, + 0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0, + 0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0), nrow=16, ncol=16, byrow=TRUE)

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(3) Execute the “gplot” command to generate the graph corresponding to the new matrix.

1

2

3

4

5

6

7

8

9 10

11

12

13

14

15 16

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The distance from each actor to

actor

1

has been remarkably reduced thanks to actor

2

.

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before after

actor1 0 0

actor2 1 1

actor3 6 2

actor4 7 2

actor5 4 2

actor6 Infinity 2

actor7 4 2

actor8 5 2

actor9 5 2

actor10 2 2

actor11 2 2

actor12 1 1

actor13 3 2

actor14 4 2

actor15 Infinity 2

actor16 3 2

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Actor2 functions as the hub of the network, or the intermediary to connect many actors by shorter distances to actor1.

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1

2

3

4

5

6

7

8

9 10

11

12

13

14

15 16

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頂点同士のつながりを支援し，また，

つながりを切断するルールの例

43

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The position of actor2 is appealing to entrepreneurs.

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1

2

3

4

5

6

7

8

9 10

11

12

13

14

15 16

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Followers of actor2 (perhaps, actor12?) may participate in the market to commercially disseminate the work of actor1 to multiple actors.

45

1

2

3

4

5

6

7

8

9 10

11

12

13

14

15 16

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Competition between Hubs may occur.

46

1

2

3

4

5

6

7

8 9 10

11 12

13

14

15 16

1 2

3 4

5 6

7

8

9

10 11

12

13 14

15

16

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Can you decide to invest labour and money to prepare yourself to become the first hub in the communication network?

47

1

2

3

4

5

6

7

8 9 10

11 12

13

14

15 16

1 2

3 4

5 6

7

8

9

10 11

12

13 14

15

16

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Uncertainties regarding competition in the market.

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(49)

Uncertainty may make the entrepreneur hesitate to make investment.

49

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Is the law able to alleviate the negative impact of free competition?

50

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A solution by giving a private right to actor2. This is the strategy employed by IP laws.

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2

12

16

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（特許権の効力）

(Eﬀect of Patent Right)

特許法第六十八条特許権者は、業として特許発明の実施をする権利を専有する。[…]

Article 68 of Patent Act The patentee has an exclusive right to work the patented

invention in the course of trade […].

52

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（複製権）

(Right of Reproduction)

著作権法第二十一条

著作者は、その著作物を複製する権利を専有する。

Article 21 of Copyright Act The author of a work has the exclusive right to reproduce the work.

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法律家がイメージするインセンティブまたはディスインセンティブは，非線形に変化

54

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階段状に変化するインセンティブの例

55

(56)

• ライセンシーがライセンサーに対して支払うロイヤルティは，

本件製品の出荷数に応じて，次の通り計算するものとする。

1. 出荷数の累計10,000ユニットまでは，売上の5%

2. 出荷数の累計10,000ユニット超15,000ユニットまでは，

売上の4％

3. 出荷数の累計15,000ユニット超30,000ユニットまでは，

売上の1％

4. 出荷数の累計30,000超については，売上の0.5%

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階段状に変化するディスインセンティブの例

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https://www.police.pref.fukuoka.jp/kotsu/unkan/004.html

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非線形に変化するインセンティブを用意する試みの例

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Personal Healthcare Records (PHR)

Electronic Healthcare Records EHR

Personal Healthcare Device PHD

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Designing a token allocated to citizens to promote their use of PHR

Purpose

•

To encourage citizens to utilize PHR on a routine or everyday basis.

•

To allocate tokens to citizens when they utilize their PHR, while inhibiting instant outflow of cash from the public health insurance budget.

•

To enable citizens to exchange tokens to publicly

traded cryptocurrencies or fiat currency to purchase medical and healthcare services.

•

To prevent citizens from spending tokens too quickly.

•

To encourage citizens to accumulate and save tokens continuously.

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•

The liquidity of a token means the exchange rate of the token against publicly traded cryptocurrency or fiat

currency.

•

The liquidity should be very low for a certain period after a relevant citizen receives a token, and should quickly

increase after such period.

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Liquidity

Time

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A network

•

We deem that the tokens allocated to one citizen belong to a single network (for the purpose of

convenience, we call this network a “wallet”).

63

Tokens Ties

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Density of a network.

•

The density of a wallet, comprised of n tokens and m ties connecting tokens is .

•

The density of a network is non-linearly increased (or, decreased) when the vertices in the network gain (or lose) incrementally additional ties with one another.

•

This suggests that we will be able to design non- linearly increasing liquidity of tokens by using non- linearly increasing or decreasing density of a wallet.

n(n −m1)/2

64

(65)

A model of tokens of which liquidity increases non-linearly

Density

65

•

At the time of the allocation of tokens to the citizen, every

token therein is connected

with one another (Density = 1).

•

Decreasing the density

incrementally by repeating the following steps:

•

to choose one vertex that is not isolated from other vertices; and

•

to isolate the chosen vertex from other vertices by

cutting off any ties that connect the chosen vertex with other vertices.

256 tokens

(66)

Density

66

(67)

Liquidity

67

• Because the density of the wallet is decreasing non-linearly, we can devise a formula to produce non-linearly increasing liquidity of the tokens.

• • • The increasing curve of liquidity can be adjusted by adopting a different value for τ.

liquidity = exp(− density

τ )

τ = 0.01

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Liquidity

68

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Code (in R)

69

library (igraph) n<-256

w01adj<-matrix(1,n,n) for (i in 1:n){

w01adj[i,i]<-0

}w01<-graph_from_adjacency_matrix(w01adj, mode=c("undirected"))

tiff (file="000.tiff")

plot(w01, vertex.size=5, vertex.label=NA) dev.off()

w01dens<-graph.density(w01) τ<-0.01

nwd<-w01dens

w01Liquid<-exp(-nwd/τ) w01cAdj<-w01adj

for (q in 1:n){

file.name<-sprintf("%03d.tiff", q) for (i in 1:n){

w01cAdj[q, i]<-0 w01cAdj[i, q]<-0

}w01c<-graph_from_adjacency_matrix(w01cAdj, mode=c("undirected"))

tiff (file.name)

plot(w01c, vertex.size=5, vertex.label=NA) dev.off()

w01cD<-graph.density(w01c)

cat(w01cD, "\n", file="w01coolingDensity.csv", append=TRUE)

nwd<-w01cD

w01cLiquid<-exp(-nwd/τ) cat(w01cLiquid, "\n",

file="w01coolingLiquidity.csv", append=TRUE) }

(70)

Thank you.

70

法律家が数学者に助けを求める理由

法律家が数学者に助けを求める理由

•

•

•

•

•

•

•

•

社会は，ネットワークある いは行列で表現できる。

Anna is good at singing and writing songs

• Anna always writes funny

songs, and sings these songs to Bob.

• Bob really enjoys Anna singing funny songs.

Indicate a person by a dot or a small circle

Cindy goes on a date with Bob

•

•

•

•

Cindy goes on a date with Bob

Bob and Dann

• Dann is a colleague of Bob at their office.

• Bob and Dann often go for a drink after work.

• Bob always sings Anna’s

songs when he is in a good mood after drinking.

• So, the melodies of Anna’s songs stick to Dann’s

memory.

Bob and Dann

Dann and Elena

• Dann is humming the

melodies of Anna’s songs, but he modifies or mutilates them funnily, when doing yard work in his backyard.

• Elena, Dann’s partner, hears

Dann’s humming Anna’s songs in the backyard.

Dann and Elena

Anna, Elena and Fujiko

•

•

Anna

Elena Fujiko

Anna, Elena and Fujiko

• A graph gives us an intuitive understanding of the relationship between actors.

• However, we have to utilize numerics in order to analyze the relationship between actors using a computer.

We can use a matrix (a “sociomatrix”) to represent the relationship between actors by numerics.

A B C D E F A

B C D

E F

Each column corresponds to one actor.

A B C D E F A

B C D

E F

Each row also corresponds to one actor.

A B C D E F A

B C D

E

F

The order of actors aligned in the rows must correspond to the order in the columns.

A B C D E F A

B C D

E

F

If actor

sends an arc to actor

, put “ 1 ” in the cell (i, j),

which means the cell on which the row of actor

meets the column of actor

.

i j

i 1

i j j

Be careful about the direction of arcs.

If actor

does not send an arc to actor

, put “0” in the cell (j, i).

i j

i 1

j 0

i j

Complete the sociomatrix.

社会は，ネットワークあるいは行列で表現できる。

頂点同士の distance を縮めることが，多くのビジネスの核心

特許法第六十八条特許権者は、業として特許発明の実施をする権利を専有する。[…]

著作者は、その著作物を複製する権利を専有する。

非線形に変化するインセンティブを用意する試みの例