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エネルギー環境論
担当教官：谷本 潤 教授

第６回講義

社会ジレンマをモデル化する－統計物理学，進化ゲーム理論と社会ジレンマ－

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Mutually-interpenetrative view overwide spatial-scales

Urban

Architecture

Human
Global scale

Urban scale
Building-block

都市境界層

-∞

106 105

104

103

長さスケール[m]Building

To elaborate the Human - Environment -Social
System, it’s important a concept of“Simultaneous” or “Bridging to various scales”.
Two physical systems
having neighboring
special scales aremutually connected
through boundary
conditions.
Small scale←Interaction→Large scale

104

101 水

湿気

光

Room

空気
熱
音

10-1
-∞

100
10-2

Human

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Environment

Social System

Mutually-interpenetrative view overmutually different systems

Human

Environ
土壌
建築
ment

Revised AUSSSM

Between physical systems

××
都市“Environmental problems”
mean social dilemmas
conflicting those three
Unless bridging,systems.

appropriate boundary
conditions MUST be given.

Human

SocialHow “bridges” are defined?
Decision making → Social

Science for complex system
Evolutionary game theory, Multi-agent simulation,
Artificial intelligence (GA, NNw etc）

Environment

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What is the Game Theory ?
Game theory is a study of strategic decision making. More formally, it is "the study ofmathematical models of conflict and cooperation between intelligent rational decisionmakers.“
John von Neumann & Oskar Morgenstern; Theory of games and economic behavior,
1944.
Game theory has been widely recognized as an important tool inmany fields; economics, political science, psychology, as well as
biology, information science and even statistical physics. Eight gametheorists, including John Nash have won the Nobel Memorial Prize in
Economic Sciences, and John Maynard Smith was awarded the
Crafoord Prize for his application of game theory to biology.

Zero-sum (Constant-sum) games(Japanese) Chess, Go. Minimax theorem (von Neumann); For every twoperson, zero-sum game with finitely many strategies, there exists a value V
and a mixed strategy for each player, such that (a) Given player 2's strategy,
the best payoff possible for player 1 is V, and (b) Given player 1's strategy,
the best payoff possible for player 2 is −V.

Non zero-sum (Non constant-sum) games
Many applications happening in real world. Social dilemma, Prisoner’s
Dilemma, Chicken games etc.

Cuba Crisis -->Chicken game?

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2 by 2 game
Agent1

Agent2Agent2 Cooperation
Agent1
Cooperation（C）
Defection（D）（C）

Defection
（D）

R，R

S，T

T，S

P，P

R；Reward，T；Temptation，S；Sucker，P；Punishment

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Application; Analytical approach concerningequilibrium (steady-state) for Nonlinear systems
2-player 2-strategy game (2 by 2 game)

Class
Dilemma?
Prisoner’s Dilemma; PD
Yes
Chicken (Snow Drift; Hawk-Dove) Yes

GID RADYes YesYes NoStag Hunt; SH

Yes

No

Yes

Trivial

No

No

No

Basic Assumption-

Infinite population.

-

One-shot game; well-mixed situation (withneither social viscosity nor assortment
among agents).

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Prisoner’s Dilemma
Agent1

Agent2Agent2 Cooperation
Agent1
Cooperation（C）
Defection（D）（C）

Defection
（D）

R，R

S，T

T，S

P，P

R；Reward，T；Temptation，S；Sucker，P；Punishment

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Prisoner’s Dilemma
Agent1C

Agent2

C

D

R, R

S, T

D
T, S P, P
R；Reward，T；Temptation
S；Sucker，P；Punishment

2R（=8）>T+S（=6）>2P（=4）

Agent2 Cooperation

Defection（D）（C）
Agent1
Cooperation
5, 5
1, 7
（C）
Equal Pareto Optimum
Defection
（D）

7, 1

3, 3
Nash Equilibrium

Gamble-Intending Dilemma (GID); Dg=T-R=7-5>0
Risk-Averting Dilemma (RAD);

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Prisoner’s Dilemma
Agent1C

Agent2

C

D

R

S

D
T
P
R；Reward，T；Temptation
S；Sucker，P；Punishment

2R（=8）>T+S（=6）>2P（=4）

Agent2 Cooperation

Defection（D）（C）
Agent1
Cooperation
5
1（C）
Equal Pareto Optimum
Defection（D）

7

3
Nash Equilibrium

Gamble-Intending Dilemma (GID); Dg=T-R=7-5>0
Risk-Averting Dilemma (RAD);

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Prisoner’s Dilemma

Player2
Dr > 0
Dg > 0 Equal Pareto Optimum

Worst preferable
for Player 1

Pareto InverseOptimum

Pareto Optimum

Equal Pareto
Inverse-Optimum

Most preferable
for Player 1

S

P

S<P<R<T

R

T
Player1

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Chicken／ Hawk–Dove Game (Maynard Smith (1982))／ Snowdrift Game
Player2
Dr < 0
Dg > 0Equal Pareto Optimum

Pareto Optimum

Worst

Most preferable
for Player 1

P
S
P<S<R<T

R

T
Player1

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ChickenAgent1C

Agent2

C

D

R

S

D
T
P
R；Reward，T；Temptation
S；Sucker，P；Punishment

2R（=8）>T+S（=6）>2P（=4）

Agent2 Cooperation

Defection（D）（C）
Agent1
Cooperation
5
3 Nash Equilibrium（C）
Equal Pareto Optimum
Defection
7（D）
Nash Equilibrium

1

Worst

Gamble-Intending Dilemma (GID); Dg=T-R=7-5>0
Risk-Averting Dilemma (RAD);

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Stag Hunt／ Inspired by Jean-Jacques Rousseau; “Discours sur l'origine
et les fondements de l'inégalité parmi les hommes” (Chapter 2)

Player2
Dr > 0Best

Worst preferable
for Player 1

Dg < 0Pareto InverseOptimum
Equal Pareto
Inverse-OptimumPS
S<P<T<R

T

R
Player1

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Stag Hunt
Agent1

Agent2Agent2 Cooperation

C

C

D

R

S

D
T
P
R；Reward，T；Temptation
S；Sucker，P；Punishment

Defection（D）（C）
Agent1
Cooperation Nash Equilibrium
71（C） Best=Equal Pareto Optimum
Defection（D）

5

3
Nash Equilibrium

Gamble-Intending Dilemma (GID); Dg=T-R=5-7<0
Risk-Averting Dilemma (RAD);

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Trivial Dilemma Free game
Player2
Best

Dg < 0Dr < 0Worst

SP
P<S<T<R

T

R
Player1

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TrivialAgent1

Agent2Agent2 Cooperation

C

C

D

R

S

D
T
P
R；Reward，T；Temptation
S；Sucker，P；Punishment

Defection（D）（C）
Agent1
Cooperation Nash Equilibrium
73（C） Best=Equal Pareto Optimum
Defection
（D）

5

1

Gamble-Intending Dilemma (GID); Dg=T-R=5-7<0
Risk-Averting Dilemma (RAD);

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Evolutionary game
2 by 2 game considered time evolution
0

1
1+Dg

-Dr

0

0

1+Dg
-Dr

1

1

0

-Dr

C

D

C

1

-Dr

D

1+Dg

0

Dg； GID
Dr； RAD
1+Dg-DrBattle field

Defection

Cooperation
fraction

Cooperation

In case if PD（Dg>0, Dr>0）

1．A focal player plays a game witha randomly selected opponent.
2．Strategy (whether C or D)
Time step
adaptation based on obtained
payoff is considered.
You never see emerging cooperation, unless some additional
mechanism for social viscosity is implemented.

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What is Social Viscosity?
・Kin selection
・Direct reciprocity
・Indirect Reciprocity
・Network Reciprocity
・Group selection

Well-mixed situation

A restricted relation among
agentsLessing Anonymity
Emerging cooperation

A Game on a network

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Let us back to the Basic Assumption again;-

Infinite population.
One-shot game; well-mixed situation (with neither social
viscosity nor assortment among agents).

Let us describe Cooperation and defection strategies by;T

e1  1 0  ; CT

e2  0 1 ; D

Also, let us define game structure, i.e. payoff matrix as below;

R S 
T P   MFurther, let us define strategy frequency among agents at a
certain time step as below; Ts  s1 s2

Fraction of CD

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By simplex constraint;s2  1  s1 .

Let us think a simple example. When a focal player who offers D,
how much of payoff expectation she can get in case of paying withanother D player as her game opponent?

 P S  0 
   P
0 1  
T P  1 
By analogy, payoff expectations of both a C and D players
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Let us consider the following system dynamics, called
Replicator Dynamics, which is thought to be a good modelfor describing the reproduction process of population dynamics foranimal species.

si TT ei  M s s  M s
siChanging rate ofstrategy i; C when i=1& D when i=2

Payoff expectations of astrategy i player paying
with an average player
at this time step

Payoff expectations ofan average player
paying with an average
player at this time step

Implying benefit brought to a player who
adopts strategy i.

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si TT
Replicator Dynamics:
 e i  M s  s  M s has three equilibriums.
si
Two obvious equilibriums are;(1,0) ; A state absorbed by C where all players offer C (C Dominate phase) .
(0,1) ; A state absorbed by D where all players offer D (D Dominate phase) .
The third one is;PS P T  S  R

R T


P T  S  R (Polymorphic phase).

A question is what dynamics would be if analytic approach is applied to the
Replicator Dynamics, which is a (nonlinear) cubic equation for s1 or s2.

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Let us describe Replicator Dynamics explicitly by substituting i=1 and 2.

si T e i  M s T s  M ssi

 s1  R  T   s1  P  S   s2  s1  s2s2  R  T   s1  P  S   s2  s1  s2s  f s , s2  and s2  f 2 s1 , s2 
s2  1  s1 , we know;

When defining
1
1 1
reminding Simplex constraint;

as well as

f1   f 2
Again, Our current target is to evaluate Eigen values of Jacobi Matrix atf1 x   f1 x 
respective three equilibrium; s*.

 x
xn 
1

f
x




 
f x*  

 
PS
R T


x x x*(1,0) (0,1)  P  T  S  R P  T  S  R 
f n x   f n x 


 x1




xn 
 x  x*Generated by Foxit PDF Creator © Foxit Software
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f1
f 2
2

 3 R  S  T  P s1
s1
s1
 2R  2 S  T  2 P s1  S  P
f1
f 2
2
 3 R  S  T  P s1

s 2
s 2
- 2R  2 S  T  2 P s1  S  P
We know two Eaigen values of0 andf1 f1s1 s2

 f1
 s
J 1
 f 2 s
 1

f1   f1
s2   s1
f 2   f1s2   s1(its eiven vector is (1,-1)) .

f1 s2 
f
 1
s2 

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Thus, what we should currently do is evaluate sings ofat respective three equilibrium; s*.

f1 f1s1 s2f1 f12
 6 R  S  T  P s1s1 s2 4R  2 S  T  2 P s1  2S  P (1) At s*  1,0  ;   2 R  2T .
Thus, for(2) Ats*  0,1 ;Thus, for(3) At 0

  2S  2 P
 0

T  R  Dg  0 .

, it must be.

P  S  Dr  0 .

R  T P  S 2, it must be

PS
R T

s*   ; P T  S  R P T  S  R 
Thus, for   0, it must be;

R  S T  P.

P  S  R  T  P  S  Dr  0  T  R  D g  0 .

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Summing up all so far, we obtain;
Game

Trait

Nash Equilibrium

class

Sing

Sing

of

of

GID;

RSD;

Dg

Dr

+

Source or sink at Equilibrium; s*
(1,0)

(0,1)

+

Sourcesink

Saddle+-SourceSource

Sink

 Dr

 Dg  Dr Dg 
Dr  Dg PD

D-dominate

ChickenPolymorphic

Stag Hunt

Bi-stable(0,1) or (1,0)

-

+

Sink

Sink

SourceTrivialC-Dominate

(1,0)

-

-

Sink

SourceSaddleR T  DrP  T  S  R   D g  Dr

 Dg 
Dr  D g 

(0,1)

 Dr

D D
r
 g

 Dg 
Dr  Dg 

PS


s*

Where
 P T  S  R

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Phase diagram of 2 by 2 games
Dg

ChickenPDDr

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Prisoner’s Dilemma, PD

Dg

ChickenPD

Dr

SourceSink

TrivialStag Hunt

All agents areabsorbed by D.

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ChickenDg

ChickenPD
D-dominate

SourceDr

SourceSink

TrivialStag Hunt
All agents areabsorbed by Internal
Equilibrium.

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Stag Hunt

Dg

Source
Sink
Sink

0

Chicken
Depending on initial
distribution, someagents are absorbed
by D and other aresabsorbed by C.
1

TrivialPD
D-dominate

Dr

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Trivial, dilemma free game

SourceDg

ChickenSink

PD
All agents areabsorbed by C.

Polymorphic

s
0

1

Dr

TrivialStag Hunt
Bi-stable

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Phase diagram of 2 by 2 games
Dg

ChickenPD

Polymorphic

D-dominate

Dr

TrivialStag HuntC-dominate

Bi-stable

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Backgrounds & Purpose
Most previous studies

The real world

Entirely
cooperation

Entirely
cooperation

Entirely
defection

Agents can offer either
cooperation (C) or defection (D)Discrete strategy

Entirely
defection

Actual options might becontinuous rather than discreteContinuous or mixed strategy

One crucial question is whether there is a considerable
difference in game equilibria between the continuous ormixed strategies and those of discrete strategies?

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Setting for continuous, and mixed strategy games
Continuous strategy
Mixed strategy
1. Strategy value: si  [0,1]
si=1
si=0

complete cooperation
complete defection

1. Strategy value: si  [0,1]
si=1
si=0

complete cooperation
complete defection（0.8） （0.5） （0.2）（0.8） （0.5） （0.2）

2. Payoff function
 ( si , s j )   Dr si  (1  Dg ) s j

Agents can only offer either
C or D according to this strategy

 ( Dg  Dr ) si s j
T(=1+Dg)

P(=0)
0

R(=1)
1.0
0.3

C when Rnd[] < si, otherwise D
Rnd[ ]: a random number

2. Payoff function
Agent jAgent i

C

D

0.7

C

1, 1

-Dr, 1+Dg

S(=-Dr)
1.0 0

D

1+Dg, -Dr0, 0

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Results
Averaged cooperation fraction
0
1
Games are played on lattices (k = 8)

C

D

C

1

-Dr

D

1+Dg

0

Dg； GID
C

Discrete strategy

Continuous strategy

1

1

0.8

0.8

0.8

0.6
Dg
0.4

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0

0

0

0 0.2 0.4 0.6 0.8 1
DrD

Mixed strategy

1

0 0.2 0.4 0.6 0.8 1
DrDr； RAD

0 0.2 0.4 0.6 0.8 1Dr