social norm, costly punishment and the evolution to ... · social norm, costly punishment and the...

Social Norm, Costly Punishment and the Evolution to Cooperation :Theory, Experiment and Simulation

Tongkui Yu1, 2, Shu-Heng Chen2, Honggang Li1*

1 Department of Systems Science, Beijing Normal University2 AI-ECON Research Center, Taiwan Chengchi University

Kung Fu Tzu: Confucianism (ethics)

Han Fei Tzu: Legalism(law and punishment）

Punish or not?

(Legalism prevailed in turbulent society )Han Fei Tzu: Legalism(law and punishment）

Punish or not?

Kung Fu Tzu: Confucianism (ethics)

(Confucianism prevailed in stable society）

Punish or not?

OutlineModel of donation game

Evolutionary Game Theoretical Analysis

Live Interactive Experiments

Agent-based Computer Simulation

Motivation

Conclusion

Cooperation is very important

not only for a society of human beings

Motivation Model Theory Experiment Simulation Conclusions

Cooperation is very important

but also for many other biological systems …


Why people cooperate?

Natural selection (Biology)Competition

Rationality (Economics)Self-interest


Cooperation

Costly punishment as a mechanism topromote cooperation People are willing to pay

a personal cost in order to punish wrongdoers.

E. Fehr, S. Gächter, Nature 415, 137 (2002)

E. Fehr, U. Fischbacher, Nature 425, 785 (2003)

D. J. Quervain, et al Science 305, 1254 -1258 (2004)

C. F. Camerer, E. Fehr, Science 311, 47 (2006)

J. Henrich et al., Science 312, 1767 (2006).


The role of costly punishment in promoting cooperation is ambiguous Positive

Fehr E. & Gachter S. (2000) Am. Econ. Rev. 90, 980-994 Fehr E. & Gachter S. (2002) Nature 415, 137–140 Gurerk, O., Irlenbusch, B. & Rockenbach, B. (2006). Science

312,108–111 Negative

Dreber, A. etc. (2008) Nature, 452, 348-351. Egas, M. & Riedl, A. (2008) PNAS, 275, 871-878. Ohtsuki, H. etc. (2009) Nature 457, 79-82. Costly punishment can hardly lead to an efficient

equilibrium The best choice to defectors is withholding help

rather than punishing them




312,108–111 Negative

Dreber, A. etc. (2008) Nature, 452, 348-351. Egas, M. & Riedl, A. (2008) PNAS, 275, 871-878. Ohtsuki, H. etc. (2009) Nature 457, 79-82.

Costly punishment can hardly lead to an efficient equilibrium The best choice to defectors is withholding help rather than

punishing them




312,108–111. Negative

Dreber, A. etc. (2008) Nature, 452, 348-351. Egas, M. & Riedl, A. (2008) PNAS, 275, 871-878. Ohtsuki, H. etc. (2009) Nature 457, 79-82.

Costly punishment can hardly lead to an efficient equilibrium The best choice to defectors is withholding help rather than

punishing them


Costly punishment as a mechanism of promoting cooperation is ambiguous In real world, costly punishment does exist


Controversy Costly punishment is less efficient but exists

Question What does costly punishment exist for ?

“Costly punishment remains one of the most thorny puzzles in human social dilemmas” (Nature 452, 297-298)

“Costly punishment requires a mechanism for its evolution”(Nature 452, 348-351)


Our argument

Ohtsuki’s analysis only focuses on the equilibrium (i.e. Cooperative Evolutionary Stable State, CESS).

Although in equilibrium, punishment is not the most efficient.

But from an initial state far away from equilibrium, costly punishment may play a different role.


Our work Extends Ohtsuki’s model by explicitly

modeling the evolution process of individuals’ strategies

Investigates the role of punishment in the route to cooperation


Our results Costly punishment works in the route to

cooperation Enlarge the attraction basin of cooperative

evolutionary stable state (CESS). Increase the evolution speed to cooperative

evolutionary stable state (CESS).


OutlineModel


Live interactive Experiments


Motivation

Conclusion

Donation Game A society with a very large population. Each individual is endowed with a binary

reputation: good (G) or bad (B).


Donation Game At each time, two players are sampled randomly. One

as donor and the other as recipient. Donor has 3 choices: cooperation (C), defection (D),

and punishment (P). Recipient does nothing. If C, donor spends a cost c (c=2) to give recipient a

benefit b (b=3); if D, no gain no loss; If P, donor spends a cost α (α=1) to give recipient a loss β (β=4) .

Recipient

Donor

C

(b,-c) (0,0)

D P

(-β,-α)

D

R

3b 2c 1 4


Donation Game

Each player has a strategy, which

determines his action (C, D, or P) when as a donor according to the reputation (G or B) of his recipient.


G BC CC DC P

Recipient’s reputation

D CD DD PP CP DP P

Strategy 1Strategy 2Strategy 3

Strategy 4

Donation Game After each interaction, the reputation of the

donor will be updated according to a ‘social norm’

The reputation update process is susceptible to errors. With probability μ (0<μ<1/2) , an incorrect reputation is assigned.


Social norm Assigns a new reputation to the donor, according to both the donor’s action (X) and

the recipient’s reputation (J).

G BC

D

P

G G

B G

B G

(Donor’s action)

X

J(Recipient’s reputation)

Punishment-optional norm


G BC

D

G G

B G

(Donor’s action)

X


(Donor’s new reputation)

Non-punishment norm

G BC

D

P

G G

B B

B G

(Donor’s action)

X


Punishment-provoking norm

Focus of this work

Within such a donation game framework, we will

compare the evolution process of strategy proportions in these 3 social norms

to understand the role of punishment in route to cooperative evolutionary stable state (CESS).


OutlineModel




Motivation

Conclusion

Strategy dynamics The driving force of individual’s strategy

evolution is the expected revenue of each strategy.

We calculate the expected revenue of each strategy

Insert them into “Replicator Dynamics”

HofbauerJ. & Sigmund K. (2003) Evolutionary game dynamics. Bull. Am. Math. Soc. 40, 479-519.


Strategy Dynamics

1 1 21 1( ) (1 )2 2

p c bx bx

2 1 21 1( ) (1 )2 2

p g c bx bx

3 1 2 31 1(0)2 2

p bx bx g

1 1 3 2 2 3;p p p p p p

'1 1 1

'2 2 2

( )

( )

x x p p

x x p p

Non-punishment norm


1 3 1 2 3

2 3 1 2 3

3 3 1 2 3

4 1 2 3 4 3 4

1 1 1( ) ( ) (1 )2 2 21 1 1( ) ( ) (1 )2 2 21 1 1 1( ) (1 )( ) ( ) (1 )2 2 2 21 1 (1 )( )2 2

p c x bx b x x

p g c x bx b x x

p g c g x bx b x x

p bx b x x g x g

1 1 4 2 2 4 3 3 4; ;p p p p p p p p p

'1 1 1

'2 2 2

'3 3 3

( )

( )

( )

x x p p

x x p p

x x p p

Punishment-optional

norm


Strategy Dynamics

1 1 4 2 2 4 3 3 4; ;p p p p p p p p p

'1 1 1

'2 2 2

'3 3 3

( )

( )

( )

x x p p

x x p p

x x p p

1 3 1 2 3

2 3 2 1 2 2 3

3 3 1 2 3

4 1 2 3 3

1 1 1( ) ( ) (1 )2 2 21 1 1( ) (1 )( )2 2 21 1 1 1( ) (1 )( ) ( ) (1 )2 2 2 21 1 (1 )( )2 2

p c x bx b x x

p g c x g bx bg x x

p g c g x bx b x x

p bx b x x x

Punishment-provoking norm


Strategy Dynamics

Evolutionary Stable State

Evolutionary Stable State (ESS): Attractive state

Given all other individuals take some strategy, the best choice for one individual is to take that strategy.

Cooperative (or Non-cooperative) Evolutionary Stable state (CESS or NESS) Most agents cooperate (defect).

Attraction basin of a CESS: All the initial states that will converge to the CESS. The larger attraction basin a CESS has, the more

probable the society will converge to this CESS.


Phase portraits of 3 social norms

0.02 3b 2c 1 4

Punishment-optional norm Punishment-provoking norm

Non-punishment norm


0.02 3b 2c 1 4

NESSNESS

NESS

Non-punishment norm



0.02 3b 2c 1 4

NESS

CESS

NESS

NESS

CESS

Non-punishment norm



0.02 3b 2c 1 4

NESS

CESS

NESS

NESS

CESS

CESS

Non-punishment norm



0.02 3b 2c 1 4

15%

NESS

CESS

NESS

NESS

CESS

CESS

Non-punishment norm



0.02 3b 2c 1 4

15%

60%

NESS

CESS

NESS

NESS

CESS

CESS

Non-punishment norm



0.02 3b 2c 1 4

15%

60% 81%

NESS

CESS

NESS

NESS

CESS

CESS

Non-punishment norm


Intuitive view of attraction basin



Intuitive view of attraction basin

NESS

CESS

CESS

Blue: Punishment-optional normRed: Punishment-provoking norm


Converge speed from the same initial point in different social norms

Blue: Punishment-optional norm

Red: Punishment-provoking norm


OutlineModel




Motivation

Conclusion


( Please choose your strategy )

Strategy 1 ( “CC” )Strategy 2 ( “CD” )Strategy 3 ( “CP” )Strategy 4 ( “DD” )

( 1 ) Subjects are requested to choose a strategy

Students of Taipei, Beijing and Chongqing



①Randomly match the subjects to form pairs, one as donor, the other as recipient;

②Calculate the payoffs of the subjects according to the donor’s strategy and the recipient’s reputation

③Update the donor’s reputation according to the applying social norm

( 2 ) System background processing



You are : donor or recipient

Your opponent’s reputation

Your opponent’s strategy

Your payoff in this period

Your reputation after this interaction

Your strategies and payoffs in last 15 periods

The strategies and payoffs in last 15 periods of a randomly selected subjects

( 3 ) Interaction results display


( 4 ) A new period starts and subjects are requested to choose strategy again


Starting from state with very few defectors (DD)

DDDD

DD

CP

The experiment results are consistent with the theoretical analysis qualitatively.

Non-Discriminable norm Punishment-optional norm Punishment-provoking norm

Starting from state with many defectors (DD)

Starting from state with many defectors (DD)


Initial strategy choice The strategy chose by a subject at the beginning of a experiment

Subjects make initial strategy choice without any idea about theexperiment and this may reflect the culture of real society

Taipei Beijing Chongqing



OutlineModel




Motivation

Conclusion


Artificial society system which realizes the individuals’ strategy update process in detail

Each period With a probability μ(learning rate),

sample one individual to be learner He will choose another

individual randomly Compare the total revenues in

the last L (memory length) periods

If the learner’s revenue is smaller than that of the learned, the learner will take the strategy of the learned.



Results Replicate the theoretical results (long memory length,

L; small learning rate, μ)

Theory Simulation



Results State with co-existence of CC, CD and CP strategy

(short memory length , L; quick learning rate , μ)

State with co-existence of CC, CD and CP strategy



Results State with co-existence of CC, CD and CP strategy

(short memory length , L; quick learning rate , μ)

State with co-existence of CC, CD and CP strategy

Initial strategy choice in experiment reflecting the culture in real society


OutlineModel




Motivation

Conclusion

Conclusion

Costly punishment works in promoting cooperation :


It can enlarge the attraction basin of CESS When the society has

few people cooperating, it can only struggle out of social dilemma by punishment

Conclusion

Costly punishment works in promoting cooperation :


It can increase theconvergence speed to CESS If the society are less

patient, it can only speed up to cooperative state by punishment.

(Legalism in

turbulentsociety )

(Confucianism in stablesociety）


Thank you!

social norm, costly punishment and the evolution to ... · social norm, costly punishment and the...

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