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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
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 …
Motivation Model Theory Experiment Simulation Conclusions
Why people cooperate?
Natural selection (Biology)Competition
Rationality (Economics)Self-interest
Motivation Model Theory Experiment Simulation Conclusions
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).
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
Costly punishment as a mechanism of promoting cooperation is ambiguous In real world, costly punishment does exist
Motivation Model Theory Experiment Simulation Conclusions
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)
Motivation Model Theory Experiment Simulation Conclusions
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.
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
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).
Motivation Model Theory Experiment Simulation Conclusions
OutlineModel
Evolutionary Game Theoretical Analysis
Live interactive Experiments
Agent-based Computer Simulation
Motivation
Conclusion
Donation Game A society with a very large population. Each individual is endowed with a binary
reputation: good (G) or bad (B).
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
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.
Motivation Model Theory Experiment Simulation Conclusions
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.
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
G BC
D
G G
B G
(Donor’s action)
X
J(Recipient’s reputation)
(Donor’s new reputation)
Non-punishment norm
G BC
D
P
G G
B B
B G
(Donor’s action)
X
J(Recipient’s reputation)
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).
Motivation Model Theory Experiment Simulation Conclusions
OutlineModel
Evolutionary Game Theoretical Analysis
Live interactive Experiments
Agent-based Computer Simulation
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.
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
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
Motivation Model Theory Experiment Simulation Conclusions
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.
Motivation Model Theory Experiment Simulation Conclusions
Phase portraits of 3 social norms
0.02 3b 2c 1 4
Punishment-optional norm Punishment-provoking norm
Non-punishment norm
Phase portraits of 3 social norms
0.02 3b 2c 1 4
NESSNESS
NESS
Non-punishment norm
Punishment-optional norm Punishment-provoking norm
Phase portraits of 3 social norms
0.02 3b 2c 1 4
NESS
CESS
NESS
NESS
CESS
Non-punishment norm
Punishment-optional norm Punishment-provoking norm
Phase portraits of 3 social norms
0.02 3b 2c 1 4
NESS
CESS
NESS
NESS
CESS
CESS
Non-punishment norm
Punishment-optional norm Punishment-provoking norm
Phase portraits of 3 social norms
0.02 3b 2c 1 4
15%
NESS
CESS
NESS
NESS
CESS
CESS
Non-punishment norm
Punishment-optional norm Punishment-provoking norm
Phase portraits of 3 social norms
0.02 3b 2c 1 4
15%
60%
NESS
CESS
NESS
NESS
CESS
CESS
Non-punishment norm
Punishment-optional norm Punishment-provoking norm
Phase portraits of 3 social norms
0.02 3b 2c 1 4
15%
60% 81%
NESS
CESS
NESS
NESS
CESS
CESS
Non-punishment norm
Punishment-optional norm Punishment-provoking norm
Intuitive view of attraction basin
Punishment-optional norm Punishment-provoking norm
Motivation Model Theory Experiment Simulation Conclusions
Intuitive view of attraction basin
Punishment-optional norm Punishment-provoking norm
Motivation Model Theory Experiment Simulation Conclusions
Intuitive view of attraction basin
NESS
CESS
CESS
Blue: Punishment-optional normRed: Punishment-provoking norm
Motivation Model Theory Experiment Simulation Conclusions
Converge speed from the same initial point in different social norms
Blue: Punishment-optional norm
Red: Punishment-provoking norm
Motivation Model Theory Experiment Simulation Conclusions
OutlineModel
Evolutionary Game Theoretical Analysis
Live interactive Experiments
Agent-based Computer Simulation
Motivation
Conclusion
Live interactive Experiments
( 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
Motivation Model Theory Experiment Simulation Conclusions
Live interactive Experiments
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
Motivation Model Theory Experiment Simulation Conclusions
Live interactive Experiments
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
Motivation Model Theory Experiment Simulation Conclusions
( 4 ) A new period starts and subjects are requested to choose strategy again
Live interactive Experiments
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)
Motivation Model Theory Experiment Simulation Conclusions
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
Live interactive Experiments
Motivation Model Theory Experiment Simulation Conclusions
OutlineModel
Evolutionary Game Theoretical Analysis
Live interactive Experiments
Agent-based Computer Simulation
Motivation
Conclusion
Agent-based Computer Simulation
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.
Motivation Model Theory Experiment Simulation Conclusions
Agent-based Computer Simulation
Results Replicate the theoretical results (long memory length,
L; small learning rate, μ)
Theory Simulation
Motivation Model Theory Experiment Simulation Conclusions
Agent-based Computer 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
Motivation Model Theory Experiment Simulation Conclusions
Agent-based Computer 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
Initial strategy choice in experiment reflecting the culture in real society
Motivation Model Theory Experiment Simulation Conclusions
OutlineModel
Evolutionary Game Theoretical Analysis
Live interactive Experiments
Agent-based Computer Simulation
Motivation
Conclusion
Conclusion
Costly punishment works in promoting cooperation :
Motivation Model Theory Experiment Simulation Conclusions
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 :
Motivation Model Theory Experiment Simulation Conclusions
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)
Motivation Model Theory Experiment Simulation Conclusions
Thank you!