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Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational Science, Japan Pukyong National University, Korea I have a solution!

Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

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Page 1: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Finding successful strategies for social dilemma using K

computerYohsuke Murase, Seung Ki Baek

RIKEN Center for Computational Science, Japan Pukyong National University, Korea

I have a solution!

Page 2: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

iterated Prisoner’s Dilemma

cooperation defection

cooperation (3,3) (0,5)

defection (5,0) (1,1)

long-term payoff

implementation error occurs with probability e<latexit sha1_base64="fSe004YKF6vMcUZEj33avSo9wuc=">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</latexit><latexit sha1_base64="fSe004YKF6vMcUZEj33avSo9wuc=">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</latexit><latexit sha1_base64="fSe004YKF6vMcUZEj33avSo9wuc=">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</latexit><latexit sha1_base64="fSe004YKF6vMcUZEj33avSo9wuc=">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</latexit>

fi ⌘ lime!0

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player A

player B

Page 3: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Strategies for IPD gameTit-For-Tat (TFT) Win-Stay-Lose-Shift (WSLS)It is guaranteed that your payoff is no less than the co-players'.

Cooperation is fragile against an error.

Cooperation is tolerant against an error.

Repeatedly exploited against AllD.

c

c

c d

c c

c d

d c

c

d

Other strategies (TF2T, ZD strategies, generous TFT…) have positives and negatives.

dc d

d

c

ddd

Page 4: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

partners or rivals?

defensibleefficient

successful

AllC AllD

generous TFT

generous ZD

WSLS

TF2T

extortionate ZD

TFT

Trigger

more cooperations more defections

It would be great if a single strategy satisfies these advantages simultaneously.

=> Nash equilibrium with a guarantee that you'll never lose.

(partner) (rivals)Efficiency

⌦cooperation

Defensibility

�⌦

???

Page 5: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

A solution for Prisoner’s Dilemma

comprehensive search in Memory-2 strategies

Yi et al., J. Theor. Biol. (2017)

4 out of 65,536 strategies satisfies the conditions

It is indeed possible to realize cooperation without exposing themselves to the risk of being exploited.

TFT-ATFT

Page 6: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

What about n-person public-goods game?

n=3 the number of defecting co-players

Murase & Baek, J.Theor.Biol. (2018)

TFT-ATFT does not work for n=3. Does a solution exist for n=3?

Page 7: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Enumeration using K-computer

22nm

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n=3, m=1 : 256n=3, m=2 : 1,099,511,627,776 (2^40)

Page 8: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Enumeration of strategies1,099,511,627,776# of m=2 strategies

Defensibility against AllD 805,306,368

Efficiency 0Efficiency

⌦cooperation

Defensibility 3,483,008Defensibility

�⌦

Page 9: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Impossibility for m=2

AllDTFTAllCAON

TF2T

memory-2 strategies

no successful strategy exists in memory-2 strategy space.

0 / 1,099,511,627,776

No solution for n=3 game? or The solution exists when m>2?

Page 10: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Memory-3 strategies

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497323236409786642155382248146820840100456150797347717440463976893159497012533375533056

=

comparable to the number of protons in the universe

Although direct enumeration is impossible,we found there are at least 256 solutions!

Page 11: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Enumeration of strategies

1,099,511,627,776# of m=2 strategies

Defensibility against AllD

Defensibility 3,483,008

805,306,368

“Partially” Successful Strategies (PS2)

“Partial” Efficiency (pcooperation> 0) 544Efficiency

⌦cooperation

Defensibility

�⌦

Distinguishability 256Distinguishability

>⌦

AllC

AllC

Page 12: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

elevating m=2 PS2 to m=3 successful strategies

24_cddccc

B

B&CB

C

B

C

B C

0_cccccc

34_dcccdc

25_cddccd

12_ccddcc

46_dcdddc

4_cccdcc

40_dcdccc

22_cdcddc

3_ccccdd

43_dcdcdd

1_cccccd

10_ccdcdc37_dccdcd

48_ddcccc 58_dddcdc

16_cdcccc

63_dddddd

23_cdcddd

41_dcdccd35_dcccdd

11_ccdcdd

29_cdddcd

44_dcddcc14_cccddc38_dccddc

53_ddcdcd50_ddccdc 26_cddcdc

56_dddccc

31_cddddd

15_ccdddd

5_cccdcd17_cdcccd

20_cdcdcc

47_dcdddd59_dddcdd

62_dddddc

61_ddddcd

60_ddddcc

55_ddcddd

51_ddccdd

Page 13: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Successful strategies indeed exists when m=3. Full cooperation is achieved

while keeping the defensibility and the distinguishability.

AllDTFTAllCWSLS

TF2T

defensible strategiesefficient strategiesEfficiency

⌦cooperation

Defensibility

�⌦

� 256

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Page 14: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

An example of successful strategies.

Page 15: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

memory length and # of players

m � n<latexit sha1_base64="En7TK770IIYn3eKi5faMAC8QzmU=">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</latexit><latexit sha1_base64="En7TK770IIYn3eKi5faMAC8QzmU=">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</latexit><latexit sha1_base64="En7TK770IIYn3eKi5faMAC8QzmU=">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</latexit><latexit sha1_base64="En7TK770IIYn3eKi5faMAC8QzmU=">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</latexit>

n=2 : TFT-ATFT (m=2) n=3 : m=3 Successful Strategies ...

for general n (>=3), we show

There is a critical memory length above which a fundamentally new class of strategies may exist.

# of players

mem

ory

leng

th

(3,3)

(2,2)

must be satisfied.

Murase & Baek, J.Theor.Biol. (2018)

Page 16: Finding successful strategies for social dilemma …...Finding successful strategies for social dilemma using K computer Yohsuke Murase, Seung Ki Baek RIKEN Center for Computational

Conclusions• A new class of Nash equilibrium strategies was found with the aid

of the K computer. • Well-computed strategies can solve social dilemma without

appealing to our moral.

• Y. Murase & S.K. Baek, “Seven rules to avoid the tragedy of the commons”, J. Theor. Biol. 449 p94 (2018) • Y. Murase & S.K. Baek, “Automata representation of successful strategies for social dilemmas”, arXiv:1910.02634,

under review • Y. Murase & S.K. Baek, “Memory-three strategies that are partners as well as rivals”, in preparation

References

Exploring a broader strategy space does make a difference.• public-goods game for n>3 • perception error • multiple choice of actions • indirect reciprocity • network reciprocity