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!

iterated Prisoner’s Dilemma

cooperation defection

cooperation (3,3) (0,5)

defection (5,0) (1,1)

long-term payoff

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player A

player B

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

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

�⌦

???

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

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?

Enumeration using K-computer

22nm

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

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

Defensibility against AllD 805,306,368

Efficiency 0Efficiency

⌦

⌦

⌦cooperation

Defensibility 3,483,008Defensibility

�⌦

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?

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!

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

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

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

�⌦

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An example of successful strategies.

memory length and # of players

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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)

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