Successful Strategies in the Tragedy of the Commons

Successful Strategies in the Tragedy of the Commons

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Yohsuke Murase

June 12, 2018
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  1. Successful strategies in the Tragedy of the Commons Yohsuke Murase,

    Seung Ki Baek RIKEN Center for Computational Science, Japan Pukyong National University, Korea GAMENET @NetSci 2018 2018 Jun. 12 Y. Murase, S.K. Baek, J. Theor. Biol. 449 p94 (2018)
  2. 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 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  3. Tit-For-Tat c c c c c c d d c

    d c c d c c c It is guaranteed that your payoff is no less than the co-players'. Cooperation is fragile against an error. Copy the co-player's previous move. c c c d c c c d d c c d It cannot exploit naive cooperator. TFT TFT TFT c c c c c c TFT ALLC
  4. Win-Stay-Lose-Shift Switch your action if you are unhappy. c c

    c c c c d d c d c c c c c c The cooperation is tolerant against an error. It can exploit naive cooperators. Weak against defectors. c c c c c c d d c c d d c c d c d d c d d d c d d d c d WSLS WSLS WSLS AllC WSLS ALLD
  5. three conditions for successful strategies Efficiency ⌦ ⌦ ⌦ cooperation

    Defensibility ⌦ Distinguishability > ⌦ AllC AllC WSLS WSLS TFT
  6. form a cooperative Nash-equilibrium Efficiency ⌦ ⌦ ⌦ cooperation Defensibility

    ⌦ & You can publicly announce your strategy. Co-players understand it is not possible to exploit you. The best thing the co-players can do is to take the same strategy. => A cooperative Nash-equilibrium is formed. ⌦
  7. form a cooperative Nash-equilibrium Efficiency ⌦ ⌦ ⌦ cooperation Defensibility

    ⌦ & You can publicly announce your strategy. Co-players understand it is not possible to exploit you. The best thing the co-players can do is to take the same strategy. => A cooperative Nash-equilibrium is formed. ⌦ ⌦ ⌦
  8. AllD TFT AllC WSLS TF2T more cooperation more defections defensible

    strategies efficient strategies Is there a strategy satisfying these conditions simultaneously? Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦
  9. solution for n=2 • Enumerate 65536 memory-2 strategies. • =>

    4 strategies are found! TFT-ATFT
  10. What about n-person public-goods game? n=3 the number of defecting

    co-players
  11. TFT-ATFT fails in 3-person game c c c c c

    c d d c d c c d c c c c c c c c c d d c d c c d d c d c c c c d d d d n=2 n=3 They suspect each other to keep defecting. After Alice recovered 'c', Bob can safely return to 'c'.
  12. Brute-force enumeration! • # of memory-2 strategies = 2^40

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

  14. Enumeration of strategies 1,099,511,627,776 # of m=2 strategies Defensibility against

    AllD 805,306,368
  15. Enumeration of strategies 1,099,511,627,776 # of m=2 strategies Defensibility against

    AllD 805,306,368 Defensibility 3,483,008 Defensibility ⌦
  16. Enumeration of strategies 1,099,511,627,776 # of m=2 strategies Defensibility against

    AllD 805,306,368 Efficiency 0 Efficiency ⌦ ⌦ ⌦ cooperation Defensibility 3,483,008 Defensibility ⌦
  17. impossibility AllD TFT AllC AON TF2T There is no defensible

    and efficient strategy for n=3 in memory-2 strategy space. 2288 ⇡ 5 ⇥ 1086 <latexit sha1_base64="59xhvHjWfcNgQtBcMOCRhvnkoys=">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</latexit> <latexit sha1_base64="59xhvHjWfcNgQtBcMOCRhvnkoys=">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</latexit> <latexit sha1_base64="59xhvHjWfcNgQtBcMOCRhvnkoys=">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</latexit> <latexit sha1_base64="59xhvHjWfcNgQtBcMOCRhvnkoys=">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</latexit> The number of m=3 strategies : Is there no solution for the three-person game? or Does the solution exist in memory-3 strategies?
  18. Enumeration of strategies 1,099,511,627,776 # of m=2 strategies Defensibility against

    AllD Defensibility 3,483,008 805,306,368 Defensibility ⌦
  19. Enumeration of strategies 1,099,511,627,776 # of m=2 strategies Defensibility against

    AllD Defensibility 3,483,008 805,306,368 “Partial” Efficiency (pcooperation> 0) 544 Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦
  20. Enumeration of strategies 1,099,511,627,776 # of m=2 strategies Defensibility against

    AllD Defensibility 3,483,008 805,306,368 “Partial” Efficiency (pcooperation> 0) 544 Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦ Distinguishability 256 Distinguishability > ⌦ AllC AllC
  21. 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) 544 Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦ Distinguishability 256 Distinguishability > ⌦ AllC AllC
  22. elevate m=2 PS2 to m=3 successful strategies • When n=2,

    a successful strategy TFT-ATFT is constructed based on TFT (partially efficient & defensible) strategy. • For n=3 PG game, we can construct a successful strategy based on PS2.
  23. 24_cddccc B B&C B 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_ccdcdc 37_dccdcd 48_ddcccc 58_dddcdc 16_cdcccc 63_dddddd 23_cdcddd 41_dcdccd 35_dcccdd 11_ccdcdd 29_cdddcd 44_dcddcc 14_cccddc 38_dccddc 53_ddcdcd 50_ddccdc 26_cddcdc 56_dddccc 31_cddddd 15_ccdddd 5_cccdcd 17_cdcccd 20_cdcdcc 47_dcdddd 59_dddcdd 62_dddddc 61_ddddcd 60_ddddcc 55_ddcddd 51_ddccdd
  24. Successful strategies are found. Mutual cooperation is reached with probability

    1 while keeping the defensibility and the distinguishability. AllD TFT AllC WSLS TF2T defensible strategies efficient strategies Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦ An example of successful strategy.
  25. 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, we proved There is a critical memory length above which a fundamentally new class of strategies may exist.
  26. Conclusions • It is indeed possible to design a strategy

    satisfying the three conditions for n=2 and n=3 PG game. • We can realize mutual cooperation without a risk of being exploited. Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦ Distinguishability > ⌦ AllC AllC social dilemma can be solved by well-engineered strategies, not by appealing to our moral. Y. Murase, S.K. Baek, J. Theor. Biol. 449 p94 (2018)
  27. Open Access! Y. Murase, S.K. Baek, J. Theor. Biol. 449

    p94 (2018)