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)

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 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 fi ⌘ lim e!0 lim T !1 1 T T 1 X t=0 F(t) i 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

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

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

three conditions for successful strategies Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦ Distinguishability > ⌦ AllC AllC WSLS WSLS TFT

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. ⌦

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. ⌦ ⌦ ⌦

AllD TFT AllC WSLS TF2T more cooperation more defections defensible strategies efficient strategies Is there a strategy satisfying these conditions simultaneously? Efficiency ⌦ ⌦ ⌦ cooperation Defensibility ⌦

solution for n=2 • Enumerate 65536 memory-2 strategies. • => 4 strategies are found! TFT-ATFT

What about n-person public-goods game? n=3 the number of defecting co-players

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'.

Brute-force enumeration! • # of memory-2 strategies = 2^40

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

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

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

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 ⌦

impossibility AllD TFT AllC AON TF2T There is no defensible and efficient strategy for n=3 in memory-2 strategy space. 2288 ⇡ 5 ⇥ 1086 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 AAACpHichVG7ThtBFD0seRAT4g00kSiyigVKZd218rCoEDQpKADHNhLG1u4yNiPvS7tjC1i5pOEHUqQKUoqIligfQMMPpOATopREoqHger0SIijkjmbmzJl77szRtUNXxorofEwbf/Dw0eOJJ7nJp1PP8vrz6Voc9CJHVJ3ADaIN24qFK31RVVK5YiOMhOXZrqjb3eXhfb0volgG/ke1F4otz+r4si0dSzHV0l+WmkmpXB4YDSsMo2DXeNtQ0hOxYVIzKb8btPQCFSkN4y4wM1BAFquB/gMNbCOAgx48CPhQjF1YiHlswgQhZG4LCXMRI5neCwyQY22PswRnWMx2ee3waTNjfT4Pa8ap2uFXXJ4RKw3M0U/6Rhd0Rsf0i67+WStJawz/sse7PdKKsJU/fFG5/K/K411h50Z1j8Lm7Ps9KbRRTr1I9hamzNClM6rf3/90UVlYn0vm6Yh+s78vdE6n7NDv/3G+ron1z8hxg8y/23EX1EpFk4rm2pvC4lLWqgnM4hVecz/eYxEfsIoqv3uAY5zguzavrWgVrTpK1cYyzQxuhda8BlBbmyw= 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 The number of m=3 strategies : Is there no solution for the three-person game? or Does the solution exist in memory-3 strategies?

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

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 ⌦

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

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

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.

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

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.

memory length and # of players m n 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 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.

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)

Open Access! Y. Murase, S.K. Baek, J. Theor. Biol. 449 p94 (2018)