Winner-weaken-loser-strengthen rule leads to optimally cooperative interdependent networks

  • Lei Shi
  • Chen Shen
  • Yini Geng
  • Chen Chu
  • Haoran Meng
  • Matjaž Perc
  • Stefano Boccaletti
  • Zhen WangEmail author
Original Paper


We introduce a winner-weaken-loser-strengthen rule and study its effects on how cooperation evolves on interdependent networks. The new rule lowers the learning ability of a player if its payoff is larger than the average payoff of its neighbors, thus enhancing its chance to hold onto its current strategy. Conversely, when a player gaining less than the average payoff of its neighborhood, its learning ability is increased, thus weakening the player by increasing the chance of strategy change. Furthermore, considering the nature of human pursue fairness, we let a loser, someone who has larger learning ability, can benefit from another network, whereas a winner cannot. Our results show that moderate values of the threshold lead to a high cooperation plateau, while too high or too small values of the threshold inhibit cooperation. At moderate thresholds, the flourishing cooperation is attributed to species diversity and equality, whereas a lacking of species diversity determines the vanishing of cooperation. We thus demonstrate that a simple winner-weaken-loser-strengthen rule significantly expands the scope of cooperation on structured populations.


Cooperation Evolutionary Game Theory Interdependent Network Winner-Weaken-Loser-Strengthen Rule Coevolution 



We are grateful to Dr. Keke Huang for useful discussions. We acknowledge support from (i) National Natural Science Foundation of China (Grants No. U1803263), the National 1000 Young Talent Plan (No. W099102), the Fundamental Research Funds for the Central Universities (No. 3102017jc03007) and China Computer Federation - Tencent Open Fund (No. IAGR20170119) to Z.W, (ii) the National Natural Science Foundation of China (Grants No. 11671348) to L.S., (iii) the Yunnan Postgraduate Scholarship Award to C.S. and (iv) the Slovenian Research Agency (Grants J1-7009, J1-9112 and P1-0403) to M.P.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Statistics and MathematicsYunnan University of Finance and EconomicsKunmingChina
  2. 2.School of SoftwareYunnan UniversityKunmingChina
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Center for Applied Mathematics and Theoretical PhysicsUniversity of MariborMariborSlovenia
  5. 5.Complexity Science Hub ViennaViennaAustria
  6. 6.CNR Institute for Complex SystemsFlorenceItaly
  7. 7.Unmanned Systems Research InstituteNorthwestern Polytechnical UniversityXi’anChina
  8. 8.School of Mechanical Engineering and Center for OPTical IMagery Analysis and Learning (OPTIMAL)Northwestern Polytechnical UniversityXi’anChina

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