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Cooperative hierarchical structures emerging in multiadaptive games

Cooperative hierarchical structures emerging in multiadaptive games. Sungmin Lee (Norwegian University of Science and Technology). & Petter Holme (Umeå University, SungKyunKwan University) Zhi -Xi Wu (Lanzhou University ). References).

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Cooperative hierarchical structures emerging in multiadaptive games

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  1. Cooperative hierarchical structures emerging in multiadaptive games Sungmin Lee (Norwegian University of Science and Technology) & PetterHolme (Umeå University, SungKyunKwan University) Zhi-Xi Wu (Lanzhou University) References) S. Lee, P. Holme, and Z.-X. Wu, PRL 106, 028702 (2011) S. Lee, P. Holme, and Z.-X. Wu, PRE 84, 061148 (2011)

  2. Introduction Tragedy of the commons Cooperation is everywhere! The most important question for game-theoretic research is to map out the conditions for cooperation to emerge among egoistic individuals. ► If the elements of payoff matrix are time-varying? ► If both the rules of the game and the interaction structure are shaped by the behavior of the agents? ► Feedback from the behavior of agents to the environment? ► Cooperation and network topology emerging from the dynamics?

  3. Classic model (Nowak-May game) M. A. Nowak and R. M. May, Nature 359, 826 (1992) : i’s payoff obtained from a game with j L×L agents are placed on 2d lattice Total payoff 1 if j is i’s neighbor 0 otherwise i Update Cooperator (C) Defector (D) Agent i adopts the strategy of the neighbor j with the highest payoff

  4. ρ 1 b bc Phase diagram ρ t = 0 t = 2 t = 3 Steady state t = 1 t M. A. Nowak and R. M. May, Nature 359, 826 (1992)

  5. If the element b is not constant? (feedback)

  6. Adaptive game L×L agents are placed on 2d lattice Payoff matrix Cooperator (C) : the density of cooperators in the population Defector (D) : representing a neutral cooperation level from the society’s perspective (set as 0.5) : the strength of feedback from the environment to the game rule

  7. Numerical results In region II*, there are two absorbing states, ρ= 0.5 or 0 (coexist or all-D). When the strength of feedback increases, coexistence of C and D increases.

  8. plus, interacting structure is shaped by the behavior of agents?

  9. Multiadaptive game Each agent has one non-local link, which can be rewired to maximize own payoff. k k j j i update i • If agent j has the highest payoff among i’s neighbors and i itself • Agent i adopts j’s strategy and rewire its non-local link to j’s non-local partner k. Example) L = 10 b0 = 8.0 b0 = 1.1 b0 = 2.3

  10. Numerical results Assuming a well-mixed case Replicator dynamics ρ=0, 1, and oscillating b=exponential decaying, exponential increasing, oscillating In region II, there are three absorbing states, ρ= 0.5, 1, 0 ( coexist, all-D, all-C) Increasing feedback strength, region I decreases and cooperation increases.

  11. Correlation between game and structure non-local link only 2.7(1) 2.7(1) Random → heterogeneous C-hubs Emergent network structure Hierarchical structure (C ~ 1 / k) All-C region Fat-taildistribution Disassortative mixing

  12. Stability of cooperation (noise)

  13. p=0: 2d & non-local links p=1: onlynon-local links p = Prob. oflocalconnection is removed (bondpercolation) The local connections are essential to support cooperation.

  14. Stability of all-C state The strategy of an agent on hub (a) or randomly selected (b) is changed to the opposite (flipping) for each time Δt = 100. α=4, β=1, b0=3.5 C → D or D→ C Due to a hierarchical structure, the system is governed by the strategy of the agent on hub. The noise doesn't spread to the whole system since it is mainly applied to nodes with low degree. The high-degree C can protect their neighbors from imitating defectors. No all-C. p = prob. of each agent mutates regardless of payoffs. By mutation, all-C state would not be evolutionary stable.

  15. Time scales α=4 Random updating Every time step only one randomly chosen agent may change his strategy. More strategy updating : strategy updating : link updating More link updating “The effect of more frequent link updating is similar to random dynamics” the random dynamics efficiently slows down strategy updating The existence of the all-C state needs a comparatively fast strategy dynamics.

  16. Summary If the element b is not constant? In region II*, ρ= 0.5 or 0 (coexist or all-D) α, coexistence  Interacting structure is shaped by agents’ behavior? In region II, ρ= 0.5, 0, and 1 (coexist, all-D, and all-C ) α, cooperation and region I Heterogeneous structure with C-hubs Fat-tailed, hierarchical structure, disassortative Stability of cooperation (noise) Local connections are essential to support cooperation All-C state would not be evolutionary stable All-C state needs a comparatively fast strategy dynamics

  17. Thank you for your attention! Sungmin Lee Zhi-Xi Wu PetterHolme References) S. Lee, P. Holme, and Z.-X. Wu, PRL 106, 028702 (2011) S. Lee, P. Holme, and Z.-X. Wu, PRE 84, 061148 (2011)

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