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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Cooperative hierarchical structures emerging in multiadaptive games & Petter Holme (Umeå University, SungKyunKwan University) Zhi-Xi Wu (Lanzhou University) S. Lee, P. Holme, and Z.-X. Wu, PRL 106, 028702 (2011) S. Lee, P. Holme, and Z.-X. Wu, PRE 84, 061148 (2011) References) Sungmin Lee (Norwegian University of Science and Technology)

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Introduction B A DC D PT CS R Payoff matrix Tragedy of the commons The most important question for game-theoretic research is to map out the conditions for cooperation to emerge among egoistic individuals. Cooperation is everywhere! ► 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?

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Classic model (Nowak-May game) j i DC D0b(>1) C01 M. A. Nowak and R. M. May, Nature 359, 826 (1992) L×L agents are placed on 2d lattice Update Agent i adopts the strategy of the neighbor j with the highest payoff Total payoff : i’s payoff obtained from a game with j 1 if j is i’s neighbor 0 otherwise i Cooperator (C) Defector (D)

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) t = 0 t = 1t = 2t = 3Steady state t ρ ρ b bcbc 1 Phase diagram M. A. Nowak and R. M. May, Nature 359, 826 (1992)

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) If the element b is not constant? (feedback)

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Adaptive game L×L agents are placed on 2d lattice Cooperator (C) Defector (D) j i DC D0b(t)b(t) C01 Payoff matrix : the density of cooperators in the population : 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

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) 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.

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) plus, interacting structure is shaped by the behavior of agents?

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Multiadaptive game Each agent has one non-local link, which can be rewired to maximize own payoff. i j k 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. i j k update L = 10 Example) b 0 = 2.3 b 0 = 8.0 b 0 = 1.1

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Numerical results Replicator dynamics Assuming a well-mixed case 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.

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Correlation between game and structure Emergent network structure 2.7(1) Hierarchical structure (C ~ 1 / k) C-hubs Random → heterogeneous Disassortative mixing All-C region non-local link only 2.7(1) Fat-tail distribution

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Stability of cooperation (noise)

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) p = Prob. of local connection is removed (bond percolation) The local connections are essential to support cooperation. p=0: 2d & non-local links p=1: only non-local links

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Stability of all-C state α=4, β=1, b 0 =3.5 The strategy of an agent on hub (a) or randomly selected (b) is changed to the opposite (ﬂipping) for each time Δt = 100. C → D or D → C 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. Due to a hierarchical structure, the system is governed by the strategy of the agent on hub. By mutation, all-C state would not be evolutionary stable. p = prob. of each agent mutates regardless of payoffs.

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Time scales Random updating Every time step only one randomly chosen agent may change his strategy. α=4 The existence of the all-C state needs a comparatively fast strategy dynamics. More strategy updating More link updating : strategy updating : link updating “The effect of more frequent link updating is similar to random dynamics” the random dynamics efficiently slows down strategy updating

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Summary If the element b is not constant? Interacting structure is shaped by agents’ behavior? Stability of cooperation (noise) α , coexistence 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 In region II*, ρ = 0.5 or 0 (coexist or all-D) Local connections are essential to support cooperation All-C state would not be evolutionary stable All-C state needs a comparatively fast strategy dynamics

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NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Thank you for your attention! Petter Holme S. Lee, P. Holme, and Z.-X. Wu, PRL 106, 028702 (2011) S. Lee, P. Holme, and Z.-X. Wu, PRE 84, 061148 (2011) References) Sungmin Lee Zhi-Xi Wu

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