Cooperative hierarchical structures emerging in multiadaptive games

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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)

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

Introduction

B A D C

D P T

C S 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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Classic model (Nowak-May game)

j i D C

D 0 b(>1)

C 0 1

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 j1 if j is i’s neighbor0 otherwise

i

Cooperator (C)

Defector (D)

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t = 0 t = 1 t = 2 t = 3 Steady state

t

ρ

ρbbc1

Phase diagram

M. A. Nowak and R. M. May, Nature 359, 826 (1992)

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If the element b is not constant?(feedback)

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Adaptive gameL×L agents are placed on 2d lattice

Cooperator (C)Defector (D)

j i D C

D 0 b(t)

C 0 1

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

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Multiadaptive gameEach 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 itselfAgent i adopts j’s strategy and rewire its non-local link to j’s non-local partner k.

i

j

k

update

L = 10Example)

b0 = 2.3 b0 = 8.0b0 = 1.1

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Numerical results

Replicator dynamicsAssuming 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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Correlation between game and structure

Emergent network structure

2.7(1)

Hierarchical structure(C ~ 1 / k)

C-hubsRandom → heterogeneous

Disassortative mixing

All-C region

non-local link only

2.7(1)

)/exp()/exp(~ 10 KKBKKAP

Fat-tail distribution

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Stability of cooperation(noise)

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p = Prob. of local connection is removed (bond percolation)

The local connections are essential to support cooperation.

p=0: 2d & non-local linksp=1: only non-local links

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Stability of all-C stateα=4, β=1, b0=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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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

)}](exp{1/[1)( 1 ij uujiP

)}](exp{1/[1)( 2 ij uujiW

: 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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SummaryIf 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-hubsFat-tailed, hierarchical structure, disassortative

In region II*, ρ = 0.5 or 0 (coexist or all-D)

Local connections are essential to support cooperationAll-C state would not be evolutionary stableAll-C state needs a comparatively fast strategy dynamics

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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