NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July) Cooperative hierarchical structures emerging in...
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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)
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?
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
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 j
1 if j is i’s neighbor
0 otherwise
i
Cooperator (C)
Defector (D)
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
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)
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
If the element b is not constant?(feedback)
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
Adaptive game
L×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
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.
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
plus, interacting structure is shaped by the behavior of agents?
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 = 10Example)
b0 = 2.3 b0 = 8.0b0 = 1.1
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.
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-hubsRandom → heterogeneous
Disassortative mixing
All-C region
non-local link only
2.7(1)
)/exp()/exp(~ 10 KKBKKAP
Fat-tail distribution
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
Stability of cooperation(noise)
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 linksp=1: only non-local links
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
Stability of all-C state
α=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.
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
)}](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
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
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-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
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