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) S. Lee, P. Holme, and Z.-X. Wu, PRL 106, 028702 (2011)
S. Lee, P. Holme, and Z.-X. Wu, PRE 84, 061148 (2011)
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
Introduction
Cooperation is everywhere!
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.
► 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?
B
D
C
D
P
T
C
S
R
A
Payoff matrix
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Classic model (Nowak-May game)
L×L agents are placed on 2d lattice
M. A. Nowak and R. M. May, Nature 359, 826 (1992)
Cooperator (C)
Defector (D)
Total payoff
i
: i’s payoff obtained from a game with j
1 if j is i’s neighbor
0 otherwise
j
D
C
D
0
b(>1)
C
0
1
i
Update
Agent i adopts the strategy of the neighbor j
with the highest payoff
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t=0
t=1
t=2
t=3
Steady state
ρ
ρ
Phase diagram
t
1
bc
b
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)
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Adaptive game
L×L agents are placed on 2d lattice
Cooperator (C)
Defector (D)
j
Payoff matrix
i
D
C
: the density of cooperators in the population
D
0
b(t)
C
0
1
: 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 game
Each agent has one non-local link, which can be rewired to maximize own payoff.
k
k
j
i
j
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)
b0 = 1.1
L = 10
b0 = 2.3
b0 = 8.0
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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.
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Correlation between game and structure
non-local link only
2.7(1)
2.7(1)
Random → heterogeneous
C-hubs
Emergent network structure
Fat-tail distribution
Hierarchical structure
(C ~ 1 / k)
P ~ A exp(  K / K 0 )  B exp(  K  / K1 )
All-C region
Disassortative mixing
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Stability of cooperation
(noise)
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p = Prob. of local connection is removed (bond percolation)
p=0: 2d & non-local links
p=1: only non-local links
The local connections are essential to support cooperation.
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
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.
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Time scales
α=4
Random updating
Every time step only one randomly chosen agent
may change his strategy.
More strategy updating
P(i  j )  1 /[1  exp{ 1 (u j  ui )}] : strategy updating
W (i  j )  1 /[1  exp{   2 (u j  ui )}] : 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.
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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
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)
Thank you for your attention!
Sungmin Lee
Petter Holme
Zhi-Xi Wu
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)
NSPCS 2012 (KIAS, Seoul, 3 Jul ~ 6 July)