1. Network Modeling of Cooperation in Public Goods Games
Elgar Pichler1 (elgar.pichler@gmail.com) Avi M. Shapiro2 (ashapiro@seas.harvard.edu)
1Department of Chemistry and Chemical Biology, Northeastern University
2Department of Applied Mathematics, Harvard University
Introduction
Well-functioning societies feature high levels of cooperation and strong
connections between individuals [2].
Many classic results of game theory show mutual defection (as opposed
to cooperation) to be a Nash Equilibrium in well-mixed populations.
We simulate the coevolutionary dynamics of PGG players (nodes) who
can choose their strategies, either to cooperate or to defect, and who
can adapt by altering their social connections (edges).
Certain network structures of individuals allow for high levels of
cooperation in PGGs even when players can change their strategy and
connections only based on their individual payoff and satisfaction, but
without specific information about neighbors’ strategies.
Public Goods Game (PGG)
PGG Description:
In a PGG, cooperators contribute a certain amount to the public good
and defectors do not. All players then receive an equal share of the sum
of all contributions multiplied by a synergy factor, r ≥ 1. We can write
the payoffs for cooperators and defectors,
πC = r
n
N
−1, πD = r
n
N
where n is the number of cooperators and N is the number of players.
Example:
At a given time step, a player
at node 1 has 3 neighbors and
thus collects a total payoff
from 4 separate PGGs.
1
2
PGG1
PGG2
PGG Networks:
PGGs as 2D cellular automata with nearest neighbor interactions only
were investigated in [3]. We generalize this PGG model by allowing
connections between arbitrary nodes in a non-planar network topology.
PGGs on such graphs have applications in peer-to-peer, mobile, and
vehicular networks.
Coevolutionary Rules
At time step t, player i may change strategy and/or neighbors with
independent probabilities based on satisfaction
si(t) = πi(t)−ai(t)+ηi(t),
where πi is the current total payoff, ηi is a Gaussian noise, and ai is
aspiration, defined
ai(t) = απi,max(t)+(1−α)πi,min(t),
which depends on greediness 0 ≤ α ≤ 1, another important control
parameter. πi,max/min reflect finite memory of past max/min payoffs.
Simulation Methodology
Coded in Python using NetworkX library, run on multicore Linux clusters
Network variables monitored: number of components, largest component
size, clustering coefficient, degree distribution
PGG variables monitored: aspiration, payoff, satisfaction, strategy
PGG consists of 1000 players and is a function of synergy r and
greediness α. Simulations were stopped at t = 20000.
Initial conditions: Barabási-Albert graph [1], all players are defectors
Simulation Results – Time Series
The model evolution is best captured by the variables cooperation, the
cooperator fraction, instability, the fraction of players changing strategies or
neighbors, and agglomeration, the average node degree. In these results,
r = 4 and α = 0.65.
High cooperation is
achieved through
instability triggering
higher agglomeration.
0 10000 20000
t
0.0
0.5
1.0
instability
cooperation
cooperation
instability
0
15
30
agglomeration
agglomeration
The evolution is
stochastic and undergoes
large but short spikes. To
observe larger trends, we
average over a moving
window of 100 time steps
in all other plots.
0.0
0.5
1.0 cooperator fraction
no time averaging
0 10000 20000
t
0.0
0.5
1.0
moving average
over 100 steps
Several simulations for
each parameter pair reveal
consistently similar
behavior, although shifted
in time.
0.0
0.5
1.0 cooperation
0.0
0.5
1.0 instability
0 10000 20000
t
0
15
30 agglomeration
Simulation Results – PGG Network Development
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249
647
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596
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239
257
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160
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474
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198
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816
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761
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993
8
176
614
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530
235
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300
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284
737
940
179
606
953
674
944
452
679
711
599
227
t = 0 t = 10000 t = 15000 t = 20000
Simulation Results – Network Topology
Over time, the network topology changes due to player (dis)satisfaction.
Average node degree increases but the distribution changes more
dramatically, and is no longer scale-free by t = 20000.
10
0
10
1
10
2
10
3
degree
10
-3
10
-2
10
-1
10
0
degree
distribution
r=4, =0.65
r=3, =0.65
0 10000 20000
t
0
15
30
average
nodedegree
r=4, =0.65
r=3, =0.65
Simulation Results – Phase Diagram
Cooperation is not only possible but dominates for a region of parameter
values r and α. The transition to a cooperative regime is sharp in time
as well as in parameter space. The following phase diagram shows the
cooperator fraction at t = 20000 averaged over 5 simulations.
cooperator fraction
5 10 15 20 25 30
r
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.0
0 10000 20000
t
0.0
0.5
1.0
r=4, =0.65
r=3, =0.65
Interestingly, the largest cooperation is found for high greediness and low
values of synergy.
Conclusions
In PGGs played on networks, stable cooperative behavior arises for a wide
range of parameters, even if initial conditions are seemingly
disadvantageous to cooperativity.
During the transition to a cooperative regime, the number of low degree
nodes decreases and the initial scale-free structure is destroyed.
As the PGG system approaches steady state – at t = 20000, our system
has not reached steady state – the expected behavior of the overall
system will stabilize while short time extreme fluctuations can still occur.
References
[1] Réka Albert and Albert-László Barabási.
Statistical mechanics of complex networks.
Reviews of Modern Physics, 74:47, 2002.
[2] Martin A. Nowak.
Five rules for the evolution of cooperation.
Science, 314(5805):1560–1563, Dec 2006.
[3] Carlos P. Roca and Dirk Helbing.
Emergence of social cohesion in a model society of greedy, mobile individuals.
Proc Natl Acad Sci USA, 108(28):11370–11374, Jul 2011.