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Evolving Cooperation in the N-player Prisoner's
Dilemma: A Social Network Model
Dept Computer Science and Software Engineering
Golriz Rezaei Michael Kirley
Jens Pfau
ACAL09 Conference – practice talk Oct 2009
Motivation
The Dilemma: - Contribution to the social community beneficial for everybody - Autonomous self-interested individuals rational, maximize their utility
“Tragedy of the Commons” [Hardin 1968 science] - Theoretical biology / Game theory they should “Defect” - Nature / Reality they “Cooperate”
Important Question in many areas :
How/Why does cooperation emerge? What about Artificial Multi-agent systems?
Frame work: N-player Dilemma games on social groups.
Distributed Artificial Intelligence (DAI) Physics (Statistical Physics) Biology (Theoretical biology, Nature) Evolutionary Computation (IEEE Trans, CEC) Multi agent systems (AAMAS)
Overview
What is a social network Brief overview of Prisoner’s Dilemma
(N-player PD game) PD on network Proposed model Evaluation by experiments Conclusion Questions
Complex Networks every where
Social Networks
Networks
Topology Function
Social ties Behaviour
Network Basics
Network graph, G(N, E), N finite set of nodes (vertices) E finite set of edges (links) G represented by N×N adjacency matrix
aij = 1 there is an edge between node i and j
aij = 0 otherwise
Examples of Social Net
Internet-Map
Red, blue, or green: departments Yellow: consultants
Grey: external expertswww.orgnet.com
Structure of an organization
Topological properties
Degree, ki , of a node
Path length, L average separation between any two nodes
Clustering coefficient, Ci , of a node
probability that two nearest neighbours of a node are also nearest neighbours of each other.
N
jjiE
0
Prisoner’s Dilemma (2 players)
(D,D) Nash Equilibrium
Ccooperate
DDefect
Ccooperate
b-c -c
DDefect
b 0
• 2 players / agents• 2 choices (C or D)• Payoff joint actions
N-Player Prisoner’s Dilemma
Natural extension
Utility [Boyd and Richerson 1988 J. Th. Biology]
Conditions defection is preferred for individuals
contribution to social welfare is beneficial for the group
Conventional EG (D,D, … all D)
0 cb
Nbc /
PD on Spatial structure
Local neighbourhood interaction
Clusters of cooperators
Enhance cooperation
Related work
• Santos Et. Al. [2009 Nature]Heterogeneous graphs (number and size of the game)
Promotes cooperation
• Ohtsuki Et. Al. [2006 Nature] Correlation cost and benefit & the underlying connectivity of agents
• Ellis & Yao [2007 IEEE CEC] Reputation mechanism reputation scores embedded in social
network
Contribution - Hypothesis
Introducing more cognitive agents (base their decision on some function of the opponents)
Incorporating “social network” into N-player PD (network evolves by cooperative behaviour)
Encourage high levels of cooperation Persist for longer Analyse the state of underlying network
Proposed model
Algorithm: Social network based N-PD modelRequire: Population of agents P, iteration = imax, players N 2
1: for i = 0 to imax do2: G = 0;3: while g = NextGame(P,G, N) do4: G = G {g}5: PlayGame(g)6: AdaptLinks(g)7: end while8: a,b = Random Sample(P)9: CompareUtilityAndSelect(a,b)10: end for
Decision
How
Game Execution
Two scenarios (cognitive abilities)Pure strategy (always cooperate/defect)
Mixed strategy (play probabilistically)
Based on a function of average links weight ( )(β generosity)(α gradient of the function)
– Agents receive corresponding payoff based on outcomes (Boyd and Richerson function)
Decision
)(gWi
Link adaptation
Agents play cooperatively form social links (reinforced)
One agent defects breaks his links with the opponents
How
slow positive / fast negative
Snapshots of the model
Self-organize social ties based on their self-interest
Strategy update cultural evolution
(a) Iteration 5 (b) Iteration 100 (c) Iteration 1000
Experimental Setup
population size = 1000 ε = 0.9 (game formation) b = 5 and c = 3 (payoff values benefit & cost) pure strategy scenario (50% pure C – 50% pure D) mixed strategy scenario (33.3% each) α = 1.5 and β = 0.1 (decision function) average20 independent trials up to 40000 iterations
What is the equilibrium state and network topology?
Experiment 1 Group size vs. Strategy
2
4
5
10 15 20
2
4
5
10
15
20
Pure strategy Mixed strategy
Rat
io o
f Coo
pera
tion
Rat
io o
f Coo
pera
tion
Time (iteration) Time (iteration)
Experiment 2 Emergent social networks
Pure strategy Mixed strategy
2
4
510 15 20
24
5
10
15
20
Clu
ste
r C
oe
ffici
en
t
Clu
ste
r C
oe
ffici
en
t
Time (iteration) Time (iteration)
Experiment 3 Final Degree Distribution
N = 2 N = 10
log(k) log(k)
log
(P(k
))
log
(P(k
))
Conclusion
– Results validate the hypothesis
Incorporating “social network” into N-player PD
encourage high levels of cooperation and persist for longer– Social nets important in promoting and sustaining
cooperation (specially with cognitive agent)– Endogenous network formation – Analysis of the emergent social networks
high average clustering broad-scale heterogeneity
– Local structure hierarchical organization of cooperation
Questions?
Thank you