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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

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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)

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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

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Complex Networks every where Social Networks Networks Topology Function Social ties Behaviour

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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 a ij = 1 there is an edge between node i and j a ij = 0 otherwise

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Examples of Social Net Internet-Map

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Red, blue, or green: departments Yellow: consultants Grey: external experts Structure of an organization

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Topological properties Degree, k i, of a node Path length, L average separation between any two nodes Clustering coefficient, C i, of a node probability that two nearest neighbours of a node are also nearest neighbours of each other.

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Prisoner’s Dilemma ( 2 players ) (D,D) Nash Equilibrium C cooperate D Defect C cooperate b-c-c D Defect b0 2 players / agents 2 choices (C or D) Payoff joint actions

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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)

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PD on Spatial structure Local neighbourhood interaction Clusters of cooperators Enhance cooperation

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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

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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

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Proposed model Algorithm: Social network based N-PD model Require: Population of agents P, iteration = i max, players N 2 1: for i = 0 to i max do 2: G = 0; 3: while g = NextGame(P,G, N) do 4: G = G {g} 5: PlayGame(g) 6: AdaptLinks(g) 7: end while 8: a,b = Random Sample(P) 9: CompareUtilityAndSelect(a,b) 10: end for Decision How

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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

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Link adaptation Agents play cooperatively form social links (reinforced) One agent defects breaks his links with the opponents How slow positive / fast negative

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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

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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 iterations What is the equilibrium state and network topology?

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Experiment 1 Group size vs. Strategy Pure strategyMixed strategy Ratio of Cooperation Time (iteration)

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Experiment 2 Emergent social networks Pure strategyMixed strategy Cluster Coefficient Time (iteration)

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Experiment 3 Final Degree Distribution N = 2N = 10 log(k) log(P(k))

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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

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Questions? Thank you

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