Promotion of cooperation on networks? The best response case Carlos P. Roca (1,2) José A. Cuesta (1) Anxo Sánchez (1,3,4) The.

http://gisc.uc3m.es/~anxo Promotion of cooperation on networks? The best response case Carlos P. Roca (1,2) José A. Cuesta (1) Anxo Sánchez (1,3,4) The Physics Approach To Risk: Agent-Based Models and Networks October 27-29, 2008 ETH Zürich, Switzerland (1)GISC, Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain (2)Chair for Sociology, in particular of Modelling and Simulation, ETH Zürich, Switzerland (3)Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Madrid, Spain (4)Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Zaragoza, Spain

http://gisc.uc3m.es/~anxo He who was ready to sacrifice his life (…), rather than betray his comrades, would often leave no offspring to inherit his noble nature… Therefore, it seems scarcely possible (…) that the number of men gifted with such virtues (…) would be increased by natural selection, that is, by the survival of the fittest. The puzzle of the emergence of cooperation Charles Darwin (Descent of Man, 1871)

http://gisc.uc3m.es/~anxo “Others with a mathematical bent are applying evolutionary game theory, a modeling approach developed for economics, to quantify cooperation and predict behavioral outcomes under different circumstances.” E. Pennisi, Science 309, 93 (2005) One of the 25 problems for the XXI century

http://gisc.uc3m.es/~anxo The hypothesis of structured populations Martin A. Nowak and Robert M. May, Nature 359, 826 (1992) Spatial structure promotes cooperation in evolutionary game theory

http://gisc.uc3m.es/~anxo 2x2 Symmetric Social Dilemmas 2 players 2 strategies: Cooperate or Defect DC C D 1 S 0T T > 1 : temptation to defect S < 0 : risk in cooperation

http://gisc.uc3m.es/~anxo Possible Social Dilemmas T S 2 0 1 Harmony (no tensions) Snowdrift / Hawk-Dove (anti-coordination) Stag Hunt (coordination) Prisoner’s Dilemma (both tensions) DC C D 1 S 0T 0 1 T > 1 temptation to defect S < 0 risk in cooperation

http://gisc.uc3m.es/~anxo Evolutionary Games on Networks Population structure: each player plays and compares payoff only with his neighbors Strategy evolution: update rules Darwinian evolution: individuals reproduce according to their fitness (payoffs earned from the game) G. Szabó and G. Fáth, Evolutionary games on graphs Phys. Rep. 446, 97 (2007).

http://gisc.uc3m.es/~anxo Update rules Proportional update: Similar to replicator dynamics on a infinite, well-mixed population Unconditional imitation: choose the strategy of the neighbor with the largest payoff if larger than yours Best response: choose the strategy that would have yielded the largest payoff given the neighbors’ strategies Pairwise comparison: …

http://gisc.uc3m.es/~anxo Evolutionary Games on Networks Standard reference: replicator dynamics on a complete network S T S 2 0 1 Harmony ( 1 ) Snowdrift / Hawk-Dove (0,, 1) Stag Hunt (0,,1) Prisoner’s Dilemma ( 0 ) DC C D 1 S 0 T 0 1

http://gisc.uc3m.es/~anxo Seminal result on spatial structure Nowak & May, Nature 359, 826 (1992)

http://gisc.uc3m.es/~anxo No clear global picture yet Subsequent work Different works, different models (networks, rules, games, time definition,…): Contradictory results

http://gisc.uc3m.es/~anxo unconditional imitationbest response replicator rule ( regular lattice, k=8, x 0 =0.5 ) The influence of the update rule

http://gisc.uc3m.es/~anxo Random networks and lattices: Replicator rule k=4k=6k=8 random network lattice

http://gisc.uc3m.es/~anxo Random networks and lattices: Unconditional imitation k=4k=6k=8 random network lattice

http://gisc.uc3m.es/~anxo Effects of spatial structure Spatial structure has a strong effect only when the clustering coefficient is high Stochastic update rules (replicator): asymmetry of effects between coordination (Stag Hunt) and anti-coordination games (Snowdrift, Hawk-Dove) Unconditional imitation: the highest promotion of cooperation, the only rule with a relevant effect on Prisoner’s Dilemma Small-world networks produce results almost identical to those of regular lattices C. P. Roca, J. A. Cuesta, A.S., arXiv/0806.1649 (2008)

http://gisc.uc3m.es/~anxo Mesoscopic structure also plays a role S. Lozano, A. Arenas, A.S., PLoS ONE 3(4): e1892 (2008) S. Lozano, A. Arenas, A.S., J. Econ. Interact. Coord., in press (2009) C. P. Roca, S. Lozano, A. Arenas, A.S., work in progress (2008) Communities modify the response PGP Social networkRandomized

http://gisc.uc3m.es/~anxo complete graph( regular lattice, k=8, x 0 =0.5 ) Hauert & Doebeli results Replicator rule: Cooperation is (mostly) inhibited!

http://gisc.uc3m.es/~anxo well mixed( regular lattice, k=8, x 0 =0.5 ) Sysi-Aho et al. results Best response: Cooperation is promoted for large r!

http://gisc.uc3m.es/~anxo The best response case Best response is “the” rule of choice for many applications in economics Best response is both deterministic (as unconditional imitation) and innovative (it reintroduces extinct strategies) Best response is a step further in “intelligence” if compared to imitation If best response leads to an equilibrium, it is a Nash equilibrium of the networked game

http://gisc.uc3m.es/~anxo Best response on well-mixed populations Well-mixedComplete graph Introduce a probability p to update strategy to avoid alternance

http://gisc.uc3m.es/~anxo Best response on random networks and lattices k=4k=8 random network lattice

http://gisc.uc3m.es/~anxo Best response on… almost anything CompleteErdös-Rènyi (4)Barabási-Albert (4) Klemm-Eguíluz (8)Small World (8)Barabási-Albert (8)

http://gisc.uc3m.es/~anxo But on a closer look… No effects on best response PD and Harmony cannot change because they have only one dominant strategy (D or C resp.) which is the only best response to any other SH and SD might in principle be affected: –SH ends up converging to one of the two equilibria –SD ends up forever switching strategies (if p=1) or converges to the mixed equilibrium (if p  1)

http://gisc.uc3m.es/~anxo Lattices: initial conditions Well mixedLattice (4)Lattice (8) X c =2/3 X c =1/3

http://gisc.uc3m.es/~anxo Cluster formation on lattices k=8, x 0 =1/3, S=-0.6, T=0.2k=4, x 0 =1/3, S=-0.6, T=0.2

http://gisc.uc3m.es/~anxo Finite size effect k=8, x 0 =1/3, S=-0.6, T=0.2 Histograms of asymptotic cooperation

http://gisc.uc3m.es/~anxo Initial conditions in other networks RandomWell mixed X c =1/3 Scale-free Effect is noticeable on other lattices No cluster effects

http://gisc.uc3m.es/~anxo Summary Assessment of the effect of the relevant topological properties for the evolution of cooperation (network clustering and degree heterogeneity) Best response analyzed in a large variety of networks proves to be independent of the social network Quantitative understanding of the dynamic mechanisms involved: initial conditions relevant Bi-dimensional parameter space: ST-plane

http://gisc.uc3m.es/~anxo What does it mean “promotion of cooperation”? Different regions, initial conditions, rules, … Replicator ruleBest response

http://gisc.uc3m.es/~anxo many thanks for your attention

http://gisc.uc3m.es/~anxo Spatial structure: Dependence on initial densities replicator dynamics unconditional imitation

http://gisc.uc3m.es/~anxo The case of snowdrift Hauert & Doebeli, Nature 428, 643 (2004) Cooperation is inhibited! (However, cf. small r)

http://gisc.uc3m.es/~anxo The case of snowdrift Sysi-Aho et al., Eur. Phys. J. B 44, (2005) Cooperation may be promoted! (cf. dependence on r)

http://gisc.uc3m.es/~anxo Cluster formation on lattices k=8, x 0 =1/3, S=-0.6, T=0.2

http://gisc.uc3m.es/~anxo Best response dynamics is largely unaffected by the existence of a social network: Relevance of the network for human-like rules? Evolutionary game theory on networks is non universal: need for rationales for models Conclusions

http://gisc.uc3m.es/~anxo Spatial structure: Local densities Payoffs Complete network Structured population SH SD

http://gisc.uc3m.es/~anxo Spatial structure: Temporal evolution ( replicator dynamics, k=8 )

http://gisc.uc3m.es/~anxo Spatial structure: Effect of network clustering latticeWatts-Strogatz small-world Replicator dynamics, k=8

http://gisc.uc3m.es/~anxo Spatial structure: Temporal evolution

http://gisc.uc3m.es/~anxo Spatial structure: Transitions in Unconditional Imitation, k=8

http://gisc.uc3m.es/~anxo E. Pennisi, Science 309, 93 (2005) An issue at the frontier between (behavioral) economics, sociology, (social) anthropology, (evolutionary) psychology, (evolutionary) biology, and… statistical mechanics One of the 25 problems for the XXI century

http://gisc.uc3m.es/~anxo many thanks for your attention

Promotion of cooperation on networks? The best response case Carlos P. Roca (1,2) José A. Cuesta (1) Anxo Sánchez (1,3,4) The.

Similar presentations

Presentation on theme: "Promotion of cooperation on networks? The best response case Carlos P. Roca (1,2) José A. Cuesta (1) Anxo Sánchez (1,3,4) The."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Promotion of cooperation on networks? The best response case Carlos P. Roca (1,2) José A. Cuesta (1) Anxo Sánchez (1,3,4) The.

Similar presentations

Presentation on theme: "Promotion of cooperation on networks? The best response case Carlos P. Roca (1,2) José A. Cuesta (1) Anxo Sánchez (1,3,4) The."— Presentation transcript:

Similar presentations

About project

Feedback