Promotion of cooperation on networks? The best response case

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

2. The puzzle of the emergence of cooperation 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. Charles Darwin (Descent of Man, 1871)

3. One of the 25 problems for the XXI century E. Pennisi, Science309, 93 (2005) “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.”

4. The hypothesis of structured populations Martin A. Nowak and Robert M. May, Nature359, 826 (1992) Spatial structure promotes cooperation in evolutionary game theory

5. C D 1 S C T 0 D 2x2 Symmetric Social Dilemmas • 2 players • 2 strategies: Cooperate or Defect T > 1 : temptation to defect S < 0 : risk in cooperation

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

7. Evolutionary Games on Networks • Darwinian evolution: individuals reproduce according to their fitness (payoffs earned from the game) • Population structure: each player playsand compares payoff only with his neighbors • Strategy evolution: update rules G. Szabó and G. Fáth, Evolutionary games on graphsPhys. Rep.446, 97 (2007) .

8. 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: • …

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

10. Seminal result on spatial structure Nowak & May, Nature 359, 826 (1992)

11. Subsequent work Different works, different models (networks, rules, games, time definition,…): Contradictory results No clear global picture yet

12. The influence of the update rule unconditional imitation best response replicator rule ( regular lattice, k=8, x0=0.5 )

13. Random networks and lattices: Replicator rule k=4 k=6 k=8 random network lattice

14. Random networks and lattices: Unconditional imitation k=4 k=6 k=8 random network lattice

15. 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)

16. Mesoscopic structure also plays a role PGP Social network Randomized 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

17. Hauert & Doebeli results ( regular lattice, k=8, x 0=0.5 ) complete graph Replicator rule: Cooperation is (mostly) inhibited!

18. Sysi-Aho et al. results ( regular lattice, k=8, x 0=0.5 ) well mixed Best response: Cooperation is promoted for large r!

19. 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

20. Best response on well-mixed populations Well-mixed Complete graph Introduce a probability p to update strategy to avoid alternance

21. Best response on random networks and lattices k=4 k=8 random network lattice

22. Best response on… almost anything Complete Erdös-Rènyi (4) Barabási-Albert (4) Klemm-Eguíluz (8) Small World (8) Barabási-Albert (8)

23. 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) But on a closer look…

24. Lattices: initial conditions Well mixed Lattice (4) Lattice (8) Xc=1/3 Xc=2/3

25. Cluster formation on lattices k=4, x0=1/3, S=-0.6, T=0.2 k=8, x0=1/3, S=-0.6, T=0.2

26. Finite size effect Histograms of asymptotic cooperation k=8, x0=1/3, S=-0.6, T=0.2

27. Initial conditions in other networks Scale-free Well mixed Random Xc=1/3 Effect is noticeable on other lattices No cluster effects

28. 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

29. What does it mean “promotion of cooperation”? Replicator rule Best response Different regions, initial conditions, rules, …

30. many thanksfor your attention

31. Spatial structure: Dependence on initial densities replicator dynamics unconditional imitation

32. The case of snowdrift Hauert & Doebeli, Nature428, 643 (2004) Cooperation is inhibited! (However, cf. small r)

33. The case of snowdrift Sysi-Aho et al., Eur. Phys. J. B44, (2005) Cooperation may be promoted! (cf. dependence on r)

34. Cluster formation on lattices k=8, x 0=1/3, S=-0.6, T=0.2

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

36. Spatial structure: Local densities Payoffs Complete network Structured population SH SD

37. Spatial structure: Temporal evolution ( replicator dynamics, k=8 )

38. Spatial structure: Effect of network clustering Replicator dynamics, k=8 lattice Watts-Strogatz small-world

39. Spatial structure: Temporal evolution

40. Spatial structure: Transitions in Unconditional Imitation, k=8

41. One of the 25 problems for the XXI century E. Pennisi, Science309, 93 (2005) An issue at the frontier between (behavioral) economics, sociology, (social) anthropology, (evolutionary) psychology, (evolutionary) biology, and…statistical mechanics

42. many thanksfor your attention