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

http://gisc.uc3m.es/~anxo

Promotion of cooperation on networks? The best response

caseCarlos P. Roca

(1,2)José A. Cuesta

(1)Anxo Sánchez

(1,3,4)

The Physics Approach To Risk: Agent-Based Models and NetworksOctober 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


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)


“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


The hypothesis of structured populations

Martin A. Nowak and Robert M. May, Nature 359, 826

(1992)

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are needed to see this picture.QuickTime™ and a

decompressorare needed to see this picture.

Spatial structure promotes cooperation in evolutionary game theory


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


Possible Social Dilemmas

T

S

20

1

-1

Harmony(no tensions)

Snowdrift /Hawk-Dove

(anti-coordination)

Stag Hunt(coordinatio

n)

Prisoner’sDilemma

(both tensions)

DC

C

D

1 S

0T

01

T > 1 temptation to defect

S < 0risk in cooperation


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 graphsPhys. Rep. 446, 97 (2007) .


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:

• …


Evolutionary Games on Networks

Standard reference:replicator dynamics on a complete network

S

T

S

20

1

-1

Harmony( 1 )

Snowdrift /Hawk-Dove

(0, , 1)

StagHunt

(0, ,1)

Prisoner’sDilemma( 0 )

DC

C

D

1

S

0

T

0 1


Seminal result on spatial structure

Nowak & May, Nature 359, 826 (1992)


No clear global picture yet

Subsequent work

Different works, different models (networks,rules, games, time definition,…): Contradictory results


unconditional imitation best responsereplicator rule

( regular lattice, k=8, x0=0.5 )

The influence of the update rule


Random networks and lattices: Replicator rule

k=4 k=6 k=8

random network

lattice


Random networks and lattices: Unconditional imitation

k=4 k=6 k=8

random network

lattice


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)


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 network Randomized


complete graph( regular lattice, k=8, x 0=0.5 )

Hauert & Doebeli results


are needed to see this picture.

Replicator rule:

Cooperation is (mostly) inhibited!


well mixed( regular lattice, k=8, x 0=0.5 )

Sysi-Aho et al. results

Best response:

Cooperation is promoted for large r!


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


Best response on well-mixed populations



Well-mixed Complete graph

Introduce a probability p to update strategy to avoid alternance


Best response on random networks and lattices

k=4 k=8

random network

lattice


Best response on… almost anything

Complete





Erdös-Rènyi (4)



Barabási-Albert (4)



Klemm-Eguíluz (8)



Small World (8)Barabási-Albert (8)


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 p1)


Lattices: initial conditions

Well mixed Lattice (4) Lattice (8)

Xc=2/3

Xc=1/3


Cluster formation on lattices

k=8, x0=1/3, S=-0.6, T=0.2 k=4, x0=1/3, S=-0.6, T=0.2


Finite size effect

k=8, x0=1/3, S=-0.6, T=0.2

Histograms of asymptotic cooperation


Initial conditions in other networks

RandomWell mixed

Xc=1/3

Scale-free

Effect is noticeable on other lattices

No cluster effects


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


What does it mean “promotion of cooperation”?

Different regions, initial conditions, rules, …

Replicator rule

Best response




many thanksfor your attention


Spatial structure: Dependence on initial densitiesreplicator dynamics

unconditional imitation


The case of snowdrift

Hauert & Doebeli, Nature 428, 643 (2004)

Cooperation is inhibited!

(However, cf. small r)


The case of snowdrift

Sysi-Aho et al., Eur. Phys. J. B 44, (2005)

Cooperation may be promoted!

(cf. dependence on r)


Cluster formation on lattices

k=8, x 0=1/3, S=-0.6, T=0.2


Best response dynamics is largely unaffectedby 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


Spatial structure: Local densities

Payoffs

Complete network

Structured population

SH

SD


Spatial structure: Temporal evolution

( replicator dynamics, k=8 )


Spatial structure: Effect of network clustering

lattice Watts-Strogatz small-world

Replicator dynamics, k=8


Spatial structure: Temporal evolution


Spatial structure: Transitions in

Unconditional Imitation, k=8


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




many thanksfor your attention

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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