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Promotion of cooperation on networks?
The best response case
Carlos P. Roca (1,2)
José A. Cuesta (1)
Anxo Sánchez (1,3,4)
(1)
(2)
(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
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The puzzle of the emergence of cooperation
Charles Darwin
(Descent of Man, 1871)
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.
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One of the 25 problems for the XXI century
E. Pennisi, Science 309, 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.”
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The hypothesis of structured populations
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Martin A. Nowak and Robert M. May,
Nature 359, 826 (1992)
Spatial structure promotes cooperation in
evolutionary game theory
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2x2 Symmetric Social Dilemmas
• 2 players
• 2 strategies: Cooperate or Defect
C
D
C
1
S
D
T
0
T > 1 : temptation to defect
S < 0 : risk in cooperation
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Possible Social Dilemmas
T > 1 temptation to defect
S
C
C
1
1
D
S
Snowdrift /
Hawk-Dove
Harmony
(no tensions)
D
T
(anti-coordination)
0
0
0
S<0
risk in cooperation
2
1
Stag Hunt
(coordination)
Prisoner’s
Dilemma
(both tensions)
-1
T
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Evolutionary Games on Networks
• Darwinian evolution: individuals reproduce
according to their fitness (payoffs earned
from the game)
• Population structure: each player plays
and compares payoff only with his neighbors
• Strategy evolution: update rules
G. Szabó and G. Fáth, Evolutionary games on
graphs
Phys. Rep. 446, 97 (2007) .
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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:
•…
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Evolutionary Games on Networks
Standard reference:
replicator dynamics on a complete network
S
C
D
C
1
S
D
T
0
1
S
Snowdrift /
Hawk-Dove
(0, , 1)
Harmony
(1)
0
0
1
Stag
Hunt
(0, ,1)
-1
2
Prisoner’s
Dilemma
(0)
T
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Seminal result on spatial structure
Nowak & May, Nature 359, 826 (1992)
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Subsequent work
Different works, different models (networks,
rules, games, time definition,…):
Contradictory results
No clear global picture yet
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The influence of the update rule
unconditional imitation
replicator rule
( regular lattice, k=8, x0=0.5 )
best response
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Random networks and lattices: Replicator rule
lattice
random network
k=4
k=6
k=8
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Random networks and lattices: Unconditional imitation
lattice
random network
k=4
k=6
k=8
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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)
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Mesoscopic structure also plays a role
PGP Social network
Randomized
Communities modify the response
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)
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Hauert & Doebeli results
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( regular lattice, k=8, x 0=0.5 )
complete graph
Replicator rule:
Cooperation is (mostly) inhibited!
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Sysi-Aho et al. results
( regular lattice, k=8, x 0=0.5 )
well mixed
Best response:
Cooperation is promoted for large r!
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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
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Best response on well-mixed populations
Well-mixed
Complete graph
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Introduce a probability p to update strategy to avoid alternance
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Best response on random networks and lattices
lattice
random network
k=4
k=8
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Best response on… almost anything
Complete
Erdös-Rènyi (4)
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Klemm-Eguíluz (8)
Small World (8)
Barabási-Albert (4)
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Barabási-Albert (8)
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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…
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Lattices: initial conditions
Xc=2/3
Xc=1/3
Well mixed
Lattice (4)
Lattice (8)
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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
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Finite size effect
Histograms of asymptotic cooperation
k=8, x0=1/3, S=-0.6, T=0.2
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Initial conditions in other networks
Random
Scale-free
Xc=1/3
Well mixed
Effect is noticeable on other lattices
No cluster effects
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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
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What does it mean
“promotion of cooperation”?
Replicator rule
Best response
Different regions, initial conditions, rules, …
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for your attention
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unconditional imitation
replicator dynamics
Spatial structure: Dependence on initial densities
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The case of snowdrift
Hauert & Doebeli, Nature 428, 643 (2004)
Cooperation is inhibited!
(However, cf. small r)
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The case of snowdrift
Sysi-Aho et al., Eur. Phys. J. B 44, (2005)
Cooperation may be promoted!
(cf. dependence on r)
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Cluster formation on lattices
k=8, x 0=1/3, S=-0.6, T=0.2
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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?
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Spatial structure: Local densities
Payoffs
Complete network
Structured population
SH
SD
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Spatial structure: Temporal evolution
( replicator dynamics, k=8 )
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Spatial structure: Effect of network clustering
Replicator dynamics, k=8
lattice
Watts-Strogatz small-world
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Spatial structure: Temporal evolution
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Spatial structure: Transitions in
Unconditional Imitation, k=8
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One of the 25 problems for the XXI century
E. Pennisi, Science 309, 93 (2005)
An issue at the frontier between (behavioral)
economics, sociology, (social) anthropology,
(evolutionary) psychology, (evolutionary) biology,
and… statistical mechanics
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many thanks
for your attention
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