Transcript Sistemas complejos en la vida cotidiana y sus modelos

Time Scales in Evolutionary Dynamics
Angel Sánchez
Grupo Interdisciplinar de Sistemas Complejos (GISC)
Departamento de Matemáticas – Universidad Carlos III de Madrid
Instituto de Biocomputación y Física de Sistemas Complejos (BIFI)
Universidad de Zaragoza
with Carlos P. Roca and José A. Cuesta
Trieste, October 19, 2006
Cooperation: the basis of human societies
Anomaly in the animal world:
• Occurs between genetically unrelated individuals
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Time Scales in Evolution 2
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Cooperation: the basis of human societies
Anomaly in the animal world:
• Shows high division of labor
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Time Scales in Evolution 3
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Cooperation: the basis of human societies
Anomaly in the animal world:
• Valid for large scale organizations…
…as well as hunter-gatherer groups
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Time Scales in Evolution 4
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Cooperation: the basis of human societies
Some animals form complex societies…
…but their individuals are genetically related
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Altruism: key to cooperation
Altruism:
fitness-reducing act that benefits others
Pure altruism is ruled out by natural selection acting on
individuals á la Darwin
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Time Scales in Evolution 6
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How did altruism arise?
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)
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Altruism is an evolutionary puzzle
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Group selection? Cultural evolution?
A man who was not impelled by any deep, instinctive feeling, to
sacrifice his life for the good of others, yet was roused to such
actions by a sense of glory, would by his example excite the same
wish for glory in other men, and would strengthen by exercise the
noble feeling of admiration. He might thus do far more good to
his tribe than by begetting offsprings with a tendency to inherit his
own high character.
Charles Darwin (Descent of Man, 1871)
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Answers to the puzzle…
Kin cooperation (Hamilton, 1964)
common to animals and humans alike
 Reciprocal altruism in repeated interactions
(Trivers, 1973; Axelrod & Hamilton, 1981)
primates, specially humans
 Indirect reciprocity (reputation gain)
(Nowak & Sigmund, 1998)
primates, specially humans

None true altruism: individual benefits in the long run
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… but only partial!

Strong reciprocity
(Gintis, 2000; Fehr, Fischbacher & Gächter, 2002)
typically human (primates?)
 altruistic rewarding: predisposition to reward others for
cooperative behavior
 altruistic punishment: propensity to impose sanctions on
non-cooperators
Strong reciprocators bear the cost of altruistic acts even if
they gain no benefits
Hammerstein (ed.), Genetic and cultural evolution of cooperation (Dahlem Workshop
Report 90, MIT, 2003)
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Time Scales in Evolution 11
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One of the 25 problems for the XXI century:
E. Pennisi, Science 309, 93 (2005)
“Others with a more 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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Evolution
• There are populations of reproducing individuals
• Reproduction includes mutation
• Some individuals reproduce faster than other (fitness).
This results in selection
Game theory
• Formal way to analyze interactions between agents who behave
strategically (mathematics of decision making in conflict
situations)
• Usual to assume players are “rational”
• Widely applied to the study of economics, warfare, politics, animal
behaviour, sociology, business, ecology and evolutionary biology
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Evolutionary Game Theory
Successful strategies spread by natural selection
Payoff = fitness
John Maynard Smith 1972
(J.B.S. Haldane, R. A. Fisher, W. Hamilton, G. Price)
•
•
•
•
•
•
Everyone starts with a random strategy
Everyone in population plays game against everyone else
Population is infinite
Payoffs are added up
Total payoff determines the number of offspring: Selection
Offspring inherit approximately the strategy of their parents: Mutation
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Time Scales in Evolution 14
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Equations for evolutionary dynamics
Quasispecies
equation
Replicator-mutator
equation
Lotka-Volterra
equation
replicator-mutator
Price equation
Game dynamical
equation
Price
equation
replicator
Price equation
Adaptive dynamics
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Case study on strong reciprocity
and altruistic behavior:
Ultimatum Games, altruism
and individual selection
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The Ultimatum Game
(Güth, Schmittberger & Schwarze, 1982)
experimenter
M euros
M-u
proposer
OK
NO
u
M-u
0
0u
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responder
Time Scales in Evolution 17
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Experimental results
Extraordinary amount of data
Camerer, Behavioral Game Theory (Princeton University
Press, 2003)
“At this point, we should declare a moratorium on
creating ultimatum game data and shift attention towards
new games and new theories.”
Henrich et al. (eds.), Foundations of Human Sociality :
Economic Experiments and Ethnographic Evidence from Fifteen
Small-Scale Societies (Oxford University Press, 2004)
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Time Scales in Evolution 18
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What would you offer?
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Rational responder’s optimal strategy: accept anything
Rational proposer’s optimal strategy: offer minimum
Experimental results
• Proposers offer substantial amounts
(50% is a typical modal offer)
• Responders reject offers below 25% with high probability
• Universal behavior throughout the world
• Large degree of variability of offers among societies (26 - 58%
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Model
A.S. & J. A. Cuesta, J. Theor. Biol. 235, 233 (2005)
......
N players
ti , oi : thresholds (minimum share
player i accepts / offers)
player i
fi : fitness (accumulated capital)
M monetary units (M=100)
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Game event
......
N players
responder
proposer
op
tr
fp+M-op
fr +op
op ≥
< tr
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Reproduction event (after s games)
......
new
player
minimum
fitness
N players
maximum
fitness
t,t’,oo’min
max
t, omax
fmin
max
fmax
mutation: t’, o’max= t, omax ± 1
(prob.=1/3)
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Time Scales in Evolution 23
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Slow evolution (large s)
N =1000, 109 games, s = 105, ti = oi =1 initial condition
accept
offer
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Fast evolution (small s)
N =1000, 106 games, s =1, uniform initial condition
accept
offer
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Adaptive dynamics (“mean-field”) results
• Results for small s (fast
selection) differ qualitatively
• Implications in behavioral
economics and evolutionary
ideas on human behavior!
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Selection/reproduction interplay
in simpler settings:
Equilibrium selection in
2x2 games
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Moran Process
Select one, proportional to fitness
Substitute a randomly chosen individual
P. A. P. Moran, The
statistical processes of
evolutionary theory
(Clarendon, 1962)
Game event
2x2 game
Choose s pairs of agents to play
the game between reproduction events
Reset fitness after reproduction
C. P. Roca, J. A. Cuesta, A. Sánchez, Phys. Rev. Lett. 97, 158701 (2006)
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Fixation probability
Probability to reach state N when starting from state i =1
1-x1
i0
x1
i 1
iN
Absorbing states
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Fixation probability
Probability to reach state N when starting from state n
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Fixation probability
Probability to reach state N when starting from state n
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Time Scales in Evolution 31
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Fixation probability
Probability to reach state N when starting from state n
Number of games s enters through transition probabilities
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Fixation probability
Probability to reach state N when starting from state n
Fitness: possible game sequences times
corresponding payoffs per population
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Example 1: Harmony game
Payoff matrix:
Unique Nash equilibrium in pure strategies: (C,C)
(C,C) is the only reasonable behavior anyway
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Example 1: Harmony game
s infinite (round-robin, “mean-field”)
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Example 1: Harmony game
s = 1 (reproduction following every game)
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Example 1: Harmony game
Consequences
•Round-robin: cooperators are selected
•One game only: defectors are selected!
•Result holds for any population size
•In general: for any s, numerical evaluation
of exact expressions
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Example 1: Harmony game
Numerical evaluation of exact expressions
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Example 2: Stag-hunt game
Payoff matrix:
Two Nash equilibria in pure strategies: (C,C), (D,D)
Equilibrium selection depends on initial condition
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Example 2: Stag-hunt game
Numerical evaluation of exact expressions
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Example 3: Snowdrift game
Payoff matrix:
One mixed equilibrium
Replicator dynamics goes always to mixed equilibrium
Moran dynamics does not allow for mixed equilibria
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Example 3: Snowdrift game
Numerical evaluation of exact expressions
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Example 3: Snowdrift game
Numerical evaluation of exact expressions
s=5
s = 100
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Example 4: Prisoner’s dilemma
Payoff matrix:
Unique Nash equilibrium in pure strategies: (C,C)
Paradigm of the emergence of cooperation problem
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Example : Prisoner’s dilemma
Numerical evaluation of exact expressions
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Results are robust
Increasing system size does not changes basins of
attrractions, only sharpens the transitions
Small s is like an effective small population, because
inviduals that do not play do not get fitness
Introduce background of fitness: add fb to all payoffs
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Background of fitness: Stag-hunt game
Numerical evaluation of exact expressions
fb = 0.1
fb = 1
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Conclusions
• In general, evolutionary game theory studies a limit situation: s
infinite! (every player plays every other one before selection)
• Number of games per player may be finite, even Poisson
distributed
• Fluctuations may keep players with smaller ‘mean-field’ fitness alive
• Changes to equilibrium selection are non trivial and crucial
New perspective on evolutionary game theory:
more general dynamics, dictated by the specific
application (change focus from equilibrium
selection problems)
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Time Scales in Evolutionary Dynamics
A. Sánchez & J. A. Cuesta, J. Theor. Biol. 235, 233 (2005)
A. Sánchez, J. A. Cuesta & C. P. Roca, in “Modeling Cooperative
Behavior in the Social Sciences”, eds. P. Garrido, J. Marro
& M. A. Muñoz, 142–148. AIP Proceedings Series (2005).
C. P. Roca, J. A. Cuesta, A. Sánchez, arXiv:q-bio/0512045
(submitted to European Physical Journal Special Topics)
C. P. Roca, J. A. Cuesta, A. Sánchez, Phys. Rev. Lett. 97, 158701 (2006)
Trieste, October 19, 2006
Time Scales in Evolution 49
http://www.gisc.es
[email protected]