Time Scales in Evolutionary Dynamics

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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
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Cooperation the basis of human societies
Anomaly in the animal world

• Occurs between genetically unrelated individuals

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Cooperation the basis of human societies
Anomaly in the animal world

• Shows high division of labor

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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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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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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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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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Game theory

• Evolution
• There are populations of reproducing individuals
• Reproduction includes mutation
• Some individuals reproduce faster than other
  (fitness). This results in selection

• 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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Equations for evolutionary 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
proposer
M-u
0
responder
u
0
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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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What would you offer?
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Experimental results
Rational responders optimal strategy accept
anything Rational proposers optimal strategy
offer minimum

• 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)
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Game event
......
N players
responder
proposer
tr fr
op fp
op
M-op
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Reproduction event (after s games)
......
N players
new player
minimum fitness
maximum fitness
t, omax fmax
t, omin fmin
t, omax fmax
(prob.1/3)
mutation t, omax t, omax 1
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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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P. A. P. Moran, The statistical processes of
evolutionary theory (Clarendon, 1962)
Moran Process
Select one, proportional to fitness Substitute a
randomly chosen individual
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
x1
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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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 Prisoners dilemma
Payoff matrix
Unique Nash equilibrium in pure strategies (C,C)
Paradigm of the emergence of cooperation problem
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Example Prisoners 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, 142148. AIP Proceedings Series
  (2005).

C. P. Roca, J. A. Cuesta, A. Sánchez,
arXivq-bio/0512045 (submitted to European
Physical Journal Special Topics)
C. P. Roca, J. A. Cuesta, A. Sánchez, Phys. Rev.
Lett. 97, 158701 (2006)