Title: Time Scales in Evolutionary Dynamics
1Time Scales in Evolutionary Dynamics
Angel Sánchez Grupo Interdisciplinar de Sistemas
Complejos (GISC) Departamento de Matemáticas
Universidad Carlos III de Madrid Instituto de
Biocomputación y FÃsica de Sistemas Complejos
(BIFI) Universidad de Zaragoza
with Carlos P. Roca and José A. Cuesta
2Cooperation the basis of human societies
Anomaly in the animal world
- Occurs between genetically unrelated individuals
3Cooperation the basis of human societies
Anomaly in the animal world
- Shows high division of labor
4Cooperation the basis of human societies
Anomaly in the animal world
- Valid for large scale organizations
as well as hunter-gatherer groups
5Cooperation the basis of human societies
Some animals form complex societies
but their individuals are genetically related
6Altruism key to cooperation
Altruism fitness-reducing act that
benefits others
Pure altruism is ruled out by natural selection
acting on individuals á la Darwin
7How 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)
8- Altruism is an evolutionary puzzle
9Group 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)
10Answers 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
11 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)
12One 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.
13Game 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
14Evolutionary 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
15Equations for evolutionary dynamics
16Case study on strong reciprocity and altruistic
behavior
Ultimatum Games, altruism and individual selection
17The Ultimatum Game
- (Güth, Schmittberger Schwarze, 1982)
experimenter
M euros
proposer
M-u
0
responder
u
0
18Experimental 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)
19What would you offer?
20Experimental 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)
21Model
A.S. J. A. Cuesta, J. Theor. Biol. 235, 233
(2005)
22Game event
......
N players
responder
proposer
tr fr
op fp
op
M-op
23Reproduction 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
24Slow evolution (large s)
N 1000, 109 games, s 105, ti oi 1 initial
condition
accept offer
25Fast evolution (small s)
N 1000, 106 games, s 1, uniform initial
condition
accept offer
26Adaptive dynamics (mean-field) results
- Results for small s (fast selection) differ
qualitatively - Implications in behavioral economics and
evolutionary ideas on human behavior!
27Selection/reproduction interplay in simpler
settings
Equilibrium selection in 2x2 games
28P. 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)
29Fixation probability
Probability to reach state N when starting from
state i 1
1-x1
x1
Absorbing states
30Fixation probability
Probability to reach state N when starting from
state n
31Fixation probability
Probability to reach state N when starting from
state n
32Fixation probability
Probability to reach state N when starting from
state n
Number of games s enters through transition
probabilities
33Fixation probability
Probability to reach state N when starting from
state n
Fitness possible game sequences times
corresponding payoffs per population
34Example 1 Harmony game
Payoff matrix
Unique Nash equilibrium in pure strategies (C,C)
(C,C) is the only reasonable behavior anyway
35Example 1 Harmony game
s infinite (round-robin, mean-field)
36Example 1 Harmony game
s 1 (reproduction following every game)
37Example 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
38Example 1 Harmony game
- Numerical evaluation of exact expressions
39Example 2 Stag-hunt game
Payoff matrix
Two Nash equilibria in pure strategies (C,C),
(D,D)
Equilibrium selection depends on initial condition
40Example 2 Stag-hunt game
- Numerical evaluation of exact expressions
41Example 3 Snowdrift game
Payoff matrix
One mixed equilibrium
Replicator dynamics goes always to mixed
equilibrium
Moran dynamics does not allow for mixed equilibria
42Example 3 Snowdrift game
- Numerical evaluation of exact expressions
43Example 3 Snowdrift game
- Numerical evaluation of exact expressions
s 5
s 100
44Example 4 Prisoners dilemma
Payoff matrix
Unique Nash equilibrium in pure strategies (C,C)
Paradigm of the emergence of cooperation problem
45Example Prisoners dilemma
- Numerical evaluation of exact expressions
46Results 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
47Background of fitness Stag-hunt game
- Numerical evaluation of exact expressions
fb 0.1
fb 1
48Conclusions
- 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)
49Time 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)