Trieste, October 19, 2006 Time Scales in Evolutionary Dynamics Angel Sánchez Grupo Interdisciplinar...

Trieste, October 19, 2006

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

Time Scales in Evolution 2

http://[email protected]


Cooperation: the basis of human societies

• Occurs between genetically unrelated individuals

Anomaly in the animal world:





• Shows high division of labor






• Valid for large scale organizations…


…as well as hunter-gatherer groups





Some animals form complex societies…

…but their individuals are genetically related




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




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)

How did altruism arise?




Altruism is an evolutionary puzzle




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)

Group selection? Cultural evolution?




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




… 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 benefitsHammerstein (ed.), Genetic and cultural evolution of cooperation (Dahlem Workshop Report 90, MIT, 2003)




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.”




Evolution• There are populations of reproducing

individuals• Reproduction includes mutation• Some individuals reproduce faster than

other (fitness). This results in selectionGame 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




• 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

John Maynard Smith 1972(J.B.S. Haldane, R. A. Fisher, W. Hamilton, G. Price)

Evolutionary Game TheorySuccessful strategies spread by natural selectionPayoff = fitness




replicator-mutator Price equationreplicator-mutator Price equation

QuasispeciesequationQuasispeciesequation

Lotka-VolterraequationLotka-Volterraequation

Adaptive dynamicsAdaptive dynamics

Game dynamical equationGame dynamical equation

replicatorPrice equationreplicatorPrice equation

Replicator-mutator equationReplicator-mutator equation

Price equationPrice equation

Equations for evolutionary dynamics




Case study on strong reciprocity and altruistic

behavior:

Ultimatum Games, altruism and individual

selection




The Ultimatum Game(Güth, Schmittberger & Schwarze, 1982)

experimenter

proposerresponder

M euros

M-u uOK

uM-u

NO

0 0




Experimental resultsExtraordinary amount of dataCamerer, Behavioral Game Theory (Princeton University Press, 2003)

Henrich et al. (eds.), Foundations of Human Sociality : Economic Experiments and Ethnographic Evidence from Fifteen Small-Scale Societies (Oxford University Press, 2004)

“At this point, we should declare a moratorium on creating ultimatum game data and shift attention towards new games and new theories.”




What would you offer?




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%)

Rational responder’s optimal strategy: accept anything

Rational proposer’s optimal strategy: offer minimum




Model

...... N players

player i

ti , oi : thresholds (minimum share

player i accepts / offers)

fi : fitness (accumulated capital)M monetary units (M=100)

A.S. & J. A. Cuesta, J. Theor. Biol. 235, 233 (2005)




op tr <

Game event

...... N players

proposer responder

op

fp

tr

fr

op

tr

≥

+M-op

+op




t, omin

fmin

t’, o’max

fmax

new player

Reproduction event (after s games)

...... N players

minimum fitness

maximum fitness

t, omax

fmax

mutation: t’, o’max= t, omax ± 1

(prob.=1/3)




N =1000, 109 games, s = 105, ti = oi =1 initial condition

accept

offer

Slow evolution (large s)




N =1000, 106 games, s =1, uniform initial condition

accept

offer

Fast evolution (small s)




Adaptive dynamics (“mean-field”) results

• Results for small s (fast selection) differ qualitatively

• Implications in behavioral economics and evolutionary ideas on human behavior!




Selection/reproduction interplay in simpler

settings:

Equilibrium selection in

2x2 games




Select one, proportional to fitnessSubstitute a randomly chosen individual

Moran Process

Game eventChoose s pairs of agents to play the game between reproduction events

Reset fitness after reproduction

2x2 game

P. A. P. Moran, The statistical processes of evolutionary theory(Clarendon, 1962)

P. A. P. Moran, The statistical processes of evolutionary theory(Clarendon, 1962)

C. P. Roca, J. A. Cuesta, A. Sánchez, Phys. Rev. Lett. 97, 158701 (2006)




Fixation probabilityProbability to reach state N when starting from state i =1

0i Ni 1i

Absorbing states

1-x11-x1 x1x1




Fixation probabilityProbability to reach state N when starting from state n





Number of games s enters through transition probabilities





Fitness: possible game sequences times corresponding payoffs per population




Example 1: Harmony game

Payoff matrix:

Unique Nash equilibrium in pure strategies: (C,C)(C,C) is the only reasonable behavior anyway




Example 1: Harmony games infinite (round-robin, “mean-field”)




Example 1: Harmony games = 1 (reproduction following every game)




Example 1: Harmony gameConsequences

•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

•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




Example 1: Harmony gameNumerical evaluation of exact expressionsNumerical evaluation of exact expressions




Example 2: Stag-hunt game

Payoff matrix:

Two Nash equilibria in pure strategies: (C,C), (D,D)Equilibrium selection depends on initial condition




Example 2: Stag-hunt gameNumerical evaluation of exact expressionsNumerical evaluation of exact expressions




Example 3: Snowdrift game

Payoff matrix:

One mixed equilibrium

Replicator dynamics goes always to mixed equilibriumMoran dynamics does not allow for mixed equilibria




Example 3: Snowdrift gameNumerical evaluation of exact expressionsNumerical evaluation of exact expressions




Example 3: Snowdrift gameNumerical evaluation of exact expressionsNumerical evaluation of exact expressions

s = 5

s = 100




Example 4: Prisoner’s dilemma

Payoff matrix:

Paradigm of the emergence of cooperation problem

Unique Nash equilibrium in pure strategies: (C,C)




Example : Prisoner’s dilemmaNumerical evaluation of exact expressionsNumerical evaluation of exact expressions




Results are robust

Increasing system size does not changes basins of attrractions, only sharpens the transitionsSmall s is like an effective small population, because inviduals that do not play do not get fitnessIntroduce background of fitness: add fb to all payoffs




Background of fitness: Stag-hunt gameNumerical evaluation of exact expressionsNumerical evaluation of exact expressions

fb = 0.1

fb = 1




• 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

Conclusions

New perspective on evolutionary game theory: more general dynamics, dictated by the specific application (change focus from equilibrium selection problems)




C. P. Roca, J. A. Cuesta, A. Sánchez, arXiv:q-bio/0512045(submitted to European Physical Journal Special Topics)

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, Phys. Rev. Lett. 97, 158701 (2006)

Time Scales in Evolutionary Dynamics

Trieste, October 19, 2006 Time Scales in Evolutionary Dynamics Angel Sánchez Grupo Interdisciplinar...

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