Fluctuation effects in evolutionary game theory

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Fluctuation effects in evolutionary game theory

• Angel Sánchez
• GISC/Matemáticas
• Universidad Carlos III de Madrid
• http//gisc.uc3m.es/anxo

With Carlos P. Roca and José A. Cuesta
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• Evolution basic principles
• There are populations of reproducing individuals.
• Reproduction includes mutation.
• Some individuals reproduce faster than other.
  This results in selection.

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Evolutionary Dynamics ...is the attempt to
invent and study mathematical equations
describing how population change over time due to
mutation and selection.
1930s Ronald Fisher, JBS Haldane, Sewall Wright
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Game Theory

• John von Neumann

Oskar Morgenstern
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What is game theory?

• Formal way to analyse 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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Game Theory

• John von Neumann

Oskar Morgenstern
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Evolutionary game theory
Fitness depends on the relative abundance of
different types.
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Evolutionarily stable strategy
If every individual of a population adopts the
evolutionarily stable strategy, then no mutant
can invade.
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Replicator dynamics
(Evolutionary game dynamics)
Taylor Jonker, Hofbauer Sigmund
Lotka Volterra equation of ecology
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Evolution to equilibria
A B A a b B c d
Payoff matrix
A B

• A dominates B agtc, bgtd
• Coexistence altc, bgtd
• Exclusion agtc, bltd

A B
A B
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Replicator mutator equation
no mutation
constant fitness
Quasispecies equation Replicator equation
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General framework
Quasispecies equation
replicator-mutator Price equation
Replicator-mutator equation
Price equation
Lotka-Volterra equation
Game dynamical equation
replicator Price equation
Adaptive dynamics
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Generic assumptions

• Everyone starts with a random strategy
• Everyone population plays game against everyone
  else
• The population is infinite
• The payoffs are added up
• The total payoff determines the number of
  offspring (Selection)
• Offspring inherit approximately the strategy of
  their parents (Mutation)
• Note similarity to genetic algorithms.

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Moran Process
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Moran Process
select one (proportional to fitness)
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Moran Process
produce offspring
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Moran Process
eliminate one
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Moran Process
add the new one
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Moran Process
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Moran is a birth-death process
probability to go from i to j
Transition matrix is tri-diagonal
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Fixation probability
probability to reach state n when starting from
state i 1
rrelative fitness of new mutant npopulation size
For a neutral mutant, r1, the fixation
probability is 1/n.
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Random gaming
H R H 6 0 R 5 1
Stag-hunt game

• Two equilibria, H and R
• Round-robin gaming H or Rselected from x0(x
  fraction of H)
• s games between death-birth

A B
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Random gaming
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Ultimatum game (s small)
N 1000, 106 games, s 1, uniform initial
condition
accept offer
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Ultimatum game (s middle)
N 1000, 108 games, s 1000, ti ai 1 initial
condition
accept offer
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Ultimatum game (s large)
N 1000, 109 games, s 105, ti ai 1 initial
condition
accept offer
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Adaptive dynamics
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Issues

• Description of social models/agent-based
  simulations in terms of game theory
• Different alternatives for the dynamics
• Analytical description available for some choices
• Results depend on the choice for the dynamics
• Analytical descriptions are mean-field like for
  different microscopic dynamics need new tools
  for others
• Comparison between theories and
  simulations/models and modeled issues

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Selection
Fitness of B 1.1
Fitness of A 1
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Selection
B out-competes A
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Quasispecies equation
Manfred Eigen Peter Schuster
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Replicator equation
Lotka Volterra equation of ecology
Hofbauer Sigmund