Genetic Algorithms and Game Theory

1
Genetic Algorithms and Game Theory

• Douglas King
• Department of General Engineering
• University of Illinois at Urbana-Champaign
• December 4, 2003

2
Overview

• What is a genetic algorithm?
• Axelrod Using the genetic algorithm to develop
  successful strategies in the iterated prisoners
  dilemma
• Riechmann Genetic algorithm as a game, itself

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What is a Genetic Algorithm?

• Search/Optimization method inspired by
  genetic/evolutionary theory
• Maintains a collection (population) of solutions
  rather than just one
• These solutions (strategies) are represented as
  strings of bits (chromosomes)
• Population evolves using three genetic operators
• Selection Survival of the fittest
• Mutation Random bit-flip (probabilistic)
• Crossover Combine two chromosomes (probabilistic)

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Axelrod Iterated Prisoners Dilemma (IPD)

• Equilibrium when both defect, but both will do
  better if they cooperate
• Background Axelrods tournaments
• TIT-FOR-TAT wins both tournaments
• Desirable strategy characteristics
• Niceness
• Vengefulness
• Forgiveness

Figure 1 Payoff Matrix
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Axelrods GA Approach

• Strategies have three-turn memory
• Strategies coded as strings of 70 bits
• 64 for the possible three-turn combinations
• 6 for the initial conditions
• Fitness determined by performance against
  Kingmakers from second tournament
• Population size of 20
• Experiments run for 50 generations

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GA Experiment Results

• GA evolves TIT-FOR-TAT-like behavior over time
• Niceness Continue to cooperate after three
  rounds of mutual cooperation
• Vengefulness Defect when opponent breaks a
  sequence of mutual cooperation
• Forgiveness Cooperate when opponent appears to
  apologize for defection

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Some Concerns

• Axelrod Would these GA-strategies do as well in
  a different environment?
• Is GA population size too small?
• Note Chromosome can only represent a small
  subset of strategies
• Memory increases chromosome size exponentially
• Nevertheless, these results show promise

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Riechmanns Analysis of the GA

• Genetic algorithm as an evolutionary game
• Many agents who interact with each other
• Fitness based on how well agents play the game
• More advanced conditions
• Population as a group of agents trying to achieve
  Nash equilibrium
• Agents play against all other agents
• HOWEVER Population does not represent every
  strategy

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Summary

• The field of genetic algorithms is closely
  related to the field of game theory
• Applications Axelrod
• Theoretical Riechmann
• Further examination of the links between these
  fields could provide a greater understanding