Evolving Strategies for the Prisoner’s Dilemma

1. Evolving Strategies for the Prisoner’s Dilemma Jennifer Golbeck University of Maryland, College Park Department of Computer Science July 23, 2002

2. Overview • Previous Research • Prisoner’s Dilemma • The Genetic Algorithm • Results • Conclusions

3. Previous Research

4. Axelrod • Robert Axelrod’s experiments of the 1980’s served as the starting point for this research • Implementation closely adheres to the configuration of his experiments • Same model for the Prisoner’s Dilemma • Minor variation in the implementation of the Genetic Algorithm

5. Prisoner’s Dilemma

6. The Prisoner’s Dilemma Model • The basic two-player prisoner’s Dilemma • Both players are arrested for the same crime • Each has a choice • Confess - Cooperate with the authorities (admit to doing the crime) • Deny - Defect against the other player (claim the other person is responsible) • No knowledge of “opponent’s” action

7. Payoff Matrix • Optimization • If both players cooperate, they each receive 3 points • If both players Defect, each receives 1 point • If there is a mixed outcome, the Defector gets 5 points and the cooperator gets 0 points

8. Iterated Game • In simulation, the endpoint of the game is unknown to the players, making it essentially an infinitely iterated game • Each player has a memory of the previous three rounds on which to base his strategy • Strategies are deterministic - for a given history h players will always make the same move • With 4 possible configurations in each round and a history of 3, each strategy is comprised of 43 = 64 moves

9. Previous Results • Axelrod tournaments • Using the three-round history model, teams submitted strategies to be competed in a round-robin tournament • Tit for Tat • Pavlov strategy, developed after these tournaments, was shown to be an effective strategy as well.

10. The Genetic Algorithm

11. The Model • Darwinian Survival of the Fittest • Genetic representation of entities • Fitness function • Select most fit individuals to reproduce • Mutate • Traits of most fit will be passed on • Over time, the population will evolve to be more fit, optimal

12. GA’s and the Prisoner’s Dilemma • Population: 20 individuals • Chromosome: 64-bit string where each bit represents the Cooperate or Defect move played for a specific strategy

14. GA’s and PD II • Fitness: Each player competes against every other for 64 consecutive rounds, and a cumulative score is maintained • Selection:Roulette Wheel selection • Reproduction: Random point crossover with replacement • Mutation rate 0.001 • Generations: 200,000 generations

15. Simulation and Results

16. Hypothesis • Past research has looked at which strategy was “best”. This research looks as what makes a “good” strategy. • Tit for Tat and Pavlov both perform very well, and share two traits • Defend against Defectors • Cooperate with other cooperators

17. Hypothesis • All populations evolve over time to possess and exhibit these two traits • This behavior evolves regardless of the initial makeup of the population

18. Five Initial Populations All “Always Cooperate (Confess)” (AllC) All “Always Defect (Deny)” (AllD) All Tit for Tat All Pavolv All Randomly generated (independently) Experiment I

19. Experiment II • Controls: Tit for Tat and Pavolv • Statistically equal performance • Support the hypothesis by showing: • Traits are not present in other initial populations • Over time, populations evolve to exhibit those traits and perform as well as Tit For Tat and Pavlov

20. Experiment II • To show that the hypothesized traits evolve, populations must demonstrate • In the presence of Defectors, evolved populations perform identically to the controls • In the presence of cooperators, evolved populations perform identically to controls

21. Part 1:Defend Against Defectors I • Mix each initial population with a small set of AllD • Tit for Tat and Pavolv (controls) perform at about 80% of maximum • All others perform significantly worse that Tit For Tat and Pavolv • AllC and Random populations perform significantly worse than their normal behavior • This shows that a priori, the AllC and random populations cannot defend against Defectors

22. Part 1: Defend against Defectors II • Evolve each population and then mix with small set of AllD • All populations now perform equally as well as each other, and as well as the TFT and Pavlov controls • Fitness at about 80% maximum

23. Part 2: Cooperate with Cooperators • As before, each startup population is mixed with a small set of AllC • TFT, Pavlov, do very well • AllC does exceptionally well • Others do significantly worse • Evolve and then add AllC • All populations perform equally as well as each other • Identical performance to TFT and Pavlov

24. Performance of Different Experiments

25. Conclusions

26. Conclusions I • Performance measures show that AllC, AllD, and random populations do not generally possess defensive or cooperative traits a priori • After evolution, all populations have changed to incorporate both traits • Evolved strategies perform as well as TFT and Pavlov, traditional “best” strategies

27. Conclusions II • In both experiments there is no statistical difference between the performance of evolved populations before and after the introduction of AllC or AllD players • Indicates that not only do the populations exhibit hypothesized traits in experimental conditions, but it is their normal behavior to do so.

28. Future Work

29. Non-deterministic Players • This work shows results for players with deterministic strategies • Much previous research has been done on stochastic strategies • Preliminary results show that the results presented here apply to stochastic strategies as well, but a formal study is necessary.