Phase transitions to cooperation in the prisoner ‘ s dilemma

1. Phase transitions to cooperation in the prisoner‘s dilemma Matthäus Kerres Matthäu Kerres

2. Introduction to Game Theory • Game theory problem: - 2 or more parties - both make a decision which effect themselves and other party Matthäus Kerres

3. Prisoner‘s Dilemma • Most profitable if everyone cooperates • Higher individual Layout non-cooperative players • Example: two parties: A, B Matthäus Kerres

4. Prisoner‘s Dilemma Matthäus Kerres

5. Replicator Equation • p(i,t) increases if: expected success > average success Matthäus Kerres

6. Stability of Games • now two strategies only: Matthäus Kerres

7. Four different Cases of Stability • Case 1: λ1 = P12 – P22 < 0 and λ2 = P21 – P11 > 0  P22 > P12 and P21 > P11 applies to prisoner dilemma, where: P21 > P11 > P22 > P12 Matthäus Kerres

8. Four different Cases of Stability • Case 2: λ1 = P12 – P22 > 0 and λ2 = P21 – P11 < 0  P22 < P12 and P21 < P11 applies to harmony game, where: P11 > P21 > P12 > P22 Matthäus Kerres

9. Harmony Game • solution cooperation is stable  ends up with cooperation by everybody Matthäus Kerres

10. Four different Cases of Stability • Case 3: λ1 = P12 – P22 > 0 and λ2 = P21 – P11 > 0  P22 < P12 and P21 > P11 applies to chicken game, where: P21 > P11 > P12 > P22 Matthäus Kerres

11. Chicken Game • both solutions unstable  cooperators coexist with defectors Matthäus Kerres

12. Four different Cases of Stability • Case 4: λ1 = P12 – P22 < 0 and λ2 = P21 – P11 < 0  P22 > P12 and P21 < P11 applies to stag hunt game, where: P11 > P21 > P22 > P12 Matthäus Kerres

13. Stag Hunt Game • no nash equilibrium • both solutions stable  full cooperation possible, depends on history Matthäus Kerres

14. Phase Transitions • Prisoners dilemma: vital interest to get to full cooperation • remember: Matthäus Kerres

15. Phase Transitions • Prisoners dilemma: vital interest to get to full cooperation • how to do that? Idea: transforming payoffs with taxes Matthäus Kerres

16. Phase Transitions • Prisoners dilemma: vital interest to get to full cooperation • how to do that? Idea: transforming payoffs with taxes Matthäus Kerres

17. Phase Transitions • Taxes: Tij = Pij0 – Pij  new Eigenvalues: λ’1 = λ1 +T22 – T12 λ’2 = λ2 +T11 – T21 • Taxes form different routes to cooperation • characterized by different kinds of phase transitions Matthäus Kerres

18. Phase Transitions • Route 1: Prisoner’s Dilemma  Harmony Game transforms system from stable defection to stable cooperation Matthäus Kerres

19. Phase Transitions • Route 2: Prisoners Dilemma  Stag Hunt Game Matthäus Kerres

20. Stag Hunt Game Matthäus Kerres

21. Phase Transitions • Route 2: Prisoners Dilemma  Stag Hunt Game bistable system: leads history dependent to cooperation or defection to reach cooperation: reduce λ2 largely negatively  p3(t) = λ1 / (λ1 +λ2) Matthäus Kerres

22. Phase Transitions • Route 3: Prisoner’s Dilemma  Chicken Game Matthäus Kerres

23. Phase Transitions • Route 3: Prisoner’s Dilemma  Chicken Game transforms system from total defection (PD) to coexistence: p3(t) = λ1 / (λ1 +λ2)  by increasing λ1 we get higher cooperation Matthäus Kerres

24. Cooperation Supporting Mechanics • group selection (competition between different populations) [1] • kin selection (genetic relatedness) [1] • direct reciprocity [2a] (repeated interaction) • indirect reciprocity [2b] (trust and reputation) • network reciprocity [1] Matthäus Kerres

25. Cooperation Supporting Mechanics • costly punishment [2c] • friendship networks [3] • time dependent taxation [6] Matthäus Kerres

26. Summary • what has to happen to create cooperation in the PD: • moving stable stationary solution away from pure defection • stabilizing unstable solutions • creating new stationary solutions Matthäus Kerres