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
Player A
Cooperate
Defect
Player B
50, 50
0, 80
80, 0
10, 10
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4. Prisoner‘s Dilemma Player A Cooperate Defect Player B 50, 50 0, 80
80, 0
10, 10
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5. Replicator Equation p(i,t) increases if:
expected success > average success
relative frequency of behavior
payoff (i = players decision, j = others decision)
expected “success”
average success
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6. Stability of Games now two strategies only:
= p(1,t): decision one, here cooperate
λ1 = P12 – P22
λ2 = P21 – P11
= p(2,t): decision two, here defect
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7. Four different Cases of Stability
λ1 = P12 – P22 < 0 and λ2 = P21 – P11 > 0
 P22 > P12 and P21 > P11
applies to prisoner dilemma, where:
P21 > P11 > P22 > P12
remember: P21 means, you choose decision 2 (defection) and the others chose 1
 choosing 1 includes much more risk
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8. Four different Cases of Stability
λ1 = P12 – P22 > 0 and λ2 = P21 – P11 < 0
 P22 < P12 and P21 < P11
applies to harmony game, where:
P11 > P21 > P12 > P22
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9. Harmony Game solution cooperation is stable
 ends up with cooperation by everybody
Player A
Cooperate
Defect
Player B
4, 4
3, 2
2, 3
1, 1
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10. Four different Cases of Stability
λ1 = P12 – P22 > 0 and λ2 = P21 – P11 > 0
 P22 < P12 and P21 > P11
applies to chicken game, where:
P21 > P11 > P12 > P22
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11. Chicken Game both solutions unstable
 cooperators coexist with defectors
Player A
Cooperate
Defect
Player B
3, 3
2, 4
4, 2
1, 1
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12. Four different Cases of Stability
λ1 = P12 – P22 < 0 and λ2 = P21 – P11 < 0
 P22 > P12 and P21 < P11
applies to stag hunt game, where:
P11 > P21 > P22 > P12
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13. Stag Hunt Game no nash equilibrium both solutions stable
 full cooperation possible, depends on history
Player A
Cooperate
Defect
Player B
3, 3
1, 2
2, 1
2, 2
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14. Phase Transitions
Prisoners dilemma: vital interest to get to full cooperation
remember:
Player A
Cooperate
Defect
Player B
50, 50
0, 80
80, 0
10, 10
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15. Phase Transitions
Prisoners dilemma: vital interest to get to full cooperation
how to do that?
Idea: transforming payoffs with taxes
Player A
Cooperate
Defect
Player B
50, 50
0, 80 – 100
80 – 100, 0
10 – 100, 10 – 100
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16. Phase Transitions
Prisoners dilemma: vital interest to get to full cooperation
how to do that?
Idea: transforming payoffs with taxes
Player A
Cooperate
Defect
Player B
50, 50
0, –20
–20 , 0
–90, –90
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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
original PD payoff
new payoff
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18. Phase Transitions Route 1: Prisoner’s Dilemma  Harmony Game
transforms system from stable defection to stable cooperation
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19. Phase Transitions Route 2: Prisoners Dilemma  Stag Hunt Game
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20. Stag Hunt Game Player A Cooperate Defect Player B 3, 3 1, 2 2, 1 2, 2
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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)
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22. Phase Transitions Route 3: Prisoner’s Dilemma  Chicken Game Player A
Cooperate
Defect
Player B
3, 3
2, 4
4, 2
1, 1
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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
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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]
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25. Cooperation Supporting Mechanics
costly punishment [2c]
friendship networks [3]
time dependent taxation [6]
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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
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