Phase transitions to cooperation in the prisoners dilemma

Phase transitions to cooperation in the prisoner‘s dilemma Matthäus Kerres Matthäu Kerres | 18. 11. 2013 | 1

Introduction to Game Theory § Game theory problem: - 2 or more parties - both make a decision which effect themselves and other party Matthäus Kerres | 18. 11. 2013 | 2

Prisoner‘s Dilemma § Most profitable if everyone cooperates § Higher individual Layout non-cooperative players § Example: two parties: A, B Player A Player B Cooperate Defect Cooperate 50, 50 0, 80 Defect 80, 0 10, 10 Matthäus Kerres | 18. 11. 2013 | 3

Prisoner‘s Dilemma Player A Player B Cooperate Defect Cooperate 50, 50 0, 80 Defect 80, 0 10, 10 Matthäus Kerres | 18. 11. 2013 | 4

Replicator Equation relative behaviorj =“success” average success payoff (ifrequency = playersofdecision, expected others decision) § p(i, t) increases if: expected success > average success Matthäus Kerres | 18. 11. 2013 | 5

Stability of Games § now two strategies only: = p(1, t): decision one, here cooperate λ 1 = P 12 – P 22 λ 2 = P 21 – P 11 = p(2, t): decision two, here defect Matthäus Kerres | 18. 11. 2013 | 6

Four different Cases of Stability § Case 1: λ 1 = P 12 – P 22 < 0 and λ 2 = P 21 – P 11 > 0 P 22 > P 12 and P 21 > P 11 applies to prisoner dilemma, where: P 21 > P 11 > P 22 > P 12 remember: P 21 means, you choose decision 2 (defection) and the others chose 1 choosing 1 includes much more risk Matthäus Kerres | 18. 11. 2013 | 7

Four different Cases of Stability § Case 2: λ 1 = P 12 – P 22 > 0 and λ 2 = P 21 – P 11 < 0 P 22 < P 12 and P 21 < P 11 applies to harmony game, where: P 11 > P 21 > P 12 > P 22 Matthäus Kerres | 18. 11. 2013 | 8

Harmony Game Player A Player B Cooperate Defect Cooperate 4, 4 3, 2 Defect 2, 3 1, 1 § solution cooperation is stable ends up with cooperation by everybody Matthäus Kerres | 18. 11. 2013 | 9

Four different Cases of Stability § Case 3: λ 1 = P 12 – P 22 > 0 and λ 2 = P 21 – P 11 > 0 P 22 < P 12 and P 21 > P 11 applies to chicken game, where: P 21 > P 12 > P 22 Matthäus Kerres | 18. 11. 2013 | 10

Chicken Game Player A Player B Cooperate Defect Cooperate 3, 3 2, 4 Defect 4, 2 1, 1 § both solutions unstable cooperators coexist with defectors Matthäus Kerres | 18. 11. 2013 | 11

Four different Cases of Stability § Case 4: λ 1 = P 12 – P 22 < 0 and λ 2 = P 21 – P 11 < 0 P 22 > P 12 and P 21 < P 11 applies to stag hunt game, where: P 11 > P 22 > P 12 Matthäus Kerres | 18. 11. 2013 | 12

Stag Hunt Game Player A Player B Cooperate Defect Cooperate 3, 3 1, 2 Defect 2, 1 2, 2 § no nash equilibrium § both solutions stable full cooperation possible, depends on history Matthäus Kerres | 18. 11. 2013 | 13

Phase Transitions § Prisoners dilemma: vital interest to get to full cooperation § remember: Player A Player B Cooperate Defect Cooperate 50, 50 0, 80 Defect 80, 0 10, 10 Matthäus Kerres | 18. 11. 2013 | 14

Phase Transitions § Prisoners dilemma: vital interest to get to full cooperation § how to do that? Idea: transforming payoffs with taxes Player A Player B Cooperate Defect Cooperate 50, 50 0, 80 – 100 Defect 80 – 100, 0 10 – 100, 10 – 100 Matthäus Kerres | 18. 11. 2013 | 15

Phase Transitions § Prisoners dilemma: vital interest to get to full cooperation § how to do that? Idea: transforming payoffs with taxes Player A Player B Cooperate Defect Cooperate 50, 50 0, – 20 Defect – 20 , 0 – 90, – 90 Matthäus Kerres | 18. 11. 2013 | 16

Phase Transitions § Taxes: Tij = Pij 0 – Pij new Eigenvalues: λ’ 1 = λ 1 +T 22 – T 12 λ’ 2 = λ 2 +T 11 – T 21 original PD payoff new payoff § Taxes form different routes to cooperation § characterized by different kinds of phase transitions Matthäus Kerres | 18. 11. 2013 | 17

Phase Transitions § Route 1: Prisoner’s Dilemma Harmony Game transforms system from stable defection to stable cooperation Matthäus Kerres | 18. 11. 2013 | 18

Phase Transitions § Route 2: Prisoners Dilemma Stag Hunt Game Matthäus Kerres | 18. 11. 2013 | 19

Stag Hunt Game Player A Player B Cooperate Defect Cooperate 3, 3 1, 2 Defect 2, 1 2, 2 Matthäus Kerres | 18. 11. 2013 | 20

Phase Transitions § Route 2: Prisoners Dilemma Stag Hunt Game bistable system: leads history dependent to cooperation or defection to reach cooperation: reduce λ 2 largely negatively p 3(t) = λ 1 / (λ 1 + λ 2) Matthäus Kerres | 18. 11. 2013 | 21

Phase Transitions § Route 3: Prisoner’s Dilemma Chicken Game Player A Player B Cooperate Defect Cooperate 3, 3 2, 4 Defect 4, 2 1, 1 Matthäus Kerres | 18. 11. 2013 | 22

Phase Transitions § Route 3: Prisoner’s Dilemma Chicken Game transforms system from total defection (PD) to coexistence: p 3(t) = λ 1 / (λ 1 + λ 2) by increasing λ 1 we get higher cooperation Matthäus Kerres | 18. 11. 2013 | 23

Cooperation Supporting Mechanics § group selection (competition between different populations) [1] § kin selection (genetic relatedness) [1] § direct reciprocity [2 a] (repeated interaction) § indirect reciprocity [2 b] (trust and reputation) § network reciprocity [1] Matthäus Kerres | 18. 11. 2013 | 24

Cooperation Supporting Mechanics § costly punishment [2 c] § friendship networks [3] § time dependent taxation [6] Matthäus Kerres | 18. 11. 2013 | 25

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 | 18. 11. 2013 | 26