Phase transitions to cooperation in the prisoner‘s dilemma Matthäus Kerres Matthäu Kerres 18.11.2013

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

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 Matthäus Kerres 18.11.2013

Prisoner‘s Dilemma Player A Cooperate Defect Player B 50, 50 0, 80 80, 0 10, 10 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

Chicken Game both solutions unstable  cooperators coexist with defectors Player A Cooperate Defect Player B 3, 3 2, 4 4, 2 1, 1 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

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 Matthäus Kerres 18.11.2013

Phase Transitions Route 1: Prisoner’s Dilemma  Harmony Game transforms system from stable defection to stable cooperation Matthäus Kerres 18.11.2013

Phase Transitions Route 2: Prisoners Dilemma  Stag Hunt Game Matthäus Kerres 18.11.2013

Stag Hunt Game Player A Cooperate Defect Player B 3, 3 1, 2 2, 1 2, 2 Matthäus Kerres 18.11.2013

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 18.11.2013

Phase Transitions Route 3: Prisoner’s Dilemma  Chicken Game Player A Cooperate Defect Player B 3, 3 2, 4 4, 2 1, 1 Matthäus Kerres 18.11.2013

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 18.11.2013

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 18.11.2013

Cooperation Supporting Mechanics costly punishment [2c] friendship networks [3] time dependent taxation [6] Matthäus Kerres 18.11.2013

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