Game Theory Fall 2018 - Mike Shor Topic 5

“Доверяй, Но Проверяй.” (Trust But Verify) – Old Russian Proverb Game Theory © Mike Shor 2018

Predicting likely outcome of a game Review Predicting likely outcome of a game Sequential (Look forward and reason back) Simultaneous (Look for simultaneous best replies) What if interaction is repeated? When is cooperation possible? Game Theory © Mike Shor 2018

Firm 2 Firm 1 Equilibrium $54K Low High 54 , 54 72 , 47 47 , 72 Prisoner’s Dilemma Equilibrium $54K Firm 2 Low High Firm 1 54 , 54 72 , 47 47 , 72 60 , 60 Cooperation $60K Game Theory © Mike Shor 2018

“Sometimes rational decisions aren't sensible!” – Ian Stewart mathematician & science fiction author Game Theory © Mike Shor 2018

Private rationality  Collective irrationality Prisoner’s Dilemma Private rationality  Collective irrationality Dominant strategies makes everyone worse off Goal: Mutually beneficial cooperative outcome Overcome incentives to cheat (A note about tacit collusion) Game Theory © Mike Shor 2018

Repeated Games Interaction between the same players playing the same (or similar) game multiple times

People think forward if: Finite Interaction Cooperation is impossible if the relationship between players is for a fixed and known length of time. People think forward if: Game length is uncertain Game length unknown Game length too long Game Theory © Mike Shor 2018

Long-Term Interaction No last period, so no rollback Use history-dependent strategies Game Theory © Mike Shor 2018

Cooperate as long as the rivals do Upon observing a defection: Trigger Strategies Begin by cooperating Cooperate as long as the rivals do Upon observing a defection: Immediately initiate a period of “punishment” – play non-cooperatively for a specified period of time Game Theory © Mike Shor 2018

Grim trigger strategy (GTS) Trigger Strategies Grim trigger strategy (GTS) Cooperate until a rival deviates Once a deviation occurs, play non-cooperatively forever Tit-for-tat (TFT) Cooperate if rival cooperated in the most recent period Don’t cooperate if rival didn’t in the most recent period Game Theory © Mike Shor 2018

“Is cooperation possible?” Trigger Strategies Tit-for-Tat is Most forgiving Shortest memory Proportional Credible, but lacks deterrence “Is cooperation easy?” Grim Trigger is Least forgiving Longest memory MAD Adequate deterrence, but lacks credibility “Is cooperation possible?” Game Theory © Mike Shor 2018

Why Cooperate against GTS? Firm 2 Low High Firm 1 54 , 54 72 , 47 47 , 72 60 , 60 Cooperate: 60 today, 60 next year, 60 … 60 Defect: 72 today, 54 next year, 54 … 54 Game Theory © Mike Shor 2018

Payoff streams (GTS) 72 cooperate 60 54 defect t t+1 t+2 t+3 time Game Theory © Mike Shor 2018

$1 tomorrow is worth less than $1 today If the interest rate is r: Discounting $1 tomorrow is worth less than $1 today If the interest rate is r: Invest $1 today to get $(1+r) next year $1 next year is worth $1/(1+r) today Annuity paying $1 today and $1 every year is worth $1+1/r today Game Theory © Mike Shor 2018

Aside: Infinite Discounting Why? Game Theory © Mike Shor 2018

Why Cooperate against GTS? Cooperate if the present value of cooperation is higher than the present value of defection. PV(cooperation) 60…60…60…60… 60 + 60/r 6/r r > < PV(defection) 72…54…54…54… 72 + 54/r 12 6/12 = 50% Game Theory © Mike Shor 2018

Why Cooperate against GTS? Cooperate if the present value of cooperation is higher than the present value of defection. In this case, cooperation is sustainable using grim trigger strategies as long as r < 50% Or, as long as $1 invested today does not return more than $1.50 next period Game Theory © Mike Shor 2018

Payoff streams (TFT) 72 defect once cooperate 60 54 defect t t+1 t+2 time Game Theory © Mike Shor 2018

Why Cooperate against Tit-for-Tat? PV(cooperation) 60…60…60…60… 60 + 60/(1+r) 13/(1+r) 13 r > and < PV(defect forever) PV(defect once) 72…47…60…60… 72 + 47/(1+r) 12 12 + 12r 1/12 = 8.3% Game Theory © Mike Shor 2018

Why Cooperate against Tit-for-Tat? Cooperate if the present value of cooperation is higher than the present value of defection. In this case, cooperation is sustainable using grim trigger strategies as long as r < 8.3% Much harder to sustain than grim trigger In this case, cooperation may not be likely Game Theory © Mike Shor 2018

Grim Trigger and Tit-for-Tat are extremes Balance two goals: Trigger Strategies Grim Trigger and Tit-for-Tat are extremes Balance two goals: Deterrence GTS is adequate punishment Tit-for-tat might be too little Credibility GTS hurts the punisher too much Tit-for-tat is credible Game Theory © Mike Shor 2018

Main Ideas from Axelrod Not necessarily tit-for-tat Doesn’t always work Don’t be envious Don’t be the first to cheat Reciprocate opponent’s behavior Both cooperation and defection Don’t be too clever Game Theory © Mike Shor 2018

Finite interaction versus infinite interaction Folk theorem Theoretical Asides Finite interaction versus infinite interaction Folk theorem Game Theory © Mike Shor 2018

Challenge of Cooperation Requires deterrence Requires credibility Summary Challenge of Cooperation Higher payoffs today versus future relationship Requires deterrence A clear, provocable policy of sufficient punishment Requires credibility Must incorporate forgiveness Looking ahead: How to be credible? Game Theory © Mike Shor 2018